to invariance of the space-time under one-parameter group actions, and of the ... ers in general relativity have taken over only those parts of Lie's theory of finite.
General Relativity and Gravitation, Vol. 16,No. 10, 1984
Projective Differential Geometry and Geodesic Conservation Laws in General Relativity. I: Projective Actions G. E. PRINCE I
School of Theoretical Physics, Dublin Institute for Advanced Studies, 10 Burlington Road, Dublin 4, Ireland M. CRAMPIN
Faculty of Mathematics, The Open University, Walton Hall, Milton Keynes, MK7 6AA, United Kingdom Received October 31,1983
A bstrac t The Lagrangian structure of the geodesic equation allows an extension of classical projective geometry to one-parameter projective group actions on R X TM (where M is the spacetime). We determine all those projective actions which arise by prolongation of oneparameter group actions on R • M. The relation between projective actions on R • TM and the equation of geodesic deviation is developed.
w
Introduction
The exact nature and role of conserved quantities in general relativity has always been problematic. On the one hand there are the first integrals of the geodesic equation, and on the other there are the conservation laws associated with the stress-energy-momentum tensor and the field equations. One would like to think of the former as geometric conservation laws somehow connected to invariance of the space-time under one-parameter group actions, and of the latter as dynamical conservation laws providing operational definitions of physi] Present address: Department of Mathematics, Royal Melbourne Institute of Technology, GPO Box 2476 V, Melbourne, Victoria, Australia, 3001. 921 0001-7701/84[ 1000-0921503 50/0 9 1984 Plenum Publishing Corporation
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PRINCE AND CRAMPIN
cal quantities. Furthermore, these two sets should be interrelated, and a geometric symmetry, for example, should lead to the identification of a dynamical one. Davis and Moss [1-3] were perhaps the first authors to pursue this line; a useful review of current ideas is to be found in [4]. However, while this overall structure is clear the exact nature of the internal workings is not. In particular there are geometric conservation laws (those associated with Killing tensors) which do not obviously arise from one-parameter group actions on the space-time, and this clouds the well-established relation with dynamical conserved quantities. It also highlights the lack of any sort of one-to-one correspondence between first integrals of the geodesic equation or dynamical conservation laws and underlying symmetries of the space-time. There have been considerable advances in general relativity in the areas of conservation laws, Hamilton-Jacobi theory, and schemes to generate new solutions from old and even new space4imes from known ones: see, for example, [4 - 8 ] . However, the many recent advances in the theory of differential equations and classical Lagrangian theory have had little impact. In particular, workers in general relativity have taken over only those parts of Lie's theory of finite continuous groups and ordinary differential equations which seemed most suited to the metric theory of gravitation, namely, isometric, affine, and conformal motions on the space-time (M, g) [9, 10]. Lie's theory as it now stands considers group actions on R • M (R for the parametrization of the trajectories) and it may be interpreted as a generalized projective differential geometry of paths. Armed with one or more of these wider group actions, some well-defined algorithms are available for the construction of first integrals through successive reduction o f order of the geodesic equation [11 ]. From the second area, that of Lagrangian formulations, there lies ready for exploitation the one-to-one correspondence between first integrals and oneparameter group actions on evolution space E = R • TN (N is the classical configuration space) essentially due to E. Cartan [ 12]. The important feature here is that the association is gained at the expense of considering symmetries other than the obviously "'geometric" ones, that is to say, group actions on N (or R XN). Our purpose, in this paper and a second one with the same general title [13], is to describe the relations between group actions and geodesic conservation laws, using the modern version of projective differential geometry and, in part, Cartan's approach. In the present paper we concentrate on the projective differential geometric aspects of the problem. The paper is arranged as follows. The notational and computational conventions required for both papers are set out in Section 2. Section 3 is a summary of classical results. Section 4 is devoted to the so-called Hamilton-Cartan approach to Lagrangian theory, explicitly formulated here to allow for parameter dependence of both symmetries and first integrals. Section 5 deals with the application
PROJECTIVE
DIFFERENTIAL
GEOMETRY
AND PROJECTIVE
ACTIONS
923
of this theory to the geodesic problem, and the development of modern projective differential geometry. Our second paper will discuss the derivation of geodesic conservation laws, using the results of this paper.
