Universita degli Studi di Perugia

DIPARTIMENTO DI FISICA E GEOLOGIA

Corso di Laurea in Fisica

Tesi di Laurea Magistrale

Black holes and supertranslation memory

Buchi neri e memoria di supertranslazione

Candidate:

Lorenzo RossiAdvisor:

Prof. Gianluca Grignani

Academic year 2015–2016

Contents

Introduction 1

I Black holes 5

1 Spherical stars 91.1 Cold stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2 Spherical symmetry . . . . . . . . . . . . . . . . . . . . . . . . 101.3 Time-independence . . . . . . . . . . . . . . . . . . . . . . . . 111.4 Static, spherically symmetric spacetime . . . . . . . . . . . . . 121.5 Tolman-Oppenheimer-Volkoff (TOV) equations . . . . . . . . 141.6 Outside the star: the Schwarzschild solution . . . . . . . . . . 171.7 The interior solution . . . . . . . . . . . . . . . . . . . . . . . 181.8 Maximum mass of a cold star . . . . . . . . . . . . . . . . . . 19

2 The Schwarzschild black hole 232.1 Birkhoff’s theorem . . . . . . . . . . . . . . . . . . . . . . . . 232.2 Gravitational redshift . . . . . . . . . . . . . . . . . . . . . . . 242.3 Geodesics of the Schwarzschild solution . . . . . . . . . . . . . 252.4 Eddington-Finkelstein (EF) coordinates . . . . . . . . . . . . . 262.5 Finkelstein diagram and the black hole region . . . . . . . . . 292.6 Gravitational collapse . . . . . . . . . . . . . . . . . . . . . . . 312.7 Detecting black holes . . . . . . . . . . . . . . . . . . . . . . . 322.8 White holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.9 The Kruskal extension . . . . . . . . . . . . . . . . . . . . . . 342.10 Einstein-Rosen bridge . . . . . . . . . . . . . . . . . . . . . . . 37

3 Asymptotic flatness 413.1 Conformal compactification and Penrose diagrams . . . . . . . 413.2 Asymptotic flatness . . . . . . . . . . . . . . . . . . . . . . . . 473.3 Definition of a black hole . . . . . . . . . . . . . . . . . . . . . 52

iii

4 Charged black holes 554.1 The Reissner-Nordstrom (RN) solution . . . . . . . . . . . . . 554.2 Eddington-Finkelstein (EF) coordinates . . . . . . . . . . . . . 574.3 Kruskal-like coordinates . . . . . . . . . . . . . . . . . . . . . 584.4 Extreme Reissner-Nordstrom solution . . . . . . . . . . . . . . 624.5 Majumdar-Papapetrou (MP) solutions . . . . . . . . . . . . . 64

5 Rotating black holes 655.1 Uniqueness theorems . . . . . . . . . . . . . . . . . . . . . . . 655.2 The Kerr-Newman (KN) solution . . . . . . . . . . . . . . . . 675.3 The Kerr solution . . . . . . . . . . . . . . . . . . . . . . . . . 685.4 Maximal analytic extension . . . . . . . . . . . . . . . . . . . 695.5 The ergosphere and Penrose process . . . . . . . . . . . . . . . 71

II Asymptotic symmetries and supertranslation mem-ory 73

6 Bondi-Metzner-Sachs (BMS) group 776.1 Asymptotic symmetries of asymptotically flat spacetimes at

null infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776.2 BMS algebra and BMS group . . . . . . . . . . . . . . . . . . 806.3 BMS transformations with Weyl rescalings . . . . . . . . . . . 876.4 More precise metric in a neighbourhood of I + . . . . . . . . . 89

7 Supertranslation memory effect 917.1 BMS vacuum transitions . . . . . . . . . . . . . . . . . . . . . 917.2 Gravitational memory effect and clock desynchronisation . . . 96

8 Final state of gravitational collapse with supertranslationmemory 1018.1 Supertranslation field in the final state . . . . . . . . . . . . . 1018.2 Supertranslation-dependent final state metric . . . . . . . . . 108

8.2.1 BMS finite diffeomorphisms . . . . . . . . . . . . . . . 1098.3 Schwarzschild metric with supertranslation memory . . . . . . 113

9 Charged black hole metrics with supertranslation memory 1159.1 Reissner-Nordstrom metric with supertranslation memory . . . 1169.2 Extreme Reissner-Nordstrom metric with supertranslation mem-

ory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1179.3 Multiple charged black holes with supertranslation memory . . 117

iv

Conclusions and outlook 119

A The Schwarzschild radius 121

B Some additional details on spherical stars 125

Acknowledgements 127

Bibliography 129

v

Introduction

The theory of General Relativity, proposed by Einstein in 1915 [1], pro-vided an elegant and effective description of every classical (i.e. non-quantum)natural phenomenon in which gravity is involved. In particular, it predictedthe existence of objects in the universe from which matter and radiation can-not escape: black holes [2],[3],[4],[5]. Their existence has been supported byseveral experimental confirmations, therefore physicists can now argue thatblack holes exist. Many peculiar properties of the physics around them havebeen discovered and perfectly understood by using General Relativity.

However, Hawking, by considering also quantum effects, discovered that ablack hole can emit particles and radiation [6], whose energy must come fromthe black hole. As a consequence, the black hole will evaporate completely.This discovery predicts that a pure quantum state describing matter at thebeginning of the gravitational collapse becomes a mixed quantum state atthe end of evaporation. According to the unitarity of time evolution, one ofthe first principles of Quantum Mechanics, this is not possible, therefore theso-called information paradox, proposed by Hawking [7], arises. The paradoxis yet to be solved in a definitive way and it is believed that its resolutionmay lead to the development of the correct theory of quantum gravity.

In this work I will obtain a result that weakens the information paradoxfor static and neutral black holes, as already worked out by Compere andLong [8], and static and charged black holes, which is my original contribu-tion. This thesis is meant to be fully understandable by everyone who hasthe background knowledge of General Relativity that I learnt in my academiccourses. In Part I I will introduce the fundamental classical theoretical re-sults about black holes in a way that is intended to be accessible to everyreader that is familiar with the basics of General Relativity (for an expla-nation of these basics see [9], for a more formal explanation see [10], for aformal and concise explanation see [11]). This part will provide all the piecesof knowledge necessary to familiarise the reader with the different possibleexamples of black holes.

Part II will be introduced by a brief explanation of the information para-

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dox, easily comprehensible by everyone who has a basic knowledge of Quan-tum Mechanics, in order to present the overall concept without the mathe-matical formalism that takes quantum effects into consideration (for a formaland concise review see [12]). The formal discussion of the Hawking paradoxis not regarded as necessary in this thesis, because the main purpose of thiswork, which will be carried out in the rest of Part II, is to present the previ-ous works that are essential to enable the reader to understand the derivationof our final result: the correct metric of the final state of a black hole aftergravitational collapse, recently proposed by Compere and Long [8]. To com-pute this result, it is necessary a formal definition of “infinity” of a spacetimeand the BMS group of transformations that leave invariant the asymptoticform of the metric at null infinity. Then, it must be shown that the passageof matter and radiation at null infinity modifies spacetime according to aparticular kind of BMS transformations, called supertranslations. This ef-fect is called supertranslation memory. The consequence is that, to obtainthe correct form of the final state of a black hole after gravitational collapse,one has to apply a finite supertranslation to the usual spacetime metric. Theform of this final state weakens the information paradox, as we will explain.

Part I is structured as follows:

• in Chapter 1 we will analyse the stars from which black holes originateby gravitational collapse,

• in Chapter 2 we will study the simplest example of spacetime with ablack hole region: the Schwarzschild spacetime,

• in Chapter 3 we will define the concept of “null”, “spacelike” and “time-like infinity”, and we will give a formal definition of a black hole,

• in Chapter 4 we will describe charged black holes,

• in Chapter 5 we will explain the importance of rotating black holes.

Part II is structured as follows:

• in Chapter 6 we will define the BMS group of asymptotic symmetriesat null infinity,

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• in Chapter 7 we will describe the supertranslation memory effect andwe will explain how it is related to a particular kind of BMS transfor-mations, called supertranslations,

• in Chapter 8 we will present the derivation of the correct metric of thefinal state after gravitational collapse for the Schwarzschild black hole,

• in Chapter 9 we will apply the method outlined in the previous chapterto charged black holes, explaining two reasons why they are believedto be important,

• finally, we will draw the conclusions and outline some open questionsabout this kind of studies, which might be answered in the foremostfuture of research.

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Part I

Black holes

This first part will be dedicated to the introduction of the most importantpieces of background knowledge about black holes. To this end, we will follow[12] precisely. I believe that these Lecture Notes by Reall exactly provide apresentation of all the most important theoretical results about black holesin a complete, formal and also concise way. This amount of precise andsynthetic knowledge is perfect for the aims of this thesis.

In Chapter 1 we will study the stars that can give origin to black holes.In Chapter 2 the simplest kind of spacetime with a black hole region, theSchwarzschild spacetime, will be analysed. In Chapter 3 we will define theconcept of “infinity” of an asymptotically flat spacetime. We will also give aformal definition of a black hole. Finally, in Chapter 4 and 5 we will analysesome more complicated spacetimes describing a black hole region.

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Chapter 1

Spherical stars

1.1 Cold stars

To understand the astrophysical significance of black holes we must dis-cuss stars. In particular, how do stars end their lives?

A normal star, such as our Sun, is supported against contracting under itsown gravity by pressure generated by nuclear reactions in its core. However,eventually the star will use up its nuclear “fuel”. If the gravitational self-attraction is to be balanced then some new source of pressure is required.If this balance is to last forever, then this new source of pressure must benon-thermal because the star will eventually cool.

A non-thermal source of pressure arises quantum mechanically from thePauli principle, which makes a gas of cold fermions resist compression (thisis called the degeneracy pressure). A white dwarf is a star in which gravityis balanced by the electron degeneracy pressure. The Sun will end its life asa white dwarf. White dwarfs are very dense compared to normal stars, e.g.a white dwarf with the same mass as the Sun would have a radius arounda hundredth of that of the Sun. Using Newtonian gravity one can showthat a white dwarf cannot have a mass greater than the Chandrasekhar limit1.4M, where M is the mass of the Sun. A star more massive than thiscannot end its life as a white dwarf (unless it somehow sheds some mass, e.g.in a supernova).

Once the density of matter approaches the nuclear density, the degeneracypressure of neutrons becomes important (at such high density, inverse betadecay converts protons into neutrons). A neutron star is supported by thedegeneracy pressure of neutrons. These stars are tiny: a Solar mass neutronstar would have a radius of around 10km (the radius of the Sun is 7×105km).Recall that validity of Newtonian gravity requires |Φ| 1, where |Φ| is the

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Newtonian gravitational potential. At the surface of such a neutron star onehas |Φ| ∼ 0.1, so a Newtonian description is inadequate: one has to useGeneral Relativity.

In this chapter we will see that GR predicts that there is a maximummass for neutron stars. Remarkably, this is independent of the properties ofmatter at extremely high density and so it holds for any cold star. As we willexplain, detailed calculations reveal the maximum mass to be around 3M.Hence a hot star more massive than this cannot end its life as a cold star(unless it somehow sheds some mass, e.g. in a supernova). Instead, the starwill undergo a complete gravitational collapse to form a black hole.

In the next sections we will show that GR predicts a maximum massfor a cold star. We will make the simplifying assumption that the star isspherically symmetric. As we will see, the Schwarzschild solution is theunique spherically symmetric vacuum solution of the Einstein equation andhence describes the gravitational field outside any spherically symmetric star.The interior of a star can be modelled using a perfect fluid and so spacetimeinside the star is determined by solving the Einstein equation with a perfectfluid source and matching onto the Schwarzschild solution outside the star.

1.2 Spherical symmetryWe want to define symmetries of a spacetime (M, g).

Definition 1. An isometry, or symmetry, of the manifold M with metricgµν is a diffeomorphism φ : M → M generated by a vector field ξ such thatLξgµν = 0, i.e.

∇µξν +∇νξµ = 0, (1.1)where ∇ is the covariant derivative associated with g. This equation for ξis called the Killing equation and vector fields that satisfy it are known asKilling vector fields.

Consider the case in which there exists a chart for which the metric doesnot depend on some coordinate xσ? . Then the vector ∂σ? = ∂

∂xσ?satisfies the

Killing equation. Conversely, if a vector ξ satisfies the Killing equation, itis always possible to find a coordinate system in which ξ = ∂σ? and in thiscoordinate system the metric components are independent on xσ? . The setof all isometries of a manifold with metric forms a group, called the isometrygroup.

Now we need to define what we mean by a spacetime (M, g) being spher-ically symmetric. Consider the unit round metric on S2:

dΩ2 = dθ2 + sin2 θdφ2. (1.2)

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The isometry group of this metric is SO(3) (actually O(3) if we includereflections), which is the group of rotations. The Killing vector fields are

R = ∂φ,

S = cosφ∂θ − cot θ sinφ∂φ,T = − sinφ∂θ − cot θ cosφ∂φ.

(1.3)

They satisfy the commutation relations that characterize the Lie algebra ofSO(3):

[R, S] = T,

[S, T ] = R,

[T,R] = S.

(1.4)

Any 1-dimensional subgroup of SO(3) gives a 1-parameter group of isome-tries, and hence a Killing vector field.

Definition 2. A spacetime is spherically symmetric if its isometry groupcontains an SO(3) subgroup whose orbits are 2-spheres. (The orbit of a pointp under a group of diffeomorphisms is the set of points that one obtains byacting on p with all of the diffeomorphisms.) In other words, a spacetime isspherically symmetric if it possesses the same symmetries as a round S2.

Definition 3. In a spherically symmetric spacetime, the area-radius functionr : M → R is defined by r(p) =

√A(p)4π where A(p) is the area of the S2 orbit

through p. (In other words, the S2 passing through p has induced metricr(p)dΩ2.)

1.3 Time-independenceDefinition 4. A spacetime is stationary if it admits a Killing vector field kµwhich is everywhere timelike, i.e. gµνkµkν < 0.

We can choose coordinate as follows. Pick an hypersurface Σ nowheretangent to kµ and introduce coordinates xi on Σ. Assign coordinates (t, xi)to the point parameter distance t along the integral curve of kµ through thepoint on Σ with coordinates xi. This gives a coordinate chart such thatk = ∂

∂t. Since k is a Killing vector field, the metric is independent of t and

hence takes the form

ds2 = g00(xk)dt2 + 2g0i(xk)dtdxi + gij(xk)dxidxj, (1.5)

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where g00 < 0. Conversely, given a metric of this form, ∂∂t

is obviously atimelike Killing vector field and so the metric is stationary.

Then we need to introduce the notion of hypersurface-orthogonality. LetΣ be a hypersurface in M specified by f(x) = 0 where f : M → R is smoothwith df 6= 0 on Σ. Then the 1-form df is normal to Σ. Indeed, let tµ be anyvector tangent to Σ, then df(t) = t(f) = tµ∂µf = 0 because f is constant onΣ. Any other 1-form n normal to Σ can be written as n = gdf + fn′ whereg is a smooth function with g 6= 0 on Σ and n′ is a smooth 1-form. Hencewe have dn = dg ∧ df + df ∧ n′ + fdn′ so (dn)|Σ = (dg − n′) ∧ df . So, if n isnormal to Σ, then

(n ∧ dn)|Σ = 0. (1.6)

Conversely, the Frobenius theorem states:

Theorem 1 (Frobenius). If n is a non-zero 1-form such that n ∧ dn = 0everywhere, then there exist functions f, g such that n = gdf so n is normalto surfaces of constant f , i.e. n is hypersurface-orthogonal.

Definition 5. A spacetime is static if it admits a hypersurface-orthogonaltimelike Killing vector field.

Static implies stationary.For a static spacetime, we know that kµ is hyersurface-orthogonal, so

when defining adapted coordinates we can choose Σ to be orthogonal to kµ.But Σ is the surface t = 0, with normal dt. It follows that, at t = 0, kµ =(1, 0, 0, 0) in our chart, i.e. ki = 0. However, ki = giµ(xk)kµ = giµ(xk)δµ0 =g0i(xk) so we must have g0i(xk) = 0. Hence, in adapted coordinates a staticmetric takes the form

ds2 = g00(xk)dt2 + gij(xk)dxidxj, (1.7)

where g00 < 0. Note that this metric has a discrete time-reversal isometry:(t, xi) → (−t, xi). So static means “time-independent and invariant undertime reversal”. The metric outside a rotating star can be stationary but notstatic because time-reversal changes the sense of rotation.

1.4 Static, spherically symmetric spacetimeWe are interested in determining the gravitational field of a time-independent

spherical object so we assume our spacetime to be stationary and sphericallysymmetric. By this we mean that the isometry group is R ⊗ SO(3), wherethe R factor corresponds to “time translations” (i.e. the associated Killing

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vector field is timelike) and the orbits of SO(3) are 2-spheres as above. Itcan be shown that any such spacetime must actually be static. (The gravi-tational field of a rotating star can be stationary but the rotation defines apreferred axis and so the spacetime would not be spherically symmetric.) Solet us consider a spacetime that is both static and spherically symmetric.

Staticity means that we have a timelike Killing vector field kµ and we canfoliate our spacetime with surfaces Σt orthogonal to kµ. One can argue thatthe orbit of SO(3) through p ∈ Σt lies within Σt. We can define sphericalpolar coordinates on Σ0 as follows. Pick a S2 symmetry orbit in Σ0 anddefine spherical polars (θ, φ) on it. Extend the definition of (θ, φ) to the restof Σ0 by defining them to be constant along (spacelike) geodesics normal tothis S2 within Σ0. Now we use (r, θ, φ) as coordinates on Σ0 where r is thearea-radius function defined above. The metric on Σ0 must take the form

ds2 = e2Ψ(r)dr2 + r2dΩ2. (1.8)

drdθ and drdφ terms cannot appear because they would break sphericalsymmetry. Note that r is not ”the distance from the origin”. Finally, wedefine coordinates (t, r, θ, φ) with t the parameter distance from Σ0 along theintegral curves of kµ. The metric must take the form

ds2 = −e2Φ(r)dt2 + e2Ψ(r)dr2 + r2dΩ2. (1.9)

The matter inside a star can be described by a perfect fluid, with energy-momentum tensor

Tµν = (ρ+ p)uµuν + pgµν , (1.10)where uµ is the 4-velocity of the fluid (a unit timelike vector: gµνuµuν = −1),and ρ, p are, respectively, the energy density and pressure measured in thefluid’s local rest frame (i.e. by an observer with 4-velocity uµ). Let us recallthat the energy-momentum tensor must satisfy the conservation equation

∇µTµν = 0. (1.11)

This is equivalent to

uµ∇µρ+ (ρ+ p)∇µuµ = 0,

(ρ+ p)uµ∇µuν = −(gµν + uµuν)∇µp.(1.12)

These are relativistic generalizations of the mass conservation equation andEuler equation of non-relativistic fluid dynamics. Note that a pressurelessfluid moves on geodesics (uµ∇µuν = 0). This makes sense physically: zeropressure implies that the fluid particles are non-interacting and hence behavelike free particles.

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Since we are interested in a time-independent situation, we assume thatthe fluid is at rest, so uµ is in the time direction:

uµ = e−Φ(∂

∂t

)µ. (1.13)

Hence, we get

Tµν =

e2Φρ

e2Ψpr2p

r2(sin2 θ)p

. (1.14)

Our assumptions of staticity and spherical symmetry implies that ρ and pdepend only on r. Let R denote the (area-)radius of the star. Then ρ and pvanish for r > R.

1.5 Tolman-Oppenheimer-Volkoff (TOV) equa-tions

We now want to solve the Einstein equations

Gµν = 8πTµν , (1.15)

whereGµν = Rµν −

12Rgµν (1.16)

is the Einstein tensor,

Rρσµν = ∂µΓρνσ − ∂νΓρµσ + ΓρµλΓλνσ − ΓρνλΓλµσ (1.17)

is the Riemann tensor,Rµν = Rλ

µλν (1.18)

is the Ricci tensor, andR = gµνRµν (1.19)

is the curvature scalar. The Christoffel symbols Γσµν are calculated by theformula

Γσµν = 12g

σρ(∂µgνρ + ∂νgρµ − ∂ρgµν). (1.20)

They are symmetric under the change µ↔ ν.

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Let us calculate Gµν with the metric (1.9). The Christoffel symbols aregiven by

Γttr = ∂rΦ, Γrtt = e2(Φ−Ψ)∂rΦ, Γrrr = ∂rΨ,

Γθrθ = 1r, Γrθθ = −re−2Ψ, Γφrφ = 1

r, (1.21)

Γrφφ = −re−2Φ sin2 θ, Γθφφ = − sin θ cos θ, Γφθφ = cos θsin θ .

Anything not written down explicitly is meant to be zero, or related to whatis written by symmetries. From these we get the following non vanishingcomponents of the Riemann tensor

Rtrtr =∂rΦ∂rΨ− ∂2

rΦ− (∂rΦ)2,

Rtθtθ =− re−2Ψ∂rΦ,

Rtφtφ =− re−2Ψ sin2 θ∂rΦ,

Rrθrθ =re−2Ψ∂rΨ,

Rrφrφ =re−2Ψ sin2 θ∂rΨ,

Rθφθφ =(1− e−2Ψ) sin2 θ.

(1.22)

Taking the contraction yields the Ricci tensor

Rtt =e2(Φ−Ψ)[∂2rΦ + (∂rΦ)2 − ∂rΦ∂rΨ + 2

r∂rΦ

],

Rrr =− ∂2rΦ− (∂rΦ)2 + ∂rΦ∂rΨ + 2

r∂rΨ,

Rθθ =e−2Ψ[r(∂rΨ− ∂rΦ)− 1] + 1,Rφφ = sin2 θRθθ,

(1.23)

and the curvature scalar

R = −2e−2Ψ[∂2rΦ + (∂rΦ)2−∂rΦ∂rΨ + 2

r(∂rΦ−∂rΨ) + 1

r2 (1− e2Ψ)]. (1.24)

Using all these results we can compute the Einstein tensor:

Gtt = 1r2 e

2(Φ−Ψ)(2r∂rΨ− 1 + e2Ψ),

Grr = 1r2 (2r∂rΦ + 1− e2Ψ),

Gθθ =r2e−2Ψ[∂2rΦ + (∂rΦ)2 − ∂rΦ∂rΨ + 1

r(∂rΦ− ∂rΨ)

],

Gφφ = sin2 θGθθ.

(1.25)

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We therefore have three independent Einstein equations: the tt component

1r2 e

−2Ψ(2r∂rΨ− 1 + e2Ψ) = 8πρ, (1.26)

the rr component

1r2 e

−2Ψ(2r∂rΦ + 1− e2Ψ) = 8πp, (1.27)

and the θθ component

e−2Ψ[∂2rΦ + (∂rΦ)2 − ∂rΦ∂rΨ + 1

r(∂rΦ− ∂rΨ)

]= 8πp. (1.28)

The φφ equation is proportional to the θθ equation, so there is no need toconsider it separately.

We notice that the tt equation involves only Ψ and ρ. It is convenient toreplace Ψ(r) with a new function m(r), given by

m(r) = 12(r − re−2Ψ), (1.29)

or equivalently

e2Ψ =[1− 2m(r)

r

]−1

, (1.30)

so that the tt Einstein equation becomes

dm

dr= 4πr2ρ. (1.31)

The rr equation givesdΦdr

= m+ 4πr3p

r(r − 2m) . (1.32)

It is convenient not to use the θθ equation directly, but instead appeal toenergy-momentum conservation (1.11). For our metric, it is straightforwardto derive that ν = r is the only non-trivial component, and it gives

dp

dr= −(ρ+ p)m+ 4πr3p

r(r − 2m) . (1.33)

Now we have 3 equations but 4 unknowns (m,Φ, ρ, p) so we need one moreequation. We are interested in a cold star, i.e. one with vanishing tempera-ture T . Thermodynamics tells us that T , p and ρ are not independent: they

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are related by the fluid’s equation of state, e.g. T = T (ρ, p). Hence the con-dition T = 0 implies a relation between p and ρ, i.e. a barotropic equation ofstate p = p(ρ). For a cold star, p is not an independent variable so we have 3equations for 3 unknowns. These are called the Tolman-Oppenheimer-Volkoff(TOV) equations.

We assume that ρ > 0 and p > 0, i.e. the energy density and pressureof matter are positive. We also assume that p is an increasing function of ρ.It this were not the case, then the fluid would be unstable: a fluctuation insome region that led to an increase in ρ would decrease p, causing the fluidto move into this region and hence further increase in p, i.e. the fluctuationwould grow.

1.6 Outside the star: the Schwarzschild solu-tion

Consider first the spacetime outside the star: r > R. We then haveρ = p = 0. For r > R (1.31) gives m(r) = RS

2 , where RS is a constant knownas Schwarzschild radius. Integrating (1.32), which is the only remaining non-trivial equation, gives

Φ = 12 ln

(1− RS

r

)+ Φ0, (1.34)

for some constant Φ0. We then have gtt → −e2Φ0 as r → ∞. The constantΦ0 can be eliminated by defining a new time coordinate t′ = eΦ0t. So withoutloss of generality we can set Φ0 = 0 and we have arrived at the Schwarzschildsolution:

ds2 = −(

1− RS

r

)dt2 +

(1− RS

r

)−1

dr2 + r2dΩ2. (1.35)

As shown in Appendix A, requiring that for large r the Schwarzschild solutionreduces to the solution of linearized theory describing the weak gravitationalfield far from a body of massM (the so-called Newtonian limit), we find that

RS = M

2 . (1.36)

The components of the above metric are singular at the Schwarzschild ra-dius r = RS = 2M , where gtt vanishes and grr diverges. A solution describinga static spherically symmetric star can exist only if r = 2M corresponds toa radius inside the star, where the Schwarzschild solution does not apply.

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Hence a static, spherically symmetric star must have a radius greater thatits Schwarzschild radius

R > 2M. (1.37)Normal stars have R 2M , e.g. for the Sun, 2M ≈ 3 km whereas R ≈7× 105 km.

1.7 The interior solutionConsider now the spacetime inside the star. Integrating (1.31) gives

m(r) = 4π∫ r

0ρ(r′)r′2dr′ +m?, (1.38)

where m? is a constant.Now Σt should be smooth at r = 0 (the centre of the star). Recall

that any smooth Riemannian manifold is locally flat, i.e. measurements ina sufficiently small region of Σt will be the same as in Euclidean space. InEuclidean space, a sphere of area-radius r also has proper radius r, i.e. allpoints on the sphere lie proper distance r from the centre. Hence the samemust be true for a small sphere on Σt. The proper radius of a sphere ofarea-radius r is

∫ r0 e

Ψ(r′)dr′ ≈ eΨ(0)r for small r. Hence we need eΨ(0) = 1 forthe metric to be smooth at r = 0. This implies m(0) = 0 and so m? = 0.

