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

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688 18. Einstein’s Equations

system .x

;:::;x

/ has not only the property that S Dfx

D 0g but also the

property that the vector ﬁeld @=@x

is orthogonal to S . Thus

D 0 for 1 

k  3. Suppose also that @=@x

D N is a unit vector on S; hN; N iD1 (i.e.,

D1). Using the Gauss–Codazzi equations, we can express G

in terms

ofthemetrictensorh

of S and the second fundamental form of S  M ,which

we denote as K

. Denote the associated Weingarten map by A W T

S ! T

The Gauss equation, (4.14) of Appendix C, implies that, for X tangent to S,

(9.1)

Ric

.X; X/  Ric

.X; X/

DhR

.N; X/X; N iC

j D1

.ƒ

A/.E

^ X/;X ^ E

;

where fE

g is an orthonormal basis of T

S. From this, we have

(9.2) S

 S

D2 Tr ƒ

A  2 Ric

.N; N /;

where S

is the scalar curvature of M ,andS

the scalar curvature of S. Compare

with (4.72) of Appendix C. There is a sign difference, since here hN; N iD1.

Since G

D Ric

 .1=2/S

,wehave

 S

D2 Tr ƒ

A  2G

 S

;

or equivalently,

(9.3) G

D

C Tr ƒ

Meanwhile, the Codazzi equation, (4.16) of Appendix C, implies

(9.4) K

jkI`

D K

`kIj

 R

k0j`

We deﬁne the mean curvature H of S  M to be H D .1=3/Tr A D .1=3/K

The identity (9.4) implies K

D K

 R

0j `

, and hence

(9.5) K

D 3H

C Ric

Since G

D Ric

 .1=2/S

D Ric

,for1  j  3,wehave

(9.6) G

D K

 3H

;1 j  3:

9. Geometry of initial surfaces 689

We have the following result:

Proposition 9.1. If S is a Riemannian 3-manifold with metric tensor h

, and

is a smooth section of S



.S/, then there exists a Lorentz 4-manifold M

that is Ricci ﬂat (i.e., satisﬁes (8.1)), for which S is a spacelike hypersurface, with

induced metric tensor h

and second fundamental form K

, if and only if

(9.7) S

 K

C K

D 0

and

(9.8) K

 3H

D 0:

Proof. We have just seen the necessity of (9.7)and(9.8). For the converse, if h

and K

are given, satisfying (9.7)and(9.8), set

(9.9)

D h

if 1  j; k  3;

D1;

D 0; otherwise.

Also, set

(9.10)

D2K

if 1  j; k  3;

D 0; otherwise:

Note that, for any metric g

satisfying (8.2) with these initial data, h

and K

do specify the ﬁrst and second fundamental forms of S Dfx

D 0g. See (4.69)

of Appendix C. The fact that this prescription yields

and

, which satisfy

the compatibility conditions of Proposition 8.3, thus follows from (9.4)and(9.6).

The index raising and covariant differentiation performed in (9.7)and(9.8)are

operations deﬁned by the metric tensor h

on S. Note that (9.7) follows from

(9.3), via the identity

Tr ƒ

A D

.Tr A/



Tr A

;

which is readily veriﬁed by using a basis that diagonalizes A. In the physics liter-

ature, (9.7) is called the Hamiltonian constraint and (9.8) is called the momentum

constraint. Together, (9.7)and(9.8) are called the constraint equations.

Note that special hypotheses about the coordinates used on M disappear in the

formulation of Proposition 9.1, which is convenient.

We can deﬁne the trace-free part of K

(9.11) Q

D K

 Hh

690 18. Einstein’s Equations

Then the system (9.7)–(9.8) becomes

 Q

C 6H

D 0;(9.12)

 2H

D 0:(9.13)

This system has been studied in [Lich1,Yo 1, OY1,CBr5,CBr6,CBY]. Follow-

ing their work, we will investigate this system more closely in the special case

when H is taken to be constant on S, which we now assume to be compact.Then

(9.13) speciﬁes that Q

is a divergence-free (trace-free, order 2, symmetric) ten-

sor ﬁeld. We show how to construct all such ﬁelds on a compact Riemannian

manifold .S; h/.

