ADM formulation of general relativity

1 Introduction

One possibility to approach quantum gravity is the hamiltonian formulation of general relativity1.As it first has been formulated as a complete theory by Arnowitt, Deser and Misner one thereforeoften calles it the ADM formalism today. But in some book and lectures it is also referred to it asthe hamiltonian formulation, or foliation of general relativity.

Starting with

Gab = Rab 12gabR = Tab()

the aim is to perform a Legendre transformation to obtain a Hamiltonian formulation of theEinstein-Hilbert Lagrangian

L =gR.

We are going to use the metric signature (+++), latin indices iklj run from 1 to 3 and abcd from0 to 3, where x0 = t. In addition we set = 1. To achieve our goal we will follow the roadmapgiven below:

1. Foliate spacetime manifold into a series of spacelike hypersurfaces.

2. Re-express the Lagrangian in terms of hij , N and Ni.

3. Find the momentum conjugate to hij , N and Ni.

4. Perform a variation of the Hamiltonian with respect to N and Ni to obtain two constraintequations, viewing N and Ni as Lagrange multipliers.

Putting all those steps together we will finally obtein the Hamiltonian constraint formulation ofrelativity.

2 Foliation of spacetime

One can show that an arbitrary globally hyperbolic manifold M can be foliated into a family of

Figure 1: Foliation of spacetime in general relativity

hypersurfaces of constant t, which we call t. Assuming spacetime (M, g) is of this topologicalstructure R we can split the metric into spatial and time components. On each hypersurfaceof constant t we can now define a pure spatial metric hij , by:

hij = gij + ninj , (1)

with unit normal vector ni of the hypersrface t. The unit normal vector is necessarily timelikewith lenght nana = 1, as the hypersurfaces are spacelike. Supposing the infinitesimal distance on ahypersurface is given by hij(t, x

i)dxidxj , the proper time of a co-moving observer is differing fromthe coordinate time by the lapse function N which yields:

d = N(t, xi)dt.

1We will refer to general relativity as relativity if not mentioned elsewise

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The distance between two infinitasimally adjecent points on a hypersurface is given by

xi2 = xi1 N i(t, xi)dt,

where Na is called the shift vector . Physically the lapse function represents the rate of flow ofproper time with respect to t:

N = tn (2)and Na represents the movement tangential to the surface t

N i = hijtj . (3)

They allow us to sew two hypersurfaces together as illustrated in figure 1. In four dimensionalspacetime the line-element ist - as we remember - given by

ds2 = (coordinate distance)2 (proper time)2. (4)Introducing our previous results we obtain for our line-element the expression:

ds2 = hij(dxi +N idt)(dxj +N jdt) (Ndt)2

= gabdxadxb.

(5)

Using this result, the components of gab can be derived.

g00 = hijNiN j N2 = NjN j N2 (6)

g0b = hijNi = Nj (7)

ga0 = hijNj = Ni (8)

gab = hij (9)

It also is quite easy to see that

g = detg = N2dethij = Nhhij , N

i and N are the new variables defining the field since they contain the same information asthe original spacetime metric. As the next step the Lagrangian given above hast to be re-expressedin terms of these variables.

3 The field equations in terms of the new variables

Now that we have definde the foliation of spacetime, we take the field variables to be hij , Ni and

N . These lead to the relations2

piij =L

hij(10)

piN =L

N(11)

piiN =L

N i(12)

To be abled to find the time-derivative Lthij we introduce the extrinsic curvature, which is definedas

Kij := hkiknj =

1

2Lnhij = 1

2N1 (Lthij LNhij) . (13)

2When looking at those equations it is important to realize that the dot does not indicate a time-derivative.Because of the canonical framework the diffeomorphism invariance of general relativity has to be maintained (e.g.the system should be coordinate-independent). As time is defined differently in each coordinate-system it is notsuited for this approach. It is necessary to differentiate to the local time that is perpendicular to the hypersurface ateach point of it and this perpendicular direction is given by the field that describes time. Therefore the Lie-derivativeis used.