Now at r = R, our interior solution must match onto the exterior Schwarzschildsolution. For r > R we have m(r) = M so continuity of m(r) determines M :

M = 4π∫ R

0ρ(r)r2dr. (1.39)

This is formally the same as the equation relating total mass to densityin Newtonian theory. But there is an important difference: in Euclideanspace of Newtonian theory, the volume element on a surface of constant t isr2 sin θdr∧dθ∧dφ and so the right hand side of (1.39) gives the total energymatter. However, in GR, the volume element on Σt is eΨ(r)r2 sin θdr∧dθ∧dφso the total energy of the matter is

E = 4π∫ R

0ρ(r)eΨ(r)r2dr, (1.40)

and since eΨ > 1 (as m > 0) we have E > M : the energy of the matter inthe star is greater than the total energyM of the star. The difference E−Mis interpreted as the gravitational binding energy of the star.

In GR there is a lower limit on the size of stars that has no Newtoniananalogue. To see this, note that the definition (1.29) implies m(r)

r< 1

2 for all

18

r. Evaluating at r = R recovers the result R > 2M discussed above. (To seethat this has no Newtonian analogue, we can reinsert factors of G and c towrite it as GM

c2R< 1

2 . Taking the Newtonian limit c→∞ the equation becomestrivial.) This lower bound can be improved. Note that (1.33) implies dp

dr≤ 0

and hence dρdr≤ 0. Using this it can be shown, see Appendix B, that

m(r)r

<291− 6πr2p(r) + [1 + 6πr2p(r)] 1

2. (1.41)

Evaluating at r = R we have p = 0 and hence obtain the Buchdahl inequality

R >94M. (1.42)

The derivation of this inequality assumes only ρ ≥ 0 and dρdr≤ 0 and nothing

about the equation of state, so it also applies to hot stars satisfying theseassumptions. This inequality is sharp: in Appendix B it is shown that starswith constant density ρ can get arbitrarily close to saturating it (the pres-sure at the centre diverges in the limit in which the inequality becomes anequality).

The TOV equations can be solved by numerical integration as follows.Regard (1.31) and (1.33) as a pair of coupled first order differential equationsfor m(r) and ρ(r) (recall that p = p(ρ) and dp

dρ> 0). These can be solved,

at least numerically, given initial conditions for m(r) and ρ(r) at r = 0. Wehave just seen that m(0) = 0. Hence just need to specify the value ρc = ρ(0)for the density at the centre of the star.

Given a value for ρc we can solve (1.31) and (1.33). The latter equationshows that p (and hence ρ) decreases as r increases. Since the pressurevanishes at the surface of the star, the radius R is determined by the conditionp(R) = 0. This determines R as a function of ρc. Equation (1.39) thendetermines M as a function of ρc. Finally we determine Φ(r) inside thestar by integrating (1.32) inwards from r = R with initial condition Φ(R) =12 ln

(1 − 2M

R

)(from (1.34)). Hence, for a given equation of state, static,

spherically symmetric, cold stars form a 1-parameter family of solutions,labelled by ρc.

1.8 Maximum mass of a cold starWhen one follows the above procedure then one finds that, as ρc increases,

M increases to a maximum value but then decreases for larger ρc. Themaximum mass will depend on the details of the equation of state of cold

19

matter. For example, taking an equation of state corresponding to whitedwarf matter reproduces the Chandrasekhar bound (as mentioned above,one does not need GR for this; it can be obtained using Newtonian gravity).Experimentally we know this equation of state up to some density ρ0 (aroundnuclear density) but we do not know its form for ρ > ρ0. One might expectthat by an appropriate choice of the equation of state for ρ > ρ0 one couldarrange for the maximum mass to be very large, say 100M. This is not thecase. Remarkably, GR predicts that there is an upper bound on the mass ofa cold, spherically symmetric star, which is independent of the form of theequation of state at high density. This upper bound is around 5M.

Recall that ρ is a decreasing function of r. Let us define the core of thestar as the region in which ρ > ρ0 where we do not know the equation ofstate, and the envelope as the region ρ < ρ0 where we do know the equationof state. Let r0 be the radius of the core, i.e. the core is the region r < r0and the envelope the region r0 < r < R. The mass of the core is defined asm0 = m(r0). Equation (1.38) (where m? = 0) gives

m0 ≥43πr

30ρ0. (1.43)

We would have the same result in Newtonian gravity. In GR we have theextra costraint (1.41). Evaluating at r = r0 gives

m0

r0<

291− 6πr2

0p0 + [1 + 6πr20p0] 1

2, (1.44)

where p0 = p(r0) is determined from ρ0 using the equation of state. Notethat the right hand side is a decreasing function of p0 so we obtain a simpler(but weaker) inequality by evaluating the right hand side at p0 = 0:

m0 <49r0, (1.45)

i.e. the core satisfies the Buchdahl inequality. The two inequalities (1.43)and (1.44) restrict r0 and m0 to a compact region of the m0 − r0 plane.Therefore, m0 is bounded. The upper bound on the mass of the core is

m0 <

√16

243πρ0. (1.46)

Hence, although we do not know the equation of state inside the core, GRpredicts that its mass cannot be indefinitely large. Experimentally, we do notknow the equation of state of cold matter at densities much higher than the

20

density of atomic nuclei so we take ρ0 = 5× 1014 gcm3 , the density of nuclear

matter. This gives an upper bound on the core mass m0 < 5M.Now, given a core with mass m0 and radius r0, the envelope region is

determined uniquely by solving numerically (1.31) and (1.33) with initialconditions m = m0 and ρ = ρ0 at r = r0, using the known equation of stateat density ρ < ρ0. This show that the total massM of the star is a function ofthe core parameters m0 and r0. By investigating (numerically) the behaviourof this function, it is found that the M is maximised at the maximum of m0.At this maximum, the envelope contributes less than 1% of the total mass,so the maximum value of M is almost the same as the maximum value ofm0, i.e. 5M.

It should be emphasized that this is an upper bound that applies forany physically reasonable equation of state for ρ > ρ0. But any particularequation of state will have its own upper bound, which will be less than theabove bound. Indeed, one can improve the above bound by adding furthercriteria to what one means by “physically reasonable”. For example, thespeed of sound in the fluid is (dp

dρ) 1

2 . It is natural to demand that this shouldnot exceed the speed of light, i.e. one could add the extra condition dp

dρ≤ 1.

This has the effect of reducing the upper bound to about 3M.

21

Chapter 2

The Schwarzschild black hole

We have seen that General Relativity predicts that a cold star cannothave a mass larger than a few times M. A very massive hot star cannotend its life as a cold star unless it somehow sheds some of its mass. Instead,it will undergo complete gravitational collapse to form a black hole. Thesimplest black hole solution is described by the Schwarzschild geometry. Sofar, we have used the Schwarzschild metric to describe the spacetime outsidea spherical star. In this chapter we will investigate the geometry of spacetimeunder the assumption that the Schwarzschild solution is valid everywhere.

2.1 Birkhoff’s theoremIn Schwarzschild coordinates (t, r, θ, φ), the Schwarzschild solution is

ds2 = −(

1− 2Mr

)dt2 +

(1− 2M

r

)−1

dr2 + r2dΩ2. (2.1)

This is actually a 1-parameter family of solutions. The parameter M takeeither sign but, as mentioned above, it has the interpretation of a mass sowe will assume M > 0.

Previously we assumed that we were dealing with r > 2M . But the abovemetric is also a solution of the vacuum Einstein equation for 0 < r < 2M .We will see below how the two regions are related.

We derived the Schwarzschild solution under the assumptions of staticityand spherical symmetry. It turns out that the former is not required.

Theorem 2 (Birkhoff). Any spherically symmetric solution of the vacuumEinstein equation is isometric to the Schwarzschild solution.

23

This theorem assumes only spherical symmetry but the Schwarzschildsolution has an additional isometry: ∂

∂tis a hypersurface-orthogonal Killing

vector field. It is timelike for r > 2M so the r > 2M Schwarzschild solutionis static.

Birkhoff’s theorem implies that the spacetime outside any spherical bodyis described by the time-independent (exterior) Schwarzschild solution. Thisis true even if the body itself is time-dependent. For example, let us considera spherical star that ”uses up its nuclear fuel” and collapses to form a whitedwarf or neutron star. The spacetime outside the star will be described bythe static Schwarzschild solution even during the collapse.

2.2 Gravitational redshiftConsider two observers A and B who remain at fixed (r, θ, φ) in the

Schwarzschild geometry. Let A have r = rA and B have r = rB where rB >rA. Now assume that A sends two photons to B separated by a coordinatetime ∆t as measured by A. Since ∂

∂tgenerates an isometry, the path of the

second photon is the same as the path of the first one, just translated in timethrough an interval ∆t.

The proper time between the photons emitted by A, as measured by A is∆τA =

√1− 2M

rA∆t. Similarly, the proper time interval between the photons

received by B, as measured by B is A is ∆τB =√

1− 2MrB

∆t. Eliminating∆t gives

∆τB∆τA

=

√√√√√1− 2MrB

1− 2MrA

> 1. (2.2)

Now imagine that we are considering light waves propagating from A toB. Applying the above argument to two successive wavecrests shows thatthe above formula relates the period ∆τA of the waves emitted by A to theperiod ∆τB of the waves received by B. For light, the period is the same asthe wavelength (since c = 1): ∆τ = λ. Hence λB > λA, which means thatthe light undergoes a redshift as it climbs out of the gravitational field.

If B is at large radius, i.e. rB 2M , then we have

1 + z ≡ λBλA

=√√√√ 1

1− 2MrA

. (2.3)

Note that this diverges as rA → 2M . We showed above that a spherical starmust have radius R > 9

4M , so (taking rA = R) it follows that the maximumpossible redshift of light emitted from the surface of a spherical star is z = 2.

24

2.3 Geodesics of the Schwarzschild solutionLet xµ(τ) be an affinely parameterized geodesic with tangent vector uµ =

dxµ

dτ. Since k = ∂

∂tandm = ∂

∂φare Killing vectors fields we have the conserved

quantities1

E = −k · u =(

1− 2Mr

)dt

dτ, (2.4)

andh = m · u = r2 sin2 θ

dφ

dτ. (2.5)

For a timelike geodesic, we choose τ to be the proper time, then E has theinterpretation of energy per unit rest mass and h is the angular momentumper unit rest mass. (To see this, we would have to evaluate the expressionsfor E and h at large r where the metric is almost flat, so that one can useresults from special relativity.) For a null geodesic, the freedom to rescale theaffine parameter implies that E and h do not have direct physical significance.However, the ratio h

Eis invariant under this rescaling. For a null geodesic

which propagates to large r (where the metric is almost flat and the geodesicis a straight line), b =

∣∣∣ hE

∣∣∣ is the impact parameter, i.e. the distance of thenull geodesic from “a line through the origin”, more precisely the distancefrom a line of constant φ parallel (at large r) to the geodesic.

Now, the t and φ component of the geodesics equation are

d

dτ

[(1− 2M

r

)dt

dτ

]= 0, d

dτ

(r2 sin2 θ

dφ

dτ

)= 0. (2.6)

They simply say again that dEdτ

= 0 and dhdτ

= 0. The θ component is

d

dτ

(r2 dθ

dτ

)− r2 sin θ cos θ

(dφ

dτ

)2

= 0. (2.7)

Eliminating dφdτ

by using the definition of h, this can be written as

r2 d

dτ

(r2 dθ

dτ

)− h2 cos θ

sin3 θ= 0. (2.8)

One can define spherical polar coordinates on S2 in many different ways.It is convenient to rotate our (θ, φ) coordinates so that our geodesic has

1In fact, dEdτ = −d(k·u)

dτ = −u(k · u) = −∇u(k · u) = −uµ∇µ(kνuν) = −uνuµ∇µkν −kνu

µ∇µuν . The firs term vanishes because the Killing equation implies that ∇µkν isantisymmetric. The second term vanishes because of the geodesic equation uµ∇µuν = 0.The same reasoning shows that also dh

dτ = 0.

25

θ = π2 and dθ

dτ= 0 at τ = 0, i.e. the geodesic initially lies in, and is moving

tangentially to, the “equatorial plane” θ = π2 . We emphasize: this is just

a choice of the coordinates (θ, φ). Now, whatever r(τ) is (and we do notknow yet), (2.8) is a second order ordinary differential equation for θ withinitial conditions θ = π

2 ,dθdτ

= 0. One solution of this initial value problemis θ(τ) = π

2 for all τ . Standard uniqueness results for ordinary differentialequations guarantee that this is the unique solution. Hence, we have shownthat we can always choose our θ, φ coordinates so that the geodesic is confinedto the equatorial plane. We shall assume this henceforth.

Choosing τ to be the proper time in the case of a timelike geodesic, andthe proper distance in the case of a spacelike geodesic gives one final equationgµνu

µuν = −σ, where σ = 1, 0,−1 for a timelike, null, or spacelike geodesic,respectively. Explicitly this equation is

−(

1−2Mr

)(dt

dτ

)2

+(

1−2Mr

)−1(dr

dτ

)2

+r2[(dθ

dτ

)2

+ sin2 θ

(dφ

dτ

)2]. (2.9)

Rearranging it gives the equation for r(τ)

12

(dr

dτ

)2

+ V (r) = 12E

2, (2.10)

whereV (r) = 1

2

(1− 2M

r

)(σ + h2

r2

). (2.11)

Hence, the radial motion of the geodesic is determined by the same equationas a Newtonian particle of unit mass and energy E2

2 moving in a 1d potentialV (r).

2.4 Eddington-Finkelstein (EF) coordinatesConsider the Schwarzschild solution with r > 2M . Let us study the

simplest type of geodesics: radial null geodesics. “Radial” means that θ andφ are constant along the geodesic, so h = 0. By rescaling the affine parameterτ we can arrange that E = 1. The geodesic equation reduces to

dt

dτ=(

1− 2Mr

)−1

,dr

dτ= ±1, (2.12)

where the upper sign is for an outgoing geodesic (i.e. increasing r) andthe lower for an ingoing one. From the second equation it is clear that an

26

Figure 2.1: Regge-Wheeler radial coordinate

ingoing geodesic starting at some r > 2M will reach r = 2M in finite affineparameter. Dividing gives

dt

dr= ±

(1− 2M

r

)−1

. (2.13)

The right hand side has a simple pole at r = 2M and hence t divergeslogarithmically as r → 2M . To investigate what is happening at r = 2M ,we need to define the “Regge-Wheeler radial coordinate” r∗ by

dr∗ = dr

1− 2Mr

⇒ r∗ = r + 2M ln∣∣∣∣ r2M − 1

∣∣∣∣ , (2.14)

where we made a choice of constant of integration. (We are interested onlyin r > 2M for now, the modulus signs are for later use.) Note that r∗ ∼ rfor large r and r∗ → −∞ as r → 2M (see Figure 2.1). Along a radial nullgeodesic we have

dt

dr∗= ±1, (2.15)

sot∓ r∗ = constant. (2.16)

Let us define a new coordinate v by

v = t+ r∗, (2.17)

so that v is constant along ingoing radial null geodesics. Now let us use(v, r, θ, φ) as coordinates instead of (t, r, θ, φ). The new coordinates are called

27

ingoing Eddington-Finkelstein (EF) coordinates. We eliminate t by t = v −r∗(r) and hence

dt = dv − dr

1− 2Mr

. (2.18)

Substituting this into the metric gives

ds2 = −(

1− 2Mr

)dv2 + 2dvdr + r2dΩ2. (2.19)

Written as a matrix we have, in these coordinates,

gµν =

−(1− 2M

r

)1 0 0

1 0 0 00 0 r2 00 0 0 r2 sin2 θ

. (2.20)

Unlike the metric components in Schwarzschild coordinates, the componentsof the above matrix are smooth for all r > 0, in particular they are smooth atr = 2M . Furthermore, this matrix has determinant −r4 sin2 θ and hence isnon-degenerate for any r > 0 (except at θ = 0, π but this is just because thecoordinates (θ, φ) are not defined at the poles of the spheres). This impliesthat its signature is Lorentzian for r > 0 since a change of signature wouldrequire an eigenvalue of gµν passing through zero.

The Schwarzschild spacetime can now be extended through the surfacer = 2M to a new region with r < 2M . The metric (2.19) is also a solution ofthe vacuum Einstein equation in this new region. The metric components arereal analytic functions of the above coordinates, i.e. they can be expanded asconvergent power series about any point. If a real analytic metric satisfies theEinstein equation in some open set, then it will satisfy the Einstein equationeverywhere. Since we know that the (2.19) satisfies the vacuum Einsteinequation for r > 2M , it must also satisfy this equation for r > 0.

Note that the new region with 0 < r < 2M is spherically symmetric. Howis this consistent with Birkhoff’s theorem? For r < 2M , if we define r∗ by(2.14) and t by (2.17), it is possible to show that the metric (2.19) transformedto coordinates (t, r, θ, φ) becomes exactly the Schwarzschild solution (2.1) butnow with r < 2M .

Note that ingoing radial null geodesics in the EF coordinates have drdτ

=−1 (and constant v). Hence such geodesics will reach r = 0 in finite affineparameter. What happens there? The simplest non-trivial scalar constructedfrom the metric is RµνρσR

µνρσ and a calculation gives

RµνρσRµνρσ ∝ M2

r6 . (2.21)

28

This diverges as r → 0. Since this is a scalar, it diverges in all charts. There-fore, there exists no chart for which the metric can be smoothly extendedthrough r = 0. r = 0 is a so-called curvature singularity, where tidal forcesbecome infinite and the known laws of physics break down. Strictly speaking,r = 0 is not part of the spacetime manifold because the metric is not definedthere.

Recall that in r > 2M , the Schwarzschild solution admits the Killingvector field k = ∂

∂t. Let us work out what this is in ingoing EF coordinates.

Denote the latter by xµ so we have

k = ∂

∂t= ∂xµ

∂t

∂

∂xµ= ∂

∂v, (2.22)

since the EF coordinates are independent of t except for v = t + r∗(r). Weuse this equation to extend the definition of k to r ≤ 2M . Note that k2 = gvvso k is null at r = 2M and spacelike for 0 < r < 2M . Hence, the extendedSchwarzschild solution is static only in the r > 2M region.

2.5 Finkelstein diagram and the black holeregion

So far we have considered ingoing radial null geodesics, which have v =constant and dr

dτ= −1. Now consider the outgoing geodesics. For r > 2M in

Schwarzschild coordinates these have t − r∗ = constant. Converting to EFcoordinates gives v = 2r∗ + constant, i.e.

v = 2r + 4M ln∣∣∣∣ r2M − 1

∣∣∣∣+ constant. (2.23)

To determine the behaviour of geodesics in r ≤ 2M we need to use EFcoordinates from the start. In this way we find that radial null geodesics iningoing EF coordinates fall into two families: “ingoing” with v = constantand “outgoing” satisfying either (2.23) or r ≡ 2M .

It is interesting to plot the radial null geodesics on a spacetime diagram.Let t∗ = v − r so that the ingoing radial null geodesics are straight lines at45 in the (t∗, r) plane. This gives the Finkelstein diagram of Figure 2.2.

Knowing the ingoing and outgoing radial null geodesics lets us draw light“cones” on this diagram. Radial timelike curves have tangent vectors thatlie inside the light cone at any point.

The ”outgoing” radial null geodesics have increasing r if r > 2M . Butif r < 2M then r decreases for both families of null geodesics. Both reachthe curvature singularity at r = 0 in finite affine parameter. Since nothing

29

Figure 2.2: Finkelstein diagram

can travel faster than light, the same is true for radial timelike curves. Itis possible to extend this result to any timelike or null curve (irrespective ofwhether or not it is radial or geodesic) in r < 2M .

Before giving the result, let us recall some definitions.

Definition 6. A vector is causal if it is timelike or null (we adopt the con-vention that a null vector must be non-zero). A curve is causal if its tangentvector is everywhere causal.

At any point of a spacetime, the metric determines two light cones in thetangent space at that point. We would like to regard one of these as the“future” light-cone and the other as the “past” light-cone. We do this bypicking a causal vector field and defining the future light cone to be the onein which it lies.

Definition 7. A spacetime is time-orientable if it admits a time-orientation:a causal vector field T µ. Another causal vector Xµ is future-directed if it liesin the same light cone as T µ and past-directed otherwise.

Note that any other time orientation is either everywhere in the same lightcone as T µ or everywhere in the opposite light cone. Hence a time-orientablespacetime admits exactly two inequivalent time-orientations.

In the r > 2M region of the Schwarzschid spacetime, we choose k = ∂∂t

as our time-orientation. (We could just as well choose −k but this is relatedby the isometry t → −t and therefore leads to equivalent results.) k isnot a time-orientation in r < 2M because in ingoing EF coordinates we have

30

k = ∂∂v, which is spacelike for r < 2M . However, ± ∂

∂ris globally null (because

grr = 0) and hence defines a time-orientation. We just need to choose thesign that gives a time orientation equivalent to k for r > 2M . Note that

k ·(− ∂

∂r

)= −gvr = −1, (2.24)

and if the inner product of two causal vectors is negative then they lie in thesame light cone. Therefore we can use − ∂

∂rto define our time orientation for

r > 0.

Proposition 1. Let xµ(λ) be any future-directed causal curve (i.e. onewhose tangent vector is everywhere future-directed and causal.) Assumer(λ0) ≤ 2M . Then r(λ) ≤ 2M for λ ≥ λ0.

This result implies that no future-directed causal curve connects a pointwith r ≤ 2M to a point with r > 2M . More physically: it is impossible tosend a signal from a point with r ≤ 2M to a point with r > 2M , in particularto a point at r =∞.

Definition 8. A black hole is defined to be a region of spacetime from whichit is impossible to send a signal to infinity (we will define “infinity” moreprecisely later). The boundary of this region is the future event horizon.

Our result shows that points with r ≤ 2M of the extended Schwarzschildspacetime lie inside a black hole. However, it is easy to show that there doexist future-directed causal curves from a point with r > 2M to r =∞ (e.g.an outgoing radial null curve) so points with r > 2M are not inside a blackhole. Hence, r = 2M is the even horizon.

It is possible to show that r decreases along any timelike or null curve(irrespective of whether or not it is radial or geodesic) in r < 2M . Hence,no signal can be sent from a point with r < 2M to a point with r > 2M ,in particular to a point with r = ∞. This is the defining property of ablack hole: a region of an “asymptotically flat” spacetime from which it isimpossible to send a signal to infinity.

2.6 Gravitational collapseConsider the fate of a massive spherical star once it exhausts its nu-

clear fuel. The star will shrink under its own gravity. As mentioned above,Birkhoff’s theorem implies that the geometry outside the star is given by theSchwarzschild solution even when the star is time-dependent. If the star is

31

Figure 2.3: Finkelstein diagram for gravitational collapse

not too massive then eventually it might settle down to a white dwarf or neu-tron star. But if it is sufficiently massive then this is not possible : nothingcan prevent the star from shrinking until it reaches its Schwarzschild radiusr = 2M .

We can visualize this process of gravitational collapse on a Finkelsteindiagram. We just need to remove the part of the diagram corresponding tothe interior of the star. By continuity, points on the surface of the collapsingstar will follow radial timelike curves in the Schwarzschild geometry. This isshown in Figure 2.3.

It is possible to show that the total proper time along a timelike curvein r ≤ 2M cannot exceed πM . (For M = M this is about 10−5s.) Hencethe star will collapse and form a curvature singularity in finite proper timeas measured by an observer on the surface of the star.

Let us note the behaviour of the outgoing radial null geodesics, i.e. lightrays emitted from the surface of the star. As the surface approaches r = 2M ,light from the surface takes longer and longer to reach a distant observer.The oserver will never see the star cross r = 2M . Equation (2.3) shows thatthe redshift of this light diverges as r → 2M . So the distant observer willsee the star fading from view as r → 2M .

2.7 Detecting black holesThere are two important properties that underpin detection methods for

black holes:

• there is no upper bound on the mass of a black hole. This contrastswith cold stars, which have an upper bound around 3M.

32

• black holes are very small. A black hole has radius R = 2M . A solarmass black holes has radius 3km. A black hole with the same mass asthe Earth would have radius 0.9cm.

There are other systems which satisfy either one of these conditions. Forexample, there is no upper limit on the mass of a cluster of stars or a cloud ofgas. But these would have size much greater than 2M . On the other hand,a neutron star cannot be arbitrarily massive. It is the combination of a largemass concentrated into a small region which distinguishes black holes fromother kinds of object.

Since black hole do no emit radiation directly, we infer their existencefrom their effect on nearby luminous matter. For example, stars near thecentre of our galaxy are observed to be orbiting around the galactic centre.From the shapes of the orbits, one can deduce that there is an object withmass 4×106M at the centre of the galaxy. Since some of the stars get closeto the galactic centre, one can infer that this mass must be concentratedwithin a radius of about 6 light hours (6 × 109km, about the same size asthe Solar System) since otherwise these stars would be ripped apart by tidaleffects. The only object that can contain so much mass in such a small regionis a black hole.

Many other galaxies are also believed to contain enormous black holes attheir centres (some with masses greater than 109M). Black holes with massgreater than about 106M are referred to as supermassive. There appearsto be a correlation between the mass of the black hole and the mass ofits host galaxy, with the former typically about thousandth of the latter.Supermassive black holes do not form directly gravitational collapse of anormal star (since the latter cannot have a mass much greater than about100M). It is still uncertain how such large black holes form.

2.8 White holesWe defined ingoing EF coordinates using ingoing radial null geodesics.

What happens if we do the same thing with outgoing radial null geodesics?Starting with the Schwarzschild solution with r > 2M , let

u = t− r∗, (2.25)

so u = constant along outgoing radial null geodesics. Now introduce outgoingEddington-Finkelstein (EF) coordinates (u, r, θ, φ). The Schwarzschild metric

33

becomes

ds2 = −(

1− 2Mr

)du2 − 2dudr + r2dΩ2. (2.26)

Just as for the ingoing EF coordinates, this metric is smooth with non-vanishing determinant for r > 0 and hence can be extended to a new regionr ≤ 2M . Once again we can define Schwarzschild coordinates in r < 2M tosee that the metric in this region is simply the Schwarzschild metric. Thereis a curvature singularity at r = 0.

