Taylor M.E. Partial Differential Equations III: Nonlinear Equations

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698 18. Einstein’s Equations

If '.x

/ D '

max

,then‰



.'/  0 at x

,so

(9.63) f



;E.'

/; '.x



C .1  /



'.x

/  b



 0:

If '.x

/>band  2 .0; 1, this requires f.x

;E.'

/; '.x

//  0,whichis

equal to (9.46)if'

max

>b, so as before we have the estimate (9.53).

Thus the ﬁxed-point theorem applies to our situation, so Proposition 9.5 is

proved. Thus, in rough parallel with Proposition 9.3, we have the following:

Proposition 9.6. Let S be a compact 3-manifold with metric tensor h

. Assume

the scalar curvature S

<0on S.LetH 2 C

.S/ be a positive function satis-

fying the hypothesis (9.43)–(9.44). Then there is a positive ' 2 C

.S/ such that

D '

, there is a Ricci-ﬂat Lorentz 4-manifold M , containing S as a

spacelike hypersurface, with induced metric

and second fundamental form

(9.64)

D Q

C H h

;

where

(9.65)

Q D2'

2

 D

1

rH/:

In particular, the mean curvature of S  M is H .

See [CBY] for a discussion of some cases where S is diffeomorphic to R

and asymptotically ﬂat, and [CO] for a further analysis, when also H D 0.Inthis

case, one can make a preliminary conformal change of metric to achieve S

D 0,

so that (9.33) becomes

' D

7

Exercises

1. Show that the identities (9.3)and(9.6)forG

imply Lemma 8.2.

2. Put together Proposition 9.4 and Birkhoff’s theorem, and make some deductions, re-

garding .Q

/ and solutions to (9.33), and symmetry properties that they cannot have.

3. Suppose that one wants to solve (1.1), namely,

(9.66) G

D 8T

Show that the equation (9.7)–(9.8)onS get replaced by

(9.67)

 K

C K

D 2;

 3H

D J

;

10. Time slices and their evolution 699

where

(9.68)  D 8T

D8T

Study this system, particularly in the case H D const.

4. Extend Proposition 9.5 as follows. Replace (9.43)–(9.44) by the hypothesis that, for

some p>3,

(9.69) kD

1

 A

kuk

;

with

(9.70) A

krH k

min

5. Note that solving the constraint equations (9.7)–(9.8) with K

D 0 is reduced to

solving (9.33) with f D 0, namely,

' D

Consider solutions to this, both on compact and on noncompact S . Relate the solution

'.x/ D 1 C

jxj

on ﬂat R

(outside the origin) to the Schwarzschild metric.

10. Time slices and their evolution

Suppose M is a Lorentz manifold whose metric satisﬁes the empty-space Einstein

equation (8.1), and on M we have a smooth function t such that  D grad t is

timelike. Thus M is foliated by the surfaces S

,onwhicht D c, called time

slices. One can choose local coordinates .t; x

/ on an open set O  M

with respect to which the metric is

(10.1) ds

D.t; x/

j;kD1

.t; x/ dx

This can be done by picking local coordinates .x

/ arbitrarily on one slice

andthentakingx

to be constant on each integral curve of  through such a

coordinate patch. The function

(10.2) .t; x/ D



h; i



1=2

is called the lapse function of this foliation. Note that



.c; x/



deﬁnes the

Riemannian metric induced on S

and that N D D 

1

@=@t is a unit

timelike normal to S

700 18. Einstein’s Equations

Each S

has a second fundamental form K

.c; x/, and, by Proposition 9.1,for

each c; K

D K

.c/ must satisfy the constraint equations (9.7)–(9.8), where

the covariant derivatives are given by the Riemannian metric on S

. Note that

(10.3)

D2K

The following identity complements the constraint equation:

Proposition 10.1. If the Einstein equation (8.1) holds, then

(10.4)

D

Ij Ik

C .Ric

C 3HK

 2K

Proof. Calculating the components R

j0k0

of the Riemann tensor of M , in coor-

dinates .x

;:::;x

/, with x

D t, one obtains, for 1  j; k  3,

(10.5) 

2

j0k0

D 

1



Ij Ik

C 

1

C K

Now (9.1) implies

(10.6) 

2

j0k0

D Ric

 Ric

C 3HK

 K

;

so, for any metric of the form (10.1), we have

(10.7)

D

Ij Ik

C .Ric

C 3HK

 2K

/  Ric

This proves (10.4).

