The Hawking Effectold.apctp.org/conferences/2009/IWGC2009/Lecture note... · 2009. 1. 15. · The...

The Hawking Effect

APCTP-NCTS International School/Workshop on

Gravitation and CosmologyJanuary 16, 2009

Larry FordTufts University

Lecture I

Reference: gr-qc/9707062

Conceptually cleanest if there exist flat “in” and “out” regions.

t out region

in region fj

Fj

non-static region

Particle creation by the gravitational field

The are positive frequency in the past but defined everywhere.fj

and are both complete sets, so we expand one in terms of the other:

fj , f∗j Fj , F

∗j

fj =∑

k

(αjkFk + βjkF ∗k )

Fk =∑

j

(α∗jkfj − βjkf∗

j )or

Orthonormality∑

k

(αjkα∗j′k − βjkβ∗

j′k) = δjj′

and∑

k

(αjkαj′k − βjkβj′k) = 0

aj =∑

k

(α∗jkbk − β∗

jkb†k)

Furthermore:

bk =∑

j

(αjkaj + β∗jka†j)

a Bogolubov transformations

t out region

in region fj

Fj

non-static region

has no particles in the past

bj |0〉out = 0

aj |0〉in = 0

|0〉in

|0〉out has no particles in the future

βjk != 0 ⇒ |0〉in != |0〉out

Particle creation!

Spacetime geometry of a black hole formed by gravitational collapse

vv 0

u

r=0

r=0

H+

I

I+

!

r=0

Figure 2:The Penrose diagram for the spacetime of a black hole formed by gravitational col-lapse. The shaded region is the interior of the collapsing body, the r = 0 line on theleft is worldline of the center of this body, the r = 0 line at the top of the diagram isthe curvature singularity, and H+ is the future event horizon. An ingoing light raywith v < v0 from I− passes through the body and escapes to I+ as a u = constantlight ray. Ingoing rays with v > v0 do not escape and eventually reach the singularity.

11

u = t− r∗

v = t + r∗

Schwarzschild null coordinates:

r∗ = r + 2M ln(

r − 2M

2M

)

Interior of the collapsing star

Schwarzschild spacetime

Essential idea of Hawking’s treatment of particle creation in this spacetime:

Consider modes which are pure positive frequency on These mode propagate through

the collapsing body and are highly redshifted in the exterior geometry before reaching

fω"m

I−.

Thus these modes have extremely high frequencies as they pass through the collapsing

body, and we may use geometric optics to find their form on I+.

vv 0

u

r=0

r=0

H+

I

I+

!

r=0

Figure 2:The Penrose diagram for the spacetime of a black hole formed by gravitational col-lapse. The shaded region is the interior of the collapsing body, the r = 0 line on theleft is worldline of the center of this body, the r = 0 line at the top of the diagram isthe curvature singularity, and H+ is the future event horizon. An ingoing light raywith v < v0 from I− passes through the body and escapes to I+ as a u = constantlight ray. Ingoing rays with v > v0 do not escape and eventually reach the singularity.

11

v = G(u)

u = g(v) = G−1(v)

Mapping between the ingoing and outgoing rays, as for the moving

mirror model of Fulling and Davies.

Fω"m

fω"m

pure (+) frequency on

pure (+) frequency on

I+.

I−.

This is dominated by modes which left I− with very high frequency, propagatedthrough the collapsing body just before the horizon formed, and then underwent alarge redshift on the way out to I+. Because these modes had an extremely highfrequency during their passage through the body, we may describe their propagationby use of geometrical optics.

A v = constant ingoing ray passes through the body and emerges as a u =constant outgoing ray, where u = g(v) or equivalently, v = g−1(u) ≡ G(u). Thegeometrical optics approximation leads to the following asymptotic forms for themodes:

fω"m ∼Y"m(θ, φ)√

4πω r×

e−iωv, on I−

e−iωG(u), on I+ (2.1)

and

Fω"m ∼Y"m(θ, φ)√

4πω r×

e−iωu, on I+

e−iωg(v), on I− ,(2.2)

where Y"m(θ, φ) is a spherical harmonic. Hawking [17] gives a general ray-tracingargument which leads to the result that

u = g(v) = −4M ln(

v0 − v

C

)

, (2.3)

orv = G(u) = v0 − Ce−u/4M , (2.4)

where M is the black hole mass, C is a constant, and v0 is the limiting value of v forrays which pass through the body before the horizon forms.