w
Notation
Though they are not differentiated mathematically, it will be convenient for us to distinguish notationally between a manifold representing space-time and one representing, say, the configuration space o f a classical mechanical system. We shall reserve the symbol M for space-time manifolds, denoting the generic manifold by N. Local coordinates will be denoted by (x a) for a space-time manifold, and (qk) for a generic manifold and for a configuration space. We shall deal also with the manifolds R X M and R • N; we think of a curve in M (or N ) as defining a graph inR • M ( o r R X N), or as a section of the fibration R • M-+R (or R • N-+ R). Here the different interpretations become plain: the R component represents time in the case o f R • N when N is a configuration space, whereas it is just a parameter in the case o f R • M when 34 is space-time. We accordingly use local coordinates (s,x a) for R • 3//but (t, qk) f o r R X N. We shall assume the usual topological and smoothness properties for all manifolds and make no further comment on such matters. As usual, TN and T*N denote the tangent and cotangent bundles of the manifold N. The natural coordinates on TN, based on coordinates (q k) for N, are (q k, v ~). The manifold E = R • TN, evolution space when N is con figuration space, has coordinates (t, qk, vk). We use a comma and the subscripts t, k, and/~ to denote partial differentiation with respect to t, qk, and vk~ respectively. An obvious extended version of the summation convention is used, so that for example
df =f,r dt +f,k dq ~ +f,~ dvk where f i s a smooth real-valued function on E. We need two projections, 7: E-+ R • N and r: E ~ N, together with their differentials (tangent maps) r , and rr,. The ring of smooth functions on N is written ~:(N); the ~(N)-module of vector fields is ~(N), while ~ *(N) represents the 1-forms, and ~ k ( N ) the kforms. These notations extend to other manifolds in the obvious way. Generic elements of these sets are f ~ ~:(N), X, Y , Z ~ ( N ) , a, ~ ~%*(N), and co C g2k(N). The symbols A and d have their usual meanings of exterior product and derivative, and we use _i for interior and multiplication, or contraction, of a vector field and a form: thus if X ~ ~ ( N ) and co E ~2k(N) then XJco E ~ k - 1 (N)
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PRINCE AND CRAMPIN
is defined by ( X Joo) ( X l , X2 . . . . .
X k - 1 ) = oo(X, X l , X2 . . . . .
Xk-1)
The natural pairing of vector fields and 1-forms is denoted by angle brackets: thus for X E %(N) and a E ~*(N), (X, a) E ~:(N). The Lie derivative along X is denoted b y 2 x and the following formulas are frequently used:
~exf = X ( f ) s
Y = IX, Y]
~ x a = X_lda + d ~_x~ = XAda~ + d(XA~) For a space-time (M, g) with underlying manifold M and pseudo-Riemannian metric g we shall denote by P the space R X TM, whose local coordinates are (s, x a, ua). We use the semicolon, and sometimes V, to denote covariance differentiation with respect to the symmetric connection determined by the metric. w
Classical Results
3.1. Projective Differential Geometry of Paths. S. Lie's ideas of invariance of differential equations have been only slowly assimilated into mathematical physics, and relativity is no exception. Lie always thought of "invariance" of an ordinary differential equation primarily as being the property that the "solution curves" (on some as yet-to-be-defined space) are mapped into themselves by some invariance transformation. It was undoubtedly the fact that this implied invariance of the functional form of the equation under the transformation that subsequently obscured the basic concept. As far as differential geometry was concerned, the idea initially appeared as the following question (Eisenhart [9], Sections 22, 46; Weyl [ 14] ): If the geodesics on an affine space M are the solutions of d2x---~a dxb dxC =0 ds 2 + Fge ds ds what "projective change of affine connection" leads to a connection with the same geodesics? The solution was that
rac- rac :=rgc+ g c+ag b, does the trick and the geodesic equation becomes
d2x a dx t' dx c dy----y-+ ('~e dg d g = 0
q; E
PROJECTIVE DIFFERENTIAL GEOMETRY AND PROJECTIVE ACTIONS 925 with the new parameter ~ given by
Y=c f e x p ( 2 f G d x a ) ds The notion was refined with the application of one-parameter group theory. The question became: What "projective" group action on M maps the geodesics into themselves? The solution to this question was that a one-parameter group ~x : M-+ M with local infinitesimal representation (to use in part the contemporary terminology and notation)
.2a =x a + ~a~x [in modern terms, the one-parameter group generated by the vector field
~a(3/~xa)] has the required effect if and only if PL(x) := rL(:O = r L ( x ) + a~ G(x) + a ~ b ( x ) for sonre ~ E ~*(M). Again there is an induced parameter change, and
v =cf exp [2f G(x, SX)dxalds and the transformed geodesic equation is
d22 a - dg 2
_ + P~,c
d.~ b d2 c dY d~
- 0
These results are obtained by appealing to the function form invariance of the geodesic equation under the equivalent coordinate transformation to the projective group action. In a more modern formulation, [10], the transformation equations for P~e and s become
and
~ds=(2ft~adxd+c) ds w h e r e ~ is the Lie derivative with respect to the vector field X (not written as -~x for reasons which will later be apparent). H e r e ~ ds should not be given its modern meaning: it signifies the first order change in the differential ds due to the group action. In a metric space the 1 -form ~ is exact ([9], Section 46) so that Ca is replaced by ~,a.
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We will show below how much simpler it is to regard the group action as being on R • M so that the transformation equation for s becomes an integral part o f the group transformation. This makes a number of new results readily accessible.
