NONLINEAR WAVE EQUATIONS FOR RELATIVITY
Maurice H.P.M. van Putten
CRSR & Cornell Theory Center
Cornell University, Ithaca, NY 14853-6801, and
Douglas M. Eardley,
I.T.P., UCSB, Santa Barbara, CA 93106
Abstract
ics (abelian and linear) and Yang-Mills theory (nonabelian and nonlinear; see, e:g:, [1]) and may complement present approaches in numerical relativity by way
of understanding or circumventing some of the computational diculties. This paper establishes a rst step
in this direction. Generally, wave equations provide a
setting appropriate for numerical implementation via
cylindrical or spherical coordinate systems, including
outgoing boundary conditions at grid boundaries, and
oer a possible setting for numerical treatment of ingoing horizon boundary conditions.
Pirani [4] presented compelling arguments to concentrate on the Riemann tensor in describing gravitational radiation. One can proceed by proposing to establish a numerical algorithm based on the divergence
of the Riemann tensor [11], Rabcd, in which the tensor
(1)
bcd = 16(r[c Td]b ? 21 gb[d rc] T )
acts as a source term. A tetrad f(e )b g-formalism
takes the Riemannian formulation into equations of
Yang-Mills type. With Lorentz gauge on the connection 1-forms, !a ,
Graviational radiation is described by canonical YangMills wave equations on the curved space-time manifold, together with evolution equations for the metric
in the tangent bundle. The initial data problem is
described in Yang-Mills scalar and vector potentials,
resulting in Lie-constraints in addition to the familiar
Gauss-Codacci relations.
1 Introduction and outline of
the approach
The asymptotic gravitational wave structure resulting
from the coalescence of astrophysical black holes or
neutron stars is receiving wide attention in connection with the gravitational wave detectors currently
under construction. A compact binary system produces strongly nonlinear gravitational waves of which
a small residue escapes to innity as radiation. The
asymptotic wave structure at a distant observer is in
quantitative relation to the system parameters of compact binaries, which makes gravitational radiation a
new spectrum for astronomical observations. The prediction of gravitational wave forms is presently pursued
by numerical simulation. The current approaches are
based the 3+1 Hamiltonian formulation by Arnowitt,
Deser and Misner [2], with the notable exception of
strategies employing null-coordinates (e:g: [7]) or systems of conservation laws [5].
The signicance of gravity waves in compact binaries, both in the process of coalescence and in radiation, suggests to focus on a description of relativity
by nonlinear wave equations. Wave equations provide a proper frame-work for the physics of nonlinear
wave motion, establish connection with electromagnet-
c := rc !c = 0
(2)
(ra denotes the covariant derivative associated with
the metric, gab ), fully nonlinear, canonical Yang-Mills
wave equations are obtained
2^ !a ? Rca!c ? [!c; ra!c ] = a :
(3)
Here, 2^ is the non-abelian generalization of the
Laplace-Beltrami wave operator. The wave equations
are also derived for the scalar, Ricci rotation coecients. Complemented by the equations of structure
for the evolution of the tetrad legs, a complete system
1
van Putten et: al:
2
of evolution equations is obtained which is amenable
to numerical implementation.
This interwoveness of wave motion and causal structure distinguishes gravity from the other eld theories.
In the present description, this two-fold nature of gravity has thus been made explicit. Essentially, gravitational waves are now propagated by wave equations on
the (curved) manifold, while the metric is evolved in
the (at) tangent bundle by the equations of structure.
Of course, such two-fold description is only meaningful
for wave motion with wave lengths above the Planck
scale, below which the causal structure is not welldened and wave motion can not be distinguished from
quantum uctuations.
Because the wave equations are second order in time,
conditions on the initial data involve both !a and
their normal Lie derivates Ln!a . In addition to the
Gauss-Codacci relations, !a and Ln!a are related
to the initial energy-momentum and b . The latter
follow from Gauss-Riemann relations, as non-abelian
generalizations of Gauss's law.
