We define an everywhere invariant space of metrics U to be one for which the set of diffeomorphisms, which leave U invariant, contains all the isometrics of.
General Relativity and Gravitation, Vol. 18, No. ll, 1986
Everywhere Invariant Spaces of Metrics and Isometrics S. T. Swift, ~ R. A. d'Inverno, 1 j . A. G. Vickers I Received November 19, 1985
We define an everywhere invariant space of metrics U to be one for which the set of diffeomorphisms, which leave U invariant, contains all the isometrics of the individual metrics in U. We also generalize Killing's equation to a new equation, the invariance equation, which has as solutions those vector fields which leave U invariant. By combining these ideas we give a new method for finding the isometrics of a given metric.
1. I N T R O D U C T I O N The large number of exact solutions to Einstein's equations which are now known has made it important to develop techniques for the classification of such solutions. Examples of these are the Petrov classification and classifications based on the isometrics of the solutions [1]. Knowledge concerning the existence of isometrics of a metric would seem to be one of the most basic pieces of information about a metric, but unlike the Petrov classification, for which an algorithm exists to find the Petrov type, there is no simple way of finding the isometrics of a given metric. Indeed, solving Killing's equations directly for a complicated metric is likely to be almost impossible. In 1978, d'Inverno and Smallwood [2, 3] developed a technique for finding some of the Killing vectors of a metric by first looking for infinitesimal transformations that left the "functional form" of the metric invariant and then specializing to find Killing vectors. In this paper we generalize this technique by introducing the concept of an "everywhere invariant" space of metrics and show how a given metric may be embedded in such a set. 1Department of Mathematics, University of Southampton, Southampton S09 5NH, England. 1093 0001-7701/86/1100-1093505.00/0 9 1986 Plenum Publishing Corporation
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The basic idea, is that rather than finding the diffeomorphisms which leave a given metric g invariant, one finds instead the diffeomorphisms which leave a class of metrics U, containing g, invariant. By choosing this class to be an everywhere invariant set one can then find the isometries of g by making suitable specializations. The advantage of this method is that the extra freedom in finding the diffeomorphisms that leave a whole set of metrics invariant, rather than just a single metric invariant, allows one to decouple the equations analogous to Killings equations. The detailed plan of the paper is as follows. Section 2 introduces the notation and basic ideas. The concept of an everywhere invariant space of metrics is introduced in Section 3, and Section 4 is an example of how the technique may be used to find an isometry of a particular metric. 2. N O T A T I O N AND BASIC IDEAS In this section we introduce the notation and some of the basic ideas we use elsewhere in the paper. M is a smooth finite-dimensional manifold and N ( M ) the group of diffeomorphisms of M. @(M) may be given the structure of a smooth manifold and the tangent space at the identity may be identified with ~(M), the space of smooth vector fields on M. @(M) may be thought of as an infinite dimensional "Lie group" in that the group operations are smooth, and ~(M) may be viewed as the associated "Lie algebra." In order to make this idea precise it is necessary to enlarge the space ~ ( M ) to spaces ~S(M) of H S maps which are locally like neighborhoods of Hilbert spaces. In particular it is possible to give ~ ( M ) the structure of an inverse limit Hilbert manifold. (See [4] for further details.) The set of (pseudo-) Riemannian metrics on M is denoted G(M) and SzM will denote the bundle of symmetric covariant 2-tensors on M. The set G(M) is thus the set ff(gzM ) of sections of S2M that induce an inner product (of the appropriate signature) on each tangent space TxM. Since G(M) is open in F(SzM ) w e may make the identification
TgG(M) ~- F(S2M). There is a natural right action of the diffeomorphism group @(M) on the space of metrics G(M) defined by
G(M) x ~(M) ~ G(M) (g, ~) ~ O*g where (~b*g)x (X, Y)= go(x)(D(~(X), Dqbx(Y)). The orbit o f g ~ G(M) under N(M) is denoted g" ~ ( M ) . The group of isometries of g is just the stabilizer subgroup of g under this action Isom(M, g) = Stab(M, g) = {~be ~ ( M ) : ~b*g= g}
