Projective Differential Geometry and Geodesic. Conservation Laws in General Relativity, Ih. Conservation Laws. G. E. PRINCE 1. School of Theoretical Physics, ...
General Relativity and Gravitation, Vol. 16, No. 11, 1984
Projective Differential Geometry and Geodesic Conservation Laws in General Relativity, Ih Conservation Laws G. E. P R I N C E 1
School o f Theoretical Physics, Dublin Institute Jbr Advanced Studies, 10 Burlington Road, Dublin 4, Ireland M. C R A M P I N
Faculty of Mathematics, The Open University, Walton Hall, Milton Keynes MK7 6AA, United Kingdom Received October 31, 1983
Abstract We examine the geodesic conservation laws associated with the projective actions discussed in our earlier paper with the same overall title. Using the Cartan formalism, a one-to-one correspondence between a class of these actions and all geodesic conservation laws is possible. In particular there is a natural geometric interpretation of Killing tensors. Homothetic motions are shown to correspond to conserved quantities on all geodesics (not just null ones). The same approach identifies homothetic Killing tensors and a universal quadratic first integral which reduces to the conformal Killing tensor case on null geodesics.
w
Introduction
In the first o f two papers on projective differential g e o m e t r y [1] we described the m a t h e m a t i c a l b a c k g r o u n d from m o d e r n Lagrangian dynamics in e v o l u t i o n space which we t h i n k appropriate to the study of symmetries and conservation laws for the geodesic spray; and developed the t h e o r y o f its symmetries, or as we call t h e m , projective actions on P = R X TM. We turn, in this second paper, to the consideration o f associated conservation laws. Our p r o b l e m , therefore, is to find functions F on P such that P ( F ) = 0, 1present address: Department of Mathematics, Royal Melbourne Institute of Technology, GPO Box 2476 V, Melbourne, Victoria, Australia 3001. 1063 0001-7701/84/I I00-1063503 50/0 9 1984 Plenum Publishing Corporation
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PRINCE AND CRAMPIN
where P is the geodesic spray. Such functions are indiscriminately called conservation laws, conserved quantities, or first integrals. Since each integral curve of I" projects onto a geodesic in M, the restriction of F to an integral curve is constant along it or the corresponding geodesic. There is always one conservation law for geodesics, namely, the constancy of the length of the tangent vector, or equivalently and more conveniently its "energy" (1/2) g('~, '~) = (1/2) gab~[a~/b . We express this in the form F(L) = 0 where L = (1/2) gaouau ~ is the Langrangian function on P from which P is derived. This fact can be useful in obtaining more interesting conservation laws; but more useful is the fact (from which it follows) that Ha(L ) --- 0, where Ha = ~/3x a P~acuC(b/~uo) is a basic horizontal vector field on P defined by the connection. For the definition and properties of the horizontal vector fields, and for much other necessary background material, we must refer the reader to our earlier paper, which is referred to throughout as paper I. The methods of associating conservation laws with symmetries differ according to whether the symmetry is of Cartan type or not. The structure of the paper corresponds simply to this division: Section 2 deals with the Cartan case, Section 3 with the rest. w
Conservation Laws Associated with Cartan Fields
The connections between symmetries of a physical theory and conservation laws are quite varied. As a general feature, some sort of reduction of order of the system is possible when a symmetry exists, and one or more first integrals appears through the quadrature corresponding to this reduction (see Prince [2] for a discussion of this process for the case of the Lie symmetries of a secondorder differential equation). The more structure the equations have, the closer the connection between symmetries and conservation laws. 2.1. Cartan Fields and First Integrals. In the case of the Euler field, the closest connection between symmetries and conservation laws occurs when the Caftan 2-form, which carries the Lagrangian structure of the system, is regarded as the fundamental geometric object of the theory. There is then a one-to-one correspondence between first integrals of the Euler field and equivalence classes of global Caftan fields, the symmetries of the Caftan 2-form. This is set out in the following Proposition (see, for example, Crampin [3] ). 2.1.1. Proposition. I f Z is a global Caftan field, with ~zOL = dr, then the function F given by F = f - (Z, OL) is a first integral of the corresponding Euler field P. Conversely, i f F is a first integral of F then there is a global Caftan field Z such that Z A dOL = dF; any two such differ by a multiple of F; each is a dynamical symmetry of P; the global Caftan field Z may be chosen to satisfy ~ z P = O.