2 Equations for Rabcd
We will work on a four-dimensional manifold, M , with
hyperbolic metric gab . In a given coordinate system
fxbg the line-element is given by
ds2 = gabdxa dxb :
(4)
The natural volume element on M is abcd =
p
?gabcd , where g denotes the determinant of the
metric in the given coordinate system, and abcd denotes the completely antisymmetric symbol. Covariant
dierentiation associated with gab will be referred to by
ra . The geometry of M is contained in the Riemann
tensor, Rabcd , which satises the Bianchi identity
3r[eRab]cd = re Rabcd + raRbacd
(5)
+rb Reacd = 0:
Using the
volume element abcd , the dual R is dened
as 12 abef Refcd . The Bianchi identity then takes the
form
ra Rabcd = 0:
(6)
Einstein proposed that the Ricci tensor Rab = Racbc
and scalar curvature R = Rcc are connected to energymomentum, Tab , through
(7)
Rab ? 21 gab R = 8Tab :
Geometry now becomes dynamical with (5) (see, e:g:,
[13])
rd Rabcd = 2r[bRa]c :
(8)
In the presence of Einsteins equations (7), therefore,
(8) becomes
E : raRabcd = 16(r[c Td]b ? 21 gb[d rc] T ):
(9)
The quantity on the right-hand side shall be referred to
as bcd . In vacuo, E has been discussed by Klainerman
[8], who refers to E (with bcd = 0) together with (5)
as the spin-2 equations. The tensor bcd is divergence
free:
rb bcd = 0;
(10)
in consequence of the conservation laws ra Tab = 0,
and in agreement with rbra Rabcd = 0:
2.1 Yang-Mills equations
In the language of tetrads, additional invariance arises
due to the liberty of choosing the tetrad position at
each space-time point. This invariance is described
by the Lorentz group, and introduces a vector gauge
invariance in the form of the connection 1-forms, governed by Yang-Mills equations as outlined below. The
formal arguments can be found in the theory of supersymmetry (see, e:g:, [1, 6, 3]).
Let f(e )b g denote a tetrad satisfying
8
= ;
< (e )c (e )c
(e )a (e )b = b ;
(11)
a
: = sign( ):
Neighboring tetrads introduce their connection 1forms, !a ,
!a := (e )c ra(e )c :
(12)
The connection 1-forms !a serve as Yang-Mills connections in the gauge covariant derivative
r^ a = ra + [!a; ];
(13)
satisfying r^ a (e )b = 0. Here, the commutator is dened by its action on tensors a1 a 1 as
[!a ; aP
1 a ]1 (14)
= i !a !a1 a 1 :
k
k
i
j
l
k
j l
l
van Putten et: al:
3
In particular, we have
[!a; !b] = !a !b ? !a!b :
(15)
In (23), @t(e )t is left undened. Dening b = (@t )b,
the four time-components
N := (e )a a
(24)
In what follows, Greek indices stand for contractions
with tetrad elements: if vb is a vector eld, then v = become freely specifyable functions. The evolution
vb (e )b , and v = v .
equations for the tetrad legs thus become
The Yang-Mills construction thus obtains for the
Bianchi identity
@t (e )b + !t (e )b = @b N + !b N :
(25)
r^ a Rab = 0;
(16) The tetrad lapse functions N are related to the famil
iar lapse and shift functions in the Hamiltonian forand for the equivalent of E
malism through
a
^
E : r Rab = b :
(17)
gat = N (e )a = (Nq N q ? N 2; Np ):
(26)
The Bianchi identity (16) introduces the representation
(cf: [13])
We will now turn to evolution equations for the connection 1-forms.
Rab = ra !b ? rb!a + [!a ; !b] ;
(18)
0
which will be central in our discussion.
The antisymmetries in the Riemann tensor introduce 3 Equations for !a
conditions on initial data on an initial hypersurface, .
We dene a Lorentzian cross-section of the tangent
If b denotes the normal to , and
bundle of the space-time manifold by [11]
ra = ?a ( crc ) + Da on ;
(19)
c := rd !d = 0:
(27)
we obtain the Gauss-Riemann relations
The Lorentz gauge (27) provides1 a complete, six-fold
b a
D Rabcd = ?cd ;
(20) connection between neighboring tetrads. The six con bDa Rabcd = 0;
straints c = 0 are incorporated in E by application
of the divergence technique [10, 12]:
where cd = bbcd . These may be considered generalizations of Gauss's law in electromagnetism. CondiE : r^ a fRab + gab c g = b :
(28)
tions (20) nd their equivalents in the tetrad formulation. To this end, we write, analogous to (19), on Recall the transformation rule for the connection
the derivative as
(see, e:g: [3])
c
(21)
r^ a = ?a ( r^ c) + D^ a :
!a = !a + ra:
(29)
The Gauss-Riemann relations (20) thus become
In the present tetrad language, this gauge trans b ^a
formation
established by consideration of
D Rab = ;
(22) two tetrads,is freadily
(e )b g and f(e )b g. The construction
bD^ a Rab = 0:
:= (e )c (e )c provides a nite transformation
v = v of a eld v = (e )b vb in the f(e )bg2.2 Evolution equations for the tetrads tetrad
representation into v in the f(e )b g-tetrad repThe denition of the connection 1-forms gives the equa- resentation. This applied to (12) obtains (29).