Note that @

/ D g

/C.@

.Using(10.3)–(10.4), we have

(10.8)

D

C .Ric

C 3HK

Taking the trace yields

(10.9) 3

D C .S

C 9H

The importance of the evolution equations (10.3)–(10.4) is highlighted by the

following result:

Proposition 10.2. If the evolution equations (10.3)–(10.4) hold for 0  t  T

and the constraint equations (9.7)–(9.8) hold at t D 0, then the Einstein equation

(8.1) holds for t 2 Œ0; T  (and hence so do the constraint equations).

10. Time slices and their evolution 701

Proof. To begin, from (10.7) we see that if (10.4) holds, then Ric

D 0,for

t 2 Œ0; T ; 1  j; k  3. Hence, in view of the form of the metric (10.1),

(10.10) Ric

D 0; for 0  t  T; 1  j; k  3:

From now on we drop the M from Ric

. It remains to show that Ric

D 0 for

0  t  T; 0  k  3.

We will obtain a ﬁrst order 4  4 system for Ric

, making use of the Ricci

identity

(10.11) Ric

M Ij

;

which gives

(10.12) Ric

DRic

 Ric

M Ij

By (10.10)wehave

(10.13) S

D Ric

M Ij

D @

Ric

Now, if 

are the connection coefﬁcients of M ,wehave

(10.14) Ric

D @

Ric

C 

Ric

 

Ric

Thus, again by (10.10),

(10.15) 1  j; k  3 H) Ric

D 

Ric

 

Ric

Hence, for 1  j  3,

(10.16) Ric

Ric



kD1

.

Ric

 

Ric

and we can replace the left side of (10.16)by

(10.17) Ric

D @

Ric

C 

Ric

 

Ric

It is convenient to rewrite the terms on the right side of (10.12)whenj D 0,

using

(10.18) Ric

D g

Ric

D g

Ric

D g

Ric

702 18. Einstein’s Equations

We obtain Ric

D g

Ric

, and hence

(10.19)

Ric

D

2

Ric

 

2



Ric

C 

2



Ric

Consequently, we obtain a 4  4 system of the form

(10.20)

Ric

D 2

2

m;kD1

Ric

C A.Ric/;

Ric

C B

.Ric/; 1  j  3;

where A.Ric/ and B

.Ric/ are linear in Ric. The system (10.20) is readily seen to

be a linear, symmetrizable hyperbolic system; compare with the treatment of (3.3)

in Chap. 16. Thus the quantities Ric

vanish identically provided they vanish at

t D 0. But the hypothesis that the constraint equations hold at t D 0 is equivalent

to G

D 0 at t D 0; 0  j  3,by(9.3)–(9.6), and together with (10.10)this

implies S

D 0 at t D 0, and hence that

(10.21) Ric

D 0 at t D 0; 0  j  3:

This ﬁnishes the proof of Proposition 10.2.

Note that if we regard the lapse function  as an unknown, as well as g

and K

, which are 3  3 symmetric matrices, then (10.3)–(10.4) is a system of

12 equations in 13 unknowns. This underdetermined property is a consequence

of one’s ability to perform an arbitrary change of t-variable (t

D t

.t/) without

affecting the foliation of M . We might insist that  D 1, thus producing a deter-

mined 12 12 system, but that would lead to breakdown in ﬁnite time, since then

(10.9) and the constraint equation (9.7) would imply

D K

 3H

This breakdown might well be due to a bad choice of coordinates rather than to

an actual breakdown for Einstein’s equations.