3.2. Group Actions on M. Table I is a summary of some known results concerning distinguished group actions on M and conservation laws. It is due to Katzin and Levine [ 15 ]. Projective one-parameter group actions, or projective collineations, as defined above, and some important special cases of them, are indicated with an asterisk. The induced parameter transformations are not included. The table makes use of the well known formulas hat) :=~-~gab = 2~(a;b) g
(hab;c + hdc;b - hbc;d )
= ~a;b c - ~dR~,ca Here R~c a are the components of the curvature tensor, of course; and in the table, Cgcd are those of the Weyl (conformal curvature) tensor. Some new results about the relations between these symmetries appear in later sections. When the abbreviations PC, CONFC, etc. are used later they refer strictly to their definition in this table and not to any of the generalizations to actions on R X M or P appearing in subsequent sections. The distinctions between isometries, homothetic and conformal motions in classical theory are important in understanding later results. We will maintain for the moment the traditional practice of representing vector fields by their components and not considering parameter transformations as an integral part o f group actions. An isometry is a projective action on M with no induced parameter transformation, which satisfies P~g = 0,
or equivalently ~a;b + ~b;a = 0
This leads to the conservation of ~ag[ a along a geodesic 7 with tangent vector "~: V
"a
"a "b
The vector field X with components ~a is called a Killing vector and the equation it satisfies is Killing's equation. A homothetic transformation is also a projective action on M involving a constant scale change of s and satisfying
~_~g = kg,
or equivalently ~(a; b) = ~1 kgab,
kER
This leads to the conservation of ~a'~a on null geodesics "~. A conformal motion is not a projective action in general. It is an angle-
PROJECTIVE DIFFERENTIAL GEOMETRY AND PROJECTIVE ACTIONS
927
Table 1. Distinguished Group Actions on M and Associated Geodesic First Integrals Group Action
Defining Equation
Geodesic First Integral .a a'r
*Motion (isometry) M
~ ~gab = O, i.e.,
*Homothetic motion HM
, ~ g a b = kgab, k ~ R , i.e.,
(1/2)kgabg/a~ b
~(a;b) = ( 1 / 2 ) k g a b
~a'5,a (null geodesics only)
*Affine collineation AC
s ~F~c = 0, i.e.,
h a b ~ a ~ 'b
*Projective collineation PC
~ F b ea _ - 6~r , e + 6 ac ~ , b , i.e.,
~(a;b) = 0
hab;c = 0 (hat) _ 4 ~ g a b ) ' i , a ~ b
hab;c = 2gab~P ,c + g a c ~ ,b + gcb~',a
*Special projective collineation SPC
As for PC, plus
As for PC, plus
~ ;ab = 0
~ ,aT a
Curvature collineation CC
.~R~c d = 0
Special curvature collineation SCC
.~ ~R ged = 0 and ( " ~ P bae ) ; d = O,
Conformal motion CM
~.~gab = 2q)gab
Special conformal motion SCM
As for CM, plus
Conformal collineation CONFC
~.~Cgc d = 0, i.e.,
Special conformal collineation SCONFC
As for CONFC plus . ~ F g e = 0, i.e.,
-
i.e., hab:c d = 0
i~ ;e T.a "g. t~3". c nab (h a, .b a) ,b 7
aY (nullgeodesics only)
As for CM, plus 4~ .a
C);ab = 0
,aT
(hab - 2Kgab)~a { b
hab;c = 2K, cgab
As for CONFC plus .a K,a'~
~:;ab = 0
preserving t r a n s f o r m a t i o n o f M satisfying
p.~g = ~g,
r ~:~(M)
This is also t h e n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n t h a t a n y null c o n g r u e n c e rem a i n null u n d e r t h e t r a n s f o r m a t i o n . So in a sense it is a p r o j e c t i v e a c t i o n w h e n
928
PRINCE AND CRAMPIN
the distinguished curves on M are the null curves (rather than the geodesics). However, on the basis that projective actions should lead to universal geodesic conservation laws, conformal actions are quite spurious. The indication is that any attempt to extend null geodesic conservation laws to all geodesics should be based on homothetic motions rather than conformal ones. w
The Hamilton-Cartan Formalism for Lagrangian Systems
This section first develops the idea of a second-order, time-dependent ordinary differential equation as a vector field on R X TN. Then we look at the case where this differential equation is the Euler-Lagrange equation, that is, the vector field is the Euler field. The so-called Cartan form plays a key role in setting up the Euler field (and also in investigations of the symmetry-conservation law duality, which is the subject of our second paper). We have borrowed extensively from the theory of autonomous differential equations (see, for example, Abraham and Marsden [16] and Bishop and Goldberg [ 17 ] ), the Hamiltonian formulations of symmetry (Sternberg [ 18 ] and Goldschmidt and Sternberg [19]), and the current literature in Lagrangian mechanics. In fact the Lagrangian treatment of mechanical and related systems is experiencing a rather belated renaissance following the rapid development in Hamiltonian mechanics. (See Sarlet and Cantrijn [20], and recent papers by Hermann [21], Crampin [22], and Crampin, Prince, and Thompson [23] .) Our formulation here is unashamedly Lagrangian and, unlike most treatments, explicitly includes "time" dependence. In this section the manifold N (of dimension n) will be regarded as the configuration space of a mechanical system described by a Lagrangian L. 4.1. Time-Dependent Systems. A classical trajectory 3': R ~ N , parametrized by time, defines a curve in R • N by t -+ (t, 7(t)), which is usefully thought of as a section of the fibration R • N ~ R, or as the graph of 3'- By adjoining to each point (t, 7(t)) in R • N t h e tangent vector 7(t) to 3` one obtains a curve t ~ 3`(t, 3`(0, ~(t)) in R • TN = E. We call this the natural lift of the curve 3` to E. If E is thought of as the bundle of 1-jets of sections of the fibration R • N-+ R then the natural lift of 3' is just the 1-jet of the section it defines. I f Z E ~(E) and ~"is an integral curve of Z then we call 9 o ~', the curve in R • N obtained by projecting f, a (parametrized) base integral curve of Z. In general a base integral curve will not be a section o f R • N-+ R; and even when it is, the original integral curve will not be the 1-jet of its base integral curve. A vector field on E whose integral curves do all have both these properties is called a second-order differential equation field:
PROJECTIVE DIFFERENTIAL GEOMETRY AND PROJECTIVE ACTIONS 929 4.1.1. D e f i n i t i o n . F r ~(E) is a (time-dependent) second-order differential equation field if its integral curves are all natural lifts of curves in N to E, that is, if its base integral curves are all sections of R X N--> R and if its integral curves are their 1-jets.