Let us deﬁne D

on vector ﬁelds by

(9.14) D

X D Def X 

.div X/h; D

W C

.S; T / ! C

.S; S



where Def is the deformation operator, .Def X/

D .X

j Ik

kIj

/=2,andwhere



denotes the bundle of second-order, trace-free, covariant tensors. A calcu-

lation yields

(9.15) D



Ddiv



Hence

(9.16) D



X Ddiv Def X C

grad div X:

The operator L D D



is a second-order, elliptic, self-adjoint operator, and

there is a Weitzenbock formula, which implies

(9.17) kD

krX k





kdiv Xk





Ric

.X/; X



Compare with formulas (4.26)–(4.31) of Chap.10.

The kernel of L, which is equal to ker D

, consists of conformal Killing

ﬁelds, as noted in (3.39) of Chap. 2. It is a ﬁnite-dimensional subspace of

.S; T /.LetE be the inverse of L on the orthogonal complement of ker

L, and zero on ker L. It follows that

(9.18) E W H

.S; T / ! H

sC2

.S; T /;

for all s 1, by the elliptic regularity results in Chap. 5,  1, and more generally

for all s 2 R, by the construction of E 2 OP S

2

.S/ in Chap. 7. Now set

(9.19) P

D D

E D



W H

.S; S



/ ! H

.S; S



9. Geometry of initial surfaces 691

The operator P

is the orthogonal projection onto the range of D

in L

,that

is, the image of D

acting on H

.S; T /,andP

D I  P

is the orthogonal

projection onto the kernel of D



in L

.S; S



/.From(9.19) it follows that

(9.20) P

W C

.S; S



/ ! C

.S; S



The set of smooth, divergence-free, trace-free, second-order, symmetric tensor

ﬁelds Q

is precisely the image of P

in (9.20).

Now, if we have a Riemannian metric h

on S and a solution Q

to (9.13),

with H constant, the scalar curvature may not satisfy (9.12). In [Yo1]and[OY1],

following Lichnerowicz, who treated the case H D 0, it was shown that the triple

;H;Q

/ leads to a new triple .h

;H;Q

/,whereh

is a conformal

multiple of h

(9.21)

D '

;

H is unchanged, and

is a smooth multiple of Q

, involving a different

power of ', as we will see below. Then (9.12)–(9.13) hold for this new triple,

provided ' satisﬁes a certain elliptic PDE, which we proceed to derive.

For these calculations, denote covariant differentiation associated to h and

by r and

r, respectively. Then

(9.22)

C .

 

C .

 

;

where the connection coefﬁcients are related by

(9.23)



D 



' C ı

'  h



Consequently, for any symmetric, second-order tensor ﬁeld

' 

The last term vanishes if

D 0. If, furthermore, Q

D Q

,then

(9.24)

D r





692 18. Einstein’s Equations

The last term here vanishes if D '

10

,sowehave

(9.25)

D '

10

H) r

D '

10

when Q

is symmetric and has trace zero. Note that (9.25) implies Q

2

Thus, if Q

is constructed as above, as an element in the range of P

,soit

has trace zero and solves r

D 0,wealsohaver

D 0 whenever h

is related to h

by (9.21). The scalar curvatures of .S; h/ and .S; h/ are related

(9.26)

D '

4

 8'

5

';

where  is the Laplace operator on .S; h/. Now we want to satisfy the analogue

of (9.12), namely,

(9.27)

D Q

 6H

D '

12

 6H

D '

12

f  6H

By (9.26), we want ' to be a smooth, positive solution to the PDE

(9.28) ' D

' 

7

D F.x;'/:

Equations of this form are discussed in  1 of Chap. 14.

If f>0on S and H ¤ 0, then Theorem 1.10 of Chap. 14 implies the solv-

ability of (9.28), which is a special case of (1.50) of Chap. 14. We thus have

Proposition 9.2. Let S be a compact, connected 3-manifold, with metric tensor

a smooth, divergence-free, trace-free section of S



.S/, and let H

be a nonzero constant. Then there exists a positive ' 2 C

.S/ such that if

(9.29)

D '

; Q

D '

2

;

then there is a Ricci-ﬂat Lorentz manifold M , containing S as a spacelike hyper-

surface, with induced metric

and second fundamental form

(9.30)

D Q

C H h

;

provided Q

D f is not identically zero on S.