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Proof From the condition ~.zO L = d f it follows that Z-JdOL = d F
where F = f - (Z, OL)
Since F is characteristic for dOL, F(F) = (F, Z_J dO L ) = - ( Z , F 2 dO L ) = 0 In order to prove the converse, we show first that given a E ~ *(P) such that (F, a) = 0 there is a Z C 3((P) such that Z A dOL -'- a. The linear map from tangent to cotangent space to P at any point p defined by interior multiplication of tangent vectors with (dOL)p has for its null space the one-dimensional subspace spanned by Fp. All the elements of the image space annihilate Fp. It follows from the regularity of L and considerations of dimension that the image space is precisely the annihilator of Fp, from which the result follows. Thus i f F is a first integral, (F, dF) = 0 and so dF = Z _JdOi for some Z E ~ (P). But then 5~zOi = d F - d(Z, OL) = d f a n d so Z is a global Caftan field. The difference of two solutions of the equation dF = Z J dOL is a characteristic vector field of dOi and thus a multiple of F. All Cartan fields are dynamical symmetries of F. Suppose finally that ZJdOL = dF and Z(s) = a, and consider Z = Z - oF; then Z-JdOL = dF and s
= ~ z r + r(o) r = - r ( o ) r + r(o) v = 0
2.2. First lntegrals Associated with Noether Symmetries. In paper I we gave all the Noether symmetries of the geodesic spray (the Caftan fields which are also point symmetries). Determination of the first integrals corresponding to these symmetries via Proposition 2.1.1 is straightforward. When one uses the explicit form for the Cartan 2-form 0 for the Lagrangian L = (1/2)gabuau ~ one finds that, when Z = o(0/0s) + ~a(O/Oxa) + T/a(o/oua),
F = f + oL - gab~au b Furthermore, it is easily shown that when Z is the prolongation of a vector field on R • M, f E Y(R • M), in other words, f is independent o f u a. The results of the calculations are set out in Table I. (Here ~a are local functions on M.) The important feature is that all the first integrals are universal; that is, defined not just on null geodesics but on all geodesics. Homothetic actions give the usual null geodesic result (Table I of paper I) since the parameter-dependent part of the first integral disappears (L = 0). This situation supports the paradigm that universal point projective actions (universal in the sense that they act on all geodesics) lead to universal constants of the motion. This parallels the case of the classical Kepler problem where positive, negative, and zero energy orbits share the same projective invariances and hence the same reductions of order and constants of the motion (Prince and Eliezer [4], Prince [5] ). It should be noted that if one commutes the symmetry of case (d) with 3/Os
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PRINCE AND CRAMPIN Table I. Noether Symmetries and First Integrals
Symmetry
(a)
--
Integrability conditions
-
as
First integral
L
a
(b)
~a _ _
,,~gab = 0
-ua~a
(c)
0 a ks - - + ~ia - as ax a
~-~gab = k g a b
k s L - ua~a (ua~a on null geodesics)
1 k s 2 ~ + s~ a ~ Os ax a
~ g a b = kgab ~a = ~ , a
(1/2) k s 2 L - sua~a + ( ~ - sua~a on null geodesics)
(d)
ax a
one obtains the symmetry of case (c), albeit with the additional restriction that ~a = $,a. Of course the commutator of two Noether symmetries is again a Noether symmetry, and 3/~)s is a Noether symmetry for any geodesic spray; it
follows that if case (d) is admitted then case (c) is also admitted automatically. Moreover, if we write F for the first integral for case (d), then OF]bs (which is in effect the Poisson bracket of L with F ) is just the first integral for case (c). A similar, though rather less interesting, remark could be made about commuting the symmetry of case (c) with O/~s. All first integrals associated with Noether symmetries are at best trivially quadratic in u a (that is, with quadratic part merely a function of L). It is necessary to consider Cartan fields which are not point symmetries to handle more complicatedly nonlinear first integrals. At this point we make contact with the theory of Killing tensors. 2.3. Homogeneous First Integrals and Killing Tensors. We remind the reader that a Killing tensor of valence p on M is a completely symmetric tensor K of type (0, p) which satisfies K(ab...e;f)