This r < 2M regions is not the same as the r < 2M region in the ingoingEF coordinates. An easy way to see this is to look at the outgoing radialnull geodesics, i.e. lines of constant u. We saw above (in the Schwarzschildcoordinates) that these have dr

dτ= 1 hence they propagate from the curvature

singularity at r = 0, through the surface r = 2M and then extend to larger. This is impossible for the r < 2M region we discussed previously sincethat regions is a black hole. It is also simple to show that k = ∂

∂uin outgoing

EF coordinates, and that the time-orientation which is equivalent to k forr > 2M is given by ∂

∂r.

The r < 2M region of the outgoing EF coordinates is a white hole: aregion which no signal from infinity can enter. The surface r = 2M is thepast event horizon: the boundary of the white hole region.

A white hole is the time reverse of a black hole. To see this, make thesubstitution u→ −v to see that the above metric is isometric to (2.19). Theonly difference is the sign of the time orientation. It follows that no signalcan be sent from a point with r > 2M to a point with r < 2M . Any timelikecurve starting with r < 2M must pass through the surface r = 2M withinfinite proper time.

White holes are believed to be unphysical. A black hole is formed from anormal star by gravitational collapse. But a white hole begins with a singu-larity, so to create a white hole one must first make a singularity. Black holesare stable objects:small perturbations of a black hole are believed to decay.Applying time-reversal implies that white holes must be unstable objects:small perturbations of a white hole become large under time evolution.

2.9 The Kruskal extensionWe have seen that the Schwarzschild spacetime can be extended in two

different ways, revealing the existence of a black hole region and a whitehole region. How are these different regions related to each other? Thisis answered by introducing a new set of coordinates. Start in the region

34

Figure 2.4: Kruskal diagram

r > 2M . Define Kruskal-Szekeres (KS) coordinates (U, V, θ, φ) by

U = −e− u4M , V = e

v4M , (2.27)

so U < 0 and V > 0. Note that

UV = −er∗2M = −e r

2M

(r

2M − 1). (2.28)

The right hand side is a monotonic function of r and hence this equationdetermines r(U, V ) uniquely. We also have

V

U= −e t

2M , (2.29)

which determines t(U, V ) uniquely. In Kruskal-Szekeres coordinates, the met-ric is

ds2 = −32M3e−r(U,V )

2M

r(U, V ) dUdV + r(U, V )2dΩ2. (2.30)

Let us now define the function r(U, V ) for U ≥ 0 or V ≤ 0 by (2.28). Thisnew metric can be analytically extended, with non-vanishing determinant,through the surfaces U = 0 and V = 0 to new regions with U > 0 and V < 0.

Let us consider the surface r = 2M . Equation (2.28) implies that eitherU = 0 or V = 0. Hence KS coordinates reveal that r = 2M is actually twosurfaces, that instersect at U = V = 0. Similarly, the curvature singularityat r = 0 corresponds to UV = 1, a hyperbola with two branches. Thisinformation can be summarized on the Kruskal diagram of Figure 2.4.

One should think of “time” increasing in the vertical direction on thisdiagram. Radial null geodesics are lines of constant U or V , i.e. lines at 45

35

Figure 2.5: Kruskal diagram for gravitational collapse. The region to theleft of the shaded region is not part of the spacetime.

to the horizontal. This diagram has four regions. Region I is the region westarted in, i.e. the region r > 2M of the Schwarzschild solution. Region IIis the black hole that we discovered using ingoing EF coordinates (note thatthese coordinates cover regions I and II of the Kruskal diagram). Region IIIis the white hole that we discovered using outgoing EF coordinates. Andregion IV is an entirely new region. In this region, r > 2M and so this regionis again described by the Schwarzschild solution with r > 2M . This is anew asymptotically flat region. It is isometric to region I: the isometry is(U, V )→ (−U,−V ). Note that it is impossible for an observer in region I tosend a signal to an observer in region IV. If they want to communicate thenone or both of them will have to travel into region II, where they will hit thesingularity.

Note that the singularity in region II appears to the future of any point.Therefore it is not appropriate to think of the singularity as a “place” insidethe black hole. It is more appropriate to think of it as a “time” at which tidalforces become infinite. The black hole region is time-dependent because, inSchwarzschild coordinates, it is r, not t, that plays the role of time.

Most of this diagram is unphysical. If we include a timelike worldlinecooresponding to the surface of a collapsing star and then replace the regionto the left of this line by the (non-vacuum) spacetime corresponding to theinterior of the star, then we get a diagram in which only regions I and IIappear (see Figure 2.5). Inside the matter, r = 0 is just the origin of polarcoordinates, where the spacetime is smooth.

Finally, let us discuss time translations in Kruskal coordinates. In Kruskal

36

coordinates

k = 14M

(V

∂

∂V− U ∂

∂U

), k2 = −

(1− 2M

r

). (2.31)

The result for k2 can be deduced either by direct calculation or by notingthat it is true for r > 2M (e.g. using Schwarzschild coordinates) and thatthe left hand side and right hand side are both analytic functions of U, V(since the metric is analytic). Hence the result must be true everywhere.

k is timelike in regions I and IV, spacelike in regions II and II, and null(or zero) where r = 2M , i.e. where U = 0 or V = 0. The orbits (integralcurves) of k in the Kruskal diagram are shown in Figure 2.6.

Figure 2.6: Integral curves of k in the Kruskal diagram

Note that the sets U = 0 and V = 0 are fixed (mapped into them-selves) by k and that k = 0 on the “bifurcation 2-sphere” U = V = 0. Hencepoints on the latter are also fixed by k.

2.10 Einstein-Rosen bridge

Recall equation (2.29): in region I we have VU

= −e t2M . Hence a surface

of constant t in region I is a straight line through the origin in the Kruskaldiagram (see Figure 2.7). These extend naturally into region IV.

37

Figure 2.7: Surface of constant t in the Kruskal diagram

Let us investigate the geometry of these hypersurfaces by using isotropiccoordinates. They are defined on a static spherically symmetric spacetime asa set of coordinates that satisfies the isotropic gauge, which means that themetric has the form [13]

ds2 = −f(ρ)2dt2 + g(ρ)2(dρ2 + ρ2dΩ2). (2.32)For the Schwarschild solution, let use define the coordinate ρ by

r = ρ+M + M2

4ρ =(

1 + M

2ρ

)2

ρ. (2.33)

Figure 2.8: Isotropic radial coordinate

Given r, there are two possible solutions for ρ. We choose ρ > M2 in

region I and 0 < ρ < M2 in region IV (see Figure 2.8). The Schwarzschild

metric in coordinates (t, ρ, θ, φ) becomes then

ds2 = −

(1− M

2ρ

)2

(1 + M

2ρ

)2dt2 +

(1 + M

2ρ

)4

(dρ2 + ρ2dΩ2), (2.34)

38

so these coordinates are isotropic.The transformation ρ → M2

4ρ is an isometry that interchanges regions Iand IV. Of course the above metric is singular at ρ = M

2 but we know thisis just a coordinate singularity. Now consider the metric of a 3d surface ofconstant t,

ds2(3) =

(1 + M

2ρ

)4

(dρ2 + ρ2dΩ2). (2.35)

Figure 2.9: Einstein-Rosen bridge

This metric is non-singular for ρ > 0. It defines a Riemannian 3-manifoldwith topology R × S2 (where R is parameterized by ρ). Its geometry canbe visualized by embedding the surface into a 4d Euclidean space. If wesuppress the θ direction, this gives the diagram in Figure 2.9. The geometryhas two asymptotically flat regions, ρ → ∞ and ρ → 0, connected by a“throat” with minimum radius 2M at ρ = M

2 . A surface of constant t in theKruskal spacetime is called an Einstein-Rosen bridge.

39

Chapter 3

Asymptotic flatness

In this chapter we will define a method to obtain a graphic representationof spacetimes that we have previously described. This will allow us to givethe rigorous definitions of “infinity” and black hole.

3.1 Conformal compactification and Penrosediagrams

Given a spacetime (M, g) we can define a new metric g = Ω2g whereΩ is a smooth positive function on M . We say that g is obtained from gby a conformal transformation1 The metrics g, g agree on the definitions of“timelike, “spacelike” and “null” so they have the same light cones, i.e. thesame causal structure.

The idea of conformal compactification is to choose Ω so that “points atinfinity” with respect to g are at “finite distance” w.r.t. the “unphysical”metric g. To do this we need Ω → 0 at “infinity”. More precisely, wetry to choose Ω so that the spacetime (M, g) is extendible, i.e. (M, g) ispart of a larger “unphysical” spacetime (M, g), which we usually refer to astheconformal compactification of M. M is then a proper subset of M withΩ = 0 on the boundary ∂M of M in M . This boundary ∂M corresponds to“infinity” in (M, g) and it is called conformal infinity. It is easiest to showhow this works by looking at some important examples.

1To be precise, a conformal transformation is defined as a change of coordinates x→ xsuch that g(x) → g(x) = Ω2(x)g(x), while the transformation considered in the text,g(x) → g(x) = Ω2(x)g(x), should be called a Weyl rescaling, because it is a physicalchange of the metric, not due to a change of coordinates. However, in GR it is almostalways called a conformal transformation and we will follow this convention.

41

Minkowski spacetime

Let (M, g) be a Minkowski spacetime. In spherical coordinates the metricis

g = −dt2 + dr2 + r2dω2. (3.1)(We denote the metric on S2 by dω2 ≡ dθ2 + sin2 θdφ2, and not by dΩ2, toavoid confusion with the conformal factor Ω.) Define retarded and advancedtime coordinates

u = t− r v = t+ r. (3.2)In what follows it will be important to keep track of the ranges of the differentcoordinates: since r ≥ 0 we have −∞ < u ≤ v <∞. The metric is

g = −dudv + 14(u− v)2dω2. (3.3)

Now define new coordinates (p, q) by

u = tan p v = tan q, (3.4)

so the range of p, q is finite:−π/2 < p ≤ q < π/2. This gives

g = (2 cos p cos q)−2[−4dpdq + sin2(q − p)dω2]. (3.5)

“Infinity” in the original coordinates corresponds to |t| → ∞ or r → ∞. Inthe new coordinates this corresponds to |p| → π/2 or |q| → π/2.

To conformally compactify this spacetime, define the positive function

Ω = 2 cos p cos q (3.6)

and letg = Ω2g = −4dpdq + sin2(q − p)dω2. (3.7)

Finally define

T = q + p ∈ (−π, π) R = q − p ∈ [0, π), (3.8)

sog = −dT 2 + dR2 + sin2 χdω2. (3.9)

Now dR2 + sin2Rdω2 is the unit radius round metric on S3. If we hadT ∈ (−∞,∞) and R ∈ [0, π] then g would be the metric of the Einstein staticuniverse (ESU) R×S3, given by the product of a flat time direction with theunit round metric on S3. The ESU can be visualized as an infinite cylinder,whose axis corresponds to the time direction. In our case the restrictions on

42

Figure 3.1: The Einstein static universe, R × S3, portrayed as a cylinder.The shaded region is conformally related to Minkowski space.

the ranges of p, q imply thatM is just the finite portion of the ESU displayedin Figure 3.1.

Let (M, g) denote the ESU. This is an extension of (M, g). The boundary∂M of M in M corresponds to “infinity” in Minkowski spacetime and it iscalled conformal infinity. The conformal infinity consists of five differentparts:

i+ = (T = π,R = 0),i0 = (T = 0, R = π),i− = (T = −π,R = 0),

I + = (T = π −R, 0 < R < π),I − = (T = −π +R, 0 < R < π).

(I + and I − are pronounced as “scri-plus” and “scri-minus”, respectively.)Note that i+, i0, and i− are actually points, since R = 0 and R = π arethe north and south poles of S3. Meanwhile I + and I − are actually nullhypersurfaces, which are parameterized by R ∈ (0, π) and (θ, φ) and hencehave the topology of cylinders R× S2 (since (0, π) is diffeomorphic to R).

It is convenient to project the above diagram onto the (T,R)-plane toobtain the Penrose diagram of Minkowski spacetime. The result is shown inFigure 3.2. Formally, a Penrose diagram is a bounded subset or R2 endowedwith a flat Lorentzian metric (in this case −dT 2 + dR2). Each point of theinterior of a Penrose diagram represents an S2. Points of the boundary can

43

Figure 3.2: The Penrose diagram for Minkowski spacetime. Light conesare at ±45 throughout the diagram.

represent an axis of symmetry (where r = 0) or points at “infinity” of ouroriginal spacetime with metric g.

Let us understand how the geodesics of g look on a Penrose diagram.This is easiest for radial geodesics, i.e. constant θ, φ. Remembering that thecausal structure of g and g is the same, we have that radial null curves ofg are null curves of the flat metric −dT 2 + dR2, i.e. straight lines at 45.These all start at I −, pass through the origin, and end at I +. For thisreason, I − is called past null infinity and I + is called future null infinity.Similarly, radial timelike geodesics, which have r = constant, start at i− andend at i+ so i− is called past timelike infinity and i+ is called future timelikeinfinity. Finally, radial spacelike geodesics, which have t = constant, startand end at i0 so i0 is called spatial infinity.

One could also plot the projection of non-radial curves onto the Penrosediagram. This projection makes things look “more timelike” w.r.t. the 2d flatmetric (because moving the final term in (3.9) to the left hand side gives anegative contribution). Hence a non-radial timelike geodesic remains timelikewhen projected and a non-radial null curve looks timelike when projected.

The behaviour of geodesics has an analogue for fields. Roughly speaking,massless radiation “comes in from” I − and “goes out to” I +. For example,consider a massless scalar field ψ in Minkowski spacetime, i.e. a solutionof the wave equation ∇a∇aψ = 0. For simplicity, assume it is sphericallysymmetric, ψ = ψ(t, r). The general spherically symmetric solution of the

44

wave equation in Minkowski spacetime is

ψ(t, r) = 1r

(f(u) + g(v)) = 1r

(f(t− r) + g(t+ r)), (3.10)

where f and g are arbitrary functions. This is singular at r = 0 (and hencenot a solution there) unless g(x) = −f(x) which gives

ψ(t, r) = 1r

(f(u)− f(v)) = 1r

(F (p)− F (q)), (3.11)

where F (x) = f(tan x). If we let F0(q) denote the limiting value of rψ onI − (where p = −π/2) then we have F (−π/2) − F (q) = F0(q) so F (q) =F (−π/2)− F0(q). Hence we can write the solution as

ψ = 1r

(F0(q)− F0(p)), (3.12)

which is uniquely determined by the function F0 governing the behaviour ofthe solution I −. Similarly, it is uniquely determined by the behaviour atI +.

Kruskal spacetime

In this case, we know that the spacetime (M, g) has two asymptoticallyflat regions. It is natural to expect that “infinity” in each of these regions hasthe same structure as in Minkowski spacetime. Hence we expect “infinity” inKruskal spacetime to correspond to two copies of infinity in Minkowski space-time. To construct the Penrose diagram for Kruskal we would define newcoordinates P = P (U) and Q = Q(V ) (so that lines of constant P or Q areradial null geodesics) such that the range of P,Q is finite, say (−π

2 ,π2 ), then

we would need to find a conformal factor Ω so that the resulting unphysicalmetric g can be extended smoothly onto a bigger manifold M (analogous tothe Einstein static universe we used for Minkowski spacetime). M is then asubset of M with a boundary that has 4 components, corresponding to placeswhere either P or Q is ±π

2 . We identify these 4 components as future andpast null infinity in region I, which we denote as I + and I −, respectively,and future and past null infinity in region IV, which we denote again by I +

and I −, respectively.

45

Figure 3.3: The Penrose diagram for the Kruskal spacetime

Doing this explicitly is fiddly. Fortunately we do not need to do it: nowthat we have understood the structure of infinity we can deduce the formof the Penrose diagram from the Kruskal diagram obtained above. This isbecause both diagrams show radial null curves as straight lines at 45. Theonly important difference is that “infinity” corresponds to a boundary of thePenrose diagram. It is conventional to use the freedom in choosing Ω toarrange that the curvature singularity at r = 0 is a horizontal straight linein the Penrose diagram. The result is shown in Figure 3.3.

In contrast with the conformal compactification of Minkowski spacetime,it turns out that the unphysical metric is singular at i±. This can be under-stood because lines of constant r meet at i±, and this includes the curvaturesingularity r = 0. Less obviously, it turns out that one cannot choose Ω tomake the unphysical metric smooth at i0.

Spherically symmetric collapse

The Penrose diagram for a spherically symmetric gravitational collapseis easy to deduce from the form of the Kruskal diagram and it is shown inFigure 3.4.

46

Figure 3.4: The Penrose diagram for a black hole formed from a collapsingstar. The shaded region contains matter and is described by the interiorsolution. The exterior region is described by the Schwarzschild solution

3.2 Asymptotic flatness

In this chapter, we will define what it means for a spacetime to beasymptotically flat. An asymptotically flat spacetime is one that “looks likeMinkowski spacetime at infinity”. In this section we will define this precisely.Infinity in Minkowski spacetime consists of I ±, i±, i0. However, we sawthat i± are singular points in the conformal compactification of the Kruskalspacetime. Since we want to regard the latter as asymptotically flat, we can-not include i± in our definition of asymptotic flatness. We also mentionedthat i0 is not smooth in the Kruskal spacetime so we will also not includei0. (However, it is possible to extend the definition to include i0, see [10] fordetails.) So we will define a spacetime to be asymptotically flat if it has thesame structure for null infinity I ± as Minkowski spacetime, which is shownin Figure 3.5.

47

Figure 3.5: The part of Penrose diagram that describes “infinity” in everyasymptotically flat spacetime

Definition 9. A time-orientable spacetime (M, g) is asymptotically flat atnull infinity if there exists a spacetime (M, g) such that

1. There exists a positive function Ω onM such that (M, g) is an extensionof (M,Ω2g) (hence if we regard M as a subset of M then g = Ω2g onM).

2. Within M , M can be extended to obtain a manifold with boundaryM ∪ ∂M

3. Ω can be extended to a function on M such that Ω = 0 and dΩ 6= 0 on∂M

4. ∂M is the disjoint union of two components I + and I −, each diffeo-morphic to R× S2

5. No past (future) directed causal curve starting in M intersects I +

(I −)

6. I ± are “complete”. We will define this below.

Conditions 1, 2, 3 are just the requirement that there exists an appropriateconformal compactification. The condition dΩ 6= 0 ensures that Ω has afirst order zero on ∂M , as in the example discussed above. This is neededto ensure that the spacetime metric approaches the Minkowski metric atan appropriate rate near I ±. Conditions 4, 5, 6 ensure that infinity hasthe same structure as null infinity of Minkowski spacetime. In particular,condition 5 ensures that I + lies “to the future of M” and I − lies “to thepast of M”.

48

Let us now see how the above definition implies that the metric mustapproach the Minkowski metric near I + (of course I − is similar). Let ∇denote the Levi-Civita connection of g and Rab the Ricci tensor of g. It ispossible to show that on M

Rµν = Rµν + 2Ω−1∇µ∇νΩ + gµν gρσ(Ω−1∇ρ∇σΩ− 3Ω−2∂ρΩ∂σΩ) (3.13)

We will consider the case in which (M, g) satisfies the vacuum Einsteinequation. This assumption can be weakened: our results will apply also tospacetimes for which the energy-momentum tensor decays sufficiently rapidlynear I +. The vacuum Einstein equation is Rµν = 0. Multiply by Ω to obtain

ΩRµν + 2∇µ∇νΩ + gµν gρσ(∇ρ∇σΩ− 3Ω−1∂ρΩ∂σΩ) = 0 (3.14)

Since g and Ω are smooth at I +, the first three terms in this equation admita smooth limit to I +. It follows that so must the final term which impliesthat Ω−1gρσ∂ρΩ∂σΩ can be smoothly extended to I +. This implies thatgρσ∂ρΩ∂σΩ must vanish on I +, i.e. dΩ is null (w.r.t g) on I +. But dΩ isnormal to I + (as Ω = 0 on I +) hence I + must be a null hypersurface in(M, g).

Now the choice of Ω in our definition is far from unique. If Ω satisfies thedefinition then so will Ω′ = ωΩ where ω is any smooth function on M thatis positive on M ∪∂M . We can use this conformal gauge freedom to simplifythings further. If we replace Ω with Ω′ then we must replace g′µν = ωgµν . Theprimed version of the quantity we just showed can be smoothly extended toI + is then

Ω′−1g′ρσ∂ρΩ′∂σΩ′ = ω−3gρσ(Ω∂ρω∂σω + 2ω∂ρΩ∂σω + ω2Ω−1∂ρΩ∂σΩ)= ω−1(2nµ∂µ lnω + Ω−1gρσ∂ρΩ∂σΩ) on I + (3.15)

wherenµ = gµν∂νΩ (3.16)

is normal to I + and hence also tangent to the null geodesic generators ofI +. We can therefore choose ω to satisfy

nµ∂µ lnω = −12Ω−1gρσ∂ρΩ∂σΩ on I + (3.17)

since this is just an ordinary differential equation along each generator ofI +. Pick an S2 cross-section of I +, i.e. an S2 ⊂ I + which intersectseach generator precisely once. There is a unique solution of this differentialequation for any (positive) choice of ω on this cross-section. We have now

49

shown that the RHS of (3.15) vanishes on I +, i.e. that we can choose agauge for which

Ω−1gρσ∂ρΩ∂σΩ = 0 on I + (3.18)

Evaluating (3.14) on I + now gives

2∇µ∇νΩ + gµν gρσ∇ρ∇σΩ = 0 on I +. (3.19)

Contracting this equation gives gρσ∇ρ∇σΩ = 0. Substituting back in weobtain

∇µ∇νΩ = 0 on I + (3.20)

In particular we havenµ∇µn

ν = 0 on I + (3.21)

so, in this conformal gauge, n is tangent to affinely parameterized (w.r.t. g)null geodesic generators of I +.

We introduce coordinates near I + as follows. In our choice of conformalgauge, we still have the freedom to choose ω on a S2 cross-section of I +.A standard result is that any Riemannian metric on S2 is conformal to theunit round metric on S2. Hence we can choose ω so that the metric onour S2 induced by g (i.e. the pull-back of g to this S2) is the unit roundmetric. Introduce coordinates (θ, φ) on this S2 so that the unit round metrictakes the usual form dθ2 + sin2 θdφ2. Now assign coordinates (u, θ, φ) to thepoint parameter distance u along the integral curve of n through the pointon this S2 with coordinates (θ, φ). This defines a coordinate chart on I +

with the property that the generators of I + are lines of constant θ, φ withaffine parameter u.

On I + consider the vector that are orthogonal (w.r.t. g) to the 2-spheresof constant u, i.e. orthogonal to ∂

∂θand ∂

∂φ. Such vectors form a 2d subspace

of the tangent space. In 2d, there are only two distinct null directions. Hencethere are two distinct null directions orthogonal to the 2-spheres of constantu. One of these is tangent to I + so pick the other one, which points intoM .Consider the null geodesics whose tangent at I + is in this direction. Extend(u, θ, φ) off I + by defining them to be constant along these null geodesics.Finally, since dΩ 6= 0 on I +, we can use Ω as a coordinate near I +. Wenow have a coordinate chart (u,Ω, θ, φ) defined in a neighbourhood of I +,with I + given by Ω = 0.

By construction we have a coordinate chart with n = ∂∂u

on I +. Hencenµ = δµu . But the definition of n implies ∂µΩ = gµνn

ν from which we deduceguµ = δΩ

µ at Ω = 0. Since (u, θ, φ) do not vary along the null geodesicspointing into M , the tangent vector to these geodesics must be proportionalto ∂

∂Ω . Since the geodesics are null we must therefore have gΩΩ = 0 for all Ω.

50

We also know that these geodesics are orthogonal to ∂∂θ

and ∂∂φ

on I + hencewe have gΩθ = gΩφ = 0 at Ω = 0.

Now consider the gauge condition (3.20). Writing this out in our coordi-nate chart, it reduces to

0 = ΓΩµν = 1

2 gΩρ(gρµ,ν + gρν,µ − gµν,ρ) = 1

2(guµ,ν + guν,µ − gµν,u) at Ω = 0(3.22)

where in the last equality we used gΩρ = gνρ(dΩ)ν = nρ = δρu. Taking µ andν to be θ or φ, we have guµ,ν = guν,µ = 0 so we learn that gµν,u = 0 at Ω = 0,i.e. the θ, φ components of the metric g on I + do not depend on u. Sincewe know that this metric is the unit round metric when u = 0, it must bethe unit round metric for all u.

We have now deduced the form of the unphysical metric g on I +:

g|Ω=0 = 2dudΩ + dθ2 + sin2 θdφ2 (3.23)

For small Ω 6= 0, the metric components will differ from the above resultby O(Ω) terms. However, by setting ν = Ω in (3.22) and taking µ to be u,θ or φ, we learn that guµ,Ω = 0 at Ω = 0 so smoothness of g implies thatguµ = O(Ω2) for µ = u, θ, φ.

Now we can write down the physical metric g = Ω−2g. It is convenientto define a new coordinate r = 1

Ω so that I + corresponds to r →∞. Aftera finite shift in r, r → r + f(u, θ, φ), the metric can be brought to the form

g = −du2 − 2dudr + r2(dθ2 + sin2 θdφ2) + . . . (3.24)

for large r, where the ellipsis refers to corrections that are subleading atlarge r. The leading term written above is simply the metric of Minkowskispacetime in outgoing Eddington-Finkelstein coordinates. If one convertsthis to inertial frame coordinates (t, x, y, z), defined by t = u+ r and (x, y, z)related to (r, θ, φ) as for spherical coordinates, so that the leading ordermetric is −dt2 + dx2 + dy2 + dz2, then the correction terms are all of order 1

r.

Hence the metric of an asymptotically flat spacetime does indeed approachthe Minkowski metric at I +.

Finally we can explain the condition 6 of our definition of asymptoticflatness. Nothing in the above construction guarantees that the range of uis (−∞,∞) as it is in Minkowski spacetime. We would not want to regard aspacetime as asymptotically flat if I + “ends” at some finite value of u. Recallthat u is the affine parameter along the generators of I + so if this happensthen the generators of I + would be incomplete. Condition 6 eliminates thispossibility.

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Definition 10. I + is complete if, in the conformal gauge (3.20), the genera-tors of I + are complete (i.e. the affine parameter extends to ±∞). Similarlyfor I −.

3.3 Definition of a black holeIn this section we want to give the rigorous definition of a black hole.

First, we need to give some other definitions regarding the causal structureof a spacetime.

