We show that, given an arbitrary shift, the lapse $N$ can be chosen so that
the extrinsic curvature $K$ of the space slices with metric $\overline g$ in
arbitrary coordinates of a solution of Einstein's equations satisfies a
quasi-linear wave equation. We give a geometric first order symmetric
hyperbolic system verified in vacuum by $\overline g$, $K$ and $N$. We show
that one can also obtain a quasi-linear wave equation for $K$ by requiring $N$
to satisfy at each time an elliptic equation which fixes the value of the mean
extrinsic curvature of the space slices.Comment: 13 pages, latex, no figure

Choquet-Bruhat, Y.

York, Jr, J. W.

English

arXiv

We show that, given an arbitrary shift, the lapse N can be chosen so that the extrinsic curvature K of the space slices with metric \overline g in arbitrary coordinates of a solution of Einstein's equations satisfies a quasi-linear wave equation. We give a geometric first order symmetric hyperbolic system verified in vacuum by \overline g, K and N. We show that one can also obtain a quasi-linear wave equation for K by requiring N to satisfy at each time an elliptic equation which fixes the value of the mean extrinsic curvature of the space slices

Choquet-Bruhat, Y

York, J W

CERN Document Server

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Submitted to C.R. Acad. Sc. Paris
Geometrical Well Posed Systems for the Einstein Equations
Yvonne Choquet-Bruhat

Gravitation et Cosmologie Relativiste
Universite Paris VI, t. 22-12
Paris 75252 France
and
James W. York
y
Department of Physics and Astronomy
University of North Carolina
Chapel Hill NC 27599-3255 USA
June 26, 1995
IFP-UNC-509, TAR-UNC-047
Abstract
We show that, given an arbitrary shift, the lapse N can be chosen so that the
extrinsic curvature K of the space slices with metric g in arbitrary coordinates of
a solution of Einstein's equations satises a quasi-linear wave equation. We give a
geometric rst order symmetric hyperbolic system veried in vacuum by g, K and N .
We show that one can also obtain a quasi-linear wave equation for K by requiring N
to satisfy at each time an elliptic equation which xes the value of the mean extrinsic
curvature of the space slices.

Electronic address: choquet@circp.jussieu.fr
y
Electronic address: york@physics.unc.edu
1
SYSTEMES GEOMETRIQUES BIEN POSES POUR LES EQUATIONS D'EINSTEIN.
Resume Nous donnons deux conditions dierentes sur le lapse pour que la courbure ex-
trinseque K des sections d'espace d'une solution des equations d'Einstein satisfasse une
equation d'onde quasi-lineaire, le shift etant arbitraire.
Version francaise abregee
Nous considerons les equations d'Einstein sur un espace temps de dimension 4, de topologie
M R dont nous ecrivons la metrique
ds
2
=  N
2
(
0
)
2
+ g
ij

i

j
dans le corepere donne par (x
i
coordonnees locales sur M , t 2 R)

0
= dt; 
i
= dx
i
+ 
i
dt:
Nous denissons l'operateur
^
@
0
sur les tenseurs d'espace dependant du temps par (L est la
derivee de Lie, @
0
la derivee pfaenne)
^
@
0
= @=@t  L