Proof. The argument above gives the result provided f.x/ > 0 on S. It remains

to weaken this condition on f . First, if the scalar curvature S

of .S; h/ is negative

on fx 2 S W f.x/ D 0gD†, then Theorem 1.10 of Chap. 14 still implies the

solvability of (9.28). On the other hand, we can make a preliminary conformal

9. Geometry of initial surfaces 693

deformation of the metric tensor of S to make its scalar curvature negative on any

proper closed subset of S,suchas†, as long as † is not all of S, so Proposition

9.2 is proved.

If Q

is taken to be zero, then (9.28) becomes

(9.31) ' D

' C

Integrating both sides, we see that if there is a positive solution ',then

.x/'.x/ dV .x/ < 0:

In particular, (9.31) has no positive solution if S

 0 on S. Here is a positive

result:

Proposition 9.3. Let S be a compact, connected 3-manifold with metric tensor

. Assume the scalar curvature of .S; h/ satisﬁes S

.x/ < 0 on S.LetH be

a nonzero constant. Then there is a positive ' 2 C

.S/ such that if h

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

hypersurface, with induced metric

and second fundamental form

(9.32)

D H h

Proof. The equation (9.31) has the same form as (1.49) in Chap.14, as the equa-

tion for the conformal factor needed to alter .S; h/, with scalar curvature S

,to

.S;

h/, with scalar curvature S

D3H

. Thus solvability follows from Propo-

sition 1.11 of Chap. 14.

If H D 0,then(9.28) becomes

(9.33) ' D

' 

7

;

where we recall that f D Q

. The solvability of (9.33) implies the identity

'dV D

7

dV , so there is no positive solution if S

 0 on S.On

the other hand, Theorem 1.10 of Chap. 14 applies if S

.x/ > 0 on S and f>0

on S, so we have the following:

Proposition 9.4. Let S be a compact, connected 3-manifold with metric tensor

. Assume the scalar curvature of .S; h/ satisﬁes S

.x/ > 0 on S .LetQ

be a smooth, divergence-free, trace-free section of S



.S/. Then there is a pos-

itive ' 2 C

.S/ such that if h

and Q

are given by (9.29), then there is a

694 18. Einstein’s Equations

Ricci-ﬂat Lorentz manifold M , containing S as a spacelike hypersurface, with

induced metric

and second fundamental form

(9.34)

D Q

;

provided Q

is nowhere vanishing. In such a case, the mean curvature of S  M

vanishes.

Next, following [CBIM], we extend Proposition 9.3 to allow H to be non-

constant, provided it does not vary too much. As before, we have a compact,

3-manifold S , with Riemannian metric tensor h, scalar curvature S

.LetH

be a smooth, real-valued function on S. We want to construct a positive ' 2

.S/ and a second-order, symmetric, trace-free tensor ﬁeld Q

on S (i.e.,

Q 2 C

.S; S



/) such that if we change the metric tensor to h

D '

and set

D '

2

,asin(9.29), then S is a spacelike hypersurface of a

Ricci-ﬂat Lorentz 4-manifold, with induced metric

and with second funda-

mental form

D Q

C H h

,asin(9.30).

In order to achieve this, we need to satisfy the Gauss–Codazzi system

(9.35)

 Q

C 6H

D 0;

 2r

H D 0;

which one converts to

(9.36)

' D

' 

jQj

7

;

 2'

D 0;

where jQj

D Q

. As before (and as noted in [Yo 3]and[CBY]), we get

from (9.35)to(9.36) via the identities (9.25)and(9.26). The ﬁrst equation in

(9.36)isthesameas(9.28).

The second equation in (9.36)is

(9.37) D



Q C 2'

grad H D 0:

We look for Q in the form

(9.38) Q D D

X C Q

; D



D 0;

where X is a vector ﬁeld on S .Asin(9.14), D

X D Def X  .1=3/.div X/h.

Pick Q

2 R.P

/.Then(9.37) becomes

(9.39) LX D2'

grad H;

9. Geometry of initial surfaces 695

where

(9.40) LX D D



X Ddiv Def X C

grad div X:

Assume that L is invertible, that is, .S; h/ has no conformal Killing ﬁelds. Thus,

given ',wesolve(9.39)forX; X D2L

1

grad H/. Then it remains to

solve for ':

(9.41) ' D

' 

jE.'