We will derive this result for the explicit case of a thin shell. The spacetime insidethe shell is flat and may be described by the metric

ds2 = dT 2 − dr2 − r2 dΩ2. (2.5)

Thus, in the interior region, V = T + r and U = T − r are null coordinates which areconstant on ingoing and on outgoing rays, respectively. The exterior of the shell is aSchwarzschild spacetime with the metric

ds2 =(

1 −2M

r

)

dt2 −(

1 −2M

r

)−1dr2 − r2 dΩ2. (2.6)

As noted above, the null coordinates here are v = t + r∗ and u = t − r∗, where

r∗ = r + 2M ln(

r − 2M

2M

)

(2.7)

is the “tortoise coordinate”. Let r = R(t) describe the history of the shell. Themetric in this three dimensional hypersurface must be the same as seen from bothsides of the shell. (The intrinsic geometry must match.) This leads to the condition

1 −(

dR

dT

)2

=(

R − 2M

R

)(

dt

dT

)2

−(

R − 2M

R

)−1(dR

dT

)2

. (2.8)

12

Asymptotic forms:

Massless scalar field modes:

Consider the case of a collapsing thin shell:

R(t)

Interior: flat spacetime

ds2 = dT 2 − dr2 − r2 dΩ2

interior time

Exterior: Schwarzschild spacetime

Intrinsic geometry must match as seen from both sides:

(dT 2 − dr2 − r2 dΩ2

)

r=R=

((1− 2M

r

)dt2 −

(1− 2M

r

)−1dr2 − r2 dΩ2

)

r=R

(1st Israel junction condition)

We need to find the function g(v).

1−(

dR

dT

)2

=(

R− 2M

R

)(dt

dT

)2

−(

R− 2M

R

)−1(dR

dT

)2

The 2nd junction condition matches the extrinsic curvature and gives the equation for ; it is not needed here.R(t)

Now consider a null ray which enters (1) the shell, passes through it, and exits (2) just before R = 2M.

v V U u

1 2

Figure 3:An ingoing ray enters the collapsing shell at point 1, passes through the origin, andexits as an outgoing ray at point 2, when the shell has shrunk to a smaller radius(dotted circle).. This is illustrated schematically in this diagram. Note that the raysin question are actually imploding or exploding spherical shells of light.

There is a second junction condition, that the extrinsic curvatures of each side of thishypersurface match [18]. This leads to the equation which determines R(t) in termsof the stress-energy in the shell. For our purposes, this equation is not needed, andwe may assume an arbitrary R(t).

There are now three conditions to be determined: the relation between the valuesof the null coordinates v and V for the ingoing ray, the relation between V and U atthe center of the shell, and finally the relation between U and u for the outgoing ray.This sequence of matchings is illustrated in Fig. 3.

• Let us suppose that our null ray enters the shell at a radius of R1, which isfinitely larger than 2M . At this point, both R/R − 2M and dR/dT are finiteand approximately constant. Thus dt/dT is approximately constant, so t ∝ T .Similarly, r∗ is a linear function of r in a neighborhood of r = R1. Thus, weconclude that

V (v) = av + b (2.9)

in a neighborhood of v = v0, where a and b are constants.

• The matching of the null coordinates at the center of the shell is very simple.Because V = T + r and U = T − r, at r = 0 we have that

U(V ) = V. (2.10)

13

u = t− r∗

v = t + r∗

U = T −R

V = T + R

Perform a sequence of matchings:

At entrance, both and are finite and non-zero, so V (v) = av + b.

dt

dT

dr∗

dr

constantsAt the center, so r = 0 V = U.