; alors que @
0
= @=@t  
i
@=@x
i
:
La courbure extrinseque K des sections d'espace M  ftg est telle que
^
@
0
g
ij
=  2NK
ij
:
Nous utilisons la decomposition 3+1 du tenseur de Ricci de l'espace temps pour obtenir
l'identite, (2.1) donnee au x2, generalisation de celle donnee dans [2] dans le cas  = 0, ou
on a pose H = K
i
i
et (ij) = ij+ ji. Cette identite donne une equation d'ondes pour K dans
les cas suivants:
1. N est choisi tel que @
0
N +N
2
H = 0.
2. N est choisi tel que H = h, une fonction donnee.
2
Nous donnons dans le cas 1 un systeme symetrique hyperbolique du premier ordre satisfait
par les espaces temps einsteiniens.
Introduction
We examine the Cauchy problem
1
for general relativity as the time history of the geometry
of a spacelike hypersurface. Constraints on the initial data, g, the space metric, and K,
the extrinsic curvature, can be posed and solved as an elliptic system by known methods.
However, the equations of motion for g and K, which are essentially the Arnowitt-Deser-
Misner canonical equations, do not, despite their usefulness, manifest mathematically the
physical propagation of gravitational eects along the light cone. These equations, of course,
contain gauge eects and cannot, therefore, yield directly a physical wave equation, that is,
a hyperbolic system with suitable characteristics. (For a review of the relevant geometry,
see, for example,
5
.)
In this paper we give two dierent methods for obtaining exact nonlinear physical wave
equations from the equations of motion of g and K. One of these, which relies on a \har-
monic" time-slicing condition, gives equations of motion completely equivalent to a rst-
order symmetric hyperbolic system with only physically appropriate characteristics. We
construct this system explicitly. Among the propagated quantities is, in eect, the Riemann
curvature. The space coordinates and shift vector are arbitrary; and, in this sense, the
system is gauge-invariant.
Our gauge invariant nonlinear hyperbolic system, being exact and always on the physical
light cone, is well suited for use in a number of problems that now confront gravity theorists.
These include large-scale computations of astrophysically signicant processes (such as black
hole collisions) that require ecient stable numerical integration, extraction of gravitational
radiation with arbitrarily high accuracy from Cauchy data, gauge-invariant perturbation
and approximation methods, and posing boundary conditions compatibly with the causal
structure of spacetime.
3
1. NOTATIONS AND 3+1 DECOMPOSITION
We consider a manifold V of dimension 4, which has the topology M  R, with M a C
1
,
3 dimensional manifold. We denote by (x; t) 2 M  R a point of V . When we take
local coordinates they will always be adapted to the product structure: x
i
, i = 1; 2; 3 are
coordinates on M and x
0
 t 2 R. We consider on V a pseudo-riemannian metric g of
lorentzian signature (  + ++). It induces a properly riemannian metric g
t
on each space
submanifold M
t
M  ftg. We denote by  and N respectively the shift and the lapse of
the foliation. We dene a coframe 

by

0
 dt; 
i
 dx
i
+ 
i
dt:
The corresponding Pfa derivatives @

are
@
0

@
@t
  
i
@
i
; @
i

@
@x
i
:
In this coframe the metric reads
ds
2
 g





  N
2
(
0
)
2
+ g
ij

i

j
:
We dene the operator
^
@
0
on t dependent space tensors by (L

is the Lie derivative with
respect to )
^
@
0

@
@t
  L

:
The extrinsic curvature K of the space manifold is
K
ij
  
1
2
N
 1
^
@
0
g
ij
:
The Ricci curvature of space time admits the 3+1 decomposition, with H = K
h
h
R
ij
=  N
 1
^
@
0
K
ij
+HK
ij
  2K
im
K
m
j
 N
 1

r
j
@
i
N +

R
ij
;
R
0
j
= N
 1
(

r
h
K
h
j
  @
j
H);
R
0
0
=  (N
 1

r
i
@
i
N  K
ij
K
ij
+N
 1
@
0
H);
where an overbar denotes a quantity relative to the space metric g  (g
ij
).
4
2. SECOND ORDER EQUATION FOR K
In the formula above R
ij
, like the right hand side giving its decomposition, is a t dependent
space tensor, the projection on space of the Ricci tensor of g. We compute its
^
@
0
derivative.
First we compute
^
@
0

R
ij
. The innitesimal variation of the Ricci curvature corresponding to
an innitesimal g variation of the space metric is, with (ij) = ij + ji (no factor of 1=2)


R
ij
=
1
2
f

r
h

r
(i
g
j)h
 

r
h

r
h
g
ij
 

r
j
@
i
(g
hk
g
hk
)g:
This expression applies to (@=@t)

R
ij
with g
ij
= (@=@t)g
ij
and to L


R
ij
with g
ij
= L

g
ij
.
Therefore, using the relation between
^
@
0
g
ij
and K
ij
, we obtain
^
@
0

R
ij
=  

r
h

r
(i
(NK
j)h
) +

r
h

r
h
(NK
ij
) +

r
j
@
i
(NH)
  

r
(i

r
h
(NK
j)h
) +

r
h

r
h
(NK
ij
) +

r
j
@
i
(NH)
 2N

R
h
ijm
K
m
h
 N

R
m(i
K
j)
m
:
We now use the expressions for R
0
i
and R
ij
to obtain the identity


ij

^
@
0
R
ij
+

r
(i
(N
2
R
j)
0
) 
 
^
@
0
(N
 1
^
@
0
K
ij
) +

r
h

r
h
(NK
ij
) +
^
@
0
(HK
ij
  2K
im
K
m
j
)
 