= 0
A Killing tensor of valence 1, of course, defines a Killing vector; Killing tensors of higher valence therefore generalize Killing vectors, at least so far as this defining differential condition is concerned. As we will demonstrate below, they also generalize the geometrical properties of Killing vectors, though this is not apparent if consideration is restricted to M alone. Killing tensors came to prominence in general relativity with the discovery by Carter [6] of an unexpected separability of the Hamilton-Jacobi equation associated with a quadratic conserved quantity for the Kerr metric. It is this inte-
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gral of the motion which allowed calculation of black hole behavior in binary systems. Considerable effort has since been put into the theory of the separability of the Hamilton-Jacobi equation and its relation to Killing tensors (see, for example, [7, 8], and also [9] for a review of the theory in general relativity and many references); but the connection between Killing tensors and group actions which we are about to describe has received little previous attention in the literature. It is convenient to mention here also the notion of a conformal Killing tensor. A conformal Killing tensor bears the same relation to a conformal motion as a Killing tensor does to a Killing vector, or isometry. Thus a conformal Killing tensor of valance p is a completely symmetric tensor H which satisfies H ( a b . :. e; f ) = h ( a b . . . a l g e r )
for some t y p e - ( 0 , p - 1) tensor h. For p = 1 this reduces to the condition for a conformal motion. We shall refer to this concept in Section 2.4. We consider now the case of first integrals which are homogeneous polynomials in the velocities ua; they are generated by Killing tensors in a natural way. Observe first of all that any completely symmetric type (0, p) tensor field K on M defines a function f K on P, homogeneous of degree p in the u a, by f x ( S , X , bl) = K a b . . " e ( X ) u a u b . . .
Ue
A type-(1, q) tensor field on M, on the other hand, symmetric in the covariant indices, may be used to define a vector field on P as follows. 2.3.1. P r o p o s i t i o n . Given a type-(1, q) tensor field J on M, symmetric in its covariant indices, there is a unique vector field J on P such that A
(i)
"lr,J(s ' x, u) = J ~ e
.
..
e(X) t l b u e
.
tie ~ .
.
8x a
(ii) ] is a variation of P (iii) (J, d s ) = 0 Proof.
I f J = a F + XaHa + p a v a with respect to the vector field basis
{ P , H a , Va} adapted to the horizontal-vertical decomposition of the tangent
spaces to R X T M = P determined by the connection, then the condition for f to be a variation of P is that pa = F(Xa) + pf)e3.buC (see paper I, Section 4.3). The three conditions of the theorem therefore effectively determine a, Xa, and laa; and in fact "]= J g c . . . e u b u c . . . ueHa + J g c . . . e; f u b u c " . . u e u f Va
In the case q = 0 this construction reduces to the prolongation of a vector field on M.
9
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PRINCE AND CRAMPIN
2.3.2. Theorem. I f K is a completely symmetric t y p e - ( 0 , p ) tensor field on M then f g is a first integral of I" if and only if K is a Killing tensor. If K is a Killing tensor and K * is the type-(1, p - 1) tensor field obtained by raising one of its indices then/s is a Cartan symmetry corresponding to - ( l / p ) f K and commutes with P./ x If J is a type-(1, q) tensor field on M, symmetric in its covariant indices, and J is a Cartan symmetry, then J , , the type-(0, q + 1) tensor field obtained by lowering the contravariant index, is completely symmetric and is a Killing tensor.
Proof.