Definition 11. Let (M, g) be a time-orientable spacetime and U ⊂M . Thechronological future of U , denoted I+(U), is the set of points of M whichcan be reached by a future-directed timelike curve starting on U . The causalfuture of U , denoted J+(U), is the union of U with the set of points of Mwhich can be reached by a future-directed causal curve starting on U . Thechronological past I−(U) and causal past J−(U) are defined similarly.

For example, let p be a point in Minkowski spacetime. Then I+(p) is theset of points strictly inside the future light cone of p and J+(p) is the set ofpoints on or inside the future light cone of p, including p itself.

Then, we need to introduce some basic topological concepts.

Definition 12. A subset S of M is open if, for any point p ∈ S, there existsa neighbourhood V of p (i.e. a set of points whose coordinates in some chartare a neighbourhood of the coordinates of p) such that V ⊂ S.

Small deformations of timelike curves remain timelike. Hence I±(U) areopen subsets of M .

Definition 13. The closure of a subset S ⊂ M , denoted by S, is the unionof the set and its limit points. The subset is closed if it contains its limitpoints, i.e. S = S. A point p ∈ S is an interior point if there exists aneighbourhood of p contained in S. The interior of S, denoted int(S) is theset of interior points of S. If S is open then int(S) = S. The boundary of Sis S = S\int(S).

Finally, we are in the position to give the definition of a black hole as aregion of an asymptotically flat spacetime from which it is impossible to senda signal to infinity. I + is a subset of our unphysical spacetime (M, g) so wecan define J−(I +) ⊂ M . The set of points of M that can send a signal toI + is M ∩ J−(I +). We define the black hole region to be the complementof this region, and the future event horizon to be the boundary of the blackhole region.

52

Definition 14. Let (M, g) be a spacetime that is asymptotically flat at nullinfinity. The black hole region is B = M\[M ∩ J−(I +)] where J−(I +)is defined using the unphysical spacetime (M, g). The future event horizonis H+ = B (the boundary of B in M), equivalently H+ = M ∩ ˙J−(I +).Similarly, the white hole region isW = M\[M ∩J+(I −)] and the past eventhorizon is H+ = W = M ∩ ˙J+(I −).

For example, in the Kruskal spacetime no causal curve from region II andIV can reach I + hence B is the union of regions II and IV (including theboundary U = 0 where r = 2M). H+ is the null surface U = 0. W is theunion of regions III and IV (including the boundary V = 0). H− is the nullsurface V = 0.

53

Chapter 4

Charged black holes

In this chapter, we will discuss the Reissner-Nordstrom spacetime, whichdescribes a charged, spherically symmetric black hole. Large imbalances ofcharge do not occur frequently in nature, so matter undergoing gravitationalcollapse would be expected to be almost neutral. Furthermore, a chargedblack hole would preferentially attract particles of opposite charge and hencegradually lose its charge. Hence charged black holes are believed not to be offundamental importance in Astrophysics and they have been studied mainlybecause they are a relatively simple model of a black hole with many peculiarfeatures, as we will see. However, there are two additional important reasonsto study charged black holes. Firstly, they have played an important role inquantum gravity, especially in String Theory. Secondly, recently evidence oftheir existence may have been found. These two reasons will be explained inmore detail in the introduction to Chapter 9.

4.1 The Reissner-Nordstrom (RN) solutionThe action for Einstein-Maxwell theory is

S = 116π

∫d4x√−g(R− F µνFµν), (4.1)

where F = dA with A a 1-form potential. Note that the normalisation ofF used here differs from the standard particle physics normalisation. TheEinstein equation is

Rµν −12Rgµν = 2

(F ρµ Fνρ −

14gµνF

ρσFρσ)

(4.2)

and the Maxwell equations are

∇µFµν = 0, dF = 0. (4.3)

55

There is a generalisation of Birkhoff’s theorem to this theory.

Theorem 3. The unique spherically symmetric solution of the Einstein-Maxwell equations with non-constant area radius function r is the Reissner-Nordstrom (RN) solution:

ds2 = −(

1 + 2Mr

+ e2

r2

)dt2 +

(1− 2M

r+ e2

r2

)−1

dr2 + r2dΩ2,

A = −Qrdt− P cos θdφ, e =

√Q2 + P 2.

(4.4)

This solution has 3 parameters: M,Q,P . It can be shown that thesecorrespond to the mass, electric charge and magnetic charge respectively(there is no evidence that magnetic charge occurs in nature but it is allowedby the equations).

Several properties are similar to the Schwarzschild solution: the RN so-lution is static, with timelike Killing vector field k = ∂

∂t. The RN solution

is asymptotically flat at null infinity in the same way as the Schwarzschildsolution.

If r is constant then the above theorem does not apply. In this case, oneobtains the so-called Robinson-Bertotti (AdS2 × S2) solution, that we willnot analyse.

To discuss the properties of this solution, it is convenient to define

∆ = r2 − 2Mr + e2 = (r − r+)(r − r−), r± = M ±√M2 − e2, (4.5)

so the metric isds2 = −∆

r2dt2 + r2

∆dr2 + r2dΩ2. (4.6)

If M < e then ∆ > 0 for r > 0 so the above metric is smooth for r > 0. ThePenrose diagram of this solution is shown in Figure 4.1. There is a curvaturesingularity at r = 0. This is a naked singularity, i.e. one from which signalscan reach I +, i.e. one that is not hidden behind an event horizon. Dy-namical formation of such a singularity is excluded by the cosmic censorshipconjecture:Cosmic censorship conjecture. Naked singularities cannot form in grav-itational collapse from generic, initially non singular states in an asymptot-ically flat spacetime obeying the dominant energy condition, i.e. −T µνV ν

is a future-directed causal vector for any future-directed timelike vector V(T is the energy-momentum tensor). (The requirement that the initial datamust be in some sense “generic” is important, as numerical experiments haveshown that finely-tuned initial conditions of the collapse are able to give rise

56

Figure 4.1: The Penrose diagram for the Reissner-Nordstrom spacetimewith M < e. There is a naked singularity at the origin.

to naked singularities.) In fact, if one considers a spherically symmetric ballof charged matter with M < e then electromagnetic repulsion dominatesover gravitational attraction so gravitational collapse does not occur. Notethat elementary particles (e.g. electrons) can have M < e but these areintrinsically quantum mechanical.

4.2 Eddington-Finkelstein (EF) coordinatesThe special case M = e will be discussed later so let us consider now

the case M > e. ∆ has simple zeros at r = r± > 0. These are coordinatesingularities. To see this, we can define Eddington-Finkelstein coordinatesin exactly the same way as we did for the Schwarzschild solution. Start withr > r+ and define

dr∗ = r2

∆dr. (4.7)

Integrating gives

r∗ = r + 12κ+

ln∣∣∣∣∣r − r+

r+

∣∣∣∣∣+ 12κ−

ln∣∣∣∣∣r − r−r−

∣∣∣∣∣+ constant, (4.8)

whereκ± = r± − r∓

2r2±

. (4.9)

57

Now letu = t− r∗, v = t+ r∗. (4.10)

In ingoing EF coordinates (v, r, θ, φ), the RN metric becomes

ds2 = −∆r2dv

2 + 2dvdr + r2dΩ2. (4.11)

This is now smooth for any r > 0 hence we can analytically continue themetric into a new region 0 < r < r+. There is a curvature singularity atr = 0. A surface of constant r has normal n = dr and hence is null whengrr = ∆

r2 = 0. It follows that the surfaces r = r± are null hypersurfaces.Then, it can be shown that r decreases along any future-directed causal

curve in the region r− < r < r+. It follows from this that no point in theregion r < r+ can send a signal to I + (since r = ∞ at I +). Hence thisspacetime describes a black hole. The black hole region is r ≤ r+ and thefuture event horizon is the null hypersurface r = r+.

Similarly, if one uses outgoing EF coordinates one obtains the metric

ds2 = −∆r2du

2 − 2dudr + r2dΩ2 (4.12)

and again one can analytically continue to a new region 0 < r ≤ r+ and thisis a white hole.

4.3 Kruskal-like coordinatesTo understand the global structure, define Kruskal-like coordinates

U± = −e−κ±u, V ± = ±eκ±v. (4.13)

Starting in the region r > r+, we use coordinates (U+, V +, θ, φ) to obtainthe metric

ds2 = −r+r−κ2

+r2 e−2κ+r

(r − r−r−

)1+ κ+|κ−|

dU+dV + + r2dΩ2, (4.14)

where r(U+, V +) is defined implicitly by

− U+V + = e2κ+r

(r − r+

r+

)(r−

r − r−

) κ+|κ−|

. (4.15)

The right hand side is a monotonically increasing function of r for r > r−.Initially we have U+ < 0 and V + > 0 which gives r > r+, but now we

58

Figure 4.2: Kruskal diagram for a part of the Reissner-Nordstrom solutionwith M > e

can analytically continue to U+ ≥ 0 or V + ≤ 0. In particular, the metric issmooth and non-degenerate when U+ = 0 or V + = 0. We obtain the diagramin Figure 4.2 that is very similar to the Kruskal diagram.

Just as for Kruskal, we have a pair of null hypersurfaces which intersect inthe ”bifurcation 2-sphere” U+ = V + = 0, where k = 0. Surfaces of constant tare Einstein-Rosen bridges connecting regions I and IV. The major differencewith the Kruskal diagram is that we no longer have a curvature singularity inregions II and II because r(U+, V +) > r−. However, from our EF coordinates,we know that it is possible to extend the spacetime into a region with r < r−.Hence the above spacetime must be extendible. Phrasing things differently,we know (from the EF coordinates) that radial null geodesics reach r = r−in finite affine parameter. Hence such geodesics will reach U+V + = −∞ infinite affine parameter so we have to investigate what happens there.

To do this, we start in region II and use ingoing EF coordinates (v, r, θ, φ)(as we know these cover regions I and II). We can now define the retardedtime coordinate u in region II as follows. First define a time coordinatet = v − r∗ in region II with r∗ defined by (4.8). The metric in coordinates(t, r, θ, φ) takes the static RN form given above, with r− < r < r+. Nowdefine u by u = t− r∗ = v − 2r∗. Having defined u in region II we can nowdefine the Kruskal coordinates U− < 0 and V − < 0 in region II using theformula above. In these coordinates, the metric is

ds2 = −r+r−κ2−r2 e

2|κ−|r(r+ − rr+

)1+ |κ−|κ+dU−dV − + r2dΩ2, (4.16)

59

where r(U−, V −) < r+ is given by

U−V − = e−2|κ−|r(r − r−r−

)(r+

r+ − r

) |κ−|κ+. (4.17)

This can now be analytically continued to U− > 0 or V − > 0, giving thediagram in Figure 4.3.

Figure 4.3: Kruskal diagram for a second part of the Reissner-Nordstromsolution with M > e

We now have new regions V and VI in which 0 < r < r−. These regionscontain the curvature singularity at r = 0 (U−V − = −1), which is time-like. Region III’ is isometric to region III and so, by introducing coordinates(U+′ , V +′) this can be analytically extended to the future to give further newregions I’, II’ and IV’ shown in Figure 4.4.

In this new diagram, I’ and IV’ are new asymptotically flat regions iso-metric to I and IV. This procedure can be repeated indefinitely, to the futureand past, so the maximal analytic extension of the RN solution contains in-finitely many regions. The resulting Penrose diagram is shown in Figure 4.5and extends to infinity in both directions. By an appropriate choice of con-formal factor, one can arrange that the singularity is a straight line.

60

Figure 4.4: Kruskal diagram for a third part of the Reissner-Nordstromsolution with M > e

Figure 4.5: The Penrose diagram for the Reissner-Nordstrom spacetimewith M > e. There are an infinite number of copies of the region outside theblack hole.

61

To understand better the physics around a RN black hole with M > e,let us imagine to be observers falling into the black hole from far away. r+is just like 2M in the Schwarzschild metric: at this radius r switches frombeing a spacelike coordinate to a timelike coordinate, and we necessarilymove in the direction of decreasing r. Witnesses outside the black hole alsosee the same phenomena that they would see outside an uncharged hole: theinfalling observer is seen to move more and more slowly, and is increasinglyredshifted. But the inevitable fall from r+ to ever-decreasing radii only lastsuntil we reach the null surface r = r−, where r switches back to being aspacelike coordinates and the motion in the direction of decreasing r can bearrested. Therefore we do not have to hit the singularity at r = 0; this isto be expected, since r = 0 is a timelike line (and therefore not necessarilyin our future). In fact we can choose either to continue on to r = 0, orbegin to move in the direction of increasing r back through the null surfaceat r = r−. Then r will once again be a timelike coordinate, but with reversedorientation, hence we are forced to move in the direction of increasing r. Wewill eventually be spit out past r = r+ once more, which is like emergingfrom a white hole into the rest of the universe. From here we can chooseto go back into the black hole (this time, a different hole than the one weentered in the first place) and repeat the voyage as many times as we like.This little story, taken by [9], can be understood by observing the Penrosediagram in Figure 4.5.

4.4 Extreme Reissner-Nordstrom solutionThe RN solution with M = e is called the extreme Reissner-Nordstrom

(RN) solution. The metric is

ds2 = −(

1− M

r

)2

dt2 +(

1− M

r

)−2

dr2 + r2dΩ2. (4.18)

In this case r+ = r− = M so this black hole has only one coordinatesingularity at r = M , which is the event horizon. Starting in the regionr > M one can define dr∗ = dr(

1−Mr

)2 , i.e.

r∗ = r + 2M ln∣∣∣∣r −MM

∣∣∣∣− M2

r −M, (4.19)

and introduce ingoing EF coordinates v = t+ r∗ so that the metric becomes

ds2 = −(

1− M

r

)2

dv2 + 2dvdr + r2dΩ2, (4.20)

62

which can be analytically extended into the region 0 < r < M , which is ablack hole region. Similarly one can use outgoing EF coordintes to uncovera white hole region. Each of these can be analytically extended across aninner horizon. The Penrose diagram is shown in Figure 4.6. The singularityat r = 0 is a timelike line, as in the previous case. So for this black hole anobserver can again avoid the singularity and continue to move to the futureto extra copies of the asymptotically flat region.

Figure 4.6: The Penrose diagram for the extreme Reissner-Nordstromspacetime. There is a singularity at the origin and an infinite number ofexternal regions.

A novel feature of this solution is that a surface of constant t is not anEinstein-Rosen bridge. Let us consider the proper length of a line of constantt, θ, φ from r = r0 > M to r = M :

∫ r0

M

dr

1− Mr

=∞. (4.21)

Hence a surface of constant t exhibits an “infinite throat”, as shown in Fig-ure 4.7.

63

Figure 4.7: Representation of a surface of constant t in the extremeReissner-Nordstrom spacetime. It presents an “infinite throat”.

4.5 Majumdar-Papapetrou (MP) solutionsA fascinating property of extremal black holes is that the mass is in

some sense balanced by the charge. More specifically, two extremal holeswith same-sign charges will attract each other gravitationally, but repel eachother electromagnetically, and it turns out that these effects precisely cancelfor an extreme RN black hole. To see this, introduce a new radial coordinateρ = r −M and assume P = 0. The extreme RN metric becomes

ds2 = −H−2dt2 +H2(dρ2 + ρ2dΩ2), H = 1 + M

ρ. (4.22)

This is a special case of the Majumdar-Papapetrou (MP) solution for the :

ds2 = −H(x)−2dt2 +H(x)2(dx2 + dy2 + dz2), A = H−1dt, (4.23)

where x = (x, y, z) and H obeys the Laplace equation in 3d Euclidean space

∇2H = 0. (4.24)

Choosing

H = 1 +N∑i=1

Mi

|x− xi|(4.25)

gives a static solution describing N extreme RN black holes of masses Mi atpositions xi (each of this is an S2, not a point).

64

Chapter 5

Rotating black holes

In this chapter we will discuss the Kerr solution, which describes a sta-tionary rotating black hole. The solution is considerably more complicatedthan the spherically symmetric solutions that we have discussed so far. Wewill start by explaining why the Kerr solution is believed to be the uniquestationary black hole solution.

5.1 Uniqueness theoremsBlack holes form by gravitational collapse, a time-dependent process.

However, we would expect an isolated black hole eventually to settle downto a time-independent equilibrium state (this is actually a very fast process,occurring on a time scale set by the radius of the black hole: microseconds fora solar mass black hole). Hence it is desirable to classify all such equilibriumstate, i.e. all possible stationary black hole solutions of the vacuum Einstein(or Einstein-Maxwell) equations.

First, we need to weaken slightly our definition of a “stationary space-time” to cover rotating black holes.Definition 15. A spacetime asymptotically flat at null infinity is stationaryif it admits a Killing vector field kµ that is timelike in a neighbourhood ofI ±. It is static if it is stationary and kµ is hypersurface-orthogonal.

It is conventional to normalize kµ so that k2 → −1 at I ±. Sometimesthe terms “strictly stationary” or “strictly static” are used if kµ is timelikeeverywhere, not just near I ±. So Minkowski spacetime is strictly static. TheKruskal spacetime is static but not strictly static (because kµ is spacelike inregions II and III).

So far, we have discussed only spherically symmetric black holes. Butrotating black holes cannot be spherically symmetric. However, they can

65

be axisymmetric, i.e. “symmetric under rotations about an axis”. For astationary spacetime we define this as follows:

Definition 16. A spacetime asymptotically flat at null infinity is stationaryand axisymmetric if

(i) it is stationary;

(ii) it admits a Killing vector field mµ that is spacelike near I ±;

(iii) mµ generates a 1-parameter group of isometries isomorphic to U(1);

(iv) [k,m] ≡ kµ∂µmν∂ν −mµ∂µk

ν∂ν = 0.

(We can also define the notion of axisymmetry in a non-stationary space-time by deleting (i) and (iv).) For such a spacetime, one can choose coordi-nates so that k = ∂

∂tand m = ∂

∂φwith φ ∼ φ+ 2π.

Now recall that a spherically symmetric vacuum spacetime must be static,by Birkhoff’s theorem. The converse of this is untrue: a static vacuum space-time does not need to be spherically symmetric, e.g. consider the spacetimeoutside a cube-shaped object. However, if the only object in the spacetimeis a black hole then we have

Theorem 4. (Israel 1967, Bunting & Masood 1987) If (M, g) isa static, asymptotically flat, vacuum black hole spacetime that is suitablyregular on, and outside an event horizon, then (M, g) is isometric to theSchwarzschild solution.

We will not attempt to describe precisely what “suitably regular” meanshere. This theorem establishes that static vacuum multi black hole solutionsdo not exist. There is an Einstein-Maxwell generalisation of this problem,which states that such a solution is either Reissner-Nordstrom or Majumdar-Papapetrou.

For stationary black holes, we have the following theorem:

Theorem 5. (Hawking 1973, Wald 1992) If (M, g) is a stationary, non-static, asymptotically flat analytic solution of the Einstein-Maxwell equationsthat is suitably regular on, and outside an event horizon, then (M, g) is sta-tionary and axisymmetric.

This is sometimes stated as “stationary implies axisymmetric” for blackholes. But this theorem has the unsatisfactory assumption that the spacetimeis analytic. This is unphysical: analyticity implies that the full spacetime isdetermined by its behaviour in the neighbourhood of a single point. If oneaccepts the above result, or simply assumes axisymmetry, then it is possibleto prove the following theorem:

66

Theorem 6. (Carter 1971, Robinson 1975) If (M, g) is a stationary,axisymmetric, asymptotically flat vacuum spacetime suitably regular on, andoutside,a connected event horizon then (M, g) is a member of the 2-parameterKerr family of solutions. The parameters are the mass M and the angularmomentum J .

These results lead to the expectation that the final state of gravitationalcollapse is generically a Kerr black hole. This implies that the final state isfully characterized by just 2 numbers: M and J . In contrast, the initial statecan be arbitrarily complicated. Nearly all information about the initial stateis lost during the gravitational collapse (either by radiation to infinity, or byfalling into the black hole), with just 2 numbers M,J required to describethe final state on, and outside the event horizon.

There is an Einstein-Maxwell generalization of the above theorem, whichstates that (M, g) should belong to the 4-parameter Kerr-Newman solutiondescribed in the next section.

5.2 The Kerr-Newman (KN) solutionThis is a rotating, charged solution of Einstein-Maxwell theory. In Boyer-

Lindquist (BL) coordinates, it is

ds2 =− ∆− a2 sin2 θ

Σ dt2 − 2a sin2 θr2 + a2 −∆

Σ dtdφ

+(

(r2 + a2)2 −∆a2 sin2 θ

Σ

)sin2 θdφ2 + Σ

∆dr2 + Σdθ2,

A =− Qr(dt− a sin2 θdφ) + P cos θ(adt− (r2 + a2)dφ)Σ ,

(5.1)

where

Σ = r2 + a2 cos2 θ, ∆ = r2 − 2Mr + a2 + e2, e =√Q2 + P 2. (5.2)

At large r, the coordinates (t, r, θ, φ) reduce to spherical polar coordinates inMinkowski spacetime. In particular, (θ, φ) have their usual interpretation asangles on S2 so 0 < θ < π and φ ∼ φ + 2π. It can be shown that the KNsolution is asymptotically flat at null infinity.

The solution is stationary and axisymmetric with two commuting Killingvector fields:

k = ∂

∂t, m = ∂

∂φ. (5.3)

67

k is timelike near infinity although, as we will discuss, it is not globallytimelike. The solution possesses a discrete isometry t→ −t, φ→ −φ, whichsimultaneously reverses the direction of time and the sense of rotation.

The solution has 4 parameters: M,a,Q, P . M can be identified with themass, Q with the electric charge, P with the magnetic charge, and a = J

M

where J can be identified with the angular momentum. When a = 0 the KNsolution reduces to the RN solution. Note that φ→ −φ has the same effectas a→ −a so there is no loss of generality in assuming a ≥ 0.

5.3 The Kerr solutionIf we set Q = P = 0 in the KN solution we get the Kerr solution of the

vacuum Einstein equation. Let us analyze the structure of this solution. Aswe did for the RN spacetime, we write

∆ = (r − r+)(r − r−), r± = M ±√M2 − a2. (5.4)

The solution with M < a describes a naked singularity, which is not allowedby the cosmic censorship conjecture. So let us assume M > a (and discussM = a later). The metric is singular at θ = 0, π but these are just the usualcoordinate singularities of spherical polars. The metric is also singular at∆ = 0 (i.e. r = r±) and at Σ = 0 (i.e. r = 0, θ = π

2 ). Starting in theregion r > r+, the first singularity we have to worry about is at r = r+. Wewill now show that this is a coordinate singularity. To see this, define Kerrcoordinates (v, r, θ, χ) for r > r+ by

dv = dt+ r2 + a2

∆ dr, dχ = dφ+ a

∆dr, (5.5)

which implies that in the new coordinates we have χ ∼ χ+ 2π and

k = ∂

∂v, m = ∂

∂χ. (5.6)

The metric is

ds2 =− ∆− a2 sin2 θ

Σ dv2 + 2dvdr − 2a sin2 θr2 + a2 −∆

Σ dvdχ

− 2a sin2 θdχdr +(

(r2 + a2)2 −∆a2 sin2 θ

Σ

)sin2 θdχ2 + Σdθ2.

(5.7)

The normal to the null hypersurface r = r+ is

ξ = k + ΩHm, (5.8)

68

whereΩH = a

r2+ + a2 . (5.9)

Just as for the RN solution, the region r ≤ r+ is (part of) the black holeregion of this spacetime with r = r+ (part of) the future event horizon H+.

In BL coordinates we have ξ = ∂∂t

+ ΩH∂∂φ. Hence ξµ∂µ(φ − ΩHt) = 0

so φ = ΩHt + constant on orbits of ξ rotating with angular velocity ΩH

with respect to to an stationary observer (i.e. someone on an orbit of k). Inparticular, they rotate with this angular velocity w.r.t. a stationary observerat infinity. Since ξ is tangent to the generators of H+1, it follows that thesegenerators rotate with angular velocity ΩH w.r.t. a stationary observer atinfinity, so we interpret ΩH as the angular velocity of the black hole.

5.4 Maximal analytic extensionThe Kerr coordinates are analogous to the ingoing EF coordinates we

used for RN. One can similarly define coordinates analogous to retarded EFcoordinates and use these to construct an analytic extension into a whitehole region. Then, just as for RN, one can define Kruskal-like coordinatesthat cover all of these regions, as well as a new asymptotically flat region, i.e.there are regions analogous to regions I and IV of the analytically extendedRN solution.

Just as for the RN solution, the spacetime can be analytically extendedacross null hypersurfaces at r = r− in regions II and III. The resulting max-imal analytic extension is similar to that of the RN spacetime except forthe behaviour near the singularity. In the Kerr case, it turns out that thecurvature singularity has the structure of a ring and by passing through thering one can enter a new asymptotically flat region, which is described bythe Kerr solution but with r < 0. As a result ∆ never vanishes, so there areno horizons in the new asymptotically flat spacetime. One also finds that mµ

becomes timelike near the singularity. The orbits of m are closed (becauseφ ∼ φ + 2π) hence there are closed timelike curves near the singularity, i.e.time travel is possible.

The Kerr solution is not spherically symmetric so one cannot draw aPenrose diagram for it. However, if one considers the submanifold of thespacetime corresponding to the axis of symmetry (θ = 0 or θ = π) then,since this submanifold is two-dimensional, one can draw a Penrose diagram

1This is a property of null hypersurfaces: they can be divided into a set of null geodesics,called the generators of the hypersurface, and the normal to the hypersurface is tangentto these generators.

69

for it. Note that this submanifold is “totally geodesic”, i.e. a geodesic initiallytangent to it will remain tangent. (The same is true for the “equatorial plane”θ = π

2 .) The resulting Penrose diagram takes the form shown in Figure 5.1.

Figure 5.1: The Penrose diagram for the Kerr spacetime with M > a. Aswith the analogous charged solution, there are an infinite number of copiesof the region outside the black hole.

When we studied the Schwarzschild solution, we saw that it describesthe metric outside a spherical star. This was a consequence of Birkhoff’stheorem. In contrast, the Kerr solution does not describe the spacetimeoutside a rotating star. This solution is expected to describe only the “finalstate” of the gravitational collapse. One cannot obtain a solution describinggravitational collapse to form a Kerr black hole simply by “gluing in” a ballof collapsing matter as we did for Schwarzschild. In particular, the spacetimeduring such collapse would be non-stationary because the collapse would leadto emission of gravitational waves.

Finally, the special case M = a is called the extreme Kerr solution. It isa black hole solution with several properties similar to those of the extremeRN solution. In particular, surfaces of constant t exhibit an “infinite throat”.