^
@
0
(N
 1

r
j
@
i
N)  N

r
i
@
j
H
 

r
(i
(K
j)h
@
h
N)  2N

R
h
ijm
K
m
h
 N

R
m(i
K
j)
m
+H

r
j
@
i
N:
3. HYPERBOLIC SYSTEM FOR g, K, N
We shall eliminate at the same time the third derivatives of N and the second derivatives
of H as follows (cf. this elimination in the case of zero shift in
2
). We compute
^
@
0

r
j
@
i
N  (
@
@t
 L

)

r
j
@
i
N:
We nd at once
@
@t

r
j
@
i
N 

r
j
@
i
@N
@t
 
1
2
g
kl
(

r
(i
@
@t
g
j)l
 

r
l
@
@t
g
ij
)@
k
N;
5
and we nd an analogous formula when the operator @=@t is replaced by L

. Finally
^
@
0

r
j
@
i
N 

r
j
@
i
^
@
0
N + f

r
(i
(NK
j)l
) 

r
l
(NK
ij
)g@
l
N:
The third order terms in N and second order terms in H in the second order equation for
K can therefore be written in terms of
C
ij
 N
 1

r
j
@
i
(
^
@
0
N +N
2
H):
We satisfy the condition C
ij
= 0 by requiring N to satisfy the dierential equation (note
that
^
@
0
N  @
0
N)
(N
0
) @
0
N +N
2
H = 0:
Using the expression for H, the equation (N
0
) reads
^
@
0
logfN(det g)
 1=2
g = 0:
We nd, generalizing the result obtained in
2
in the case of zero shift, that the general solution
of this equation is
N = 
 1
det(g)
1=2
;
^
@
0
 = 0:
The second equation is a linear dierential equation for the scalar density , depending only
on . The above choice of N is called an algebraic gauge. A possible choice if  does not
depend on t is to take  independent of t and such that L

 = 0. In practice, one may
simply regard (N
0
) as a dierential equation to be solved simultaneously with (K
0
) below.
Taking into account the equation (N
0
) satsied by N we see that the Einstein equations
R

= 

imply the wave equation for K
(K
0
) NK
ij
= Q
ij
+
ij
;
where we now set
K
ij
=  N
 2
^
@
0
^
@
0
K
ij
+

r
h

r
h
K
ij
;
6
Qij
  K
ij
@
0
H + 2g
hm
K
m(i
^
@
0
K
j)h
+ 4Ng
hl
g
mk
K
lk
K
im
K
jh
+(2

r
(i
K
j)l
)@
l
N   2HN
 1
@
i
N@
j
N   2@
(i
N@
j)
H   3@
h
N

r
h
K
ij
 K
ij

r
h

r
h
N  N
 1
K
ij
(

r
h
N)@
h
N +N
 1
K
h(i

r
j)
N@
h
N
+(

r
(i
@
h
N)K
j)h
+ 2N

R
h
ijm
K
m
h
+N

R
m(i
K
j)
m
  2H

r
j
@
i
N;

ij

^
@
0

ij
+

r
(i
(N
2

j)
0
):
An immediate consequence is the following theorem.
Theorem In algebraic gauge and with the denition K
ij
  (2N)
 1
^
@
0
g
ij
, the system


ij
 
ij
= 0 is a quasi-diagonal third order system for g
ij
with principal operator @
0
. This
system is hyperbolic if g is properly riemannian and N
2
> 0. (In practice, one can regard the
denition of K
ij
as an equation (g
0
) for
^
@
0
g
ij
, and then consider the dierential equations
(N
0
), (K
0
) and (g
0
) to be solved simultaneously.)
The usual local in time existence theorem for a solution of the Cauchy problem results
from the Leray theory of hyperbolic systems
4
.
4. VERIFICATION OF THE ORIGINAL EINSTEIN EQUATIONS
We set


 (R

  

) 
1
2
g

(R   ):
Suppose we have solved the equations


ij

^
@
0
R
ij
+

r
(i
(N
2
R
0
j)
) = 
ij
; (4.1)
that is, equivalently, doing a few computations,
r
0

ij
+r
(i
(N
2

0
j)
)   g
ij
r
h
(N
2

0h
) + f
ij
= 0; (4.2)
where f
ij
is linear and homogeneous in 

.
We suppose that the stress energy tensor T

 