On computing P(fK) one finds that P(fK) = Kay...
e; f uaub
. 9 9 ueuf
and therefore P(fK) = 0 if and only if K(ab...
e; f) = 0
When P ( f K ) = O, dfK
= Ha(&) = gbc"
o,3a + V a ( f K )
0a
. . f ; a l g bble . . . H f ( . o a + p g a b "
,. ell b . . . Ueo a
I f Z = XaHa + laaVa is a Cartan symmetry corresponding t o - ( 1 / p ) f K then Z l gaoO a A oab = -gabXb Oa + gaMa~ coa = _ 1 dfK P
whence gab~k b = K a b "
gabld b = -_
1
.. ell b 9 . . U e
Kbe"
. t, u C . . . u f
.. f;a u
P
and therefore ~ka =
t.ta
Ka~c. . . e -u b u e
= - - -1
gab K b c . . .
. . . H e
f; h blbtlc
. . . uf
P = K%c,..
e; f u b u c
9 . 9 ueuf
by the Killing tensor condition. Thus Z = K*. We give the proof of the final assertion for the case q = 1, for simplicity; the general case is essentially similar. Then ] =Jgu~'Ha +Jg;cUbUCVa
and
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] 3 dO = Jab; c ubue ~ a - Jab ubOa /x
The exterior derivative of J_/dO is found to contain the terms
JabO ~ A 0 ~ + (J~b, c + Jbc~ a + J ~ , b) u C ~ A 0 b the other terms being linearly independent of these. Thus if this exterior derivative vanishes, then J~a = Jaa, and J(aa; c) = 0. The remaining terms in the exterior derivative vanish as a consequence of these conditions. Thus J . is symmetric and is a Killing tensor. 9 We have therefore succeeded in finding a projective action on P corresponding to a Killing tensor field on M. (Sommers [ 10] noted a similar correspondence, but working in the cotangent bundle T*M rather than P; Crampin [ 11 ] has given another version appropriate to Hamiltonian dynamics.) This projective action is not, of course, a prolonged action in general. The value of having a vector field description of Killing tensors is that, for example, the actions they induce on the metric may be calculated. To be precise it is the action of the vector field not on the metric on M, but on the tensor field obtained by pulling it up to P, which will be calculated. We shall not distinguish notationally between these two tensors, however. Not surprisingly the Killing tensor projective action is, in a sense, isometric. This is established in the next theorem, for which we need the following.
2.3.3. Proposition.
For any Z c ~(P),
p) = Z(L) - gf
89
zP, r)
Proof. Note that L =
g(r, r )
(Indeed, this holds for any second-order differential equation, not just the geodesic spray.) Thus
Z(L) = 89 2.3.4. Corollary.
P) + g(
zr, r)
9
I f Z is a variation of P then 89( s
P) = Z(L) + 2LP(o)
where o = (Z, ds). I f Z commutes with P then 89( ~ z g ) ( r , P) = Z(L)
Proof. When Z is a variation of P then s is (-P(a)) times a second-order differential equation. I f Z commutes with P then i~ z P = O. 9 This result applied to the Killing tensor symmetry gives the following theorem.