R(T ) ≈ 2M + A(T0 − T )At exit,(

dt

dT

)2

≈(

R− 2M

2M

)−2(dR

dT

)2

≈ (2M)2

(T − T0)2

t ∼ −2M ln(

T0 − T

B

), T → T0

r∗ ∼ 2M ln(

r − 2M

2M

)∼ 2M ln

[A(T0 − T )

2M

]Similarly,

u = t− r∗ ∼ −4M ln(T0 − T

B′

)so

U = T − r = T −R(T ) ∼ (1 + A)T − 2M −AT0

However,

Putting everything together yields our final result:

u = g(v) = −4M ln(

v0 − v

C

)

constant

constant

We derived this result for the special case of a thin shell, but it agrees with Hawking’s more general

argument.

We can show that our result is general by dividing an arbitrary star into a sequence of thin shells.

Only the exit from the last shell will give a logarithmic dependence; all others will

give linear functions.

• We now consider the exit from the shell. We are interested in rays which exitwhen R is close to 2M . Let T0 be the time at which R = 2M . (Note that thisoccurs at a finite time as seen by observers inside the shell.) Then near T = T0,

R(T ) ≈ 2M + A(T0 − T ), (2.11)

where A is a constant. If we insert this into Eq. (2.8), we have that

(

dt

dT

)2

≈(

R − 2M

2M

)−2(dR

dT

)2

≈(2M)2

(T − T0)2, (2.12)

which implies

t ∼ −2M ln(

T0 − T

B

)

, T → T0. (2.13)

Similarly, as T → T0, we have that

r∗ ∼ 2M ln(

r − 2M

2M

)

∼ 2M ln[

A(T0 − T )

2M

]

, (2.14)

and hence that

u = t − r∗ ∼ −4M ln(T0 − T

B′

)

. (2.15)

(Again, B and B′ are constants.) However, in this limit we have that

U = T − r = T − R(T ) ∼ (1 + A)T − 2M − AT0. (2.16)

Combining these results with Eqs. (2.9) and (2.10) yields our final result, Eq. (2.3).Although we have performed our explicit calculation for the special case of a thin shell,the result is more general, as is revealed by Hawking’s derivation. We can understandwhy this is this case; the crucial logarithmic dependence which governs the asymptoticform of u(v) comes from the last step in the above sequence of matchings. This stepreflects the large redshift which the outgoing rays experience after they have passedthrough the collapsing body, which is essentially independent of the interior geometry.We could imagine dividing a general spherically symmetric star into a sequence ofcollapsing shells. As the null ray enters and exits each shell, each null coordinate isa linear function of the preceeding one, until we come to the exit from the last shell.At this point, the retarded time u in the exterior spacetime is a logarithmic functionof the previous coordinate, and hence also a logarithmic function of v, as given byEq. (2.3).

From Eq. (2.2), we see that the out-modes, when traced back to I−, have theform

Fω"m ∼

e4Miω ln[(v0−v)/C], v < v0

0, v > v0.(2.17)

14

Thus the form of near isI−.Fω"m

Fω"m =∫ ∞

0dω′

(α∗ω′ω"mfω′"m − βω′ω"mf∗ω′"m

)

Expansion of the out-modes in terms of the in-modes and Bogolubov coefficients:

αω′ω"m = αω′"m,ω"m βω′ω"m = βω′"−m,ω"mHere

fω"m =Y"m(θ,φ)√

4πω re−iωvand I−.on

Thus we can find the Bogolubov coefficients by inverse Fourier transforms:

α∗ω′ω"m =12π

√ω′

ω

∫ v0

−∞dv eiω′v e4Miω ln[(v0−v)/C]

βω′ω"m = − 12π

√ω′

ω

∫ v0

−∞dv e−iω′v e4Miω ln[(v0−v)/C]and

α∗ω′ω"m =12π

√ω′

ωeiωv0

∫ ∞

0dv′ e−iω′v′

e4Miω ln(v′/C)

βω′ω"m = − 12π

√ω′

ωeiωv0

∫ ∞

0dv′ eiω′v′

e4Miω ln(v′/C)

orv′ = v0 − v

Integrands are analytic except on the (-) real axis, where the branch cut of the logarithm is located.

Figure 4:The closed contour of the integration in Eq. (2.23) is illustrated. The fact that thisintegral vanishes implies that the integrals along each of the dotted segments areequal, which implies the first equality in Eq. (2.24).