Second-order differential equation fields may be characterized in terms of the differential of the projection r as follows: 4.1.2. P r o p o s i t i o n . F r ~(E) is a second-order differential equation field if and only if, for all (t, q, v) E E
a r,F(t,q,v ) = ~ + V In local coordinates a second-order differential equation field takes the form
p=~+v k
+fk av k
where the f g are.(in general) functions of all 2n + 1 variables. The integral curves of F are parametrized by t and are solutions of the differential equations Ok =Ug,
~k = f k ( t , q , v )
that is to say, the system of second-order differential equations (in the ordinary sense)
/i k =Yk(t, q, 4) Conversely, any system of second-order differential equations determines such a vector field. We can also characterize a second-order differential equation field E by the conditions (F, d q k - v k d t ) = O,
(F, dr) = 1
A particular second-order differential equation field satisfies, in addition, (F, dv k - f k dr) = 0
This suggests the use of the following natural dual local bases of vector fields and 1-forms on E: F, bq k by k"
and
(dr, dq k _ ok dt, dv k _ f k dt}
The 1-forms d q k - v k d t (which are part of this basis of 1-forms for any secondorder differential equation field), and any linear combinations of them over ~=(E), are called contact 1-forms. The distinctive property of the contact 1-forms
930
PRINCE AND CRAMPIN
is that they annihilate the tangent vectors to naturally lifted curves in E. We denote the contact 1-form dq x - v k dt by cok and the 1-form dv k - f k dt by ~bk. 4.2. The Cartan Form and the Euler Field. The extremals of the variational problem with regular Lagrangian L are the base integral curves of the Euler-Lagrange equation, now represented by the so-called Euler vector field F. The key result in the Cartan-Hamilton formulation is the following one (see Goldschmidt and Sternberg [19] Section 3). 4.2.1.
Definition,
Given a function L on E, the 1-form 0L = L d t + L,~co k
is called the Cartan 1-form of the (Lagrangian) function L. 4.2.2. Proposition. I f L is a regular Lagrangian (so that the matrix (L,r~t~) is everywhere nonsingular), then there is a unique vector field F on E such that P_ldOL = 0
and
(P, dt) = 1
This vector field is a second-order differential equation field, and the equations satisfied by its integral curves are the Euler-Lagrange equations for L. For any 2-form O the vector fields X satisfying X_IO = 0 are called the characteristic vector fields of O. In an odd-dimensional manifold any 2-form has nonzero characteristic vector fields. The regularity of L is equivalent to the condition that dOL have only one dimension's worth (over 5(E)) of characteristic vector fields. The Euler field P is a characteristic vector field of dOL and is further determined by the normalization condition (In, dt) = 1. It follows that any other characteristic vector field of dOL takes the form hP for some h E ~:(E). The coefficients f k in the expression for the Euler field as a second-order differential equation field are determined by the equation L,l~thf k = L,m - L,~nn vn - L,&t 4.3. Symmetries o f Lagrangian Systems. The geometrization of physical theory allows a powerful scheme for discovering symmetry. The usual approach is to identify the "fundamental geometric object(s)" of the theory and to introduce the concept of symmetry by considering one-parameter group actions on the appropriate manifold which leave these objects "invariant." The obvious choice of fundamental geometric object in the present case is the Euler vector field P. However, since the exterior derivative of the Cartan 1-form (which one might as well call the Cartan 2-form) also plays a fundamental role, its symmetries are also important; they turn out to be a special class of symmetries of P. Lie's approach to symmetry for a second-order equation P was to consider group actions on R X N which mapped the parametrized based integral curves
PROJECTIVE D I F F E R E N T I A L GEOMETRY A N D PROJECTIVE ACTIONS
931
into themselves. In order to marry this idea up with the properties of F and 0 L we must extend, or prolong, the action on R X N to an action on E. Such a prolongation may be defined using the contact 1-forms. We give the construction in terms of a vector field, the infinitesimal generator of a one-parameter group action on R • N. 4.3.1. Proposition. For any X E ~(R X N ) there is a unique vector field X (0 on E such that (i) 7 , X (1) = X (ii) for every contact 1-form a , ~ x ( i ) a is also a contact 1-form Proof. Set X = a(3/at) + ~k(a/aq~), where ~k and ~ are functions o f q and t. Then to satisfy condition (i), X O) must have the form XO )
a at
+
av k
where the functions T/k may depend on all 2n + 1 variables. Now E X (1) (dq k _ V k dt) = d~ k _ ~?k dt - v k do and this will be a linear combination of the basic contact 1-forms COk if and only if 7?k = ( a ~ k Vj + a ~ k ~ Thus X (1) is uniquely determined, and evidently satisfies condition (ii) for any contact 1-form, not just co x . 4.3.2. Definition. For X E ~ ( R • N ) the vector field X O) defined in Proposition 4.3.1 is called the (first) prolongation o f X (to R • TN). The expressions occurring in the formula for r~k are what would classically be called the total time derivatives of ~x and or; one might thus write ~k = ~k _ vk 6