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PRINCE AND CRAMPIN
2.3.5. Theorem. The Cartan symmetry Z corresponding to the Killing tensor K, as given in Theorem 2.3.2, has the property (Yezg)(r, r ) = o
Proof. Since [Z, I'] = O, it suffices to show that Z(L) = O. But since I-Ia(L ) = O, Z(L) = K%c
,..e;fbl
b Uc . . . ueufgahUh
= 0
Some of the non-Cartan point symmetries of the geodesic spray have quadratic first integrals associated with them (Table I of paper I): habuau b in the case Of an affine collineation, and (hab - 4~gab) u a u b in the case of a special projective collineation. It is easily verified that in either case the appropriate type(0, 2) tensor is a Killing tensor (of valence 2). The corresponding Cartan field is of course quite distinct from the point symmetry in each case. We will show how these first integrals may be derived from the point symmetries, by different means from those being employed here, in the following section. In the case of a projective collineation which is not special, where the same quadratic first integral occurs as in the special case but there is no point symmetry associated with it, the Cartan field corresponding to the Killing tensor is apparently the only type of symmetry which generates the integral. Finally, we mention the role of Killing tensors in defining Jacobi fields. Caviglia, Zordan, and Salmistraro [ 12] have recently shown that Killing tensors may be used to construct Jacobi fields as follows: i f K is a Killing tensor then along a geodesic 3, the vector field Kab... e'~ b . . . "ye(o/~xa) is a Jacobi field. They also point out that the converse does not hold: a completely symmetric tensor field K, for which Kab... ca/b . . . " ~ e ( ~ / o x a ) is always a Jacobi field, is not necessarily Killing but satisfies a weaker condition. These results fit into the present framework as follows. We showed in paper I that if Z is a vector field defined along an integral curve of P by dragging along a given vector then rr,Z is a Jacobi field along the geodesic obtained by projecting the integral curve. Thus a vector field Z on P which commutes with P defines a Jacobi field on every geodesic by restriction and projection. Now if K is a Killing tensor t h e n / ( * commutes with P by Theorem 2.3.2, and its projection is a Jacobi field; but this projection is just Ka~c... c~/b~/c . . . g/e(~/Oxa), by Proposition 2.3.1. But in the converse case we know only that K * commutes with P, which is not equivalent to K * being a Cartan field, so K is not necessarily Killing. A
2.4. Generalized Homothetie Motions. The Cartan approach may be used to find vector fields which correspond to nonhomogeneous first integrals, and thus generalize transformations more complicated than isometries. The analog of homothetic transformations is a case in point. The idea is to obtain generalized homothetic motions on P by finding a suitable generalization of the first integral
t071
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sL - ~a4/a for a homothetic motion (given in the table) and then using Proposition 2.1.1 to obtain the corresponding projective action on P. We use the Killing tensor case as a guide in constructing first integrals generalizing the homothetic ones. If ~a (~/~x a) and ~7a (b/Ox a) are Killing vector fields on M, with corresponding first integrals ~a~/a and rTa~,a along a geodesic 3', then ~a1"lb4[a4[b is trivially a first integral, quadratic in ~,a and it corresponds to the Killing tensor whose components are ~(arlb). In view of Theorem 2.3.2 one might motivate the definition of aKilling tensor (of valence 2) by seeking to generalize this observation by relaxing the condition that the coefficients of the quadratic first integral be generated by two Killing vectors. Now the product of the first integrals corresponding to two homothetic motions ms(bibs) + ~a(b/~xa) and ns(~/~s) + rla(~/~x a) is mns2L z - sL(n~a + mrla) z/a + ~arlbqaqb In order to generalize this we assume the existence of a first integral of similar form but general coefficients and find the conditions the coefficients must satisfy. 2.4.1.
Theorem.
The function F = ks2L 2 - 2sL~au a + HabUaU b
is a first integral of P if and only if ~ a ( o / o x a ) is a homothetic motion on M and Hab, the components of a symmetric tensor field on M, satisfy H(ab ; c) = ~ (agbc )
In this case the vector field Z = o + XaHa Xa = H~u b _ sL~a,
+ 12 a V a ,
where
a = ks2 L
- S~abl a
(and Z is a variation of P), is a Cartan symmetry corresponding to -(1/2) F. Furthermore (• z g ) ( F , P) = (ksL - ~au a) g ( r , r') Proof.
On computing I'(F) one finds that F(F) = 2ksL 2 - 2sL~a; bUaUb - 2L~au a + Hab; cuaubu e
which vanishes if and only if ~'(a; b) = 89kgat~ H(ab; c) = ~(agbc) The remainder of the proof consists of straightforward computations, m Note that F reduces to Hao'i'a'i, b on null geodesics. In fact the condition on the Hab shows them to be the components of a conformal Killing tensor, which
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PRINCE AND CRAMPIN
one would expect to be associated with a quadratic conserved quantity on null geodesics. In the particular case in which ~'a = $,a for some function $ on M , Hat, $gat, are the components of a Killing tensor. The integral F is then simply the sum of the quadratic integral arising from the Killing tensor, and the integral of case (d) of the table multiplied by 2L. A special case of this construction leads to a slightly simpler first integral: by taking k = 0 we obtain the first integral HabUaU b - 2sL~cuC = HabUa Ub - Sgat,~cua ub u c
announced by Prince [13]
w
;~a(~/Oxa)must now be a Killing vector field.