Everywhere lnvariant Spaces of Metrics and lsometries
1095
Let ~bt be a 1-parameter subgroup of isometries; then, differentiating ~b*(g) = g gives
d.(g) Lx g==--~
t=0
=0
(1)
where X e if(M) is the vector field d X ( x ) = Z 4,(x) ,~o
Let K(M, g ) = { X ~ ( M ) : L x g = O } denote the set of Killing vector fields on M. The 1-parameter groups of isometries give rise to Killing vector fields and conversely the flow of a Killing vector field gives rise to a 1-parameter group of isometrics. Rather than consider the stabilizer subgroup of a single metric g we wish to consider the stabilizer subgroup of a set of metrics U. Let U_~ G(M); then we define Stab(M, U) = {~b~ ~ ( M ) : ~b*(U) ___U} Let ~bt be a 1-parameter subgroup of Stab(M, U) and let g be some fixed element of U; then we may define a smooth curve t ~ g, in U by g , = ~b*(g)
(2)
Lx(g) = Y~ I~ U ~ F(S2M)
(3)
differentiating w.r.t, t we obtain
let K(M, U ) = { X ~ ( M ) : Lx(g)6 TgU, Vg6 U}. The set K(M, U) is the required generalization of Killing vector fields. In most useful situations U will be parameterized in some convenient way by some infinite-dimensional Fr6chet manifold F, and we have a Fr6chet differentiable bijection 7: F ~ U and write 7 ( f ) = g. For example, we could take F = {f: ~ 2 x S 2 ~ [ ~ + : f i s smooth} and
7(f) = ~ =fdt 2 - f
~dr2 _ r2(dO2 + sin 20d~ 2)
Then U are the metrics of the same "functional form" as the Schwarzschild metric and Stab(M, U) those diffeomorphisms that preserve the functional form (see [2] for further details). In this situation we can obtain an equation which the vector fields in K(M, U) must satisfy. We first note that there exists a map ~u: F x Stab(M, U) ~ F defined by 7t(f, qt) = 7 l(~b*(7(f)))
(4)
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Swift, d'lnverno, and Vickers
i.e., ~b* = g. Note also hu(f, id.) ---f
(5)
Let V( U) ~_-~(M) be those vector fields that generate elements of Stab(M, U). Let Z e V(U) generate ~b,e Stab(M, U) and let f be any fixed element of F. Define f, = ~(f, ~,); then, by (5), f a - - f . By the definition o f f . f 29 ~?g---x
d
~
df, t~0
t=0
Lxg = DTj(I~)
where
dz t=0
(6) (7)
We call (6) the invariance equation for U. Now suppose that X~7~(M) satisfies (6) for all f e F . Let $, be the 1-parameter group generated by ?(; then ~,*(Lxg) = ~,* [DvA~)] d~(t)
~ - - - ~ = ~b*[DTf(p)]
(8) where
.(t) = ~,*~
(9)
(10)
Regarding (9) as a first-order O.D.E. for a(t), (9) has a unique solution satisfying a(0) = ~, which is seen to be given by a(t) = g. Thus f 29
O*g=geU
(11)
and hence ~, e Stab(M, U) and X~ V(U). Thus
K(M, ~(F)) = {X ~ ~(M): Z,,(7(f) = D~,~(~), Vf e F} i.e., the vector fields which leave U = 7(F) invariant are those satisfying the invariance equation. In the special situation that F has the structure of a linear space, then (6) reduces to the "functional form invariance" (F.F.I) equation of d'Inverno and Smallwood [2]. Although the invariance equation looks harder to solve than Killing's equation, the fact that it must be solved for arbitrary f ~ F means that one
Everywhere Invariant Spaces of Metrics and Isometries
1097
can decouple the equations so that they can in fact be solved more easily than Killing's equation (see Sect. 4 below). In some situations it is easier to consider the components of the metric in a frame rather than in terms of local coordinates. We end this section by deriving a frame version of the invariance equation. To avoid problems concerning the existence of global sections, we work in some suitable open set W ~ M. We write g in the form f
/"
f
g = rl~bO~| 0 b
(12)
where {0~} is a basis for T * M and depends smoothly on x ~ W. N o w define
z~:F~AI(W)
by
f
~ " ( / ) = 0"
(13)
f
where we may suppose 0" have been chosen so that ~" depends smoothly on f Then the invariance equation gives ' a rlab(LxOf ) | ~b f f + ,labO~| (LxOfb )=q~b(Dr f(t~)| fOa) + tlab[O~| (Dz~(#) ]
(14) f
~ rl~bQ~ | 0~ + ~ b 0 a | ~
= 0
(15)
f
where s a = L x O~ - Dr"I(t~ ). ~ . fb If we now define f2~b by (2 = Q b0 , (15) then becomes q,.b f2ca + r/acf2cb = 0
(16)
and we see that (2ab must be in the Lie algebra of the Lorentz group. Note that the invariance equation will be satisfied if f
LxOa= Dv"f(#)
(17)
3. EVERYWHERE INVARIANT SPACES OF METRICS In general there will be ticular metric g will leave a order to use the ideas of the we now turn our attention property.