/ C Q

7

;

where

(9.42) Eu D2D

ı L

1

.u grad H/; E 2 OPS

1

This has a slightly more complicated form than (9.28), though of course it reduces

to (9.28)whenH is constant.

We will establish the following:

Proposition 9.5. Let S be a compact 3-manifold with metric tensor h. Assume

the scalar curvature S

<0on S.LetH 2 C

.S/ be a given positive function.

Then we can ﬁnd a positive ' 2 C

.S/, solving (9.41), with Q

D 0,provided

(9.43) kD

1

 A

kuk

;

with

(9.44) A

krH k

min

As a preparation to proving this, we ﬁrst obtain some a priori estimates for a

positive solution ' 2 C

.S/ to (9.41), with Q

D 0,thatis,to

(9.45) ' D f



x; E.'

/; '



;

where

(9.46) f



x; E.'

/; '



' 

jE.'

7

Note that if '.x

/ D '

min

>0,then'.x

/  0,soif(9.45) holds, then

f.x

;E.'

/; '.x

//  0. Hence

(9.47)

H.x

'.x





S



'.x

/ 

'.x

696 18. Einstein’s Equations

if S

<0on S. This implies

(9.48) '  a

on SI a

H

2

max

We next derive an upper bound on a solution to (9.48), using the hypothesis

(9.43)–(9.44). Suppose '.x

/ D '

max

.Then'.x

/  0,soif(9.45) holds, then

f.x

;E.'

/; '.x

//  0.Now

(9.49)



;E.'

/; '.x



D '

7

max

H.x

max



jE.'

max

The hypothesis (9.43)–(9.44) implies

(9.50) kE.'

 2A

krH k

min

max

;

(9.51)

H.x

max



jE.'

 '

max

;

 D



min

 A

krH k



>0;

and hence

(9.52) f



;E.'

/; '.x



 '

max

If the left side of (9.52)is 0, this requires '

max

C.1=8/S

max

 0. Thus,

if jS

j, a solution to (9.45) must satisfy

(9.53) '  a

on S; a



8

=4

min

 A

krH k

For the rest of the discussion, we modify the formula (9.46), replacing it by

(9.54) f



x; E.'

/; '



.'/ 

jE.'

˛.'/ C

;

where

(9.55)

.'/ D '; '  a

;

˛.'/ D '

7

;' a

9. Geometry of initial surfaces 697

In addition, we require the functions  and ˛ to be smooth and monotonic on R,

for ˛ to be linear on '  0,for.'/  ' on R,andfor.'/ to be some positive

constant (say 

)for'  a

=2.

To prove Proposition 9.5, we will use the Leray–Schauder ﬁxed-point theorem,

in the following form (cf. Theorem B.5 in Chap. 14):

Theorem. Let V be a Banach space, F W Œ0; 1 V ! V a continuous, compact

map such that F.0;v/ D v

, independent of v. Suppose there exists M<1 such

that, for all .; x/ 2 Œ0; 1  V ,

(9.56) F.;x/ D x H) kxk <M:

Then F

W V ! V , given by F

.x/ D F.1;x/, has a ﬁxed point.

We will apply this to V D C.S/ and

(9.57) F.;'/ D .  1/

1



‰



.'/  '



;

where, picking b D .a

C a

/=2,weset

(9.58) ‰



.'/ D .1  /.'  b/ C f



x; E.'

/; '



;

with f as in (9.54). Note that

(9.59) F.0;'/ D.  1/

1

b D b:

Also,

(9.60) F.;'/ D ' ” ' D f



x; E.'

/; '



C .1  /.'  b/:

To check (9.56), we need to estimate '

max

and '

min

whenever  2 Œ0; 1 and '

satisﬁes (9.60). The case  D 0 is clear, so we may assume >0.If'.x

/ D

min

,then‰.'/  0 at x

,so

(9.61) f



;E.'

/; '.x



C .1  /



'.x

/  b



 0:

If '.x

/<band  2 .0; 1, this requires f.x

;E.'

/; '.x

//  0, or (in place

of (9.47))

(9.62)

H.x

min





S



.'

min

This forces '

min

 .=12/H

2

max

.'

min

/, which in turn forces '

min

>0.Since

.'/  ' and S

>0,thisimplies(9.47). Therefore, again we get the estimate

(9.48).