tions of structure [13]
0
00
@[a (e )b] = (e )[b !a] :
1
Conceptually, c = f (!a ; gab) will also serve its purpose,
(23) where f ( ; ) depends analytically on its arguments.
van Putten et: al:
4
3.1 Existence of Lorentzian crosssection
The inhomogeneous Gauss-Riemann relations are
gauge covariant, so that we are at liberty to consider
initial data satisfying both (38) and the gauge choice
To proceed, we consider in an open neighborhood (37), whence
N () of with Gaussian normal coordinates f; xpg
( crc = 1) the innitesimal Lorentz transformation
c = ( cr^ c)c = 0 on :
(39)
1
= + 2 2 in N ():
(30) In (28) c satises a homogeneous Yang-Mills wave
It is convenient to employ language of scalar and vector equation (cf: [10, 12]),
potentials, and Aa , respectively, dened in
2^ c := r^ cr^ c c = 0:
(40)
!a = a + Aa :
(31)
It follows that the scalar elds c are solutions to an
Denoting the eect of (30) via (29) by a superscript initial
value problem for homogeneous wave equations
(r), we have
(40) with trivial Cauchy data (39). Consequently,
(r) = !
!a
(32)
a + a in N ();
c = 0 in D+ ();
(41)
so that
(r) = ? in N ():
(33) which establishes that solutions to E are solutions to
E.
Using a geodesic extension of the normal b o , con- We now elaborate further on (28).
straints (27) become
c = rc !c = _ + Dc !c on : (34) 3.2 Canonical wave equations
In Aa , this obtains
In Lorentz gauge (27), the divergence equation E
obtains wave equations for the connection 1-forms
c = _ + Dc ( c + Ac )
(35)
c
_
through the representation of the Riemann tensor (18).
= + K + Dc A Indeed, by explicit calculation, we have
on : Consequently, we have the transformations
9
2^ !a ? Rca!c ? [!c; ra!c ] = a :
(42)
>
(r) = =
(
r
)
(36)
_ = _ ? > on :
Here, we have used c = 0, so that r^ a c = rac . 2^
;
(r)
Aa = Aa
is used to denote the Yang-Mills wave operator r^ c r^ c.
By choice of = c , the constraints (27) transform The Ricci tensor Rab in (42) is understood in terms of
Tab using Einstein's equations.
into
we can obtain a system of scalar equations
c(r) = c ? = 0:
(37) forSimilarly,
the Ricci rotation coecients, ! = (e )a !a .
Notice that bringing the tetrad in Lorentz gauge (37) This involves a number of manipulations, the result of
by (30) is achieved by proper second time-derivative which is
@2 (E )b =0 of its legs, leaving the tetrad position and
2^ ! ? R! ? ! [! ; ! ]
its rst time-derivative as invariants at = 0.
(43)
?[! ; @! ] = :
We shall now establish that E maintains the
Lorentz gauge (27) in the future domain of dependence of . The inhomogeneous Gauss-Riemann re- The two-fold nature of gravity can now be expressed
lation (20) is implied by antisymmetry of the Riemann as
tensor in its coordinate indices, and gives
Separation Theorem. 1 Gravitational wave motion
0 = bfr^ a(Rab + gabc ) ? b g
is governed by canonical wave equations on M . In re(38) sponse to the wave motion, the metric on M evolves in
= b(D^ a Rab ? b ) + ( br^ b )c
b
^
the tangent bundle of M by the equations of structure.
= ( rb)c :
00
0
00
j
00
van Putten et: al:
The coupling of matter to the connections is accounted for by bcd . It is of interest to note that bcd
contains vorticity; a detailed enumeration of the nature
of bcd falls outside the scope of this dicsussion.