no guarantee that an isometry of some parset of metrics U containing g, invariant. In previous section to find Killing vector fields, to those spaces of metrics U that have this
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Swift, d'Inverno, and Vickers
Definition. Let U be a subspace of the space of metrics (of a given signature) on M. U is said to be everywhere invariant if and only if all isometries of all the metrics in U are contained in the set of diffeomorphisms leaving U invariant. That is Vge U
~be Stab(M, g)=~r
U)
(18)
Our main result will be to show how to construct everywhere invariant spaces of metrics. It is worth remarking that in certain situations one can show the converse implication also holds. For example, when U is a convex compact subset of G(M) one can use the Schauder fixed point theorem [5] to show that ~b*(U)c_ U implies that there exists some g e U such that ~b*g=g and thus ~ b ~ S t a b ( M , U ) ~ 3 g ~ U such that ~be Stab(M, g). For our purposes (18) is the important implication, and we will not pursue the converse result here. Let H be the group of bijections of G(M). (We are only concerned with the algebraic properties of this group. In a wider context it would be of importance to study the differentiable and topological structure of the group of "diffeomorphisms" of G(M).) The group H gives rise to a leftaction on G(M) in the obvious way.
H x C(M)
C(M)
(h, g) -~ h(g) A subgroup of H, which will be important, consists of those elements that cover diffeomorphisms = {h
H:
Rather than consider stabilizer subgroups under the right action of ~ ( M ) on G(M), it will be more convenient to consider the corresponding subgroups for the left action of ~ * ( M ) on G(M). Isom*(M, g) = Stab*(M, g) = {~b* e @*(M): r
= g}
Stab*(M, U ) = {~b* e @*(M): ~b*(U) _ U} We now show how to construct everywhere invariant spaces of metrics U.
Proposition Let S be a subgroup of H such that the normalizer of S in H contains
~*, N(S) ~_@*; then S ' g, the orbit of g under S, is everywhere invariant. Proof Let ~ be some fixed element of G(M) and T = S- ~ = { g e G(M): g = s(~), s e S}. Then, given any g e T we have g=sl(~)
where
sleS
let (19)
Everywhere Invariant Spaces of Metrics and Isometrics
Now let r
1099
6 Stab*(M, g). Then r (r
But ~ * c N ( S ) , so r 1 6 2
g
sl)(~) = sl(~)
(20)
for some s2eS and thus (s2or
= sl(~) r
~- S2 1 " S l ( g ) ~-- s 3 ( g )
(21)
Now let ~ be any element of T, then = s4(~)
for some
s4 9 S
(22)
But ~ * c N(S), so (r
s4)(~) = (sso r
for some
= (s5 o s3)(~) = ~(g)
s5 ~ S
by equation (22)
(say)
Thus r 1 4 9 T and therefore r Stab*(M, T). We have shown that r e S t a b * ( M , g ) ~ r e S t a b * ( M , T), thus proving the T is everywhere invariant. We now use this proposition to give an example of an everywhere invariant set of metrics. A generalization of a conformal structure on a manifold may be given as follows. Let f : TM--* TM be a smooth map such that ~ o f = ~ z a n d f x , the restriction of f to the fiber Tx, is an invertible linear map f~: T x M ~ T~M. We s a y f i s an automorphism of TM and denote the set of such maps Aut(TM). Now let g be some fixed metric and define } by
/g~(x, Y) = gx[f(X), f( Y)]
w h e r e f ~ Aut(TM)
We say a family of metrics of the above form is a generalized conformal structure on M. We now have the following corollary to proposition 1.