70

5.5 The ergosphere and Penrose processIn BL coordinates, let us consider the norm of the Killing vector field k:

k2 = gtt = −∆− a2 sin2 θ

Σ = −(

1− 2Mr

r2 + a2 cos2 θ

). (5.10)

Hence k is timelike in region I if and only if r2 − 2Mr + a2 cos2 θ > 0, i.e. ifand only if r > M +

√M2 − a2 cos2 θ. Hence k is spacelike in the following

region outside H+:

r+ = M +√M2 − a2 < r < M +

√M2 − a2 cos2 θ. (5.11)

This region is called the ergoregion. Its surface is called the ergosphere (seeFigure 5.2). The latter instersects H+ at the poles θ = 0, π.

Figure 5.2: Horizon structure around the Kerr solution

A stationary observer is someone with 4-velocity parallel to k. Suchobservers do not exist in the ergosphere because k is spacelike there. Anycausal curve in the ergosphere must rotate (relative to observers at infinity)in the same direction as the black hole.

Consider a particle with 4-momentum P ν = µuν (where µ is the rest massand uν is the 4-velocity). Let the particle approach a Kerr black hole alonga geodesic. The energy of the particle according to the stationary observerat infinity is the conserved quantity along the geodesic

E = −k · P. (5.12)

Suppose that the particle decays at a point p inside the ergosphere into twoother particles with 4-momenta P µ

1 and P µ2 . From the equivalence principle,

we know that the decay must conserve 4-momentum (because we can usespecial relativity in a local inertial frame at p) hence

P µ = P µ1 + P µ

2 ⇒ E = E1 + E2, (5.13)

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where Ei = −k ·Pi. Since kµ is spacelike within the ergoregion, it is possiblethat E1 < 0. We must then have E2 = E + |E1| > E. It can be shown thatthe first particle must fall into the black hole and the second one can escapeto infinity. The remarkable result is that this particle emerges from theergoregion with greater energy than the particle that was sent in. Energy isconserved because the particle that falls into the black hole carries in negativeenergy, so the energy (mass) of the black hole decreases. This Penrose processis a method for extracting energy from a rotating black hole.

How much energy can be extracted in this process? A particle crossingH+ must have −P · ξ ≥ 0 because both P µ and ξµ are future-directed causalvectors. But ξ = k + ΩHm hence

E − ΩHL ≥ 0, (5.14)

where E is the energy of the particle and

L = m · P (5.15)

is its conserved angular momentum. Hence we have L ≤ EΩH (recall our

convention a > 0 so ΩH > 0). The particle carries energy E and angularmomentum L into the black hole. If the black hole now settles down to aKerr solution then this new Kerr solution will have slightly different massand angular momentum: δM = E and δJ = L. Therefore

δJ ≤ δM

ΩH

= 2M(M2 +√M4 − J2

JδM. (5.16)

This is equivalent to δMirr ≥ 0, where the irreducible mass is

Mirr =[

12(M2 +

√M4 − J2

)] 12

. (5.17)

Inverting this equation gives

M2 = M2irr + J2

4M2irr

≥M2irr. (5.18)

Hence in the Penrose process it is not possible to reduce the mass of theblack hole below the value of Mirr: there is a limit to the amount of energythat can be extracted.

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Part II

Asymptotic symmetries andsupertranslation memory

In Part I we described some fundamental results about black holes ob-tained without considering quantum effects. Taking quantum effects intoconsideration, Hawking was able to demonstrate that black holes evaporateand eventually disappear [6] emitting particles and radiation, the so-calledHawking radiation (for a review see [12]). This is a semi-classical propertyof black holes since the gravitational field is not quantized and every back-reaction on the metric is neglected.

Let us now consider gravitational collapse of matter to form a black holewhich then evaporates away completely, leaving thermal radiation. It shouldbe possible to arrange that the collapsing matter is in a definite quantumstate, i.e. a pure state rather than a density matrix, which means that itis described by many parameters. According to the uniqueness theorems inSection 5.1, the most general black hole formed after gravitational collapseis a Kerr black hole, i.e. it is described by the value of just two param-eters: mass and angular momentum. This apparent loss of information isnot particularly alarming because we could think that some information ishidden behind the event horizon, so that we cannot extract it. However,after the black hole evaporation the information seems truly lost. In fact,the final state is a mixed state, i.e. only describable in terms of a densitymatrix. Evolution from a pure state, before the collapse, to a mixed state,after the evaporation, is impossible according to the usual unitary time evo-lution in quantum mechanics. Namely one can not reconstruct the quantumstate of what has fallen into the black hole by performing measures on theemitted particles, since the outgoing radiation is in a mixed state and thusthe S−matrix of the process is non-unitary. Another way of saying this is:information about the initial state appears to be permanently lost in blackhole formation and evaporation. This unexpected failure of the basic lawsof Quantum Mechanics is usually known as the information loss paradox [7].Conceivably, the outgoing Hawking radiation responsible for evaporation maysomehow encode information about what state was originally used to makea black hole, but how this could happen is still unclear. Understanding thisparadox is considered by many to be a crucial step in building a sensibletheory of quantum gravity.

In this second part we will present a review of some recent research papersin order to show that this paradox is significantly weakened because of aneffect called supertranslation memory. In Chapter 6 we will study mainly [14]in order to define the BMS group of transformations that leave invariant themetric in a neighbourhood of the null infinity. In Chapter 7 we will review[15], in which an important effect related to the BMS group is presented.Finally, in Chapter 8 we will finally use all the previous knowledge to derivethe form of the final state of a black hole after the gravitational collapse.

75

Compere and Long [8] obtained this state for the Schwarzschild case. InChapter 9 I will apply their method to spacetime solutions describing chargedblack hole regions. As we will see, the emergence of a new field in the finalstate result will significantly weaken the information paradox.

76

Chapter 6

Bondi-Metzner-Sachs (BMS)group

In this chapter we will consider asymptotically flat spacetimes and we willdefine the asymptotic symmetries at null infinity as the group of transforma-tions that leaves invariant the metric in a neighbourhood of I +. A totallyanalogous definition can be given for I −. This group of transformationswas originally studied by Bondi, van der Burg, Metzner [16] and Sachs [17],which is the reason why it is called the BMS group. Following mainly [14]and [18], we will derive the form of the vector fields generating the groupand we will see that it is possible to consider an extension of the group thatinvolves singular transformation and Weyl rescalings. Finally, following [19],we will present a more precise form of the metric near I +, which will beuseful later in this work.

6.1 Asymptotic symmetries of asymptoticallyflat spacetimes at null infinity

In this section we will follow [14] to define the asymptotic symmetries ofasymptotically flat spacetimes at future null infinity, namely diffeomorphismsthat leave invariant the metric near I +. Everything is completely analogousfor I −.

In order to study properties of I +, we introduce, in a neighborhood ofI +, what is known as a retarded Bondi coordinate system: (x0 = u, x1 =r, x2 = θ, x3 = φ). These coordinates are chosen by respecting the followingproperties:

1. The hypersurfaces described by u = const. are null.

77

2. The radial coordinate r is the affine parameter along the null geodesicsof the constant u hypersurfaces.

3. The coordinates (θ, φ) on an S2 cross-section of I + are such that theunit round metric on this S2 takes the usual form dθ2 + sin2 θdφ2 up toa conformal transformation.

The coordinates defined in Chapter 3 are a particular example of Bondicoordinates. As discovered by Bondi, van der Burg and Metzner for axi-symmetric asymptotically flat spacetimes [16] and generalized by Sachs [17],the metric of every asymptotically flat spacetime solving Einstein’s equationsin Bondi coordinates can be written in the form:

ds2 = e2β V

rdu2 − 2e2βdudr + gAB(dxA − UAdu)(dxB − UBdu) (6.1)

where A,B,C, ... = 2, 3 and β, V , UA, gAB det(gAB)−1/2 are 6 functions of thecoordinates, with det(gAB) = r4b(u, θ, φ) for some fixed function b(u, θ, φ).These functions satisfy some fall-off conditions clearly detailed by Barnichand Troessaert [14]. The first one is

gABdxAdxB = r2γABdx

AdxB +O(r) (6.2)where the 2-dimensional metric γAB is conformal to the usual metric of the2-sphere, γABdxAdxB = e2ϕγABdx

AdxB = e2ϕ(dθ2 + sin2 θdφ2), with ϕ =ϕ(u, xA). The determinant condition reads b(u, xA) = det γAB = e4ϕ sin2 θ.

The rest of the fall-off conditions are given by

β = O(r−2), V

r= −2r∂uϕ− e−2ϕ + ∆ϕ+O(r−1), UA = O(r−2). (6.3)

Here DA denotes the covariant derivative associated to γAB. We denote byΓABC the associated Christoffel symbols and by ∆ = DADA the associatedLaplacian. Note that gABgBC = δAC and the condition on the determinantimplies

gAB∂rgAB = 4r−1

gAB∂ugAB = γAB∂uγAB = 4∂ϕgAB∂CgAB = γAB∂C γAB = γAB∂CγAB + 4∂Cϕ,

(6.4)

where γABγBC = δAC = γABγBC . In terms of the metric and its inverse, thefall-off conditions readguu = −2r∂uϕ− e−2ϕ + ∆ϕ+O(r−1), gur = −1 +O(r−2), guA = O(1),grr = 0 = grA, gAB = r2γAB +O(r),guu = 0 = guA, gur = −1 +O(r−2),grr = 2r∂uϕ+ e−2ϕ − ∆ϕ+O(r−1), grA = O(r−2), gAB = r2γAB +O(r)

(6.5)

78

With the choice ϕ = 0, that we will use in the main part of this work,Sachs studies the vector fields generating diffeomorphisms that leave invari-ant the form of the metric (6.5), namely the metric in a neighborhood ofI+, with these fall-off conditions [20]. More precisely, he finds the generalsolution to the asymptotic Killing equations

Lξgrr = 0, LξgrA = 0, LξgABgAB = 0, (6.6)Lξgur = O(r−2), LξguA = O(1), LξgAB = O(r), Lξguu = O(r−1), (6.7)

which define the asymptotic Killing vectors. If the metric of an asymptoticallyflat spacetime is written in the form (6.5), we say that the metric is in theBMS gauge, in the sense that it is possible to perform a diffeomorphism,generated by an asymptotic Killing vector, that leaves invariant the formof the metric (6.5), i.e. that transforms from a Bondi coordinate system toanother Bondi coordinate system.

For arbitrary ϕ, the general solution to (6.6) is given byξu = f,

ξA = RA + IA, IA = −∂Bf∫∞r dr′(e2βgAB),

ξr = −12r(ψ + χ− ∂BfUB + 2f∂uϕ),

(6.8)

with ∂rf = 0 = ∂rRA and where ψ = DAR

A, χ = DAIA. This gives the

expansion ξu = f, ξA = RA − 1

r∂Bfγ

BA +O(r−2),ξr = −r

(f∂uϕ+ 1

2 ψ)

+ 12∆f +O(r−1).

(6.9)

Using the formula for the Lie derivative of the metric, Lξgµν = ∇µξν +∇νξµwhere ∇µ is the covariant derivative associated to gµν , the first equation of(6.7) implies that

∂uf = f∂uϕ+ 12 ψ ⇐⇒ f = eϕ

[T + 1

2

∫ u

0du′e−ϕψ

], (6.10)

with T = T (θ, φ). The second one requires ∂uRA = 0 and thus RA =RA(θ, φ). The third one implies that RA is a conformal killing vector of γAB(and thus also of γAB), i.e. it satisfies the conformal Killing equation

LRγAB = DCRC γAB or, equivalently, DARB + DBRA = (DCR

C)γAB,(6.11)

where the indices on RA are lowered with γAB. The last of equations (6.7) isthen satisfied without additional conditions. For the computation, one uses

79

that ∆ = e−2ϕ∆ and ψ = ψ + 2RA∂Aϕ, with ∆ = DADA and ψ = DARA

where DA is the covariant derivative associated to γAB, and the followingproperties of conformal Killing vectors RA of the unit 2-sphere,

2DBDCRA = γCADBψ+γABDCψ−γBCDAψ−γBCDAψ+2RCγBA−2RAγBC .(6.12)

This implies in particular ∆RA = −RA and also ∆ψ = −2ψ.

6.2 BMS algebra and BMS groupBy definition, the algebra BMS is the semi-direct sum of the Lie algebra

of conformal Killing vectors RA ∂∂xA

of the unit 2-sphere with the abelianideal consisting of functions T (xA) on the 2-sphere. We denote the genericelement of this algebra by (R, T ) where R indicates RA. The bracket isdefined through

[(R1, T1), (R2, T2)] = (R, T )R = RB

1 ∂BRA2 −RB

1 ∂BRA2 , (6.13)

T = RA1 ∂AT2 −RA

2 ∂AT1 + 12(T1∂AR

A2 − T2∂AR

A1 ).

Let I + = R × S2 with coordinates u, θ, φ. On I +, consider the scalarfield ϕ and the vector fields

ξ(ϕ,R, T ) = f∂

∂u+RA ∂

∂xA, (6.14)

with f given in (6.10) and RA an u-independent conformal Killing vector ofS2. These vector fields are the exact Killing vectors of I +. It is straight-forward to check that they form a faithful representation of the BMS Liealgebra for the standard Lie Bracket, [ξ1, ξ2]µ ≡ ξν1∂ν ξ

µ2 − ξν2∂ν ξ

µ1 , i.e.

[ξ(ϕ,R1, T1), ξ(ϕ,R2, T2)] = ξ(ϕ, R, T ) (6.15)

.Consider now the modified Lie bracket

[ξ1, ξ2]M ≡ [ξ1, ξ2]− δgξ1ξ2 + δgξ2ξ1, (6.16)

where δgξ1ξ2 denotes the variation in ξ2 under the variation of the metricinduce by ξ1, δgξ1gµν = Lξ1gµν .

80

Theorem 7. Spacetime vectors ξ of the form (6.8), with RA(xB) a conformalKilling vector of the 2-sphere and f(u, xB) satisfying (6.10), i.e. generatingvectors of infinitesimal diffeomorphisms that preserve the metric near I + ,provide a faithful representation of the BMS Lie algebra when equipped withthe modified Lie bracket [·, ·]M .

If we denote by ξ the vector field ξ of the form (6.8) with R replaced byR and T replaced by T , this result tells us that

[ξ1, ξ2]M = ξ, (6.17)

with the obvious notation ξi = ξ(ϕ,Ri, Ti), i = 1, 2. Let us prove it. For theu component, there is no modification due to the change in the metric andthe result follows by direct computation: [ξ1, ξ2]uM = f , where f correspondsto f in (6.10) with T replaced by T and R by R. f is ξu so this gives theresult for the u component. Then, by evaluating Lξgµν , we find

δξϕ = 0,δξβ = ξα∂αβ + 1

2 [∂uf + ∂rξr + ∂AfU

A],δξU

A = ξα∂αUA + UA(∂uf + ∂BfU

B)− ∂BξAUB

−∂uξA − ∂rξA Vr + ∂BξrgABe2β.

(6.18)

It follows thatδgξ1ξ

A2 = −∂Bf2

∫∞r dr′e2β(2δξ1βgAB + Lξ1gAB),

δgξ1ξr2 = −1

2 [DA(δgξ1ξA2 )− ∂Af2δξ1U

A].(6.19)

We obtainlimr→∞

[ξ1, ξ2]AM = RA. (6.20)

We also need∂rξ

A = gABe2β∂Bf. (6.21)After some computations we obtain

∂r([ξ1, ξ2]AM) = gABe2β∂B f . (6.22)

From (6.22) and (6.21) we see that ∂r([ξ1, ξ2]AM) = ∂rξA. This equation,

together with (6.20), gives the result for the A components. Finally, for ther component, we need

∂r

(ξr

r

)= −1

2(∂rχ− ∂Bf∂rUB). (6.23)

81

We then findlimr→∞

[ξ1, ξ2]rMr

= −12( ˆψ + 2f∂uϕ), (6.24)

where ˆψ corresponds to ψ with RA replaced by RA, and

∂r

([ξ1, ξ2]rM

r

)= −1

2(∂r ˆχ− ∂B f∂rUB), (6.25)

where ˆχ corresponds to χ with f replaced by f . As before, from (6.25) and(6.23) we see that ∂r

( [ξ1,ξ2]rMr

)= ∂r

(ξr

r

). This last result and (6.24) give the

result for the r component, and the proof is complete.In terms of the standard complex coordinates z = eiφ cot θ

2 , z = e−iφ cot θ2

(here the bar denotes complex conjugation), the metric on the sphere isconformally flat

dθ2 + sin2 θdφ2 = 2γzzdzdz, γzz ≡2

(1 + zz)2 , (6.26)

and, since conformal Killing vectors are invariant under conformal rescalingsof the metric, the conformal Killing vectors of the unite sphere are the sameas the conformal Killing vectors of the Riemann sphere. In coordinates (z, z)the conformal Killing equation for RA = (Rz, Rz) on the 2-sphere becomes∂zR

z = 0 and ∂zRz = 0. Therefore, the general solution to the conformal

Killing equation is Rz = R(z), Rz = R(z), whit R and R independent func-tions of their arguments. The standard basis for conformal Killing vectorson the 2-sphere is chosen as

ln = −zn+1 ∂

∂z, ln = −zn+1 ∂

∂z, n ∈ Z. (6.27)

Hence R(z) and R(z) are linear combinations of zn+1 and zn+1, respectively.Note that, since n ∈ Z, the number of indipendent conformal transformationsis infinite. With the common choice ϕ = 0 (i.e. γAB = γAB and DA = DA),the asymptotic Killing vector fields that generate infinitesimal diffeomor-phisms of this kind on the spacetime metric with coordinates (u, r, z, z) are

ξR =12uDAR

A∂u −12r[DAR

A − u

2rDADA(DCR

C) +O(

1r2

)]∂r

+[RA − u

2rDA(DBR

B) +O(

1r2

)]∂A, RA = (R(z), R(z))

(6.28)

82

The action of the diffeomorphisms generated by (6.28) on the coordinates is

u′ = u− 12uDAR

A

r′ = r − 12r[DAR

A − u2rDAD

A(DCRC) +O

(1r2

)]z′ = z −

[R(z)− u

2rDz(DBR

B) +O(

1r2

)]z′ = z −

[R(z)− u

2rDz(DBR

B) +O(

1r2

)],

(6.29)

where RA = (R(z), R(z)). These transformations are called superrotationsbecause they generalize standard (Lorentz) rotations, as we will see below.

Let us consider now transformations generated by the arbitrary functionT (z, z). These are called supertranslations because they generalize standardtranslations, as we will see below. Let us choose to expand T (z, z) in termsof

Tm,n = (1 + zz)γzzzmzn, m, n ∈ Z. (6.30)In terms of the basis vectors ll ≡ (ll, 0) and Tm,n ≡ (0, Tm,n), the commutationrelations for the complexified BMS algebra read

[lm, ln] = (m− n)lm+n, [lm, ln] = (m− n)lm+n, [lm, ln] = 0,

[ll, Tm,n] =(l + 1

2 −m)Tm+l,n, [ll, Tm,n] =

(l + 1

2 − n)Tm,n+l.

(6.31)

With the common choice ϕ = 0 (i.e. γAB = γAB and DA = DA), the asymp-totic Killing vector fields that generate supertranslations on the spacetimemetric with coordinates (u, r, z, z) are

ξT = T∂u + 12

[DADAT +O

(1r

)]∂r −

[1rDAT +O

(1r2

)]∂A

= T∂u +[DzDzT +O

(1r

)]∂r

−[

1rDzT +O

(1r2

)]∂z −

[1rDzT +O

(1r2

)]∂z, T = T (z, z),

(6.32)

where we used DADAT = (1 + zz)2∂z∂zT = 2DzDzT in the r component.The action of the diffeomorphisms generated by (6.32) on the coordinates is

u′ = u− Tr′ = r −

[DzDzT +O

(1r

)]z′ = z +

[1rDzT +O

(1r2

)]z′ = z +

[1rDzT +O

(1r2

)],

(6.33)

83

where T = T (z, z). Note that the transformation of the retarded Bondi timeu is the generalization of a translation; it is a local translation, being depen-dent on the angular coordinates. This explains the name “supertranslations”.

Depending on the space of functions under consideration, there are thenbasically two options which define what is actually meant by BMS group. Thefirst choice consists in restricting oneself to the globally well-defined trans-formations on the unit or, equivalently, the Riemann sphere. This singles outthe global conformal transformations and the associated group is isomorphicto the six-dimensional group SL(2,C)/Z2, which is itself isomorphic to theproper, orthochronous Lorentz group SO(3, 1)+.

Note that the basis vectors ln and ln are non-singular at z = 0, z = 0only for n ≥ −1. To study the other ambiguous point of the Riemannsphere, z = ∞, z = ∞, let us perform the substitution z = − 1

ω, z = − 1

ω

and study ω, ω → 0. Using ∂z = (−ω2)∂ω and ∂z = (−ω)2∂ω, we haveln = −(−1/ω)n−1∂ω, ln = −(−1/ω)n−1∂ω. These vectors are non-singular atω = 0, ω = 0 only for n ≤ 1. Therefore, the only globally well-defined basisvectors (6.27) are

l−1, l0, l1, l−1, l0, l1, (6.34)which form a six-dimensional group, as they should. Hence, the “globalchoice” implies that R(z) and R(z) are linear combinations of 1, z, z2 and1, z, z2, respectively.

With the choice ϕ = 0, we are able to write the finite Lorentz transfor-mation of the coordinates in a neighbourhood of I + = R×S2 with complexparameters (a, b, c, d), which is generated by (6.14) with R(z) and R(z) linearcombinations of 1, z, z2 and 1, z, z2, respectively, and T (z, z) = 0 [21]:

u′ = Ku,

z′ = az+bcz+d ,

z′ = az+bcz+d ,

K = 1+zz(az+b)(az+b)+(cz+d)(cz+d) ,

ad− bc = 1.

(6.35)

The remaining transformations of the “global” BMS group, the super-translations, form an infinite-dimensional abelian normal subgroup, denotedby ST . The factor group obtained by quotienting the “global” BMS groupby the ST group is isomorphic to SO(3, 1)+, i.e.

BMS = SO(3, 1)+ n ST. (6.36)

As we said before, every supertranslation is generated by a T (z, z) that canbe expanded into Tm,n(z, z). However, if we restrict ourselves to globally

84

well-defined transformations on the Riemann sphere, we must require thatT (z, z) is regular on the Riemann sphere. All the Tm,n are regular at z = 0but only those with m ≤ 1 and n ≤ 1 are regular at z = ∞. Therefore, inthe “globally well-defined” case T (z, z) is a linear combination of Tm,n(z, z)with m ≤ 1, n ≤ 1. In this case, it is more common to expand T (z, z) interms of the spherical harmonics in stereographic coordinates:

Ylm(z, z) = αlm∑n

(−1)nzl−nzl−m−nn!(l −m− n)!(l − n)!(m+ n)! , (6.37)

where

αlm = (−1)m√

2l + 14π (l +m)!(l −m)! l!

(1 + zz)l (6.38)

and the summation index n takes the value betweenmax(0,−m) andmin(l, l−m). It is not difficult to check that all the spherical harmonics are globallywell-defined on the Riemann sphere. There exists a unique four-dimensionalsubgroup of ST which is a normal subgroup of the BMS group and it is calledthe asymptotic translation group. In the case of Minkowski spacetime, thisfour-dimensional subgroup consists precisely of the exact translational sym-metries of Minkowski spacetime. The terms with l = 0, 1 in the expansion ofT (z, z) into spherical harmonics give the transformations of this asymptotictranslation group. A similar feature for the rotations and boosts does not ex-ist. There exists no normal subgroup of the BMS group which is isomorphicto the Poincaré group. However, we will show below that the BMS algebracontains as subalgebra the Poincaré algebra.

With the choice ϕ = 0, we are able to write the finite supertranslation ofthe coordinates on I + = R× S2, which is generated by (6.14) with RA = 0and T (z, z) an arbitrary regular function on S2:

u′ = u− T (z, z),z′ = z,

z′ = z.

(6.39)

The second possible choice consists in focusing on local properties andallowing the set of all, not necessarily invertible conformal transformationson the 2-sphere. Everything is the same as for the “globally well-defined” caseexcept that RA∂A can be linear combinations of ln and ln for all n ∈ Z andT (z, z) can be a linear combinations of Tm,n(z, z) for all m,n ∈ Z. Therefore,singular superrotations and singular supertranslations are allowed.

To conclude this section, let us show that the BMS algebra containsas subalgebra the Poincaré algebra, which we identify with the algebra of

85

exact Killing vectors of the Minkowski metric equipped with the standardLie bracket.

Indeed, these vectors form the subspace of spacetime vectors (6.8) forwhich

1. β = 0 = UA = ϕ, Vr

= −1, gAB = r2γAB,

2. the relations in (6.7) hold with 0 on the right hand sides.

The former implies in particular that IA = −1rγAB∂Bf , while a first conse-

quence of the latter is that the modified Lie bracket reduces to the standardone.

Besides the previous conditions that RA is an u-independent conformalKilling vector of the 2-sphere, i.e. LRγAB = (DCR

C)γAB, and f = T + 12uψ

with ∂uT = 0 = ∂rT , we find the additional constraints

DA∂Bψ +DB∂Aψ = γAB∆ψ, (6.40)

DA∂BT +DB∂AT = γAB∆T, ∂AT = −12∂A(∆T ). (6.41)

In the coordinates z, z, these constraints are equivalent to ∂3zR = 0 = ∂3

z Rand ∂2

z T = 0 = ∂2z T , where T = (1 + zz)γzzT , so that the complexified

Poincaré algebra is spanned by the generators

l−1, l0, l1, l−1, l0, l1, T0,0, T1,0, T0,1, T1,1. (6.42)

The first six span the algebra of the proper, orthocronus Lorentz groupSO(3, 1)+, as seen before, and the last four span the algebra of the asymptotictranslation group. The non vanishing commutation relations read

[l−1, l0] = −l−1, [l−1, l1] = −2l0, [l0, l1] = −l1,[l−1, T1,0] = −T0,0, [l−1, T1,1] = −T0,1, [l−1, T0,1] = −T0,0, [l−1, T1,1] = −T1,0,

[l0, T0,0] = 12T0,0, [l0, T0,1] = 1

2T0,1, [l0, T1,0] = −12T1,0, [l0, T1,1] = −1

2T1,1,

[l0, T0,0] = 12T0,0, [l0, T0,1] = −1

2T0,1, [l0, T1,0] = 12T1,0, [l0, T1,1] = −1

2T1,1,

[l1, T0,0] = T1,0, [l1, T0,1] = T1,1, [l1, T0,0] = T0,1, [l1, T1,0] = T1,1. (6.43)

In particular for instance, the generators for translations can be written as12(T1,1 + T0,0) = 1, 1

2(T1,1 − T0,0) = cos θ, 12(T1,0 + T0,1) = sin θ cosφ, 1

2i(T1,0 −T0,1) = sin θ sinφ, that are linear combinations of the four lowest sphericalharmonics.