The Theorem suggests some computational approximations for weakly nonlinear gravity waves. Firstly,
consider equations with uncoupled (prescribed or
xed) metric, with given Laplace-Beltrami wave operator, 2g . Thus, we have (in vacuo)
2^ g !a ? [!c; ra !c] = 0;
(44)
with the metric gab as a background eld providing the
underlying causal structure. The approximate radiation then follows from the uctuations in the metric as
would follow from the equations of structure. Secondly,
small amplitude waves allow us to linearize, thereby
obtaining the purely abelian wave equations
2 !a = 0;
(45)
where refers to the Lorentz metric. Numerical experiments must show the validity of such approximations.
4 Initial value problem
Initial data for the wave equations are !a and its
Lie derivative Ln!a on . These data must satisfy
certain constraints on , commensurate with the initial distribution of energy-momentum and the GaussRiemann equations (22). We shall express these equations in terms of scalar and vector potentials (31).
The projection tensor, hab, onto is
hab = gab + a b:
(46)
The covariant derivative induced by hab shall be denoted by D a , i:e:, D a hcd = 0. In what follows,
( crc )f is also denoted by f_. The unit normal is extended geodesically o , so that crc b = 0 and
( crc )hab = hab ( crc). Thus, _ = ( crc ) and
A_ a = ( c rc)Aa . We further dene
:= _ + K ;
(47)
Aa := LnAa = A_ a + Kac Ac :
0
0
4.1 Expressions for Aa
5
Extrinsic curvature, Kab , of can be dened `static' by
projection of the unit normal following parallel transport over ,
Kab = D a b;
(49)
1
and `dynamic' by the Lie derivative, 2 Lnhab , of the
projection operator hab with respect to a Gaussian normal coordinate, n. With a normal tetrad, its (extrinsic) helicity, i:e, the twist in a strip swept out by the
integral curves of a leg (E )b passing through an integral curve of leg (E )b is
H := b (E )c rc(E )b on :
(50)
For ; 6= n, H describes twist in when 6= ,
and bending when = . It follows that
H hh = ?K
(51)
in view of (48): a (E)a = ? on . The normal
( = n) and tangent ( 6= n) extrinsic helicities are
now
H = b(E )c f!c (E )b g
n if = n;
(52)
= ?
An if 6= n:
By (51), the symmetry of Kab and An = ?An , we
have for ; 6= n the symmetry
A[ ]n = 0:
(53)
If 6= n, (E )a = hab(E )b , and so
Aa = hba (E )c rb(E )c
(54)
= (E )c [hcd hba rb](E )d
(E )c D a (E )c :
If also 6= n, D a acts in (54) on tangent legs (E )b,
the result of which is determined by hab . We shall
write Aa = Aa h h for this intrinsic part of Aa .
In the normal tetrad, therefore, Aa falls into two
groups: (i) the extrinsic part An = K , and the (ii)
intrinsic part Aa , with which we have
A = A + 2K[ n] on :
(55)
4.2 Expressions for Rab
On , Rab contains intrinsic, three-dimensional, and
Consider a normal tetrad, f(E )b g, in which one of extrinsic parts by substitution of (31):
the legs is (initially) everywhere normal to the initial
Rab = ra !b ? rb!a + [!a; !b]
hypersurface:
(56)
= 2[b D^ a] + 2[bA_ a]
(E=n )b = b on :
(48)
+2D[a Ab] + [Aa ; Ab] :
van Putten et: al:
6
By Einsteins equations, we have
Rab = 8(Tab ? 21 gab T ) 8T~ab :
(64)
(57)
Combining the results above, we have the constraints
where D^ a Kb = Da Kb + Aa Kb . Notice that the on the Lie derivative of Aa in
intrinsic, three-curvature (3) Rabcd of associated with
D^ + A = ?8T~b b;
(65)
hab satises
(3)
Rab = D a Ab ? D b Aa + [Aa ; Ab] : (58) together with the familiar Gauss-Codacci relations
(3)R + K 2 ? K K ab = 16T ;
ab
nn
It follows that
(66)
a Kab ? D b K = 8hab T~an:
D
c
d
ha hb Rcd = 2D[a Ab] + [Aa; Ab ]
We shall refer to (65)-(66) as the energy-momentum
+2Ka[ K ]b + 4D^ [a Kb][ n]
(59)
constraints.