Conservation Laws Associated with Alternative Lagrangians
We turn now to the question of how one associates conservation laws with dynamical symmetries which are not Cartan fields. Here the correspondence is not so clear cut; however, there are several possibilities for generating conserved quantities by using the fact that a dynamical symmetry, under appropriate conditions, leads to an alternative Lagrangian (see paper I). We consider Lie symmetries first. For a Lie symmetry X (1), where X = a(O/~s) + ~a(~/~xa) E ~ ( R X M ) , the alternative Lagrangian L' generated by X (1) is
L' = X ( 1 ) ( L ) + L 6
(ignoring a total time derivative term) = ~1 h ab u a u b - L 6 + g a b u a ~ b , s
The Lie symmetries of the geodesic spray (which are listed in Table II of paper I) have components o and ~a which are polynomial functions of s-indeed, at worst quadratic functions of s. This fact may be exploited to derive a first integral. For suppose that L' is linear in s, for simplicity, and that one writes p
L' =Lo + L ' l s where the coefficients are independent of s. The corresponding energy function E ' has a similar expression
8' =8; where E'=u a OL'_L,
~L~ and E i, = u a bu--2 - Li,,
i = 0, I
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Now since L' is a Lagrangian for P, even if it is possibly not a regular one, E ' satisfies Off
r(~') +~7-~ = 0 in the usual way. Thus (P(Eo) +E~ + L ' I ) + P ( E ~ ) s = 0 Since this holds for all s, it follows that E~ is a first integral. Clearly, a similar argument applies in the general case to show that the coefficient of the highest power of s in the expression for E ' is a first integral. It may be possible, in particular cases, to obtain a conservation law from the remaining terms, which reduce to P(Eo) + u a -~L; -y=u in the linear case. It is simplest to describe how this may occur in the context of a specific example. Consider, then, the vector field X on R X M given by
X= Us ~ + ~ 0x~ where ~, ~a (local functions o n M ) satisfy
~P~c
= ~ ba~ , c
+
~,
a
~;aa = 0 Then X (1) is a Lie symmetry and corresponds to a special projective collineation [see (d) of Table II of paper I]. The alternative Lagrangian generated by X 0 ) is L' = (L
- Lt~) - ( L ~ , aUa) s
It follows that E , = ~ t~n,,a IL/au b - L ~ ) -
(2L~,aua) s
and therefore that 2Lff, a ua is a first integral. But since L is a first integral, so is ~, aUa. The other equation derived from P ( E ' ) + (OL'/Os) = 0 is P ( } habUaU b - L I ~ ) - 3 L ~ , a ua = 0
Now L gg, alg a = F ( L 1~1), and so this may be written p(1 habuau a _ 4L~) = 0
whence (1/2) habUaU b - 4Lt~ = (1/2)(hab - 4gab r ) uau b is also a first integral.
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PRINCE AND CRAMPIN
Turning to projective actions on P in general, we have the following result, by virtue of the fact that P(L) = 0.