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This definiton is analogous to the covariant derivative along a curve, which is nothing but theparallel transport along a curve. It is possible to rewrite Einsteins field equations in terms ofthis curvature and finally give the equations of motion refering to the variables (10) - (12). Wellaccomplish those to things during the next pages, starting with the vaccum field equations.

As we already introduced a 3-dimensional metric, a number of quantities in 3 dimensions have tobe defined. Therefore first the curvature tensor is expressed in terms of a dual vector field and ofthe derivative associated with hab. Starting with the 4-dimensional tensor

(ab ba)V c = RcdabV d

we indentify the changes given below

gab hija DaV d d

Rdabc (3)Rdabcand define

(3)Rabcdd = (DaDb DbDa)c. (14)

The operation Da is called the exterior derivative, is defined as

DcTa1...ak

b1...bl = ha1d1 . . . hbl

elhchffT d1...dke1...el . (15)

From this we obtain that

DaDbc = Da

(hbdhc

edc)

= hafhb

ghckf

(hdgh

ekde

)= ha

fhbdhc

efde + hceKabndde + hbdKacnede(16)

using the relation that

habhc

dbhde = habhcdb (gde + ndne) = Kacne, (17)because bgde = 0 and habbnd = Kad. One can calculate that in addition the following holds:

hbdnede = hbdd(nee) hbdedne = Kbee (18)

As the second term on the right-hand of Eq. (16) is symmetric in a and b it will vanisch in Eq.(14). By this we finally achieve

(3)Rabcd = ha

fhbghc

khdjRfgkj KacKbd +KbcKad (19)

One can also conclude

Rabcdhachbd = Rabcd(g

ac + nanc)(gbd + nbnd)

= R+ 2Racnanc

= 2Gacnanc

(20)

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Gabnanb = 12Rabcdh

abhcd

=1

2gdmRacb

mhabhcd

=1

2(hdm ndnm)Racbmhabhcd

=1

2(hdm ndnm)hfahgchkbhmj ((3)Rfgkj +KfkKgj KgkKf j)habhcb

=1

2(hdm ndnm)hfkhgdhmj ((3)Rfgkj +KfkKgj KgkKf j)

=1

2hfkgj (

(3)Rfgkj +KfkKg

j KgkKf j)

=1

2((3)R+KkkK

jj KjkKjk) = 0

From the second line in Eq. (20) we can derive the Ricci scalar as

R = 2(Gabnanb Rabnanb) (21)

If one now calculates Rabnanb in terms of the extrinsic curvature the Lagrangian densitiy can be

written, using L =gR.

Rabnanb = Racb

bnanb

= na(ac ca)nc= (ana)(cnc) (cna)(anc)a(nacnc) +a(naanc)= K2 KacKac Divergence terms

(22)

In total we now have derived the Einstein field equations in vacuum depending the extrinsiccurvature:

Gijninj =

1

2((3)RKijKij +K2)

Rijninj = K2 KijKij

(23)

Combining those two equations and attaching them into the Lagrangian yields

L =hN

((3)R+KijK

ij +K2)

=hN

[(3)R+

(1

4N2(Lthij LNhij)(LthijLNhij)KiiKii

)] (24)and we are finally capable to deduce the equations of motion (the conjugate momenta):

piij =L

Lthij=hN

(1

2N2(Lthij LNhij)

LthijK2

)=hN

(N1Kij 1

4N2(Lthijhij LNhijhij)2

)=hN(N1Kij N1Khij)

=h(Kij Khij)

(25)

and furthermore

piN =L

N= 0 (26)

piiN =L

Ni= 0. (27)

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4 Discussion of the Lagrange multipliers

We want to point out what the above found equations mean. Thereby N and N i turn out to beLagrange multipliers. To find out the means well have a look at a simple example of the classicalHamiltonian to give an adequate answer:

Consider a pendulum with coordinates (x, y) and the normalization condition x2 + y2 = l2.The Lagrangian for such a pendulum is, as we already know, given by L = 1

2(x2 + y2)mgy

and the variation priciple leads to

S = Ldt =

(L

xx+

L

yy

)dt

Since x2 + y2 = l2 the constraint (l2 x2 y2) can be added without exerting a change inaction. If further on x, y and are considered dynamical variables

p =L

= 0 (28)

always holds. As a conclusion of this results we gain, that if a constraint is hidden in aLagrangian, or the Hamiltonian for that matter, the conjugated momentum to the dynamicalvariable (which displays a degree of freedom) turns out to be 0. The retrieved constraint thenis, that if the conjugated momentum is 0, also

H

= 0. (29)

The conclusion from LN

= 0 and LNi

= 0 therefore are that these variables have to be constraintsof the system and are the so-called primary constraints.

5 The Hamiltonian and the Hamiltonian constraints

The Hamiltonian density can now be defined by the degrees of freedom hij :

H = piij hij L= piij(2KijN + LNhij)

hN((3)R+KijK

ij K2).(30)

For the exterior derivative holds

Dihjk = 0, (31)

which is analogous to the covariant derivative cgab = 0. In addition the Lie-derivative of thespatial metric can (in analogy to the Lie-derivative of spacetime metric) be written as

LV hij = DiVj +DjVi. (32)By this we are abled to rewrite the Hamiltonian in terms of Di and piij and to do so well first givethe extrinsic curvature in terms of piij and hij :

Kij = h1/2

(1

2piklh

klhij piij).

This can be shown by inserting it into Eq. (25). The Hamiltonian thereby becomes

H = piij(2KijN +DiNj +DjNi)hN((3)R+KijK

ij K2)=hN((3)RKijKij +K2 + 2h1/2piijKij) + piij(DiNj +DjNi)

(33)

We now further develop the right-hand side of Eq. (33). To do so we are going to need the followingrelations:

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KijKij = h1

(piijpi

ij piklhklhijpiij + 34pi2)

(34)

piijKij = h1/2

(piijpi

ij 12piklh

klhijpiij

)(35)

K2 =1

4h1pi2. (36)

If Eq. (34) is inserted into Eq. (33) it becomes

H =hN

((3)R+ h1piijpiij 1

2h1pi2

) 2NjDi(h1/2piij) + 2Di(h1/2Njpiij).

(37)

The last terms build just a boundary term in the integral, used to obtain the Hamiltonian and cantherefore be ignored. According to this for Lagrange multipliers holds H = 0, so that we get theHamilton constraint

H

N= (3)R+ h1piijpiij 1

2h1pi2 = 0 (38)

and the diffeomorphism or Gauss constraint

H

Nj= Di

(hpiij

)= 0. (39)

Those to have in fact first been formulated by Dirac, which is why he should also be mentioned atthis point. Finally one can perform an analysis of the problems they raise and which can partlybe solved by introduction of new variables, the so-called Ashtekar variables.

6 Problems associated with the constraint equations

The now complete ADM formulation of general relativity can be quantized by a process introducedby Dirac. For his one first calculates the Poisson-brackest

{hij(x), pikl(y)} = 12

(ki

lj +

kj li

)3(x y)

and the variables turn into operators

hij hij (40)piij i~

hij(41)

As, following the constraint equations, H = 0 always holds the Schrodinger equation reduces toH| = 0. Here we have

H =hN

H

N 2hNi

H

Ni= 0

and therefore can conclude the Wheeler-De Witt equations

H

N| = 0 (42)

H

Ni| = 0. (43)

The problem is that the first of those equations is ahighly singular functional differential equationfor which until now no physical solutions have been found.

There are of course other steps possible to be taken into quantum gravity as e.g. loops.

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