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6.3 BMS transformations with Weyl rescal-ings

More generally, one can also consider the transformations that leave theform of the metric (6.5) up to a conformal rescaling of gAB, i.e. up to a shiftof ϕ by ω(u, xA). These rescalings are sometimes called Weyl rescalings ongAB

1. They are generated by spacetime vectors satisfying

Lξgrr = 0, LξgrA = 0, LξgABgAB = 4ω, (6.44)Lξgur = O(r−2), LξguA = O(1), LξgAB = 2ωgAB +O(r),

Lξguu = −2r∂uω − 2ω−2ϕ + 2ω∆ϕ+O(r−1). (6.45)

Equations (6.44), (6.45) imply that the vectors ξ are given by (6.8) and(6.10) with the replacement ψ → ψ − 2ω:

ξu = f,

ξA = RA + IA, IA = −∂Bf∫∞r dr′(e2βgAB),

ξr = −12r(ψ − 2ω + χ− ∂BfUB + 2f∂uϕ),

(6.46)

with ∂rf = 0 = ∂rRA and where ψ = DAR

A, χ = DAIA, as before. This

gives the expansionξu = f, ξA = RA − 1

r∂Bfγ

BA +O(r−2),ξr = −r

(f∂uϕ+ 1

2 ψ)

+ 12∆f +O(r−1).

(6.47)

Moreover,

∂uf = f∂uϕ+ 12(ψ − 2ω)⇐⇒ f = eϕ

[T + 1

2

∫ u

0du′e−ϕ(ψ − 2ω)

]. (6.48)

With this replacement, the vector fields ξ(ϕ, ω, T,R) = f ∂∂u

+ RA ∂∂xA

onI + = R × S2 equipped with the modified bracket provide a faithful repre-sentation of the extension of the BMS algebra defined by elements (R, T, ω)and bracket [(R1, T1, ω1), (R2, T2, ω2)] = (R, T , ω), with R, T as in (6.13) andω = 0. Using the notation ξi = ξ(ϕ, ωi, Ri, Ti), i = 1, 2 and ˆξ = ξ(ϕ, ω =0, R, T ), this result tells us that

[ξ1, ξ2]M = ˆξ. (6.49)1This name is not very appropriate since a Weyl rescaling is not a coordinate transfor-

mation but a local change of the metric: g → g = Ω(x)2g.

87

Let us prove it. The A components of ξ are not influenced by the intro-duction of ω, therefore the result for the A component automatically followsfrom the previous result on the ξ vectors without ω. For the u componentwe need,

δgξ1f2 = ω1f2 + 1

2eϕ∫ u

0du′e−ϕ[−ω1(ψ2 − 2ω2) + 2RA

2 ∂Aω1]. (6.50)

Then, at u = 0 we get

[ξ1, ξ2]uM∣∣∣u=0

= eϕ|u=0 T , (6.51)

while direct computation shows that ∂u([ξ1, ξ2]uM) = f∂uϕ+12DBR

B = f∂uϕ+12

ˆψ. Now, from (6.10) with the replacement ψ → ψ − 2ω, we have ∂uξu =∂uf = f∂uϕ+ 1

2 ψ−ω. Therefore, remembering that ω = 0, ∂u ˆξu = f∂uϕ+ 12

ˆψ.Hence,

∂u([ξ1, ξ2]uM) = ∂uˆξu. (6.52)

Equations (6.51) and (6.52) give the result for the u component and thisconcludes the proof.

Following the same reasoning as before, one can also show the followingtheorem.

Theorem 8. Spacetime vectors ξ of the form (6.8) with the replacement ψ →ψ−2ω, i.e. generating vectors of infinitesimal diffeomorphisms that preservethe metric near I + up to a conformal rescaling of gAB, provide a faithfulrepresentation of the BMS Lie algebra when equipped with the modified Liebracket [·, ·]M .

Indeed, we have ξ = ξ+IA ∂∂xA

+ξr ∂∂r. Furthermore, [ξ1, ξ2]uM = [ξ1, ξ2]uM =

f = ξ and this gives the result for the u component. Then, the variationsof β, UA are still given by (6.18) (even when we introduce ω). We then havelimr→∞[ξ1, ξ2]AM = RA, ∂rξA = gABe2β∂Bf and, after some computations,∂r([ξ1, ξ2]AM) = gABe2β∂B f . As before, these three equations give the resultfor the A components. Finally, for the r component, we find limr→∞

[ξ1,ξ2]rMr

=−1

2( ˆψ + 2f∂uϕ), while ∂r(ξr

r

)= −1

2(∂rχ − ∂Bf∂rUB) and ∂r

( [ξ1,ξ2]rMr

)=

−12(∂r ˆχ− ∂B f∂rUB). Reasoning as before, the last three equations give the

result for the r component and conclude the proof.

88

6.4 More precise metric in a neighbourhoodof I +

By analyzing the Einstein’s equations it is possible to find the fall-offconditions to higher orders than those considered in Section 6.1 used to definethe BMS group. With the additional hypothesis that ϕ does not depend onu, i.e. ϕ = ϕ(xA) (we can make this hypothesis because we are interested inthe case ϕ = 0), in Bondi coordinates (u, r, z, z) we have [19]

gAB = r2γAB + rCAB +DAB + 14 γABC

CDC

DC + o(r−ε), (6.53)

for some symmetric tensors CAB, DAB, whose indices are raised with γAB,satisfying CA

A = 0 = DAA, which implies Czz = 0 = Dzz, and Czz is the

complex conjugate of Czz (this must be true because gABdzAdzB must bereal at every order in r). In addition, ∂uDAB = 0 and NAB ≡ ∂uCAB iscalled the Bondi news tensor and it characterizes the outgoing gravitationalradiation. Furthermore,

β = − 132r

−2CABC

BA −

112r

−3CABD

BA + o(r−3−ε), (6.54)

guA = 12DBC

BA + 2

3r−1[(ln 3 + 1

3)DBDBA + 1

4CABDCCCB +NA

]+ o(r−1−ε),

(6.55)

where NA(u, xA) is called the angular momentum aspect;V

r= −e−2ϕ + ∆ϕ+ r−12M + o(r−1−ε), (6.56)

where DA denotes the covariant derivative associated to γAB, as before.M(u, xA) is called the Bondi mass aspect (later we will denote it by mB)and it defines the local energy at retarded time u at the angle on S2 denotedby (z, z). These quantities are related by the components (u, u) and (u,A)of the Einstein’s equations on I +, which also give the evolution equationsof the mass and angular momentum aspects in retarded time u. If there isno flux of matter on I +, these evolution equations read

∂uM = −18N

ABN

BA + 1

4∆(e−2ϕ − ∆ϕ) + 14DADCN

CA, (6.57)

∂uNA = ∂AM+12C

BA∂B(e−2ϕ − ∆ϕ) + 1

16∂A[NBCC

CB ]− 1

4DACCBN

BC

− 14DB[CB

CNCA −NB

CCCA ]− 1

4DB[DBDCCCA − DADCC

BC ].(6.58)

89

In terms of the metric these fall-off conditions read

guu = −e−2ϕ + ∆ϕ+ 2Mr

+ o(r−1−ε),gur = −1 + CABC

BA

16r2 + CABCBA

6r3 + o(r−3−ε),guA = 1

2DBCBA + 2

3r

[(ln 3 + 1

3)DBDBA + 1

4CABDCCCB +NA

]+ o(r−1−ε),

grr = 0,grA = 0,gAB = r2γAB + rCAB +DAB + 1

4 γABCCDC

DC + o(r−ε).

(6.59)Using Czz = 0, we get CA

BCBA = CzzC

zz+CzzC zz but also Czz = 14(1+zz)4Czz

and C zz 14(1 + zz)4Czz, hence CA

BCBA = 2CzzCzz = 2CzzC zz. Moreover,

DBCBz = DzCzz and DBC

Bz = DzCzz. The same results are obviously valid

also for DAB and NAB. Therefore, with the choice ϕ = 0, we obtain

guu = −1 + 2Mr

+ o(r−1−ε),gur = −1 + CzzCzz

8r2 + CzzCzz

3r3 + o(r−3−ε),guz = 1

2DzCzz + 2

3r

[(ln 3 + 1

3)DzDzz + 14CzzDzC

zz +Nz

]+ o(r−1−ε),

guz = 12D

zCzz + 23r

[(ln 3 + 1

3)DzDzz + 14CzzDzC

zz +Nz

]+ o(r−1−ε),

grr = 0,grz = 0,grz = 0,gzz = rCzz +Dzz + o(r−ε),gzz = r2γzz + 1

4γzzCzzCzz + o(r−ε),

gzz = rCzz +Dzz + o(r−ε),(6.60)

If there is no flux of matter on I +, the evolution equations of M and NAB

read∂uM = −1

4NzzNzz + 1

4(D2zN

zz +D2zN

zz), (6.61)

∂uNz = ∂zM + 116∂z[CzzN

zz + CzzNzz]− 1

4(DzCzzNzz +DzCzzN

zz)

− 14Dz[CzzNzz − CzzN zz]− 1

4[∆DzCzz −D2zC

zz −DzDzCzz],

(6.62)

∂uNz = ∂zM + 116∂z[CzzN

zz + CzzNzz]− 1

4(DzCzzNzz +DzCzzN

zz)

− 14Dz[C zzNzz − CzzN zz]− 1

4[∆DzCzz −D2zC

zz −DzDzCzz].

(6.63)

90

Chapter 7

Supertranslation memory effect

In this chapter we will review [15] to introduce the gravitational memoryeffect, according to which the passage of a finite pulse of radiation or otherforms of energy through a region of spacetime produces a gravitational fieldwhich moves nearby detectors. The final positions of a pair of nearby detec-tors are generically displaced relative to the initial ones according to a simpleformula.

According to Bondi, Metzner, van der Burg [16] and Sachs [17], the clas-sical vacuum in general relativity is highly degenerate. The different vacuaare related by the supertranslations. In this chapter we will show that thepassage of radiation through a region induces a transition from one suchvacuum to another. An explicit formula (involving moments of the radia-tion energy flux) is derived for the supertranslation which relates the initialand final vacua. Moreover, relative positions and clock times of a familyof detectors stationed in the vacuum are shown to be related by the samesupertranslation. This observation provides a concrete operational meaningto BMS supertranslations.

The relative spatial displacement of nearby detectors following from theradiation-induced supertranslation is precisely the gravitational memory ef-fect. We will find that a certain family of nearby detectors undergo, inaddition to the spatial memory displacement, a relative time delay. We willrefer to the two effects combined as the supertranslation memory effect.

7.1 BMS vacuum transitions

We saw in Section 6.4 that the metric of an asymptotically flat spacetimein retarded Bondi coordinates, with the choice ϕ = 0, takes the asymptotic

91

form (6.60) or, equivalently,

ds2 =− du2 − 2dudr + 2r2γzzdzdz

+ 2mB

rdu2 + rCzzdz

2 + rCzzdz2 +DzCzzdudz +DzCzzdudz + . . . ,

(7.1)

where subleading terms are suppressed by powers of r1. We are consideringonly the first next-to-leading order terms. The Bondi mass aspect mB andCzz are related by the constraint equation Guu = 8πMTMuu on I +

∂umB = 14[D2

zNzz +D2

zNzz]− Tuu,

Tuu = 14NzzN

zz + 4π limr→∞

[r2TMuu ],(7.2)

where Nzz = ∂uCzz is the Bondi news , TMµν is the matter stress-energytensor and Tuu is the total energy flux through a given point on I +. Wesee that the Bondi mass decreases with retarded time due to gravitationalradiation and null matter leaving the bulk through null infinity. As we foundabove, the asymptotic form of the metric (7.1) is preserved by infinitesimalsupertranslations (6.33). Taking into account only the first next-to-leadingorder terms, the infinitesimal supertranslation is

u′ = u− T,r′ = r −DzDzT,

z′ = z + 1rDzT,

z′ = z + 1rDzT,

(7.3)

where T = T (z, z). The generating vector fields are given by (6.32)

ξT = T∂u +DzDzT∂r −1r

(DzT∂z +DzT∂z). (7.4)

The variation of the asymptotic data under a supertranslation is (we denoteLξT by LT )

LTmB = T∂umB, (7.5)LTCzz = TNzz − 2D2

zT. (7.6)

1In particular, the corrections 14r2CzzC

zzdudr+γzzCzzCzzdzdz contribute to the Ein-

stein’s equations at the same order.

92

Remember that two spacetimes related by supertranslations are physi-cally inequivalent, because BMS transformations are asymptotic symmetriesand not exact symmetries.

Consider a spacetime which, before some retarded initial time ui, is in astate asymptotically well-approximated by Schwarzschild, i.e.

mB = Mi, Czz = 0, for u < ui, (7.7)

while, after some retarded final time uf , it is nearly asymptotically Schwarzschild,i.e.

mB = Mf , Czz = constant 6= 0, for u > uf . (7.8)

During the intermediate interval ui < u < uf the Bondi news and/or thetotal energy flux Tuu is nonzero on I +2. For nonzero Mf the late timegeometry could for example be a stable star or a black hole.

The initial and final regions of I + before ui and after uf are in thevacuum in the sense that Nzz = 0: the radiative modes are unexcited.Christodoulou and Klainerman showed that [22] Nzz ∼ |u|−3/2 for |u| → ∞,in the case of spacetimes with small non-linear deviations from Minkowskispacetime called Christodoulou-Klainerman (CK) spacetimes, i.e. spacetimesof the same kind as those that we are analyzing but with final vanishing Bondimass Mf = 0. We assume the same fall-off condition to be valid in our caseof massive final state.

There exist other subtle constraints on Czz. Define the Weyl tensor com-ponent Ψ0

2 asΨ0

2 ≡ − limr→∞

(rCuzrzγzz), (7.9)

where Cµνρσ is the usual Weyl tensor. With the metric (7.1)

Ψ02 = −mB + 1

4CzzNzz + 1

4γzz(∂zDzCzz − ∂zDzCzz). (7.10)

Now denote by I +− the points on I + with u→ −∞. This region is the

past of I +. Similarly, denote by I ++ the points on I + with u → ∞. This

region is the future of I +. The constraints, found for CK spacetimes [24]and extended to final massive states, are

Ψ02

∣∣∣I +−

= −Mi (7.11)

2Generic spacetimes may have long time radiation tails outside this interval, but for ourpurposes in this chapter making the radiation flux outside the interval arbitrarily small isenough. In the next chapter we will assume passage of matter and radiation during thewhole gravitational collapse of a star, which starts at ui → −∞ and ends at uf → +∞.

93

andΨ0

2

∣∣∣I +

+= −Mf (7.12)

Remembering that Nzz|I +±, the real parts of these relations imply

mB|I +−

= Mi, (7.13)

mB|I ++

= Mf , (7.14)

as we have already imposed (so it is not necessary to take into considerationthese constraints). The imaginary parts yield

[∂zDzCzz − ∂zDzCzz]∣∣∣I +±

= 0, (7.15)

which is equivalent to

[D2zC

zz −D2zC

zz]∣∣∣I +±

= 0. (7.16)

This is also equivalent to [D2zCzz −D2

zCzz]|I +±

= 0 and the general solutionis

Czz|I +±

= −2D2zC(z, z) or Czz|I +

±= −2D2

zC(z, z). (7.17)

Since the Bondi news Nzz = ∂uCzz vanishes in the region u < ui and u > uf ,Czz does not depend upon u in this region, therefore we can extend thissolution to the entire region. In general, the two C(z, z) in u < ui andu > uf are different.

We discovered that the vacuum is not unique. It is characterized by anyu-independent Czz that can be written in the form (7.17). We will call C(z, z)the asymptotic supertranslation field.

Comparison with (7.6) evaluated in the region u < ui and u > uf (whereNzz = 0) implies that the different vacua are related by supertranslationsgenerated by T (z, z) under which C → C ′ = C + T . The supertranslationwhich relates the initial and final vacua can be determined by integrating theconstraint (7.2) over the transition interval ui < u < uf :

Mi −Mf = −14[D2

zCzz −D2

zCzz]u=ui

u=uf +∫ uf

uiduTuu. (7.18)

Using (7.16), we get

D2z(Czz(uf )− Czz(ui)) = 2

∫ uf

uiduTuu + 2(Mf −Mi). (7.19)

94

Finally, defining ∆mB = Mf −Mi and ∆Czz = Czz(uf ) − Czz(ui), we canwrite this equation as

D2z∆Czz = 2

∫ uf

uiduTuu + 2∆mB. (7.20)

We now want to translate this equation into one for the supertranslation∆C ≡ C ′ − C = T which produces such ∆Czz. We write D2

z∆Czz =(γzz)2D2

z∆Czz = −2(γzz)2D2zD

2z∆C, where in the second equality we used

(7.17). Using this result in (7.20) and multiplying by −12(γzz)2, we have the

desired equation

D2zD

2z∆C = −(γzz)2

( ∫ uf

uiduTuu + ∆mB

). (7.21)

The solution of this equation can be found in terms of the Green functionG(z, z; z′z′) for the differential operator D2

zD2z as follows:

∆C(z, z) =∫d2z′γz′z′G(z, z; z′, z′)

(∫ uf

uiduTuu(z′, z′) + ∆mB

), (7.22)

whereD2zD

2zG(z, z; z′, z′) = −γzzδ2(z − z′) + . . . , (7.23)

where δ2(z − z′) = δ(z − z′)δ(z − z′) and the remaining terms vanish whenmultiplied by

∫ ufuiduTuu+∆mB and integrated in d2z′ = dz′dz′. It is straight-

forward to verify that (7.22) is solution of (7.21). According to Stromingerand Zhiboedov [15], the Green function is

G(z, z; z′, z′) = − 1π

sin2 Θ2 log sin2 Θ

2 ,

sin2 Θ(z, z; z′, z′)2 = |z − z′|2

(1 + z′z′)(1 + zz′) .(7.24)

The corresponding ∆Czz = −2D2z∆C, still according to the same paper by

Strominger and Zhiboedov, is

∆Czz(z, z) = 2π

∫d2z′γz′z′

z − z′

z − z′(1 + z′z)2

(1 + z′z′)(1 + zz)3

(∫ uf

uiduTuu(z′, z′) + ∆mB

).

(7.25)To summarize, the passage of radiation through I + changes the vacuum

by a supertranslation. The supertranslation relating the initial and finalvacua is given in (7.22) by an integral of the total radiation flux over thetransition interval.

95

7.2 Gravitational memory effect and clock desyn-chronisation

In this section we will relate the supertranslations of the vacuum to thegravitational memory effect and the clock desynchronization due to the pas-sage of energy near null infinity. Towards this end we introduce a family ofobservers or detectors at large r that travel along worldlines at fixed radiusand angle:

XµBMS(s) = (s, r0, z0, z0), (7.26)

where r0 is large. We refer to them as BMS detectors. The assertion thatBMS diffeomorphisms are physically nontrivial, i.e. they are not exact sym-metries of the spacetime, is equivalent to the statement that it is meaningfulto discuss observations at a fixed value of z near I +, i.e. different observersat different values of z related by a BMS diffeomorphism would see differ-ent physical phenomena. Such observations are convenient as they behavesimply under the action of BMS transformations.

Let us consider what happens to a pair of these detectors when theyencounter a pulse of radiation passing to I +. Let us assume that they areat the same r0. Denote the initial angular coordinates of detector 1 by (z1, z1)and the initial angular coordinates of detector 2 by (z2, z2). We also assumethat the relative angular coordinate δz ≡ z1 − z2 is an infinitesimal of thesame order as 1

r0. So the coordinates of the detectors are

Xµ1 = (s, r0, z1, z1), Xµ

2 = (s, r0, z2, z2). (7.27)

Now we denote by γ(λ) the curve of minimal length, i.e. a geodesic pa-rameterized by λ, between X1 and X2. The vector describing the angularseparation between the two BMS detectors, namely the vector whose integralcurve is γ(λ), is given by the usual formula

ζµ = dXµ(λ)dλ

= (0, 0, δz, δz). (7.28)

The two detectors are initially separated by the norm of ζ evaluated at one ofthe two points, i.e. L2 = gµν(X1)ζµζν . In the initial state of our spacetime,i.e. with u < ui, mB = Mi and Czz = 0, we get

L = 2r0 |δz|1 + z1z1

. (7.29)

The pulse of radiation induce a change in the metric given by ∆Czz and∆mB, although the coordinates of the detectors and, consequently, ζµ do not

96

change. Computing the separation with the metric after uf gives, as we canread directly from (7.1),

L′2 = 4r20 |δz|

2

(+z1z1)2 + r0δz2∆Czz + r0δz

2∆Czz

= L2 + r0δz2∆Czz + r0δz

2∆Czz

= L2(

1 + r0δz2∆Czz + r0δz

2∆CzzL2

),

(7.30)

where ∆Czz = ∆Czz(z1, z1). So

L′ = L

√1 + r0δz2∆Czz + r0δz2∆Czz

L2 . (7.31)

Since L2 is order r20 and r0 is large, we can use the approximation

L′ ' L

(1 + r0δz

2∆Czz + r0δz2∆Czz

2L2

)

= L+ r0δz2∆Czz + r0δz

2∆Czz2L .

(7.32)

We get

∆L = L′ − L = r0δz2∆Czz(z1, z1) + r0δz

2∆Czz(z1, z1)2L , (7.33)

where ∆Czz(z1, z1) is given by (7.25). This is precisely the standard formulafor the gravitational memory effect discovered by Zeldovich and Polnarev[23].

Not only will the distances between BMS detectors be shifted, but if theyare equipped with initially synchronized clocks they will no longer be syn-chronised after the passage of radiation. This can be checked by sending alight ray from detector 1 to detector 2, stamping it with the time at detector 2and then returning it to detector 1. If the clocks remained synchronised, thetime stamp from detector 2 would be exactly midway between the light emis-sion and reception times at detector 1. A light ray emitted from (s, r0, z1, z1)will travel to (s + δu12, r0, z2, z2) in an infinitesimal retarded time intervalδu12 obeying ds2 = gµν(X1)Υµ

12Υν12 = 0, where now Υµ

12 = (δu12, 0, δz, δz)is the tangent vector to the trajectory of the light ray from detector 1 todetector 2. So, with the metric (7.1) after uf ,

−(δu12)2 + 2r20γzzδzδz+r0∆Czzδz2 + r0∆Czzδz2

+Dz∆Czzδu12δz +Dz∆Czzδu12δz = 0,(7.34)

97

where we omitted the term 2(Mi+∆mB)r0

(δu12)2 because it is an infinitesimal ofhigher order for large r0. On the other hand, on the return trip, the changein z has the opposite sign, i.e. the tangent vector to the trajectory of thelight ray from detector 2 to detector 1 is Υµ

21 = (δu21, 0,−δz,−δz). So, theretarded time interval δu21 obeys

−(δu21)2 + 2r20γzzδzδz+r0∆Czzδz2 + r0∆Czzδz2

−Dz∆Czzδu21δz −Dz∆Czzδu21δz = 0.(7.35)

Extracting δu12 and δu21, we obtain

δu12 − δu21 = Dz∆Czz +Dz∆Czz. (7.36)

Since this is nonzero, the clocks are not synchronised.An alternative way of computing the gravitational memory and clock

desynchronisation is as follows. The proper distance and time delay calcu-lated above is invariant under all diffeomorphisms, including BMS transfor-mations. We may therefore eliminate all ∆Czz and ∆Czz terms in the latetime metric by the inverse of the BMS supertranslation (7.22), which byconstruction obeys

2D2zT = ∆Czz, (7.37)

so that T = −∆C. This transformation will have the effect of resetting allthe clocks and relabelling the positions of the family of BMS observers by(7.3), i.e. the coordinates of BMS detectors will be

Xµ1,2 = (s+ ∆C, r0 +DzDz∆C, z1,2 −

1r0Dz∆C, z1,2 −

1r0Dz∆C), (7.38)

where ∆C and its derivatives are evaluated at (z1,2, z1,2). However, thissupertranslation transforms the initial separation vector ζ = δz∂z + δz∂zaccording to

δT ζ = LT ζ = [ξT , ζ], (7.39)

where ξT is given by equation (7.4) with T = −∆C:

ξT = −∆C∂u −DzDz∆C∂r + 1r

(Dz∆C∂z +Dz∆C∂z). (7.40)

. The u component of the commutator is

[ξT , ζ]u = ξνT∂νζu − ζν∂νξuT

= δz∂z∆C + δz∂z∆C= δzDz∆C + δzDz∆C.

(7.41)

98

It is evident that every derivative acting on ζ vanishes because it is a constantvector. So the only contribution come from the ζν∂νξT terms. This is truefor every component of the commutator. The r component is then

[ξT , ζ]r = −ζν∂νξrT= (δz∂z + δz∂z)DzDz∆C= δzD2

zDz∆C + δzD2

zDz∆C.

(7.42)

The last equality is not obvious but one can easily prove it by direct compu-tation. Similarly, the z component is

[ξT , ζ]z = −(γzz

rD2z∆Cδz+ 1 + zz

2r [2zDz∆C+(1+zz)DzDz∆C]δz), (7.43)

and the z component is

[ξT , ζ]z = −(γzz

rD2z∆Cδz+ 1 + zz

2r [2zDz∆C+(1+zz)DzDz∆C]δz). (7.44)

So we have

δT ζ = LT ζ =(δzDz∆C + δzDz∆C)∂u + (δzD2zD

z∆C + δzD2zD

z∆C)∂r

−(γzz

rD2z∆Cδz + 1 + zz

2r [2zDz∆C + (1 + zz)DzDz∆C]δz)∂z

−(γzz

rD2z∆Cδz + 1 + zz

2r [2zDz∆C + (1 + zz)DzDz∆C]δz)∂z.

(7.45)

Now we are able to calculate L′2 = gµν(ζ + δT ζ)µ(ζ + δT ζ)ν |X1, i.e. the norm

of ζ + δT ζ at X1, given by (7.38). Remember that gµν in this formula is themetric (7.1) with Czz = 0. After neglecting the terms with Mi + ∆mB asbefore because they have higher order in 1

r0(w.r.t. the other terms), we find

that L′ is the same as (7.32) so the change in the proper distance is

∆L = r0δz2∆Czz(z1, z1) + r0δz

2∆Czz(z1, z1)2L , (7.46)

which agrees with the one above. The extra terms in δT ζ cancel the changeof the metric of the flat space evaluated at the shifted point.

