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Killing vectors and maximal slicing in general relativity
Volume 77A, number 2,3
PHYSICS LETTERS
12 May 1980
KILLING VECTORS AND MAXIMAL SLICING IN GENERAL RELATIVITY Niall ó MURCHADHA Physics Department, University College, Cork, Ireland Received 20 February 1980
In solutions to Einstein’s equations, killing vectors and maximal slices are locked together. If an asymptotically flat spacetime has a maximal slice and a killing vector which lies in the maximal slice at infinity, then the killing vector lies in the slice everywhere. For closed, compact, without boundary solutions one has a more general result. In this case if one has a slice with the trace ofthe extrinsic curvature constant (not necessarily zero) then all killing vectors must lie in the slice. The only exception is closed flat space. This means that maximal slices form a natural foliation of any spacetime with a symmetry.
Spacelike three-slices of a four-dimensional manifold can be characterized by the intrinsic three-metnc i/ and the extrinsic curvature K’J. These objects form the initial data for solutions to Einstein’s equations, and completely determine the spacetime [1]. They are not independent but must satisfy the constraints. There are several reasons for considering initial data for which K = trKi/ = constant, and, in particular choosing K = 0. These are the maximal slices. The K = constant condition decouples the initial value constraints and simplifies them [2]. It is a good foliation condition in the sense that if one has one slice on which trK = constant, one has a complete family of •such slices [3]. It is probable that all solutions to Einstein’s equations possess such slices [4,5]. Maximal slices are particularly favourable for solutions with singularities because they avoid the singularity [6,7]. The maximal slicing condition has been widely used in numerical computations in general relativity [7,8]. These calculations have invariably been done with symmetric solutions and the question arises whether the maximal slices respect the symmetry. This paper shows that rotational killing vectors automatically lie in the maximal slices of asymptotically flat spacetimes. Further, the technique used extends to compact manifolds with a K = constant slice. In this case any
killing vector that the manifold possesses must lie in the slice and therefore must be spacelike. The only compact, without boundary, spacetime which has a slice for whichK = constant and possesses a timelike killing vector is closed flat spacetime. Let us start with initial data for a non-vacuum solution to Einstein’s equations and use the Einstein dynamical equations to compute the rate of change of K in a direction tM. t’1 is related to the normal to the slice i~1~Lvia the lapse N and shift functions !V1 where t~-~ = Ni~M+ NM, with N = = 0 and NMnM = 0. We get
c~/6t(K)= 47rN(T** + T(3)) —
v2N + NKiIK.. + K 1,
IV’1~
(1)
m
where T** is the local source energy density and T~3~ is the trace of the source stresses. T** ÷T(3~~ 0 if the source satisfies the dominant energy condition [9]. Let us assume that t~is a killing vector at the slice and that K = constant. Then (1) reduces to ‘~f~t(K)0 N {4ir (T**+ T(3)) + K”K~.} v2N. —
(2)
/
Eq. (2) has the form V2N AN = 0 A > 0 —
103
Volume 77A, number 2,3
PHYSICS LETTERS
which satisfies a mm-max principle [10]. If the slice is asymptotically flat and the killing vector lies in the slice at infinity, i.e., N 0 at then eq. (2) has a unique solution N 0. Of course, N 0 at ~ means that the killing vector in question must be spacelike at infinity. Therefore it must either be a translational or rotational killing vector. The result is trivial when we have a translational killing vector because then the spacetime must be flat. Therefore rotational killing vectors must lie in the maximal slice and hence are everywhere spacelike. If the slice is closed, compact, without boundary equation (2) again has a unique solution N 0, ~ without any precondition on the killing vector. Hence all killing vectors for such solutions must be spacelike and lie in any K = constant slice. The only exception is when —*
12 May 1980
If the source satisfies the dominant energy condition this implies Kit 0 and therefore the slice is flat.
‘~,
References
—~
A
=
104
41r(T**
+
r(3)) +K’JK~J
0
.
(3)
[1] R. Arnowitt, S. Deser and C.W. Misner in: Gravitation: an introduction to current research, ed. L. Wilten (Wiley, New York, 1962). [2] N. O’Murchadha and J.W. York, Phys. Rev. D10 (1974) 428. [3] J.W. York, Phys. Rev. Lett. 28 (1972) 1082. [4] M. Cantor et al., Commun. Math. Phys. 49 (1976) 187; N.ó Murchadha, to be published. [5] Y. Choquet-Bruhat, C.R. Acad. Sci. Paris 280A (1975) 169. [6] F. Estabrook et al, Phys. Rev. D7 (1973) 2814. [7] K. Maeda, to be published. [8] L. Smarr et al., Phys. Rev. D14 (1976) 2443; L. Smarr, Ann NY Acad. Sci. 302 (1977) 569. [9] S. Hawking and G. Ellis, The large scale structure of spacetime (Cambridge, 1973). [10] See, for example, M. Protter and H. Weinberger, Maximum principles in differential equations (PrenticeHall, NJ, 1967).