Notice also that we may write n k = r'(~ k) - v ~ r ( o )
for any second-order differential equation F, an expression we shall have reason to generalize below. The necessary and sufficient condition that a vector field X on R X N be the generator o f a one-parameter group of symmetries of the second-order differential equation F, in the sense of Lie, is that x 0) p = hP
for some h E Y(E)
932
PRINCE AND CRAMPIN
where X (1) is the prolongation of X to E. These vector fields are known as Lie or point symmetries. One might expect that ~x(1)P = 0 would suffice to account for such actions; however, stretching of the parametrization of the base integral curves turns out to be a nontrivial requirement so allowance has to be made for this. This scheme may be fitted into the one outlined above by considering all symmetries of F, that is, vector fields Z ~ ~ (E) (not necessarily prolongations) such that~ z F = hP, h E ~(E). These vector fields are known as dynamical symmetries.
4.3.3. Definition. A vector field Z such that gzP=hP,
h E~:(E)
is called a dynamical symmetry of P. I f Z is a prolongation, it is a Lie or point symmetry. The following remarks are in order. (i) The point symmetries form a Lie algebra of (finite) dimension r; in fact r ~< n 2 + 4n + 3. This Lie algebra corresponds to an r-dimensional Lie group of symmetries of the system. This result is often known as "Lie's counting theorem"; see Anderson and Davidson [24]. (ii) The dynamical symmetries do not form a finite-dimensional algebra, and their properties are analogous to those of the point symmetries of a firstorder differential equation; see for example Ince [25[. (iii) I f Z has local expression o(a/Ot) + ~k(a/Oqk) + rlk(O/a v k) then the condition !~z P = hP amounts to h = - r(o) n k = r(~ k) -
vkp(o)
z(f k) = r(~ k ) - fkr(o)
Note that if ~k and o are independent of v, so that Z is projectable onto a vector field X on R • N, then the second of these equations shows that Z is in fact the prolongation of X. (v) It is sometimes convenient to consider vector fields Z which satisfy just the condition which generalizes the prolongation condition, r/k = I'(~ e) - v kF(o). It is clear from the explicit calculation of ~z P that this is the necessary and sufficient condition for s F to be a multiple of a second-order differential equation, in general different from F itself. The multiplier is - F(a), as before. Given any second-order differential equation I" we call a vector field Z for which ffzP is a multiple of second-order differential equation by some element o f ~ ( E ) a variation of F. The one-parameter group generated by a variation of F maps
PROJECTIVE DIFFERENTIAL GEOMETRY AND PROJECTIVE ACTIONS 933 the integral curves of P into curves which, up to first order in the group parameter and to change in parametrization of the curves, are natural lifts of curves in N. A prolongation is a variation of every second-order differential equation, and only prolongations have this property: see, for example, Sarlet and Cantrijn [20]. The definition and remarks above apply to any second-order differential equation field. When dealing with an Euler field, one may also consider the symmetries of the Cartan 2-form. 4.3.4.
A vector field Z E ~ ( E ) is called locally Caftan if
Definition.
~-Z dOL = 0
and globally Cartan if for some f@ ~(E) ~ zOr = d f (Gotdschmidt and Sternberg [i 9] call these local and global Hamiltonian vector fields, but since we are dealing with Lagrangian theory and because this name may lead to some confusion with the Hamiltonian vector fields of symplectic geometry, we have adopted the name Caftan.) For a globally Caftan vector fieldS?_zOL is exact, whereas for a merely locally Cartan field it is merely closed. Alternatively one may say that for a locally Cartan vector field, Z, Z_JdOL is closed, and for a globally Cartan field it is exact: in fact in the latter case Z_JdOL = dF 4.3.5. Proposition. cal symmetries of F.
where F = f - (Z, OL) and ~ zOL = df Both local and global Cartan vector fields are dynami-
Proof. Suppose that $-z dOL = O. Then (•z F ) A dOL =~ z (F_JdOL ) - I~J~z dOL = 0
and SO~zF is a characteristic vector field of dOL, whence ~ z F = hV
for some h ~ Y(E)
The converse to this proposition does not hold. The discussion of dynamical symmetries which are not Caftan is taken up below (Theorem 4.3.8). 4.3.6. Proposition. The set of locally Cartan vector fields forms a Lie algebra e . The set of globally Caftan vector fields is an ideal o f e . Proof. If Y, Z are locally Cartan then ~-[y, zldOL = [~.y,~-z]dO L = 0
934
PRINCE AND CRAMPIN
Furthermore, if Z is globally Cartan, with Z J dOL = dF, then [ Y , Z ] ddOL : P g ( Z - J d O L ) : d ( Y ( F ) )
so that [Y, Z] is also globally Cartan. A global Caftan vector field which is also a point symmetry generates conserved quantities in the manner familiar from Noether's theorem, as we will show in our second paper. Accordingly, we call such vector fields Noether symmetries of F; and it is convenient to include local as well as global Cartan fields in this definition. 4.3.7. Definition. A Cartan field which is also a point symmetry is called a Noether symmetry of F.