To compare the time delay of the two detectors with this second methodwe can follow the same reasoning. We perform a supertranslation of the met-ric given by T = −∆C to eliminate all the terms with ∆Czz. The same su-pertranslation changes Υ12 and Υ21 by a quantity δTΥ12 = LTΥ12 = [ξT ,Υ12]

99

and δTΥ21 = LTΥ21 = [ξT ,Υ21], respectively. δTΥ12 is identical to δT ζ calcu-lated above, δTΥ21 can be obtained by δTΥ12 substituting δz → −δz, δz →−δz. Then, to find δu12 we require gµν(Υ12 + δTΥ12)µ(Υ12 + δTΥ12)ν |X1

= 0,where X1, given by (7.38), is the supertranslation of the original coordinatesof detector 1. To find δu21, instead, we requiregµν(Υ21 + δTΥ21)µ(Υ21 + δTΥ21)ν |X1

= 0. Then, we extract and subtractδu12 and δu21. Using the identity [Dz, Dz]Dz∆C = −Dz∆C, we obtain

δu12 − δu21 = Dz∆Czz +Dz∆Czz, (7.47)

which agrees with the one above. We note that only the components u, r ofδTΥ12 and δTΥ21 contribute to this effect.

In conclusion the effects of a radiation pulse passing through I + on a fam-ily of BMS observers is characterized by the induced supertranslation (7.22).The observations may be equivalently described as leaving the worldlines un-changed and supertranslating the metric, or leaving the metric unchangedand supertranslating the observers. In either case the observers measure thegravitational memory effect as well as clock desynchronisation.

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Chapter 8

Final state of gravitationalcollapse with supertranslationmemory

In Chapter 7 we argued that the passage of radiation and matter at nullinfinity changes the metric according to a supertranslation. In this chapterwe will present the work of Compere and Long [8], who applied this dis-covery to argue that the correct final state of gravitational collapse for theSchwarzschild spacetime is not (2.1) but it must involve a supertranslation.In fact, during the collapse the passage of both radiation and matter takesplace at null infinity. Therefore, in the final state the asymptotic supertrans-lation field C(z, z) will appear in the metric. The presence in this final stateof the asymptotic supertranslation field weakens the information paradox.

In this chapter we will first derive a conservation equation that allowsobtaining the supertranslation field by knowing the details of the collapse.Then, we will analyse the method, proposed by Compere and Long for theSchwarzschild spacetime, to obtain the final state of collapse with supertrans-lation field.

8.1 Supertranslation field in the final stateIn Chapter 7.1 we found an angular-dependent conservation of energy

equation (7.21) that connects the asymptotic supertranslation field beforeand after the passage of a non-zero radiation flux between the retarded timesui and uf . In this section we want to extend this conservation law to thecase of gravitational collapse leading to a massive final state. It will allowus to relate the amplitude of the asymptotic supertranslation field in the

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final state as compared with the initial state with the leading energy profileof the collapsing matter and radiation at null infinity. We will then drawthe consequences of this conservation law in the case of the collapse of idealspherical and non-spherical null shells.

To find the conservation equation for the gravitational collapse, first wehave to match the metric of the past null infinity I −, where the flux of nullmatter and gravitational radiation comes from, to the metric of the futurenull infinity I +, where null matter and gravitational radiation escape at.They have to be the same at the spatial infinity i0. We know that the metricin a neighbourhood of I + in Bondi coordinates (u, r, z, z) is (7.1)

ds2 =− du2 − 2dudr + 2r2γzzdzdz

+ 2mB

rdu2 + rCzzdz

2 + rCzzdz2 +DzCzzdudz +DzCzzdudz + . . . .

(8.1)

Similarly, we can define another set of coordinates (v, r, w, w), that are calledadvanced Bondi coordinates, such that the metric in a neighbourhood of I −

in these coordinates is [24]

ds2 =− dv2 + 2dvdr + 2r2γwwdwdw

+ 2mB

rdv2 + rCwwdw

2 + rCwwdw2 −DwCwwdvdw −DwCwwdvdw + . . . .

(8.2)

Let us define the asymptotic past and future of I − similarly to the definitionof I +

− and I ++ given in Section 7.1. Denote the asymptotic past of I − (i.e.

the points of I − with v → −∞) as I −− , and its asymptotic future (i.e. the

points of I − with v → ∞) as I −+ . We identify points on I +

− and I −+ via

the antipodal map w = −z−1, w = −z−1 (i.e. the map that associates eachpoint on the 2-sphere with the opposite point with respect to the centre ofthe sphere).

The junction conditions at spatial infinity, where I +− and I −

+ meet, werefound in [24] for CK spacetimes:

limu→−∞

mB(u, z, z) = limv→+∞

mB(v, w, w)

limu→−∞

Czz(u, z, z) = − limv→+∞

Cww(v, w, w).(8.3)

We have already found (7.2) that, at future null infinity, the Bondi massaspect mB(u, z, z) and the field Czz(u, z, z) are related through the (u, u)component of the Einstein equation to the total energy flux at null infinity

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Tuu(u, z, z) as

∂u

(mB −

14(D2

zCzz +D2

zCzz))

= −Tuu,

Tuu = 14NzzN

zz + 4π limr→∞

[r2TMuu ].(8.4)

We can see that the Bondi mass decreases with retarded time due to gravi-tational radiation and null matter leaving the bulk through null infinity.

At past null infinity, the same reasoning leads to

∂v

(mB −

14(D2

wCww +D2

wCww)

)= +Tvv,

Tvv = 14NwwN

ww + 4π limr→∞

[r2TMvv ],(8.5)

where Nww = ∂vCww. We can see that the Bondi mass increases with ad-vanced time due to gravitational radiation and null matter entering the bulkthrough null infinity.

Let us now consider the gravitational collapse of a massive body. Wemay include both incoming gravitational radiation and null matter flux fromI − and initial matter at past timelike infinity i−. Gravitational radiationand null matter escapes at I +. We assume that the spacetime reaches astationary final state asymptotically flat at large |u| and |v|. As we mentionedin the previous chapter, precise boundary conditions for the formation ofsuch a final state were formulated by Christodoulou and Klainerman [22]for spacetimes with small non-linear deviations from Minkowski spacetime.Such boundary conditions lead to a final state at I +

+ with vanishing Bondimass. Since we want to consider a massive final state, we need more generalboundary conditions. As we did in Section 7.1, we assume the same fall-offconditions on radiative fields as [22]. In particular, we take the news tensorNzz = ∂uCzz to obey Nzz ∼ |u|−

32 and Nww ∼ |v|−

32 for |u| , |v| → ∞.

Let us integrate (8.4) between −∞ and +∞. We denote the final sta-tionary mass by Mstat = limu→∞mB(u). We will show in a moment thatit does not depend upon z, z. Similarly, the Bondi mass at u = −∞ is thetotal mass of the system. Mtot = limu→−∞mB(u). Using (8.4), the energyradiated away is therefore

Mtot −Mstat = −14[D2

zCzz +D2

zCzz]∞u=−∞ +

∫ ∞−∞

duTuu. (8.6)

This is exactly the same equation that we would obtain taking the limitui → −∞ and uf → +∞ in (7.18). Now, as explained in Section 7.1, from

103

the vanishing of the imaginary part of the Weyl tensor component Ψ02 at I +

± ,we also have the boundary conditions (7.16) (found for CK spacetimes andextended to the case of a final massive stationary spacetime):

[D2zC

zz −D2zC

zz]∣∣∣I +±

= 0. (8.7)

In Section 7.1 we argued that the solution of this boundary condition is(7.17), Czz = −2D2

zC at both I +± , where C is the asymptotic supertrans-

lation field. In the final state, the Einstein equation Guz = 8πTuz implies∂zmB = −1

4∂z(D2zC

zz−D2zC

zz) after assuming that the matter stress-energytensor falls off sufficiently fast. The boundary condition (8.7) implies thatthe Bondi mass in the final state is a constant independent of z, z, as claimedpreviously1.

Let us define the differential operator

D = 14D

2(D2 + 2). (8.8)

Using the identity DC = (γzz)2D2zD

2zC, we can rewrite (8.6) as

Mtot −Mstat = [DC]∞u=−∞ +∫ ∞−∞

duTuu. (8.9)

Then, we can integrate (8.5) between v = −∞ and v = +∞. The initialBondi mass is denoted asMin = limv→−∞mB(v). The final Bondi mass coin-cides withMtot after using the junction condition (8.3). We impose boundaryconditions at I −

± analogous to (8.7), which allows to identify Cww = −2D2wC

at both I −± . Therefore,

Mtot −Min = −[DC]∞v=−∞ +∫ ∞−∞

dvTvv. (8.10)

The antipodal map leaves the operator D2, and therefore D, invariant.Using the junction condition (8.3), we can now subtract (8.9) and (8.10) toobtain

D(Cstat −∆C∞ − Cin) = Min −Mstat +∫ ∞−∞

dvTvv(w, w)−∫ ∞−∞

duTuu(z, z),(8.11)

where Cstat(z, z) = limu→∞C(u, z, z), Cin(w, w) = limv→−∞C(v, w, w) and∆C∞ = limu→−∞C(u, z, z)− limv→+∞C(v, w, w). Let us pause to interpretthis equation. The difference of the final asymptotic supertranslation field at

1Using the same argument, we can obtain the same result also for Mf and Mi inSection 7.1. However, we never used this fact in that section.

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angle (z, z) and the initial asymptotic supertranslation field at the antipodalangle (w, w) minus a boundary term at spatial infinity is dictated by thetotal energy flux and initial mass coming in minus the final mass and energyflux going out.

Let us now discuss the boundary term at spatial infinity ∆C∞ in more de-tail. If one starts with a configuration with no initial supertranslation fieldCin = 0 and generic incoming radiation, the conservation equation (8.10)implies that limv→+∞C(v, w, w) 6= 0 and therefore ∆C∞ 6= 0 as a conse-quence of the junction condition (8.3). However, Compere and Long in [8]make the ad hoc assumption that the boundary term at spatial infinity van-ishes: ∆C∞ = 0. They argue that this assumption is reasonable because thejunction condition (8.3) on Czz and Cww were found for CK spacetimes, i.e.small non-linear deviation of Minkowski spacetime, while they are interestedin the gravitational collapse of black holes which is a different setting froma Minkowski spacetime. This ad hoc assumption does not contradict any-thing we know but it will need to be assessed by different considerations. Itconstitutes the main caveat of the remaining of this section.

Let us now discuss the collapse of a spherically symmetric null shell, i.e.a spherically symmetric shell that obeys the null energy condition,

TMµνVµV ν ≥ 0 (8.12)

for any null vector V . The process is described by the Vaidya metric

ds2 = −(

1− 2MΘ(v)r

)dv2 + 2dvdr + r2dΩ2, (8.13)

where Θ(v) is the Heaviside step function. This metric generalizes theSchwarzschild solution and describes a stationary (non-static) and nonro-tating metric outside a spherically symmetric star. This is a solution of theEinstein’s equations only in the presence of a purely ingoing stress-energytensor given by

TMµν = Mδ(v)4πr2 δvµδ

vν , (8.14)

which satisfies the null energy condition.Comparing (8.2) and (8.13), we find that Cww(v, w, w) = 0 and mB(v) =

MΘ(v). From the first of these two equations, we have Nww = 0 and also−2D2

wC = 0 at I −± , so we can choose Cin = 0 From the second equation we

have Min = 0. Therefore, the initial state, i.e. when v → ∞, is the globalvacuum state with Cin = 0 and Min = 0. From Nww = 0 we have Tvv =4π limr→∞ r

2TMvv = Mδ(v), so∫∞−∞ dvTvv = M . There is neither outgoing

radiation, i.e. Nzz = 0, nor outgoing matter, i.e. TMuu = 0. Therefore,

105

Tuu = 0. The final state is a black hole of mass Mstat = M . In thatvery particular case and under the assumption ∆C∞ = 0, the conservationequation (8.11) reduces to

DCstat = 0. (8.15)Now, the property of spherical harmonics D2Ylm = −l(l + 1)Ylm impliesDYlm = (l−1)l(l+1)(l+2)

4 Ylm, which is 0 for l = 0, 1. Therefore, the only smoothzero modes of the operator D are the four lowest spherical harmonics, soCstat can only be a linear combination of Y0,0, Y1,−1, Y1,0, Y1,1. However, as wenoticed in Section 6.2, the four lowest spherical harmonics form the genera-tors of the translations, therefore moving from an initial state with Cin = 0to a final state with Cstat given by a linear combination of the four lowestspherical harmonics can be seen as a translation on the coordinates. If wechoose to use the center-of-mass frame centered at r = 0, we cannot performthis translation anymore (if we did it, we would obtain a frame that is notthe center-of-mass frame). Hence, we deduce that the choice of the center-of-mass frame fixes Cstat = 0. There is therefore no non-trivial asymptoticsupertranslation field in the collapse of a spherically symmetric null shell.The collapse lead to the Schwarzschild black hole.

Let us now take a non-spherical shell. Its energy is instead

Tvv =(MP in(w, w)

4πr2 +O(r−3))δ(v). (8.16)

The profile of the energy density on the spherical shell admits the harmonicdecomposition MP in(w, w) = M + M

∑l≥1,m PlmYlm. We normalized the

zero mode in order to uniquely define the mass. More precisely, we fixed∫d2ΩP in = 1 where d2Ω = sin θdθdφ.The null energy condition requires that Tvv ≥ 0 at all angles. This im-

plies P in(w, w) ≥ 0 and it thereby constraints the coefficients of the higherharmonics Plm as ∑

l≥1,mPlmYlm(θ, φ) ≥ −1 (8.17)

for any (θ, φ). For simplicity, we assume that the initial state is Min =Cin = 0. In general, the non-linearity of Einstein’s equations will leadto gravitational wave emission at I +. Let us denote by

∫∞−∞ duTuu =(

MP out(z,z)4πr2 + O(r−3)

)the total leading order outgoing energy flux profile.

The conservation equation (8.11) reads as

DCstat = M(P in(w, w)− P out(z, z))−Mstat. (8.18)

In the ideal case where the outgoing radiation is negligeable, P out = 0, thefinal mass is the initial mass, Mstat = M , and the zeroth spherical harmonic

106

in P in cancels out with the last term in (8.18). Now, the presence of higherharmonics in Tvv implies that there is a non-trivial profile for the supertrans-lation field final state. In this ideal case without outgoing radiation, (8.18)reduces to

DCstat = M∑l≥1,m

PlmYlm(w, w). (8.19)

When we pass from coordinates w, w to z, z via the antipodal map, usingthe property of spherical harmonics Ylm(π − θ, φ + π) = (−1)lYlm(θ, φ), i.e.Ylm(w, w) = (−1)lYlm(z, z), the equation becomes

DCstat = M∑l≥1,m

(−1)lPlmYlm(z, z). (8.20)

Using the property of spherical harmonics DYlm = (l−1)l(l+1)(l+2)4 Ylm, we find

that the solution is

Cstat(z, z) =∑l≤1,m

C(0)lm Ylm(z, z)

+M∑l≥2,m

(−1)l 4(l − 1)l(l + 1)(l + 2)PlmYlm(z, z).

(8.21)

The coefficients C(0)lm label the four zero modes of the differential operator D

which are the 4 lowest spherical harmonics. We have already argued thatthey must vanish if we choose the center-of-mass frame of the system. Thenull energy condition constraint (8.17) leads to non-trivial constraints onCstat.

Let us consider two simple toy models which we will use in the nextsection. If the non-sphericity of the ingoing null shell is only modelled by thel = 2,m = 0 spherical harmonic and in the center-of-mass frame, the finalsupertranslation field is

Cstat = αM

6 (3 cos2 θ − 1), −12 ≤ α ≤ 1, (8.22)

where the amplitude α has been constrained by the null energy condition(8.17). To find the constraints we notice that in our case PlmYlm = P20Y20 =α(3 cos2 θ − 1) and that 2 and −1 are the largest and the smallest value of3 cos2 θ − 1, respectively. The result follows by using (8.17) for these twolimit values.

In the case where P in is 1 plus a combination of l = 2, m = ±1 harmonics,we instead have

Cstat = αM

6 sin 2θ cos(φ+ δ), −1 ≤ α ≤ 1, δ ∈ R. (8.23)

107

As a summary, the spherical collapse of a non-spherical null shell leadsto a final state which admits a non-trivial supertranslation field which canbe determined from the “angle-dependent energy balance conservation law”(8.11). The collapse of a spherically symmetric null shell is analytic andallows to identify that the final state supertranslation field vanishes. In moregeneral cases, detailed numerical simulations would be necessary to computefor each individual collapse and subsequent evolution of the system the valueof the final state asymptotic supertranslation field. The Schwarzschild blackhole is therefore an extremely fine-tuned final state of gravitational collapse.It admits no supertranslation field. It only applies to black holes that formedin a spherically symmetric fashion such as the Vaidya spacetime. We nowturn to our second question: what is the metric of a generic classical finalstate of gravitational collapse? To address this question, first we have tocompute the finite form of a supertranslation.

8.2 Supertranslation-dependent final state met-ric

The classical final state after gravitational collapse is by definition a sta-tionary spacetime. Uniqueness theorems presented in Section 5.1 imply thatthe spacetime metric is diffeomorphic to the Kerr metric, i.e. it is linked tothe Kerr metric (5.7) via a diffemorphism. In this thesis, for technical simplic-ity, we will consider only static spacetimes. In this section, following [8], wewill construct a final state diffeomorphic to the Schwarzschild metric. In thenext chapter, we will apply the same reasoning to the Reissner-Nordstrom,extreme Reissner-Nordstrom and Majumdar-Papapetrou solution. The sameproblem for the stationary case, which takes the angular momentum intoaccount, presents many technical difficulties and it has yet to be solved.

To reach our goal the main difficulty will be to find the finite form of thediffeomorphism that gives a supertranslation of the coordinates. Then, thisdiffeomorphism will be applied to the Schwarzschild to obtain the final stateof the gravitational collapse, which will be dependent on the asymptoticsupertranslation field C(z, z). We will present the procedure followed byCompere and Long in [8] to work out the finite supertranslation for the staticcases. We emphasize that applying a large finite diffeomorphism is not aphysical process: it is a convenient solution generating technique which allowsto get the final state of gravitational collapse with a non-trivial asymptoticsupertranslation field.

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8.2.1 BMS finite diffeomorphisms

In this section we will present the derivation, found by Compere and Long[8], of the diffeomorphism which generates a supertranslation field profilelabelled by an arbitrary smooth function C(z, z) on the sphere. In a crucialtechnical step, as we will see, we will use the existence of the enhanced BMSalgebra with Weyl transformations described in Chapter 6.

Let us start with the global Minkowski vacuum written in retarded andstereographic coordinates (us, rs, zs, zs),

ds2 = −du2s − 2dusdrs + 4r2

s

(1 + zszz)2dzsdzs. (8.24)

The metric falls into the BMS gauge, i.e. (us, rs, zs, zs) are Bondi coordinates.As we showed in Chapter 6, the infinitesimal diffeomorphism which generatedBMS supertranslations and Lorentz transformations reads as

ξT,R =[T + 1

2uDARA

]∂u −

12r[DAR

A − 1rDAD

A

(T + 1

2uDCRC

)+O

(1r2

)]∂r

+[RA − 1

rDA

(T + 1

2uDBRB

)+O

(1r2

)]∂A,

(8.25)

where T (z, z) is an arbitrary real smooth function on the sphere and R(z, z)are the globally well-defined conformal Killing vectors on the unit roundsphere. The subleading terms in the radial expansion are uniquely fixedin BMS gauge. Around the Minkowski vacuum (8.24) this radial expansionexactly stops at the order indicated but it is infinite around Minkowski space-time transformed by an arbitrary finite diffeomorphism which preserves theBMS gauge.

The problem at hand is to find the corresponding finite diffeomorphismat each order in the radial expansion and then sum it to obtain a closed formexpression. The task is not straighforward since an arbitrary function T (z, z)is present, the metric and connection on the sphere matters as well and theradial expansion is only known so far as a series expansion.

In order to facilitate this daunting task, it is in fact very useful to firstconsider a new system of coordinates. Minkowski spacetime can be foliatedby complex plane sections as

ds2 = −2ducdrc + 2r2cdzcdzc. (8.26)

109

The two metric (8.24) and (8.26) are related by the finite coordinate trans-formation

rc =√

2rs1 + zszs

+ us√2,

uc = 1 + zszs√2

us −zszsu

2s

2rc,

zc = zs −zsus√

2rc, zc = zs −

zsus√2rc

.

(8.27)

One can uniquely invert this coordinate transformation as

rs = 1√2

√[uc + rc(1 + zczc)]2 − 4rcuc,

us = 1√2

[uc + rc(1 + zczc)]− rs,

zs =1√2 [uc − rc(1 + zczc)] + rs

zcus, zs =

1√2 [uc − rc(1 + zczc)] + rs

zcus.

(8.28)

The infinitesimal diffeomorphism generator of BMS and Weyl transfor-mations is given by (6.47) as an asymptotic series expansion. It takes asimple form around the vacuum metric (8.26) because there is no angulardependence. The task is then reduced to find the finite combined BMS trans-formation and Weyl rescaling around the metric (8.26). Compere and Longargue that, after some algebra involving a guess for the finite transformation,the right coordinate change reads as

rc = ∂z∂zW

∂zG∂zG+

√√√√ r2

(∂uW )2 + (∂2zG∂zW − ∂zG∂2

zW )(∂2z G∂zW − ∂zG∂2

zW )(∂zG)3(∂zG)3

,

uc = W ∂zW∂zW

∂zG∂zGrc

,

zc = G− ∂zW

∂zGrc, zc = G− ∂zW

∂zGrc,

(8.29)

where W (u, z, z) is an arbitrary function of u, z, z, G(z) is a combinationof 1, z, z2 and G(z) is its complex conjugate. The function W characterisesthe Weyl rescalings and supertranslations. The function G(z) characterizesthe Lorentz transformations. One can generalise G(z) to be an arbitrarymeromorphic function, i.e. smooth except for a set of isolated points, whichthen generates superrotations. One can also generalise W (u, z, z) to be a

110

singular function on the sphere which then generates singular Weyl rescalingsand singular supertranslations.

The resulting metric in (u, r, z, z) coordinates takes the asmptotic BMSform defined in Chapter 6 (here we use stereographic coordinates):

ds2 = e2β V

rdu2 − 2e2βdudr + gAB(dzA − UAdu)(dzB − UBdu), (8.30)

where the fields satisfy the asymptotic fall-off conditions detailed in Chap-ter 6. In particular, the metric gAB in stereographic coordinates takes theform

gABdzAdzB = r2e2ϕ2γzzdzdz +O(r), (8.31)

where ϕ(u, z, z) depends upon W and G. Here, we are interested in the caseϕ = 0. To obtain this conformal factor we must choose

W = 1√γzz

(u+ C(z, z))√∂zG∂zG. (8.32)

One can therefore generate the vacuum with an arbitrary BMS supertrans-lation field C(z, z) and superrotation field G(z) in (u, r, z, z) coordinates bystarting with the global Minkowski spacetime (8.24) written in (us, rs, zs, zs)coordinates, then using first the transformation (8.28) to switch to (uc, rc, zc, zc)coordinates then (8.29) to reach the (u, r, z, z) coordinates.

Supertranslation diffeomorphism in isotropic coordinates

Let us specialise to supertranslations only by setting the superrotationfield to the identity G(z) = z. We then have

W = 1 + zz√2

[u+ C(z, z)] (8.33)

and we obtain the vacuum metric

ds2 = −du2−2dud(√

r2 + U+12(D2+2)C

)+((r2+2U)γAB+

√r2 + UCAB

)dzAdzB,

(8.34)where CAB = −(2DADB − γABD

2)C and U = 18CABC

AB. One can checkthat under a supertranslation (6.32), the metric remains in BMS gauge andchanges according to the shift ∆TC(z, z) = T (z, z).

Isotropic coordinates are obtained by setting

t = u+ ρ,

ρ =√r2 + U + E, E(z, z) = 1

2D2C + C − C(0,0),

(8.35)

111

where C(0,0) denotes the lowest spherical harmonic mode of C. The metricthen reads

ds2 = −dt2 + dρ2 + [((ρ− E)2 + U)γAB + (ρ− E)CAB]dzAdzB. (8.36)

This metric can be compared with the original global Minkowski vacuumwritten in isotropic coordinates

ds2 = −dt2s + dρ2s + ρ2

sγABdzAs dz

Bs , (8.37)

where ts = us + ρs. Following the chain of coordinate transformations, wecan finally relate these two coordinate systems by the change of coordinates

ts = t+ C(0,0),

ρs =√

(ρC + C(0,0))2 +DACDAC,

zs = (z − z−1)(ρ− C + C(0,0)) + (z + z−1)(ρs − z∂zC − z∂zC)2(ρ− C + C(0,0)) + (1 + zz)(z∂zC − z−1∂zC) ,

zs = (z − z−1)(ρ− C + C(0,0)) + (z + z−1)(ρs − z∂zC − z∂zC)2(ρ− C + C(0,0)) + (1 + zz)(z∂zC − z−1∂zC) .

(8.38)

This is the effect of finite supertranslations on isotropic coordinates (ts, ρs, zs, zs).The metric (8.36) is written in isotropic gauge defined in Section 2.10.

The generator of supertranslations in that gauge can be written as

ξ(stat)T = T(0,0)∂t−(T−T(0,0))∂ρ+

CABDBT − 2DAT(ρ− 1

2(D2 + 2)(C − C(0,0)))

2[(ρ− 1

2(D2 + 2)(C − C(0,0)))2− U

] ∂A,

(8.39)where T(0,0) denotes the lowest spherical harmonic mode of T .

Let us convert stereographic coordinates into spherical coordinates viathe stereographic map z = eiφ cot θ

2 . A time translation is generated byC = C(0,0) = 1. A spatial translation is generated by

Ctranslation = ax sin θ cosφ+ ay sin θ sinφ− az cos θ, (8.40)

which is a linear combination of the l = 1 spherical harmonics. The trans-formation (8.38) then precisely coincides with the transformation law of thespherical coordinate system centered at the origin to a new spherical coor-dinate system centered at the translated origin. The transformation law ofthe radius follows from Pythagoras’ theorem. For a generic supertranslation,the transformation is still given by (8.38) but where C is now an arbitrarycombination of higher spherical harmonics. We will therefore refer to the

112

transformation rule of the radius (8.38) as the supertranslation Pythagorianrule. All supertranslations except the lowest harmonic mode are purely spa-tial.