PHYSICS LETTERS
12 May 1980
KILLING VECTORS AND MAXIMAL SLICING IN GENERAL RELATIVITY Niall ó MURCHADHA Physics Department, University College, Cork, Ireland Received 20 February 1980
In solutions to Einstein’s equations, killing vectors and maximal slices are locked together. If an asymptotically flat spacetime has a maximal slice and a killing vector which lies in the maximal slice at infinity, then the killing vector lies in the slice everywhere. For closed, compact, without boundary solutions one has a more general result. In this case if one has a slice with the trace ofthe extrinsic curvature constant (not necessarily zero) then all killing vectors must lie in the slice. The only exception is closed flat space. This means that maximal slices form a natural foliation of any spacetime with a symmetry.
Spacelike three-slices of a four-dimensional manifold can be characterized by the intrinsic three-metnc i/ and the extrinsic curvature K’J. These objects form the initial data for solutions to Einstein’s equations, and completely determine the spacetime [1]. They are not independent but must satisfy the constraints. There are several reasons for considering initial data for which K = trKi/ = constant, and, in particular choosing K = 0. These are the maximal slices. The K = constant condition decouples the initial value constraints and simplifies them [2]. It is a good foliation condition in the sense that if one has one slice on which trK = constant, one has a complete family of •such slices [3]. It is probable that all solutions to Einstein’s equations possess such slices [4,5]. Maximal slices are particularly favourable for solutions with singularities because they avoid the singularity [6,7]. The maximal slicing condition has been widely used in numerical computations in general relativity [7,8]. These calculations have invariably been done with symmetric solutions and the question arises whether the maximal slices respect the symmetry. This paper shows that rotational killing vectors automatically lie in the maximal slices of asymptotically flat spacetimes. Further, the technique used extends to compact manifolds with a K = constant slice. In this case any
killing vector that the manifold possesses must lie in the slice and therefore must be spacelike. The only compact, without boundary, spacetime which has a slice for whichK = constant and possesses a timelike killing vector is closed flat spacetime. Let us start with initial data for a non-vacuum solution to Einstein’s equations and use the Einstein dynamical equations to compute the rate of change of K in a direction tM. t’1 is related to the normal to the slice i~1~Lvia the lapse N and shift functions !V1 where t~-~ = Ni~M+ NM, with N = = 0 and NMnM = 0. We get
c~/6t(K)= 47rN(T** + T(3)) —
v2N + NKiIK.. + K 1,
IV’1~
(1)
m
where T** is the local source energy density and T~3~ is the trace of the source stresses. T** ÷T(3~~ 0 if the source satisfies the dominant energy condition [9]. Let us assume that t~is a killing vector at the slice and that K = constant. Then (1) reduces to ‘~f~t(K)0 N {4ir (T**+ T(3)) + K”K~.} v2N. —
(2)
/
Eq. (2) has the form V2N AN = 0 A > 0 —
103
Volume 77A, number 2,3
PHYSICS LETTERS
which satisfies a mm-max principle [10]. If the slice is asymptotically flat and the killing vector lies in the slice at infinity, i.e., N 0 at then eq. (2) has a unique solution N 0. Of course, N 0 at ~ means that the killing vector in question must be spacelike at infinity. Therefore it must either be a translational or rotational killing vector. The result is trivial when we have a translational killing vector because then the spacetime must be flat. Therefore rotational killing vectors must lie in the maximal slice and hence are everywhere spacelike. If the slice is closed, compact, without boundary equation (2) again has a unique solution N 0, ~ without any precondition on the killing vector. Hence all killing vectors for such solutions must be spacelike and lie in any K = constant slice. The only exception is when —*
12 May 1980
If the source satisfies the dominant energy condition this implies Kit 0 and therefore the slice is flat.
‘~,
References
—~
A
=
104
41r(T**
+
r(3)) +K’JK~J
0
.
(3)
[1] R. Arnowitt, S. Deser and C.W. Misner in: Gravitation: an introduction to current research, ed. L. Wilten (Wiley, New York, 1962). [2] N. O’Murchadha and J.W. York, Phys. Rev. D10 (1974) 428. [3] J.W. York, Phys. Rev. Lett. 28 (1972) 1082. [4] M. Cantor et al., Commun. Math. Phys. 49 (1976) 187; N.ó Murchadha, to be published. [5] Y. Choquet-Bruhat, C.R. Acad. Sci. Paris 280A (1975) 169. [6] F. Estabrook et al, Phys. Rev. D7 (1973) 2814. [7] K. Maeda, to be published. [8] L. Smarr et al., Phys. Rev. D14 (1976) 2443; L. Smarr, Ann NY Acad. Sci. 302 (1977) 569. [9] S. Hawking and G. Ellis, The large scale structure of spacetime (Cambridge, 1973). [10] See, for example, M. Protter and H. Weinberger, Maximum principles in differential equations (PrenticeHall, NJ, 1967).