There remains the question of the transformation properties of dOL under non-Caftan dynamical symmetries. Under certain conditions, such a symmetry generates a new Lagrangian for F. 4.3.8.
Theorem.
(Prince [26]) Suppose that
~q k +~
~v k
is a dynamical symmetry of the Euler field F of the regular Lagrangian L; and suppose further that L,~ei(~n,& - v n o , & ) = L , ~ n ~ ( ~ n , ~ - v n a , ~ )
Then there is a function K such that LzdOL = dO K
and
F_JdOK = 0
(Thus K is an alternative Lagrangian for F, though it may not be regular.) The condition is necessary and sufficient. Proof. The property LzdO L = dOK is equivalent to the local existence of a function F such that Z-IdOL = d F + OK
Using the coordinate expression OK = K d t + K,~co k
one finds that ZAdO L = ((L,krh - L , m ~ ) ( Z , COk ) + L,l~rh ( Z , Ok )} corn _ L,t~/n (Z, a) k ) (orn d F = F ( F ) d t + F , k co k + F , ~ O k
PROJECTIVE D I F F E R E N T I A L GEOMETRY AND PROJECTIVE ACTIONS
935
whence F(F) = - K, F,k = - K ~
F, tl =-L,~i,:n(Z,w m ) + (L,rn~ -
L,krh) (Z, (.~m) + L,llrh (Z,
0m
)
These equations, surprisingly, require only the conditions
F,~m =F,m~ for their integrability; and these are the conditions stated in the theorem. We make several remarks on this result. (i) The condition is also equivalent to the local existence o f f C g(E) such that ~ zOL = OK + dr, and
f=f- K = Z(L) + Lr(o) - r(f)
(ii) The condition of the theorem is always satisfied when n = 1, and when Z is a prolonged point symmetry. (iii) It is not possible to renormalize Z by Z -+ Z + hF so that the condition is satisfied. In fact the symmetries of this class all lead to the same equivalent Lagrangian K. In particular an element of this class w i t h ~ z F = 0 has the associated integrability condition L , ~ ~n,r h = L,/n,i ~n,~ (iv) Sarlet [27] has recently shown that every pair of equivalent Lagrangians leads to an equivalence class of dynamical symmetries in this way. (v) A geometric interpretation of the result of the time-independent case is given in Crampin [22]. w
Projective Differential Geometry on R • TM
Any symmetric connection on the manifold M provides a second-order differential equation field on R X TM = P whose integral curves project onto the geodesics of the connection in M. This second-order differential equation is called the geodesic spray of the connection and is given in local coordinates by F=--+ua--3s Ox a
FZcu~u e Ou a
936
PRINCE
AND
CRAMPIN
where the P~c are the connection coefficients, and the geodesics 7: R ~ M satisfy ~a
"b "c
a]a+lb~7 Y =0 the overdot indicating differentiation with respect to the affine parameter s. We will apply the methods of Section 4 to the study of geodesic sprays, concentrating of course on the case of the spray of a metric connection. Before proceeding, however, it is worthwhile to introduce new local bases for {~(P) and ~*(P).