Corollary A generalised conformal structure U is an everywhere invariant set of metrics.
Proof Let S = {h 9 H: hg =)~ for some i s Aut(TM)}. Write ~ =97(g) where jT9 S; then, by direct calculation, one can show ( r o)7)(g) = (ko r Where k e Aut(TM) is defined by kr
= De.oleo (Dr
-~.
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Swift, d'Inverno, and Vickers
Thus ~*c_N(S), and by the proposition U=S.g, is everywhere invariant. In some situations, allowing f to be any automorphism of TM gives an invariant set U which is unnecessarily large. In these situations it is sometimes possible to restrict S to some subgroup in such a way that the orbit is still everywhere invariant. For instance one can recover the usual conformal structure by demanding that fx is proportional to the identity; another example is to require fx to be symmetric with respect to gx. The exact restriction will depend on the form of the initial metric g, thus the group may be restricted so that the orbit of a Minkowskian metric consists of metrics with the same "functional form" as a Schwarzchild metric. Provided one excludes flat space, this set of metrics is everywhere invariant (see also Section 4 below). We end this section by describing a completely different method of constructing an everywhere invariant set of metrics. This involves finding a slice for the right action of ~ ( M ) on G(M). Such slices were shown to exist in the Riemannian case by Ebin [6] and Palais [7] and in the Lorentzian case by Marsden and Isenberg [8]. In either case one can prove the following theorem:
Theorem Let M be a compact n-dimensional smooth manifold. Then for each
~ G(M) there exists a submanifold S of G(M) containing ~, called a slice through ~, which is diffeomorphic to a ball in Hilbert space, such that: (i) (ii) (iii)
r e Isom(M, ~) =~ r
= S
~,) ~, a local cross-section Z: @(M)/Isom(M, ,~)~ ~ ( M ) , defined on a neighborhood W of the identity coset such that if F: W x S ~ G(M) is defined by F(~o, s)=z(co)*S, then F is a homeomorphism onto a
eel(M)
and r 1 6 2 1 6 2
neighborhood of ~. We show that a slice S is an everywhere invariant set of metrics. Let so by (ii), q~e Isom(m, ~) and thus by (i), r S. This is true for any g e S, so the slice S is everywhere invariant. The proof of the slice theorem is constructive, so in principle it could be used to construct everywhere invariant sets of metrics. However, in practice this is too difficult to carry out, and so this method cannot be used to construct actual examples.
g e S and CeIsom(m, g). Then •*g=g and hence r 1 6 2
Everywhere Invariant Spaces of Metrics and Isometries
llO1
4. ISOMETRIES OF pp-WAVES It is sometimes possible to show that a set of metrics is everywhere invariant even if the condition of proposition 1 9 " __c_N ( S ) is not satisfied. In fact, the condition given is too strong and it is only necessary to show that all elements of R = U g ~ I s o m * ( M , g ) lie in N ( S ) . Of course, Isom*(M, g) is what one wants to find, so that R is not known in advance. However, in some situations one has sufficient knowledge to show that R c_ N ( S ) even though one does not know R explicitly. We illustrate these ideas by using the concept of everywhere invariance to find the isometrics of plane-fronted gravitational waves with parallel rays (pp-waves). The metric g for a pp-wave may be written in the form = 2d{ d e - 2du dv - 2 H du 2
or alternatively writing ~
~ = (x + iy)
= dx 2 + dy 2 - 2du dv - 2 H du 2 H
In order to ensure that g represents a vacuum solution of Einsteins, equation H must be the real part of an arbitrary complex function of u and H 3. However, since we are only interested in the isometries of g, we will allow H to be arbitrary. Let U be the set of such metrics excluding flat H space. Then, using the obvious notation 7 ( H ) = g, the invariance equation H L x g = DTH(/~) can be written 2X1,1 = 0
(23)
X 2 , 1 -t- X 1 , 2 ---~ 0
(24)
- - X 4 , 1 -~- X 1 , 3 = 0
(25)
-X3,1 + X1,4 ----0
(26)
2X2,2 = 0
(27)
-2HX3,2 - X4,2 + X2,3 = 0
(28)
-X3,2 + X2,4 = 0
(29)
-- 2HX3,1
- 2X(H) - 4 H X 3, 3 - 2X4, 3 ~" - 2 #
(30)
2HX3,4
- - Jr'4, 4 ~-- 0
(31 )
2X3,4 = 0
(32)
-- X3,3 --
H o
and Killing's equations are obtained by setting # =0. Let X ~ K ( M , g ), where H o ~ O ; then X satisfies (23)-(32) with H = H o and kt=0.