The mean number of particles created into mode ω"m is now given by

Nω"m =∑

ω′

|βω′ω"m|2 =1

e8πMω − 1. (2.27)

This is a Planck spectrum with a temperature of

TH =1

8πM, (2.28)

which is the Hawking temperature of the black hole.To show that these created particles produce a steady flow of energy to I+, we need

to use either an analysis involving wavepackets [17], or else the following argument[19]. Note that the modes are discrete only if we regard the system as being enclosedin a large box, which we may take to be a sphere of radius R. Then in the limit oflarge R we have

∑

ω

→R2π

∫ ∞

0dω. (2.29)

The total energy of the created particles is

E =∑

ω"m

ωNω"m =R2π

∑

"m

∫ ∞

0dω ωNω"m. (2.30)

This quantity would diverge in the limit that R → ∞. However, this simply reflectsa constant rate of emission over an infinitely long time (when backreaction of the

16

Thus we can close the contour in the lower-half plane and write

∮

Cdv′ e−iω′v′

e4Miω ln(v′/C) = 0

∫ ∞

0dv′ e−iω′v′

e4Miω ln(v′/C) = −∫ ∞

0dv′ eiω′v′

e4Miω ln(−v′/C−iε)

= −e4πMω

∫ ∞

0dv′ eiω′v′

e4Miω ln(v′/C)

v′ → −v′

ln(−v′/C − iε) = −πi + ln(v′/C)

|αω′ω"m| = e4πMω|βω′ω"m|Thus

Total energy radiated:

E =∑

ω"m

ωNω"m =R2π

∑

"m

∫ ∞

0dω ωNω"m

Enclose in a sphere of radius∑

ω

→ R2π

∫ ∞

0dωUse

L =E

R =12π

∑

!m

∫ ∞

0dω ωNω!mLuminosity:

Effect of scattering by the spacetime curvature:

Γ!m = transmission probability

L =12π

∑

!m

∫ ∞

0dω ω

Γ!m

e8πMω − 1Now

Black Hole Thermodynamics

Generalized Second Law:

∆Sblack hole + ∆Smatter ≥ 0

∆Sblack hole =14

A A = horizon area/! G

T =! κ

2πκ = surface gravity

First Law: dM = T dSblack hole + dW

Schwarzschild black holes:

A = 16πM2 T =1

8πMdW = 0

Effects of charge and angular momentum:

a black hole has been derived in a variety of ways by several people, so

its prediction seems to be a clear consequence of our present theories of

quantum mechanics and general relativity.

That of course was my own personal view as I was finishing my Ph.D. in 1976,

heavily influenced by discussions with Hawking and with a few others, but without in

any way being a claim to a completely balanced and broad view of what others might

have been thinking at the time. However, I thought it might be of at least some

historical interest to present here this biased viewpoint of the important historical

development of black hole emission. For other viewpoints, see [40, 41, 42].

2 Hawking Emission Formulae

For the Kerr-Newman metrics [43, 44], which are the unique asymptotically flat

stationary black holes in Einstein-Maxwell theory [45, 46, 47, 48, 49], one can get

explicit expressions for the area A, surface gravity κ, angular velocity Ω, and elec-

trostatic potential Φ of the black hole horizon in terms of the macroscopic con-

served quantities of the mass M , angular momentum J ≡ Ma ≡ M2a∗, and charge

Q ≡ MQ∗ of the hole [50], using the value r+ of the radial coordinate r at the event

horizon as an auxiliary parameter:

r+ = M + (M2 − a2 − Q2)1/2 = M [1 + (1 − a2∗ − Q2

∗)1/2],

A = 4π(r2+ + a2) = 4πM2[2 − Q2

∗ + 2(1 − a2∗ − Q2

∗)1/2],

κ =4π(r+ − M)

A=

1

2M−1[1 + (1 −

1

2Q2

∗)(1 − a2∗ − Q2

∗)−1/2]−1,

Ω =4πa

A= a∗M

−1[2 − Q2∗ + 2(1 − a2

∗ − Q2∗)

1/2]−1,

Φ =4πQr+

A= Q∗

1 + (1 − a2∗ − Q2

∗)1/2

2 − Q2∗ + 2(1 − a2

∗ − Q2∗)

1/2. (4)