3.1. Theorem. For any projective action Z on P, Z(L) is a first integral of F. If Z(L) = 0 and ~ z F = 0 then (Z, 0L) is a first integral. Proof. It follows from the c o n d i t i o n ~ z P = hF that [Z, F] (L) = 0 and so P(Z(L)) = 0. I f ~ z P = 0 then
P(Z, OL ) = (Z, ~- rOL) = (Z, dL ) = Z(L ) and so if Z(L) = 0, (Z, 0L) is a first integral. " If Z(L) = 0, but ~ z r ~ O, then by adding a suitable multiple of F^to Z one may construct a vector field Z for whichs 2 P = 0; it is still true that Z(L) = 0. This theorem is related to the well-known result that for a Jacobi field X along a geodesic 7, g(V~X, ~/) is constant along % For if Z is a projective action which actually commutes with P, the projection X of its restriction to an integral curve of P is a Jacobi field along the projected curve, a geodesic 7- Suppose that Z = XaHa + laa Va with respect to the basis {P, Ha, Va} (it may be assumed without essential loss of generality that Z has no F component, since Z commutes with I" and so if it did the coefficient would have to be constant along any integral curve). Since Z is a projective action,/s a = P(Xa) + I'~c Xbu c. The projected vector field along a geodesic 7 is Xa(O/Oxa) = X. Now Ha(L) = 0, and therefore Z(L) = ga~uala b; along the geodesic,/~a = (V,?x)a, and so the constancy of Z(L) implies the constancy of g(VqX, ~). If Z(L) should be zero, then the first integral (Z, OL) reduces to g(X, ~) along any geodesic. In the case of a Jacobi field defined by a symmetric tensor field K, and Z = K*, one finds that when K is a Killing field, Z(L) = 0 (Theorem 2.3.5), which leads to the conservation of Kay.. " e'~a~tb . . . ~e as before; but in the more general case the conserved quantity takes the form g(V~X, +) and therefore involves the covariant derivative of K. /x
w
Summary
We have developed a geometric formulation of symmetries and conservation laws for geodesics in which a number of previously disparate elements-Killing tensors, Jacobi fields, etc.-have a natural place. A major advantage of the scheme is that the geometry is essentially straightforward since it involves computations with vector fields and forms and little else; the only complication is that it takes place on the space R • TM rather than on the space-time manifold M directly. We have pursued similar ideas in another direction, namely, the derivation of Raychaudhuri's equation, in another paper [ 14]. Possible future developments include the application of the theory to specific metrics (with the help of
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REDUCE); the inclusion o f conformal transformations and conformal Killing tensors in the scheme; the imposition and exploitation of a suitable affine and metric structure on P itself; the investigation of space-times with particular features by our methods, which might perhaps clarify the relation between symmetry and Petrov type, for example. We hope to treat at least some of these matters in future papers.
A cknowledgmen t
We wish to thank Frans Cantrijn and Willy Sarlet for making helpful and penetrating criticisms of an early draft of both papers.
References
1. Prince, G. E., and Crampin, M. (1984). Gen. ReL Gray., 16: 921-942. 2. Prince, G. E. (1981). Lie symmetries of differential equations and dynamical systems, (Ph.D. thesis, La Trobe University, Bundoora, Victoria, Australia). 3. Crampin, M. (1977). Int. J. Theor. Phys., 16, 741. 4. Prince, G. E., and Eliezer, C. J. (1981). J. Phys. A: Math. Gen., 14,587. 5. Prince, G. E. (1983). J. Phys. A: Math. Gen., 16, L105. 6. Carter, B. (1968).Phys. Rev., 174, 1559. 7. Benenti, S., and Francaviglia, M. (1980). In GeneralRelativity and Gravitation, One Hundred Years After the Birth of Albert Einstein, Vol. 1, A. Held, ed. (Plenum, New York). 8. Woodhouse, N. M. J. (1975). commun. Math. Phys., 44, 9. 9. Kramer, D., Stephani, H., MacCallum, M., and HerR, E. (1980). Exact Solutions of Einstein "sField Equations (Cambridge University Press, Cambridge). 10. Sommers, P. (1973). J. Math. Phys., 14,479. 11. Crampin, M. (1980). Rep. Math. Phys., 18, 2. 12. Caviglia, G,, Zordan, C., and Salmistraro, F. (1982). Int. J. Theor. Phys., 21,391. 13. Prince, G. E. (1983). Phys. Lett., 97A, 133. 14. Crampin, M., and Prince, G., (1984). Gen. Rel. Gray., 16: 675-689.