Instead of trying to specify .t; x/ a priori, one might impose an extra equa-

tion involving .In[CBR] the following approach is taken. Let



.x/



be an

arbitrary Riemannian metric on S,andset

(10.22) .t; x/ D e.t; x/

1=2

g.t; x/

1=2

;

where e D det.e

/,andg D det.g

/.Ifweset

(10.23) k

D K

;kD k

D 3H;

10. Time slices and their evolution 703

a computation yields, for a metric of the form (10.1), the identity

(10.24)

k

C 3k

`.i

Ric

 2R

 2k Ric

 2a

 4kr

 k

`.i

 a

 3ka

 4

2

D k

`.i

Ric

C 2k Ric

r





 @

Ric

;

when  satisﬁes (10.22), where

(10.25) a

D 

1

;

and r

denotes the Levi–Civita connection on S

, associated with the metric ten-

sor



.t/



,andk

is given by

(10.26) k

D 

2

 g

`;m

;

and we use the notation

(10.27) f

.ij /

D f

C f

Now, when the right side of (10.24) is required to vanish, we can couple the

resulting equation to (10.3), obtaining the system

D2k

;(10.28)

k

D3k

`.i

Ric

C 2R

C 2k Ric

C 2a

k C 4kr

C k

`.i

(10.29)

C a

C 3ka

C 4

2

As before,  is given by (10.22). This system has a hyperbolic character (compare

with the discussion of the system (8.37)). The initial-value problem is well posed,

and one has ﬁnite propagation speed. Furthermore, as shown in [CBR], if such a

system is satisﬁed, then if we use the Lorentz metric (10.1), the Einstein tensor

D G

for this metric satisﬁes a homogeneous linear hyperbolic system of

the following form (here, 1  j; k; `  3):

(10.30)

C 

 g

/ D 0;

G

C f



; r

/ D 0;

D 0:

Here, f

is linear in its arguments. In the middle equation, 0  ;   3, while

1  j; k; `  3. Note that the last equation in (10.30)isjustpartof(1.55), a

consequence of the Bianchi identity.

704 18. Einstein’s Equations

Suppose now that the system (10.28)–(10.29) is satisﬁed and also that, at t D 0,

the constraint equations (9.7)–(9.8)andthe(10.4) hold. The constraint equations

give directly that G

D 0 at t D 0,for0  j  3.Inviewof(10.7), the equation

(10.4) implies Ric

D 0 at t D 0,for1  j; k  3.Now

D

2

Ric

1j;k3

Ric

;

and Ric

D G

 .1=2/

, so we deduce that S

D 0 at t D 0, and hence

(10.31) G

D 0; at t D 0; 0  j; k  3:

Now the identity (1.55) (a consequence of Bianchi’s identity) implies

(10.32) @

kD1

D 0 on M; 0  j  3:

In concert with (10.31), this implies

(10.33) @

D 0; at t D 0;

for 0  j  3. Thus all the Cauchy data for (10.30)vanishatt D 0. Thus, under

our current hypotheses, we have a solution to Einstein’s equations G

D 0 on M .

Another approach makes use of “maximal slicing,” namely, requiring H D 0.

In view of (10.9), this requires  D S

.Now,using(9.7), we see that when

H D 0; S

D K

DjKj

, so we hence have the lapse equation

(10.34)  DjKj

:

This has no nontrivial solution if S is compact, but it does if S is unbounded and

“asymptotically ﬂat.” The use of the evolution equations with maximal slicing

plays an important role in [CK].

The use of the evolution equations, with various approaches to the lapse func-

tion, and also some variants, involving a “shift vector,” has played an important

role in numerical work. A number of papers on this can be found in [EFH]. We

also mention the recent work [CBY2], which has implications for both the theo-

retical and the numerical study of Einstein’s equations.

Exercises

1. Derive the curvature identity (10.5).

2. Show that if S is a three-dimensional Riemannian manifold, with metric tensor g

then

ij k`

D g

Ric

C g

Ric

 g

Ric

 g

Ric



 g

References 705

3. Rewrite the proof of Proposition 10.2 in a coordinate-invariant manner. Show that Ric

deﬁnes a t -dependent family of 0-forms r on S

,thatRic

;1 j  3,deﬁnesa

family of 1-forms  on S

,andthat(10.20) can be written in the form

r D 2

2

ı C A.r; /;

 D

dr C B.r; /;

where ı W ƒ

/ ! ƒ

/ is determined by the Riemannian metric on each slice S

Here, each S

is identiﬁed with a single slice S

, via the vector ﬁeld .

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