= (3) Rab + 4D^ [a Kb][ n]
In going from a rst order Hamiltonian description
+2Ka[ K ]b :
to a second order description, it is not surprising to enThis last equation generalizes a similar expression ob- counter also constraints on the Lie derivative, reecting
the conditions that the Gauss-Codacci relations also
tained in [9], where ; 6= n is considered.
must be satised on a future hypersurface in N ().
Using (55), the latter may be reduced to
Rab = 2[bfD^ a] + A_ a] g
+2D[a Ab] + [Aa ; Ab]
+4D^ [a Kb][ n] + 2Ka[ K ]b ;
0
4.3 Energy-momentum relations
Constraints related to the initial distribution of energymomentum are obtained by consideration of the Ricci
tensor. To this end, consider the two expressions (56)
and (59) for Rab of the previous section in
hca bRcb = ?D^ a ? Aa ;
(60)
hca hdb Rcdn = 2D^ [a Kb] :
Consequently, the elements of the Ricci tensor become
bRb = bhc Rcb
= ?D^ ? A ;
(61)
hdb Rdn = 2hdb ha D^ [a Kd]
= D a Kab ? D b K:
The second expression (59) for the Riemann tensor also
also gives
h h hca hdb Rab = (3) Rabcd + Ka Kb
(62)
?Ka Kb ;
resulting in the familiar identity
16Tnn = 2(Rab ? 21 gab R) a b
= hac hbdRabcd
= ha hb Rab
(63)
= (3)R + K 2 ? Ka K a
= (3)R + K 2 ? Kab K ab :
0
0
4.4 Gauss-Riemann relations
The Gauss-Riemann relations (22) can similarly be obtained in terms of and Aa . The antisymmetry
of the Riemann tensor in its coordinate indices implies
bDa Rab = Da ( bRab ) ? Rab K ab
= Da ( bRab ):
(67)
By (56), it follows that
bRab = ?Da ? A_ a ? Kbc Ac
?[!a ; ]
= ?Da ? [Aa ; ]
(68)
?A_ a ? Kbc Ac
= ?D^ a ? Aa :
Therefore, the rst, inhomogeneous Gauss-Riemann
constraint in (22) reads
bD^ a Rab = bDa Rab + [Aa ; bRab]
= Da ( bRab ) + [Aa ; bRab]
(69)
= D^ a ( bRab )
= ?D^ a D^ a ? D^ a Aa
= ;
where the symmetry of the extrinsic curvature tensor
was used in going from the third to the fourth equation.
0
0
van Putten et: al:
7
These six constraints on LnAa arise in view of the The condition on ensures that the second Liefunctional relationship
constraint remains satised. This shows that by choosing
Aa = Aa (hpq ; @r hpq ):
(70)
= (78)
Because 12 Lnhab = Kab and Ka = Aan , six degrees
= ?K on of freedom in
we obtain
(71)
LnAa = LnAa + 2Aan[ n]
(r) = 0 on :
(79)
are constraint by (70).
To summarize, the constraints on the initial data in With this restricted Lorentz gauge, the Lie-constraints
reduce to
terms of potentials are
D^ a Aa = ? ; on :
(80)
Gauss-Codacci:
_ + D^ Ac = 0:
0
(3)
R + K 2 ? Kab K ab = 16Tnn
D a Kab ? D b K = 8hab T~an
c We remark it does not appear to be feasible to ensure = 0 throughout D+ () by suitable choice of
restricted Lorentz gauge on the initial data.
Lie-constraints:
Acknowledgment. The rst author greatfully acknowledges the warm hospitality at ITP, and very
D + A = ?8T~b b
(73) stimulating discussions with S. Chaudhuri, C. Cutler,
D^ a D^ a + D^ a Aa = ?
D. Cherno, G. Cook, S.L. Shapiro, S.A. Teukolsky.
Lorentz gauge:
This work has been supported in part by NSF Grant
PHY
94-08378 and the Grand Challenge Grant NSF
+ D^ a Aa = 0:
(74) PHY 93-18152/ASC 93-18152 (ARPA supplemented).
The Cornell Theory Center is supported by NSF, NY
State, ARPA, NIH, IBM and others.
4.5 Restricted Lorentz gauge
Restricted gauge transformations for the connection in
analogy to the same problem in electromagnetics, can References
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(75)
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r) = A
A(a
a
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(72)
0
0
0
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9
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=
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> on : (77)
= _ ? = ;
0
0
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8