5.1. Horizontal VectorFieMs. Given any curve p i n M a n d vector field V along it, there is a corresponding curve s ~ (s, p(s), V(s)) in P. When Vis parallel along P the corresponding curve in P is said to be horizontal. The necessary and sufficient condition for this is that the tangent vector to the curve in P should have the form ~/~s + ~aH a where the vector fields H a on P are given by H a := ~x a - rabcU C Ou b
and the ~a are functions along O. The vector fields H a thus define the parallel transport of the connection in terms appropriate to P. They are called the basic horizontal vector fields of the connection for the local coordinate system. The vector fields of the form ~a(~/~ua) a r e said to be vertical, and the vector fields Va = O/Ou a form a local basis of vertical vector fields. Then {P, Ha, Va} is a local basis of vector fields on P, adapted to the connection. Note that P = O/Os + uaHa ; the linear independence of these vector fields is thus clear. The dual basis for 1-forms is .[ds, coa, 0 a} where coa = dx a - u a d s are contact 1-forms as before, and
oa
:=
dua + I'Ve uC dxb
The following properties of the basis vector fields are important:
[Ha,Hb ] = - R&~ua Vc [V~,Hb]
= - P acb Ve
[va, vb] = 0 From these formulas the brackets of P and H a or Va may easily be found, and the exterior derivatives of the 1 -form basis calculated. It is often more convenient to express vector fields and 1-forms in terms of this new basis rather than the natural basis used in Section 4. For example, if X = ~a(o/oxa) is a vector field on R • M (with no D/Os component) then its prolongation to P, when written in terms o f Ha and Va, is X (1) ~- ~aHa + (ub~a,b + r~c~buC)Va
PROJECTIVE D I F F E R E N T I A L G E O M E T R Y AND PROJECTIVE ACTIONS
937
which one might write, in an obvious notation, x(l) =
~aHa + Vu ~a Va
In these definitions we have adapted to P ideas of horizontal vector fields on TM described, for example, by Yano and Ishihara [28] and in a more general context by Crampin [22]. The appropriate generalizations for an arbitrary second-order differential equation field are given by Crampin et al. [23]. 5.2. The Lagrangian Structure o f the Geodesic Equation. We now assume that the connection is that determined by a pseudo-Riemannian metric g on the spacetime M. The geodesic spray is then the Euler field of the Lagrangian L given by L ( s , x , u ) = l gx(U,U)= fgab
aub
The Cartan form 0 (we drop the distinguishing suffix since we deal with just the one Lagrangian now) is given in local coordinates by 0 =Lds +gabUbCo a and its exterior derivative is dO = gabO a A O~b It is clear that P = O/as + ualta is indeed the Euler field for this Cartan form. 5.3. Projective Actions on P. The dynamical symmetries of the geodesic spray F will be called projective actions on P. The vector field Z E ~(P) which generates a projective action therefore satisfies ~ z P = hP,
h E g(P)
I f Z has components (a, ~a, r/a) and (o, Xa, #a) with respect to the coordinate basis and the basis {P, Ha, Va} , respectively, so that Xa = ~a _ uao, then the conditions for it to generate a projective action are ~a = F(U)
+
PgcXbU C
P(/a a) + P~,c#bU c + Rf, caU~Xcu a = 0 and h is given by h = - r (o)
These equations are too intractable to be tackled in full generality. We therefore consider special cases, in particular the projective actions generated by the prolongations of vector fields on R X M, that is to say, the Lie symmetries of the geodesic spray. In this case the equations are completely solvable; Table II gives
938
PRINCE AND CRAMPIN Table I1. Lie Symmetries o f the Geodesic Spray
Symmetry (a)
a ~a__ ~x a
Integrability Conditions
Comparison with Table I
*~P~c = 0
AC
(b) s~ a Ox~-g
~(a;b) = 0
M
(C) ~ - ~s
a;a b = 0
M
(d) ~s ~-s+ ~ a ax a
;ab = 0
~b=
(e)
lks2
2
~-~-+ a ~
~s
S~ ~xa
SPC, SCC
~ a ;ab
~-~gab = kgab
HM
an exhaustive list of these projective actions. (All the functions which appear in it are functions on M if not otherwise specified.) This analysis supercedes that described in Section 3. The parameter transformations of the old theory induced by the transformations appearing in the table have been incorporated in the explicit s dependence of the projective transformations of the new theory. Of course the new theory includes only those transformations of the old theory for which the induced parameter transformations are one-parameter grouplike. It is particularly interesting to note the restriction this places on PCs: according to case (d) of the table the only PCs surviving are those for which the 1-form t) a d x a defining the change in the connection coefficients is exact, and for which moreover ff;a~ = 0. The exactness of Ca d x a iS worth remarking because our calculations do not assume the existence of a metric, but apply to any symmetric connection. We find, therefore, that only SPCs have one-parameter grouplike induced parameter transformations. This contradicts the assertion of Katzin and Levine [29] that all proper PCs may have this property, and casts doubt on results derived from this assertion by Katzin, Levine, and Davis [30]. Again, determination of all Cartan symmetries is infeasible, but those generated by prolongations-the Noether symmetries-may be found. These Noether symmetries are listed in Table III. 5.4. The Geodesic Deviation Equation. In this section we explore the relation between the projective actions on P and the solutions of the equation of
PROJECTIVE D I F F E R E N T I A L GEOMETRY AND PROJECTIVE ACTIONS Table II|.