The spatial part of the diffeomorphism (8.38) takes a much simpler formwhen it transforms the original flat metric in Cartesian coordinates (xs, ys, zs)to the final one in spherical coordinates (ρ, zA) with zA = θ, φ. In fact, thetransformation that changes the metric

dx2s + dy2

s + dz2s (8.41)

intodρ2 + [((ρ− E)2 + U)γAB + (ρ− E)CAB]dzAdzB (8.42)

is

xs = (ρ− C + C(0,0)) sin θ cosφ+ csc θ sinφ∂φC − cos θ cosφ∂θC,ys = (ρ− C + C(0,0)) sin θ sinφ− csc θ cosφ∂φC − cos θ sinφ∂θC,zs = (ρ− C + C(0,0)) cos θ + sin θ∂θC.

(8.43)

After defining x = ρ sin θ cosφ, y = ρ sin θ sinφ, z = ρ cos θ, a spatial transla-tion generated by (8.40) leads to a new Cartesian coordinate system centeredat the translated origin, i.e. xs = x−ax, ys = y−ay, zs = z−az, as expected.

Now that we obtained a finite form of a supertranslation, we are ready toapply it to some of the metrics describing the black holes studied in Part I,namely to those black hole metrics for which the definition of isotropic coor-dinates is possible. The result will be a metric dependent on the asymptoticsupertranslation field, whose presence weakens the information paradox.

8.3 Schwarzschild metric with supertransla-tion memory

Following Compere and Long [8], we will now apply the results of theprevious chapter to obtain the final state of gravitational collapse for theSchwarzschild solution. We start with the Schwarzschild metric in Schwarzschildcoordinates (ts, rs, zs, zs) (2.1)

ds2 = −(

1− 2Mrs

)dt2s +

(1− 2M

rs

)−1

dr2s + r2

sdΩ2s, (8.44)

113

where dΩ2s = 2γzszsdzsdzs. Let us use isotropic coordinates (ts, ρs, zs, zs),

defined in Section (2.10) by rs = ρs(1 + M

2ρs

)2, to obtain

ds2 = −

(1− M

2ρs

)2

(1 + M

2ρs

)2dt2s +

(1 + M

2ρs

)4

(dρ2s + ρ2

sdΩ2s). (8.45)

We can now apply the finite supertranslation (8.38) to obtain the desiredfinal state of the Schwarzschild spacetime equipped with a supertranslationfield:

ds2 = −

(1− M

2ρs

)2

(1 + M

2ρs

)2dt2+(

1+ M

2ρs

)4(dρ2+[((ρ−E)2+U)γAB+(ρ−E)CAB]dzAdzB

),

(8.46)where ρs(ρ) =

√(ρC + C(0,0))2 +DACDAC and auxiliary quantities are de-

fined by

CAB(z, z) = −(2DADB − γABD2)C,

U(z, z) = 18CABC

AB,

E(z, z) = 12D

2C + C − C(0,0).

(8.47)

The metric depends not only upon the parameter M but also upon theasymptotic supertranslation field C(z, z), therefore it is characterized bymore than just one parameter. This fact weakens the information paradox.The field C(z, z) is physically fixed by the details of the collapse accordingto the conservation equation (8.11), as we have seen in Section 8.1.

114

Chapter 9

Charged black hole metricswith supertranslation memory

Although it is true that large imbalances of charge do not occur veryfrequently in nature, charged black holes are worth studying for two mainreasons. To understand the first one, we must say that a famous calcula-tion, performed by Strominger and Vafa applying String Theory to extremeReissner-Nordstrom black holes [25], gave a result coherent with QuantumField Theory in curved spacetimes1. This mentioned work is one of the mostimportant reasons for physicists to believe that String Theory may be thecorrect theory of quantum gravity. Therefore, we can argue that chargedblack holes had a fundamental importance in the physics of fundamentalinteractions and, in particular, in String Theory. Given that physicists aretrying to solve the information paradox in order to extract hints potentiallyuseful for the development of a theory of quantum gravity, and that StringTheory is one of the best candidates for such unifying model, it is believedthat methods that weaken the information paradox related to charged blackholes are worth being analysed.

The second important reason to study charged black holes is that somerecent experimental evidence might be explained by theoretical models de-scribing charged black holes [26]. More precisely, some numerical calculationsinvolving charged black holes seem to give results that are coherent with thevalue of the electromagnetic luminosity detected by the Fermi GBM group[27]. This detection appears to be coincident to the first direct detectionof gravitational waves [28] produced by the merger of a black hole binary.This suggests that such binary might have been composed of charged black

1The result is the value of the Bekenstein-Hawking entropy of a black hole (this topiccan be studied in [12]).

115

holes, therefore black holes of this kind may exist in nature and they may berelevant not only from a theoretical perspective.

For all these reasons, in this chapter we will compute a calculation, anal-ogous to the one described in the previous section and originally performedby Compere and Long [8], for the case of charged black holes: Reissner-Nordstrom, extreme Reissner-Nordstrom and Majumdar-Papapetrou. In thefuture it might be important to extend these results to Kerr and Kerr-Newman black holes.

9.1 Reissner-Nordstrom metric with super-translation memory

Let us start with the Reissner-Nordstrom spacetime with M > e0:

ds2 = −(

1 + 2Mrs

+ e2

r2s

)dt2s +

(1− 2M

rs+ e2

r2s

)−1

dr2s + r2

sdΩ2s. (9.1)

To apply the procedure outlined in the previous chapter, it is first neces-sary to write the solution in isotropic gauge. The isotropic radial coordinatefor RN spacetime is defined by

rs =(

1 + M + e

2ρs

)(1 + M − e

2ρs

)ρs. (9.2)

Given rs there are two possible choices for ρs:

ρs = 12(rs −M ±

√r2s − 2Mrs + e2

). (9.3)

When e = 0, we want this to reduce to the definition of ρs used for theSchwarzschild spacetime, i.e. we want that ρs > M

2 in region I and 0 < ρs <M2 in region IV. Therefore, we must choose the upper sign in region I and thelower sign in region IV. In isotropic coordinates (ts, ρs, zs, zs) the RN solutionwith M > e is

ds2 = −

(1− M2−e2

4ρ2s

)2

(1 + M+e

2ρs

)2(1 + M−e

2ρs

)2dt2s+(

1+M + e

2ρs

)(1+M − e

2ρs

)(dρ2

s+ρ2sdΩ2

s).

(9.4)

116

Finally, applying the finite supertranslation (8.38), we obtain the finalstate of the RN spacetime equipped with the supertranslation field:

ds2 =−

(1− M2−e2

4ρ2s

)2

(1 + M+e

2ρs

)2(1 + M−e

2ρs

)2dt2

+(

1 + M + e

2ρs

)(1 + M − e

2ρs

)(dρ2 + [((ρ− E)2 + U)γAB + (ρ− E)CAB]dzAdzB

).

(9.5)

As for the Schwarschild case, this state depends not only upon the mass Mand the charge e, but also upon the field C(z, z).

9.2 Extreme Reissner-Nordstrom metric withsupertranslation memory

We can also obtain the final state of an extreme (i.e. M = e) RN space-time.

The isotropic radial coordinate for the extreme RN solution is given byrs = ρs +M . The extreme RN solution in isotropic coordinates is

ds2 = −(

1 + M

ρs

)−2

dt2s +(

1 + M

ρs

)2

(dρ2s + ρ2

sdΩ2s). (9.6)

Applying the finite supertranslation (8.38), the metric becomes

ds2 = −(

1+Mρs

)−2

dt2+(

1+Mρs

)2(dρ2+[((ρ−E)2+U)γAB+(ρ−E)CAB]dzAdzB

).

(9.7)Once again it is evident that the metric depends upon the asymptotic super-translation field C(z, z).

9.3 Multiple charged black holes with super-translation memory

We will now apply the method to the Majumdar-Papapetrou solutionpresented in Section 4.5. The form of the metric (4.23) is given in Cartesiancoordinates:

ds2 = −H(xs)−2dt2s +H(xs)2(dx2s + dy2

s + dz2s), (9.8)

117

where we can choose

H(xs) = 1 +N∑i=1

Mi∣∣∣xs − xs(i)∣∣∣ . (9.9)

Therefore, to obtain the final state after gravitational collapse of this solution,we can apply the finite supertranslation on Cartesian coordinates (8.43). Theresult is

ds2 = −H(xs)−2dt2+H(xs)2(dρ2+[((ρ−E)2+U)γAB+(ρ−E)CAB]dzAdzB

),

(9.10)with xs, ys, zs given by (8.43). Once again, the asymptotic supertranslationfield appears in a non-trivial way.

All the examples of Chapter 9, describing charged black holes, show anon-trivial asymptotic supertranslation field C(z, z), therefore the Reissner-Nordstrom, extreme Reissner-Nordstrom and Majumdar-Papapetrou space-time after gravitational collapse are defined by much more information thanjust the values of mass and charge. As a consequence, the loss of informationis significantly less relevant than what was originally believed by Hawking.

Since the study of charged black holes has already been important to givecredibility to String Theory as a good candidate for a theory of quantumgravity, it is believed that a deeper study of the information paradox inspacetimes with charged black holes may provide further details about theconstruction of a ”theory of everything”. In addition to this, as we previouslyargued, experimental evidences seem to suggest that charged black holes maybe important also in Astrophysics, i.e. to study some real natural phenomenain the universe.

118

Conclusions and outlook

In summary, in Part I of this thesis we have presented, following mainly[12], some fundamental features of black holes. We analysed some character-istics of the stars that give origin to black holes via gravitational collapse,for instance, the possible values of their masses. Then, we gave the formaldefinition of “infinity” for an asymptotically flat spacetime. This allowed usalso to give the formal definition of a black hole. We have also analysed thevarious exact solutions of the Einstein’s equations describing different kindsof black holes: static, charged and rotating.

In Part II we then moved on to more recent studies. Following [14], wedefined and analysed the BMS group, i.e. the group of asymptotic symme-tries that leave invariant the metric near the null infinity. Then, following[15], we studied an important application related to asymptotic symmetries:the supertranslation memory effect, according to which spacetime changesbecause of the passage of matter and radiation at infinity in the same wayas under supertranslations. Finally, we applied all these pieces of knowledgeto argue, following [8], that the generic static final state of collapsing matterand radiation is not described by the Schwarzschild metric but instead bythe metric (8.46), which possesses an asymptotic supertranslation field. Thisfield can be obtained from the details of the gravitational collapse via theconservation equation (8.11), assuming that ∆C∞ = 0. Given the impor-tance of charged black holes in String Theory, their potential relevance forthe discovery of the ultimate theory of quantum gravity, and the possibil-ity, suggested by [26], that they are existing parts of the universe, I appliedthe same reasoning to the Reissner-Nordstrom, extreme Reissner-Nordstromand Majumdar-Papapetrou metric to obtain the final states (9.5), (9.7) and(9.10), respectively.

The form of these final states weakens the black hole information para-dox, because they are characterized, in addition to the usual parameters(mass and charge), also by an additional asymptotic supertranslation fieldC(z, z), which means that the amount of lost information after the black holeevaporation is considerably reduced with respect to the same spacetime met-

119

ric without asymptotic supertranslaton field. In addition, it can be shown[8] that the presence of this field allows to compute an infinite number ofconserved charges, which can be considered as other parameters of a blackhole (exactly as mass, charge and angular momentum).

To conclude this work it is necessary to mention some possible furtherresearch that may give a better understanding of the topics analysed in thisthesis. First of all, it must be said that it would be very important to obtainthe final state of gravitational collapse for Kerr and Kerr-Newman blackholes. In fact, the vast majority of black holes that have been detected canbe modelled by the Kerr solution. In particular, the gravitational waves,recently detected for the first time [28], were produced by the merger oftwo rotating black holes with masses of approximately 36M and 29M;the former had an angular momentum smaller than 0.7 [29]. They gaveorigin to a black hole with a mass of approximately 62M and an angularmomentum of approximately 0.67. Furthermore, as stated by the uniquenesstheorems presented in Section 5.1, the Kerr and Kerr-Newman spacetime isalso the most general solution with a black hole region, therefore it wouldbe of fundamental importance to generalise the results obtained in the lasttwo chapters to rotating black holes. In doing so, the main obstacle is thefact that it is not possible to define isotropic coordinates for such stationary(non-static) spacetimes. Thus, one could consider the possibility of a directapproach. In fact, in principle one could write the Kerr metric in Bondicoordinates (i.e. in BMS gauge), apply an infinitesimal supertranslation tothese coordinates, iterate the procedure to find the diffeomorphism at eachorder in the radial expansion, and finally sum it to obtain the closed form ofthe finite supertranslation. Unfortunately, this method presents some lengthycalculations that make it not easily applicable; a different solution should befound.

In addition to these technical difficulties, there are other possible ways toimprove the study that we carried out. Firstly, the assumption of Section 8.1that the boundary term at spatial infinity ∆C∞ is 0 should be proved. Sec-ondly, much of the classical and quantum structure of the final state withsupertranslation memories remains to be understood. Finally, the way toextract information about the pure quantum state, before the collapse ofa star forming a black hole, from the state after the collapse has yet to befound, namely the information loss paradox is yet to be solved. Some expertsbelieve that this kind of studies may lead to a complete theory of quantumgravity.

120

Appendix A

The Schwarzschild radius

In this appendix we will prove that the parameter of the Schwarzschildradius RS, which appears in the Schwarzschild solution (1.35), must corre-spond to half of the mass M of the object that produces the gravitationalfield.

We need to define the Newtonian limit of a theory of gravity, whichconsists of the following three requirements:

1. massive particles are moving slowly (with respect to the speed of lightc = 1),

2. the gravitational field is weak,

3. the gravitational field is static.

This is the situation that occurs far from the gravitational source. To makethe theory consistent, the equation of motion (i.e. the geodesic equation)of a massive particle in this limit must be the same as in the Newtoniantheory of gravity. This requirement gives information about the form of thett component of the metric for a generic solution in the Newtonian limit.Requiring that the gtt component of the Schwarzschild solution reduces tothe gtt component of a generic consistent solution in the Newtonian limit willgive the desired result.

As we have already seen in Section 1.3, 3. implies, in particular, that

∂tgµν = 0. (A.1)

2. means that we can decompose the metric into the Minkowski form plus asmall perturbation:

gµν = ηµν + hµν , |hµν | 1. (A.2)

121

We work in inertial coordinates, so ηµν is the canonical form of the metric.(The “smallness condition” on the metric perturbation hµν does not reallymake sense in arbitrary coordinates.) 1. implies that

dxi

dτ 1, (A.3)

where τ and xµ(τ) are, respectively, the proper time and the trajectory ofa massive particle. This tells us that the motion is non-relativistic so theproper time is approximable with the inertial time coordinate, t ≈ τ . Sodtdτ≈ 1 and (A.1) gives

dxi

dτ dt

dτ. (A.4)

In GR, the motion of free particles (i.e. those under the action of the gravi-tational field only) is described by the geodesic equation

d2xµ

dλ2 + Γµρσdxρ

dλ

dxσ

dλ= 0. (A.5)

For a massive particles, one can choose the proper time as parameter, λ = τ .We now require that, with these assumptions, the geodesic equation reducesto the equation of motion for particles in the Newtonian theory of gravity,i.e.

~a = −~∇φ or, equivalently d2xi

dt2= −∂iφ, (A.6)

where φ is the solution of the Poisson equation ∇2φ = 4πρ (ρ is the densityof the gravitational source). (A.4) tells us that (A.5) reduces to

d2xµ

dτ 2 + Γµtt( dtdτ

)2= 0. (A.7)

(A.1) tells us that

Γµtt = 12g

µλ(∂tgλt + ∂tgtλ − ∂λgtt)

= −12g

µλ∂λgtt.(A.8)

(A.2) implies thatgtt = ηtt + htt. (A.9)

Moreover, from the definition of the inverse metric, gµνgνσ = δµσ , we find thatto first order in h,

gµν = ηµν − hµν , (A.10)

122

where hµν = ηµρηνσhρσ. In fact, we can use the Minkowski metric to raiseand lower indices on an object of any definite order in h, since the correctionswould only contribute to higher orders. In other words, we can think of hµνjust as a symmetric (0,2) tensor field propagating in Minkowski space andinteracting with other fields.

Putting (A.8) and (A.9) together gives

Γµtt = −12η

µλ∂λhtt, (A.11)

so the geodesic equation (A.7) becomes

d2xµ

dτ 2 = 12η

µλ∂λhtt( dtdτ

)2. (A.12)

Using ∂thtt = 0, the µ = t component of this is just

d2t

dτ 2 = 0, (A.13)

i.e. dtdτ

is constant (as we already knew). To examine the spacelike compo-nents, recall that the spaelike components of ηµν are just those of a 3 × 3identity matrix. We therefore have

d2xi

dτ 2 = 12( dtdτ

)2∂ihtt. (A.14)

Dividing both sides by(dtdτ

)2has the effect of converting the derivative on

the left hand side from τ to t, leaving us with

d2xi

dt2= 1

2∂ihtt. (A.15)

This equation is the same as (A.6) once we identify

htt = −2φ, (A.16)

or in other wordsgtt = −(1 + 2φ). (A.17)

Now for a spherical source of mass M at the origin of our coordinatesystem, which is the case we are interested in, ρ = Mδ(~x) and the solutionof the Poisson equation in spherical coordinates is

φ = −Mr, (A.18)

123

so we have found that the Newtonian limit of the gtt component for a genericspherically symmetric solution around a spherical body of mass M is

gtt = −(1− 2M

r

). (A.19)

It is evident that taking the Newtonian limit of the Schwarzschild solution(1.35) consists of taking r large. Hence we must require that gtt in (1.35) hasthe form (A.19) for large r. But the forms are already the same; we needonly to identify

RS = 2M. (A.20)

This concludes the proof.

124

Appendix B

Some additional details onspherical stars

In this appendix we will give a detailed proof and explanation about twostatements made in Chapter 1.7. First, we want to show that the limit (1.41)can be obtained by using only ρ ≥ 0 and dρ

dr≤ 0. Using the TOV equations

(1.31) for dmdr

and (1.32) for dΦdr

to eliminate ρ and p from the TOV equation(1.33) for dp

dr, we get

d

dr

(e−Ψr−1 d

drζ)

= eΨζd

dr

(mr3

), (B.1)

where Ψ is given by (1.29) and ζ = eΦ. Now the formula relating m and ρ,i.e. (1.38) with m? = 0, is the same as the formula for the mass of a ball ofmatter of radius r in Euclidean space. In Euclidean space m

r3 is proportionalto the average density, so also in our space this must be true. Therefore,dρdr≤ 0 implies d

dr

(mr3

)≤ 0. Hence, from (B.1) we get d

dr

(e−Ψr−1 d

drζ)≤ 0.

By integrating this equation from r1 to r and using the TOV equation fordΦdr, it is straightforward to show that, for any r1 ≤ r,

dζ

dr(r1) ≥

(1− 2m(r)

r

)− 12(m(r)r3 + 4πp(r)

)ζ(r)r1

(1− 2m(r1)

r1

)− 12

. (B.2)

Then, ddr

(m(r)r3

)≤ 0 implies m(r1)

r1≥ m(r) r

21r3 for r1 ≤ r. Using this and then

performing the integration in r1, we find that

∫ r

0r1

(1− 2m(r1)

r1

)− 12

dr1 ≥r3

2m(r)

[1−

(1− 2m(r)

r

) 12]. (B.3)

125

Integrating (B.2) with respect to r1 from 0 to r and using (B.3), we easilyget

ζ(0) ≤ ζ(r)

1−(

12 + 2πr3p(r)

m(r)

)[(1− 2m(r)

r

)− 12

− 1]. (B.4)

By definition ζ is everywhere positive, in particular ζ(0) > 0. So, from (B.4)we finally obtain the lower bound (1.41)

m(r)r

<291− 6πr2p(r) + [1 + 6πr2p(r)] 1

2.

This is the announced result.As we argued in Chapter 1.7, evaluating (1.41) at r = R and using

p(R) = 0 gives the Buchdahl inequality

R >94M.

In Chapter 1.7 we also said that static, spherically symmetric stars withconstant density ρ, i.e. made of incompressible fluid, can get arbitrarily closeto saturating this inequality. We now want to prove this statement. Weassume that the density is constant ρ? out of the surface of the star, afterwhich it vanishes,

ρ(r) =

ρ?, r ≤ R,

0, r > R.(B.5)

Specifying ρ(r) explicitly takes the place of an equation of state, since p(r)can be determined from (1.33). It is then straightforward to perform theintegration in (1.38) to get

m(r) =

43πr

3ρ?, r ≤ R,43πR

3 r > R.(B.6)

Then, integrating (1.33) yields

p(r) = ρ?

[R√R− 2M −

√R3 − 2Mr2

√R3 − 2Mr2 − 3R

√R− 2m

]. (B.7)

The pressure increases near the core of the star, as one would expect. Indeed,for a star of fixed radius R, the central pressure p(0) would need to be infiniteif the mass saturated the Buchdahl inequality, i.e. M = 4

9R. Obviously, thissituation cannot occur physically but the pressure can be arbitrarily large.Therefore, the mass can get arbitrarily close to the limit value 4

9R.

126

Acknowledgements

É per me un piacere poter ringraziare tutte le persone che hanno resopossibile non soltanto la stesura di questa tesi, ma anche la mia crescitaculturale e spirituale. Mi piacerebbe menzionare ciascuno di loro e spiegarein breve il contributo che ognuno ha dato al miglioramento del mio percorsodi studi o della mia persona o di entrambe. Tuttavia, se lo facessi, questocapitolo di ringraziamenti risulterebbe più lungo della tesi stessa. Ciò è perdire che nel mio percorso ho avuto la fortuna di essere accompagnato edaiutato da numerose persone, ma potrò qui citarne solo alcune. Spero cheagli altri possa bastare di sapere che sono tutti ben impressi nella mia mentee che sono molto grato ad ognuno di loro.

I primi ringraziamenti li rivolgo alle figure accademiche e, in particolare,al Prof. Gianluca Grignani, relatore di questa tesi, che mi ha supportatoe consigliato in molti momenti, anche difficili, dei miei studi universitari eha sopportato con pazienza le mie frequenti domande, rispondendo ad esseanche quando non riguardavano i suoi corsi. Desidero ringraziare poi il Dott.Andrea Marini per la notevole generosità dimostrata nell’aiutarmi, talvoltaanche impiegando una grande quantità del suo tempo, a trovare risposte alledomande che mi ponevo. Una menzione va anche ai molti altri miei professori,tra i quali desidero citare in particolare il Prof. Sergio Scopetta, la Prof.ssaGiuseppina Anzivino, il Prof. Simone Pacetti e la Prof.ssa Marta Orselli. Iprimi tre, per gli innumerevoli consigli in ambito accademico elargiti senzarisparmiarsi e per aver acconsentito a scrivere una lettera di presentazionedella mia persona a tutte le università a cui ho chiesto di essere ammesso perproseguire i miei studi. La quarta, non tale per ordine di importanza, peraver dimostrato reale interesse per la mia carriera universitaria e averle datoun contributo significativo con i suoi importanti suggerimenti.

Per la parte scolastica ed educazionale, non posso che iniziare ringraziandola Prof.ssa Stefania Carletti, la quale ha fatto nascere in me l’interesse versolo studio della Fisica e mi ha sempre incoraggiato a credere nelle mie ca-pacità di studente. Andando molto indietro con gli anni, desidero ancheringraziare chi mi ha fatto scoprire il piacere della conoscenza, svolgendo il

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suo mestiere di insegnante ed educatrice con grande passione: la MaestraMaria Grazia Becchetti. Ero piccolo, ma non così piccolo da non ricordareche la sua passione e impegno nello svolgere il suo ruolo sono stati esemplarie per me fondamentali. Sono grato anche agli altri insegnanti che mi hannoaiutato a creare un personale metodo di studio, rivelatosi indispensabile perl’ottenimento di tutti i miei risultati accademici. Non vorrei tralasciare nep-pure gli educatori, i cui insegnamenti di vita si sono rivelati essenziali più diuna volta. Ringrazio specialmente Enrico Milletti per non aver mai smesso diinteressarsi a me e di consigliarmi sin da quando ero appena un adolescente.

Un pensiero speciale lo riservo alla mia famiglia. Il più grande di tuttii ringraziamenti va ai miei genitori Carlo e Valeria. É principalmente mer-ito loro se sono oggi consapevole che una vita sana e completa è l’effettodell’equilibrio di molte componenti e che la dedizione verso i doveri di stu-dente è solo una di queste. Per loro in particolare, non mi è possibile elencaretutti i modi in cui hanno contribuito a farmi diventare la persona e lo stu-dente che sono, ma posso dire che mi sento veramente fortunato ad essere lorofiglio. Dedico un grande ringraziamento e auguro tutto il meglio possibile amia sorella Letizia per avermi sopportato in ogni secondo della sua vita e,allo stesso tempo, per aver alleviato le mie giornate durante i lunghi e intensiperiodi di studio con il suo spirito solare e le sue risate. Ringrazio di cuorei miei nonni paterni, Gildo e Maria, per essere stati, dal primo all’ultimogiorno che li ho visti, di grande ispirazione con il loro esempio di come siapossibile, con generosità e spirito di sacrificio, affrontare con gioia ogni os-tacolo della vita. Ringrazio di cuore anche i miei nonni materni, Damiano eLuigina, per essere stati in moltissime occasioni dei secondi genitori e aversempre desiderato il meglio per me e mia sorella, come continuano ancora afare. Un caloroso ringraziamento va anche a tutti gli altri membri della miafamiglia che mi sono stati vicino.

Infine, un grazie sincero a tutti i miei amici e compagni di studio e diver-timento presenti e passati, per avermi fatto trascorrere questi anni di duro ecostante lavoro in modo molto più leggero e spensierato di quello che sareb-bero stati senza di loro. Grazie, in particolare, a coloro che mi hanno sempremostrato affetto e che “ci sono sempre stati”: Matteo Busti, Chiara Capacci,Riccardo Ercoli e Nicola Martini.

Come detto, molti altri meriterebbero una menzione per aver spontanea-mente dedicato anche uno solo dei secondi delle loro vite a me. A tutti loro,e a chi ho già nominato, non posso che promettere che farò di tutto affinchè illoro contributo sia valorizzato. Spero un giorno di poter ricambiare il favorededicandogli ben più di un secondo della mia vita. Grazie.

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