Here a∗ = a/M = J/M2 and Q∗ = Q/M are the dimensionless angular momen-

tum and charge parameters in geometrical units (G = c = k = 4πε0 = 1 but for this

without setting h = 1, so that mass, time, length, and charge all have the same units,

and angular momentum has units of mass or length squared; e.g., the angular mo-

mentum of the sun is J# ≡ a∗#M2# = (0.2158±0.0017)M2

# = 47.05±0.37 hectares =

116 ± 1 acres [51, 52]. However, we shall return to Planck units for the rest of this

9


















r+ = M + (M2 − a2 − Q2)1/2 = M [1 + (1 − a2∗ − Q2

∗)1/2],

A = 4π(r2+ + a2) = 4πM2[2 − Q2

∗ + 2(1 − a2∗ − Q2

∗)1/2],

κ =4π(r+ − M)

A=

1

2M−1[1 + (1 −

1

2Q2

∗)(1 − a2∗ − Q2

∗)−1/2]−1,

Ω =4πa

A= a∗M

−1[2 − Q2∗ + 2(1 − a2

∗ − Q2∗)

1/2]−1,

Φ =4πQr+

A= Q∗

1 + (1 − a2∗ − Q2

∗)1/2

2 − Q2∗ + 2(1 − a2

∗ − Q2∗)

1/2. (4)






# = 47.05±0.37 hectares =


9


















r+ = M + (M2 − a2 − Q2)1/2 = M [1 + (1 − a2∗ − Q2

∗)1/2],

A = 4π(r2+ + a2) = 4πM2[2 − Q2

∗ + 2(1 − a2∗ − Q2

∗)1/2],

κ =4π(r+ − M)

A=

1

2M−1[1 + (1 −

1

2Q2

∗)(1 − a2∗ − Q2

∗)−1/2]−1,

Ω =4πa

A= a∗M

−1[2 − Q2∗ + 2(1 − a2

∗ − Q2∗)

1/2]−1,

Φ =4πQr+

A= Q∗

1 + (1 − a2∗ − Q2

∗)1/2

2 − Q2∗ + 2(1 − a2

∗ − Q2∗)

1/2. (4)






# = 47.05±0.37 hectares =


9

Kerr-Newman black holes

dW = Ω dJ + Φ dQ

Emission rate proportional to

Γexp(ω −mΩ− qΦ)∓ 1)

bosons

fermions

Extreme black holes: M2 = a2 + Q2

T = 0

Third Law: Cannot reach T=0 in a finite number of steps

Possible Quantum States:

1) Unruh vacuum: black hole radiating into empty space

2) Hartle-Hawking vacuum: black hole in thermal equilibrium in a box

3) Boulware vacuum: quantum state for a field near a static star

Back reaction of the radiation on the black hole

Use 〈Tµν〉 for the quantum field and the semiclassical Einstein equations :

Gµν = 8π 〈Tµν〉

Negative energy flux across the horizon

〈Trt〉 is finite

The Transplankian Issue

Modes which give the dominant contribution leave I− with

ω ≈M−1 exp(M2/M2Pl)

Possible alternative: nonlinear dispersion relation

Jacobson, Unruh

Mode regeneration

Violates local Lorentz symmetry

Final State and Information Puzzles

Possible final states:

1) Singularity

2) Stable remnant

3) Nothing (flat spacetime)I+

I−

Is information lost? (and unitarity violated?)

Some say YES: Hawking (1976)

Some say NO: Hawking (2004)

Can interactions cause information to leak out slowly during evaporation?

‘t Hooft

Black Hole Fluctuations

Flux fluctuations Wu & LF

Mass and area fluctuations Bekenstein, Hu & Roura

Horizon fluctuations

What does this mean?

Can horizon fluctuations modify the outgoing radiation?

Svaiter &LF,Thompson &LF

2) Black holes provide an elegant link between gravity thermodynamics and quantum theory

Summary

3) Several questions remain unanswered

What is the final state?

Is information lost?

Are transplankian modes needed?

Can horizon fluctuations produce significant modifications?

1) Black holes create a (filtered) thermal spectrum of particles

The Hawking Effectold.apctp.org/conferences/2009/IWGC2009/Lecture note... · 2009. 1. 15. · The...

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