Noether Symmetries of the Geodesic Spray Integrability Conditions
Symmetry
(a)
0 -3s
(b)
~a
939
-
Comparison with Tables I and II
(c) o f Table II
9~gab = 0
M; (a), (b) of Table 1I
bxa (c)
ks ~Ts + ~a-ox a
~.~gab = kgab
HM;(d) of TablelI
(d)
l ks2 Y + s~ a ~ 2 bs ~x a
~ ; g a b = kgab
HM; (e) of Table II
~a = 7a,a
geodesic deviation (that is, Jacobi fields). As a starting point, we take a standard definition of these fields (see for example Milnor [31 ] ). A vector field X along a geodesic 3' is called a Jacobi field if it satisifies the equation o f geodesic deviation (the Jacobi equation)
In coordinates, X = ~a(~/axa), and the equation becomes D2~ a ds 2 + R ~ c d ~ b ~ c ~ d = 0
One key phase in this definition is "a vector field X along a geodesic." The vector field may, for example, be the restriction of a globally defined vector field o n M . But it need not be so: it need be defined only on a particular geodesic. Note that since the definition of a prolongation may be stated in terms of differentiation along the integral curves of any second-order differential equation, it may be adapted in an obvious way to give a prolongation o f a vector field along a geodesic to one along the integral curve of the spray which projects onto it. 5.4.1. Theorem. Suppose that p is an integral curve of P and that Z is a vector field defined along p such t h a t ~ r Z = 0. Then the vector field 7r,Z along the geodesic 3' = g ~ P is a Jacobi field. Conversely, suppose that 3` is a geodesic and X a Jacobi field along it, then ~ r X 0) = 0 along the integral curve p (s) = (s, 3`(s), ~(s)) o f P. Proof. We set Z = oF + XaHa + 11a V a and compute .~rZ, keeping in mind that along p the u a coordinates are those of p, that is, ~,a if zr o p = 3`. Then
940
PRINCE AND CRAMPIN f r Z = di` + (Xa + Xb~Ci`g~ - Ua)H~ + 0 7 + X ~ b 7"an~aa + ~bi"g~u~)V "
=0 if and only if
d=0,
ua=x a+x~Crg~
~ta + Xc'~b~[dR~cd + g[bi`gctlC = 0 that is,
6=0 DX a [da
~ - -
ds D 2 xa
ds 2
~-R ~ c a ' ~ X c Xa = 0
Now 7r,Z = ((Z, d )
+ "~a(Z, d s ) ) - ~ f i = (X a + ":/%)
8x a
and because Xa are the components of a Jacobi field and since 6 = 0 and ~ is trivially a Jacobi field, it follows that n , Z is a Jacobi field. The converse is straightforward since o = 0 for a Jacobi field. We could, without loss of generality, have taken a = 0 in the proof, because Z -* Z - ( Z , ds) P leaves ~ r Z = 0 unaltered. However, as we have already pointed out, in general we wish to distinguish between Lie and dynamical symmetries and this transformation does not preserve the former. Since from any dynamical symmetry one may obtain a symmetry which commutes with F we have the following. 5.4.2.
Corollary.
9
Every projective action on P is associated with a Jacobi
field for every geodesic. zx
P r o o f . S u p p o s e ~ z P = hi" for some projective action Z on P; t h e n ~ r Z = 0 A where Z = Z - { Z , ds) i". The restriction o f Z to any integral curve p of i" defines
(by projection) a Jacobi field along rro p. 9 Lie symmetries (on R X M) are associated with Jacobi fields by successive prolongation, normalization, and projection. To be specific: 5.4.3.
Corollary.
I f X = a ( ~ / ~ s ) + ~ a ( O / 8 x a ) is a Lie symmetry, that is,
~x(1)i" = - i"(o)I', then
2 - - (U - ,~ao) -~a ox
is a Jacobi field along the geodesic 3".
PROJECTIVE DIFFERENTIAL GEOMETRY AND PROJECTIVE ACTIONS 941 Several remarks are in order. (i) This corollary covers some results of Manoff [32], who had to generalize the Jacobi equation to accommodate PCs, etc. [because ~a(O/3xa) alone is not a Jacobi field]. (ii) In our second paper, on conservation laws, we will show how the traditional results relating Jacobi fields and geodesic first integrals fit into our scheme. (iii) We will not make any attempt here to write down a Jacobi equation on R X M, that is, a description o f the separation of the parametrized base integral curves of P, although such an attempt appears warranted for those Lie symmetries with 3/3s components. (iv) The association of a Jacobi field with a vector field "dragged" along the geodesic spray, so that it may be thought of as connecting neighboring integral curves of the spray, generalizes in a very natural way the usual concept of Jacobi fields as connecting neighboring curves in a geodesic congruence on M.
w
Summary
In this paper we have described the classical background to, and the aspects of Euler-Lagrange theory on evolution space relevant to, the study of projective transformations of geodesics. We have exhibited the Lie symmetries o f the geodesic spray; these are the most general projective actions on P = R X TM arising by prolongation from one-parameter group actions on R X M, and therefore define projective transformations which have the one-parameter group property with respect to the associated transformation of the affine parameter. It is significant that the Lie symmetries include only the special projective collineations (in the terminology o f Katzin and Levine [15 ] ), and not projective collineations in general. We have also determined the Noether symmetries of the geodesic spray, which are those Lie symmetries which also leave the Cartan 2-form invariant. Finally, we have elucidated the relationship between dynamical symmetries and Jacobi fields. In a second paper we will take up the topic of conserved quantities associated with dymamical symmetries of the geodesic spray.
References 1. 2. 3. 4.
Davis, W. R., and Moss, M. K. (1963).Nuovo Cimento, 27, 1492. Davis, W. R., and Moss, M. K. (1965).Nuovo Cirnento, 38, 1531. Davis, W. R., and Moss, M. K. (1965).Nuovo Cimento, 38, 1558. Mclntosh, C. B. G. (1980). In Gravitational Radiation, Collapsed Ob/eets and Exact Solutions (Lecture Notes in Physics No. 124, Springer, Berlin). 5. Benn, I. M. (1982).Ann. Inst. HenriPoinear& XXXVII, 67. 6. Boyer, C. P., Kalnins, E. G., and Miller, Jr., W. (1978). Cornmun. Math. Phys., 59, 285. 7. Benenti, S., and Francaviglia, M. (1980). In General Relativity and Gravitation, One
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