 
1
2
g

 satises the conservation
laws r

T

= 0. We deduce then from the Bianchi identities the equations
7
r


= 0: (4.3)
From (4.2) and (4.3) we obtain for 

a quasi diagonal third order system with principal
operator the hyperbolic operator
^
@
0
. The vanishing for t = 0 of 

results from the
constraints satised by the original initial data

0
j
j
t=0
= 0; 
00
j
t=0
= 0;
and the determination of
^
@
0
Kj
t=0
by the equation

ij
j
t=0
= 0:
We then deduce from (4.2) and (4.3) the vanishing for t = 0 of the derivatives of order  2
of 

. It results therefore, from the uniqueness theorem of Leray[4] for hyperbolic systems,
that a solution of (4.1) with initial data satisfying the constraints satises the full Einstein
equations 

= 0.
5. FIRST ORDER SYSTEM (vacuum)
The preceding results can be extended without major change to dimensions greater than
4. We will show that in dimension 4 a solution of the vacuum Einstein equations, together
with the gauge condition (N
0
), satises a rst order symmetric system, hyperbolic if g is
properly Riemannian and N
2
> 0. Such a system could be useful to establish a priori
estimates relevant to global problems, as well as to the physical applications mentioned in
the Introduction.
We have obtained for the unknowns g, K, and N the equations
^
@
0
g
ij
=  2NK
ij
; (5.1)
^
@
0
(N
2
) =  2NH(N
2
); (5.2)
NK
ij
= Q
ij
: (5.3)
8
To obtain a rst order system we take as additional unknowns:
N
 1
^
@
0
K
ij
 L
ij
;

r
h
K
ij
M
hij
; @
i
logN  a
i
;
N
 1
^
@
0
@
i
logN  a
0i
; a
ij
=

r
i
@
j
logN:
We take as equation (5.3
0
)
^
@
0
K
ij
= NL
ij
: (5:3
0
)
The equation (5.3) gives
^
@
0
L
ij
  N

r
h
M
hij
= N(HL
ij
  Q
ij
): (5.4)
In three space dimensions, the Riemann tensor is a linear function of the Ricci tensor. Using
the equation R
ij
= 0 to express

R
ij
, we write Q
ij
as a polynomial in the unknowns and in
g
ij
. (L
ij
is the essential piece of the spacetime Riemann tensor R
o
ioj
.)
We have the following lemma proved for instance by using the commutativity
^
@
0
@
i
= @
i
^
@
0
on components of tensors
Lemma For an arbitrary covariant vector u
i
we have
^
@
0

r
h
u
i
=

r
h
^
@
0
u
i
+ u
l
f

r
(h
(NK
i)
l
) 

r
l
(NK
ih
)g:
and an analogous formula for tensors with additional terms for each index.
Using this lemma we see that L
ij
and M
hij
must satisfy the equation
^
@
0
M
kij
 Nr
k
L
ij
= N

a
k
L
ij
+M
k(i
l
K
j)l
+K
l(i
M
j)k
l
 K
l(i
M
l
j)k
(5.5)
+K
l(i
(K
l
j)
a
k
+ a
j)
K
l
k
  a
l
K
j)k
)

:
On the other hand, (5.2) implies
^
@
0
a
i
=  N(Ha
i
+M
ik
k
) (5.6)
while (5.2) and the lemma yield
^
@
0
a
ji
 Nr
j
a
i0
= Na
l

M
(ij)
l
 M
l
ij
+ a
(i
K
j)
l
  a
l
K
ij

+Na
j
a
i0
: (5.7)
9
Now we deduce the value of
^
@
0
a
0i
from (5.2) and the Einstein equation
R
0
0
  fN
 1

r
h

r
h
N  K
ij
K
ij
+N
 1
@
0
Hg = 0:
Indeed these two equations imply
@
0
@
0
N = N
2

r
h

r
h
N  N
3
K
ij
K
ij
+ 2N
3
H
2
:
Hence, by dierentiation and use of the Ricci formula and the denitions of a
hi
and a
0i
, we
obtain
^
@
0
a
0i
 Nr
k
a
ki
= N

 R
k
i
a
k
+ a
i
(H
2
  2K
kl
K
kl
+ 2a
k
a
k
+ 2a
k
k
) (5.8)
+2a
k
a
k
i
+HM
ik
k
  2K
kl
M
ikl