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Swift, d'Inverno, and Vickers
Equations (32), (25), and (28) imply that X 3 depends only on u. X thus satisfies (23)-(32) with any H and # given by (30). Therefore X s K(M, ~o) implies that X s K ( M , U), and so that U is everywhere invariant. We may now find solutions to (23)-(32) for arbitrary (nonzero) H by using the decoupling method, i.e., using the fact that A H § B = 0 for all H implies A = 0 and B = 0. The solution of the invariance equation is found to be
X=[-ib~ +fl(u)]
+[-ib(+fl(u))]
_+(cu+d) o---u
+ [(fi'(u) ~ +/~'(u) ~ - cv + a(u)] ~3-~ where b, c, d are real constants, a(u) is a real-valued function, and fl(u) is a complex-valued function. The corresponding elements of Stab(M, U) are found by finding the integral curves of X and are given by
~: ( ~, ~, U, V)~ { ei~ ~ -I- h(bl) ], e-i~[ ~-[- fi(bl) ], a - l(U + Uo), air + h'(u) ~ + l~'(u) ~ + g(u)] where c~, a, Uo~ ~, g is real valued and h is complex-valued. The change in the metric is ~b*(gH) = ~ where
O = a Z [ H - h"(u) ~ - h'(u) ~ + h'(u) h'(u) - g'(u)] By now specializing by restricting the particular form of H, the possible isometries and Killing vector fields may be found. This is done in Table 21.1 of Kramer, Stephani, MacCallum, and Herlt [-9] where the possible Killing vector fields are listed. Note that it is only due to the everywhere invariance of the set U that this technique can be guaranteed to give a / / t h e Killing vector fields.
5. C O N C L U S I O N In this paper we have demonstrated a technique that may be used to find all the Killing vector fields of a given metric. We have extended and clarified the original work of d'Inverno and Smallwood on "functional form invariance" of a family of metrics by examining the left action of the automorphism group of G(M) on G(M) rather than just considering the effect of diffeomorphisms on G(M). The new idea is to introduce the concept of an everywhere invariant set of metrics U for which the set of dif-
Everywhere Invariant Spaces of Metrics and Isometries
1103
feomorphisms which leave U invariant contains all the isometries of the individual metrics of U. Proposition 1 gives a partial characterization of such sets, but as the example of Section 4 shows there exist examples of everywhere invariant sets that are not of this form. It would clearly be desirable to have a more complete characterization of such sets and have a better understanding of the exact relationship between Stab(M, U) and Stab(M, g) for g e U, and some possible directions for future work have been considered by the present authors [10]. Nevertheless, we believe that this method will prove useful in finding and understanding the isometries of both individual metrics and families of metrics.
REFERENCES 1. Kinnersley, W. (1974). Paper presented at the Seventh International conference on General Relativity, Tel-Aviv, Israel. 2. d'Inverno, R. A. and Smallwood, J. (1978). Gen. Rel. Gray., 9, 195-214. 3. d'Inverno, R. A. and Smallwood, J. (1978). Gen. Rel. Gray., 9, 215-225. 4. Omori, J. (1968). Proc. Symp. Pure Maths. XV, 167-184. 5. Schwartz, J. T. (1969). Nonlinear Functional Analysis (Gordon and Breach, New York). 6. Ebin, P. G. (1968). Proc. Symp. Pure Maths., XV, 11-40. 7. Palais, R. S. (1960). Ann. Math., 73, 295-323. 8. Isenberg, J. and Marsden, J. (1982). Phys. RED., 89, 179-222. 9. Kramer, D., Stephani, H., MacCallum, M., Herlt, E. (1980). Exact Solutions of Einstein's Field Equations (Cambridge University Press, Cambridge). 10. Swift, S. T., d'Inverno, R. A., Vickers J. A. G. (1984). Functional Form Invariance in General Relativity" and Geometry, University of Southampton Preprint.
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