:
We use again the equation R
ij
= 0 in (5.8) to replace

R
ij
by its values in terms of the
unknowns.
The right hand sides of the equations (5.1), (5.2), (5.3
0
), (5.4), (5.5), (5.6), (5.7), and
(5.8) are polynomial in the unknowns and g
ij
; they do not depend on their derivatives. We
eliminate the rst derivatives of g appearing on the left hand sides for instance by introducing
on M an a priori given metric e which may depend on t but is such that
^
@
0
e
ij
= 0:
We denote by D the covariant derivative in the metric e. We deduce then from (5.1) the
following equation for G
hij
 D
h
g
ij
:
^
@
0
G
hij
=  2Nfa
h
K
ij
+M
hij
+ S
m
h(i
K
j)m
g: (5.9)
The tensor S, the dierence of the connections of g and e, is given by
S
m
ij
=
1
2
g
mh
(G
(ij)h
 G
hij
):
We thus obtain a rst order dierential system, equivalent to a
symmetric system, hyperbolic if N
2
> 0 and g
ij
is properly riemannian. Its characteristics
are the physical light cone and the \time" axis @
0
.
10
6. MIXED HYPERBOLIC ELLIPTIC SYSTEM FOR K, g, N WHEN H IS GIVEN
Another procedure to reduce the second order equation for K obtained above to a quasi
diagonal system with principal part the wave operator is to replace in the term

r
i
@
j
H the
mean curvature H by an a priori given function h; this method was used in [3], with h = 0,
in the asymptotically euclidean case. With this replacement the equations 

ij
= 
ij
take
the form
NK
ij
= P
ij
+ 
ij
(6.1)
where P
ij
depends on K and its rst derivatives and on g, N and @
0
N together with their
space derivatives of order  2. When N and 
ij
are known the above equations together
with (5.1) are again a third order quasi-diagonal system for g, hyperbolic if N > 0 and g
properly riemannian. On the other hand, the equation R
0
0
= 
0
0
together with H = h
implies the equation

r
i
@
i
N = (K
ij
K
ij
  
0
0
)N =  @
0
h: (6.2)
This equation is an elliptic equation for N when g, K and  are known. Note that
for energy sources satisfying the \strong" energy condition we have  
0
0
 0 as well as
jKj
2
 K
ij
K
ij
 0. These properties are important for the solution of the elliptic equation.
The hyperbolic system that we have constructed (for given N and ) has local in time
solutions for Cauchy data in local Sobolev spaces, like the one obtained in algebraic gauge.
But now N is determined by an elliptic equation which has to be solved globally on each
space slice M
t
at time t. An iteration procedure leading to a local in time solution has been
used in the vacuum asymptotically at case with h = 0 [3]. We give below a theorem which
applies to the case of a compact M and which can also be proven by iteration. (Further
theorems with h 6= 0 on asymptotically at spaces can also be given.)
Theorem. Let (M;e) be a given smooth compact riemannian manifold. Let there be
given on M  I, I  [0; T ], a \pure space" smooth vector eld  and a function h such that
11
(H
k
is the usual Sobolev space relative to (M;e))
h 2
\
2k3
C
3 k
(I;H
k
); @
0
h  0; @
0
h 6 0; h(0; :)  0; or h(0; :)  0; h 6 0:
There exists an interval J  [0; `], `  T such that the system (6.1), with 
ij
= 0, (5.1),
(6.2) with 
0
0
= 0, has one and only one solution on M  J ,
g 2
\
1k3
C
3 k
(J;H
k
); K 2
\
0k2
C
2 k
(J;H
k
); N 2
\
0k2
C
2 k
(J;H
2+k
)
with N > 0 and g uniformly equivalent to e, taking the initial data
g
ij
(0; :) = 
ij
2 H
3
; K
ij
(0; :) = k
ij
2 H
2
if  is a properly riemannian metric uniformly equivalent to e and k:k 6 0.
It can be proved by using the Bianchi identities that if the initial data satisfy the con-
straints the constructed solution satises the original Einstein equations.
J.W.Y. acknowledges support from the National Science Foundation of the U.S.A., grants
PHY-9413207 and ASC/PHY 93-18152 (DARPA supplemented).
12
REFERENCES
1: Y. Choquet-Bruhat and J.W. York, \The Cauchy Problem," in General Relativity and
Gravitation, edited by A. Held, Plenum, 1979.
2: Y. Choquet-Bruhat and T. Ruggeri, Comm. Math. Phys. 89, 269 (1983).
3: D. Christodoulou and S. Klainerman, The Global Stability of the Minkowski Space,
Princeton, 1992.
4: J. Leray, \Hyperbolic dierential equations," I.A.S. Princeton 1952.
5: J.W. York, \Kinematics and Dynamics of General Relativity," in Sources of Gravitational
Radiation, edited by L. Smarr, Cambridge, 1979.
13


Geometrical Well Posed Systems for the Einstein Equations

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