On Killing Tensors and Constants of Motion - Sommers P.

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On Killing tensors and constants of motionPaul SommersCitation: J. Math. Phys. 14, 787 (1973); doi: 10.1063/1.1666395View online: http://dx.doi.org/10.1063/1.1666395View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v14/i6Published by theAmerican Institute of Physics.Additional information on J. Math. Phys.Journal Homepage: http://jmp.aip.org/Journal Information: http://jmp.aip.org/about/about_the_journalTop downloads: http://jmp.aip.org/features/most_downloadedInformation for Authors: http://jmp.aip.org/authors

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On Killing tensors and constants of motionPaul SommersCenter for Relativity Theory, The University of Texas at Austin, Austin, Texas 78712(Received 29 November 1972)Some general properties belonging to constants of motion for geodesics and charged particle orbitsare derived. A constant of motion for geodesics is seen to be a function on the cotangent bundlewhich has vanishing Poisson bracket with the "energy function" determined by the metric tensor.The resulting algebraic structure on the set of constants of motion is closely related to the Liealgebra of Killing tensors. Each constant of motion is shown to provide a family of mappings ofgeodesics into geodesics. Constants of motion for charged particles also possess a Lie algebrastructure. The relationship of Killing tensors to charged particle constants of motion is derived. Thelinear and quadratic constants of motion for charged particle orbits in the charged Kerr metricillustrate the results. Examples of valence 2 Killing tensors are given in an appendix.

1. INTRODUCTIONThe analysis of physical processes in curved spacetimes requires a knowledge of the trajectories of testparticles. The calculation of these trajectories is aformidable task in general, but it is feasible if constantsof motion are known. For geodesics! or charged particle orbits, the norm of the momentum is conserved. Ifthree other constants of motion are known, the momentum may be computed algebraically at each point ofthe trajectory, thereby reducing the orbit problem tofirst order differential equations. A constant of motionwhich is linear in the momentum exists whenever themetric admits a one-parameter group of isometries.The origins of higher degree constants of motion aremore obscure. Some examples are given in AppendixB.The Kerr metric provides a physically important example of a quadratic constant of motion. Because thismetric may be identified as the field of a rotating blackhole, calculations in black hole physics rely heavily onthe properties of geodesics and charged particle orbitsin the Kerr metric. The discovery of the quadraticconstant of motion 2 has made it practicable, for example, to construct astrophysical models of matter accretion onto black holes and to calculate radiation patternsfrom test particles near black holes, without restrictingconsideration to the equatorial plane. This quadraticconstant of motion arises from a Killing tensor ofvalence 2.3Motivated by this example, the present paper discussessome properties which are common to constants ofmotion of al l degrees. Section 2 reviews some aspectsof Killing tensors, including the generalization of theLie algebra of Killing vectors to a Lie algebra of Killingtensors of all valences. In Sec. 3 a constant of motionfo r geodesics is regarded as a function on the cotangentbundle which is constant along the integral curves ofthe geodesic spray. This leads immediately to a Liealgebra of the constants of motion for geodesics. Inaddition, each constant of motion is seen to determinea vector field on the cotangent bundle which commuteswith the geodesic spray, thereby providing a one-parameter family of transformations of geodeSics into geodesics. Section 4 shows that constants of motion forcharged particle orbits enjoy the same properties.Charged particle constants of motion fo r the chargedKerr metric are discussed.2. KILLING TENSORSIn a Riemannian or pseudo-Riemannian space, a Killingtensor is a completely symmetric tensor Kab . cwhich satisfies the Killing equation

787 J. Math. Phys., Vol. 14, No.6, June 1973

(2.1)where Va denotes co variant differentiation. A constantof motion fo r geodesics is associated with any suchKilling tensor. To be explicit, suppose P is the covariant form of the tangent to a geodesic congruence,so that PmVmp a = O. Then Kab . . cP;P b" p c is constant along the geodesics:Pmvm(Kab . cPaPb " 'P c )

= PmPaPb Pc V (mKab . c) = O. (2.2)The metric tensor trivially satisfies the Killing equatiDn,in consequence of which the norm of the tangent to ageodesic is conserved. The symmetrized outer productof Killing vectors also satisfies the Killing equation, theconstant of motion being simply the product of thoseassociated with the Killing vectors individually. Todistinguish these trivial cases, a Killing tensor is saidto be reducible if it can be written as a fixed sum ofsymmetrized outer products of lower vale'nce Kill ingtensors and the metric tensor.3 Otherwise it is irre-ducible.The set of Killing tensors on a space enjoys a Lie algebra structure which is a generalization of the Li e algebra of Killing vectors using the Lie bracket as multiplication. One first introduces a Lie algebra on the set ofcompletely symmetric contravariant tensors of allvalences. Le t sa b .. . c and Tab . . .d be symmetric tensors of valence m and n, respectively. Their skew product, pb .. . , is a symmetric tensor of valence m + n - 1and is given by4pb . .. = mSr(b . . corTd f) - nTr(b . do rse . .. ) .Suppressing indices, this will be written (2.3)

p = [S, T]. (2.4)The product, so defined, is antisymmetric in Sand T, islinear in each slot, is unchanged if ordinary partialderivatives are replaced by covariant derivatives, reduces to the Lie derivative of T along S if S is a vectorfield, satisfies the Jacobi identity and the followingLeibnitz rule: I f T n V denotes the symmetrized outerproduct of tensors T and V, then

[S,Tn V]=[S,T]n V+[S,V]nT. (2.5)Geroch5 ha s pointed out that a Lie algebra of Killingtensors can then be defined as the subalgebra of symmetric tensors Ka b c which commute with the metrictensor gab,

Copyright 1973 by the American Institute of Physics 787



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788 Paul Sommers: On Killing tensors and constants of mQtion

rK,g] = O. (2.6)This equation is identical to the Killing equation whencovariant differentiation is used in the product definition.One easily verifies that this subset is closed under theoperations n, [ , ], and addition of two tensors of equalvalence.3. CONSTANTS OF MOTION FOR GEODESICSA function F, which depends both on position in a manifold M and on a vector P" in the cotangent space at thepoint of M, should properly be regarded as a function onthe cotangent bundle, T*M, of M. Since the cotangentbundle to any manifold is endowed with a canonicalHamiltonian symplectic structure nail [cf. Appendix A],indices 6 of tensors on T*M will be lowered via naB andraised vi a its inverse, nail. In particular, the gradientof a differentiable function F may be regarded as avector field,

(3.1)This field may then ac t on another differentiable functionG,

(3.2)the result being the Poisson bracket of F and G. OnT*M, then, scalar functions may be combined by addition, multiplication, or Poisson bracket multiplication.A constant of motion fo r geodesics is a scalar functionK on T*M which is constant along th e curves of T* Mobtained by lifting geodesics from M. Because a uniquegeodesic is associated with a particular covariant vecto r at a particular point in M, th e lifted geodesics forma simple congruence of curves on T*M. The geodesicspraY,ga is the vector field tangent to this congruence.As shown in Appendix A,ga is given by

go. = a l l o ~ , (3.3)where g is the energy function l

(3.4)The condition that K be a constant of motion fo r geodesics is therefore

(3.5)Constants of motion are the functions which commutewith the energy function g. From the properties of thePoisson bracket it follows that the sum, product, orPoisson bracket of two constants of motion is again aconstant of motion.The bracket operation for symmetric tensors, as definedin Sec. 1, is closely related to the Poisson bracket. Afunction on T* M ,

(3.6)ma y be associated with any contravariant symmetrictensor Fa b C on M. I f F and G are the functionsassociated, respectively, with Fab . C and Gab . . d,then the tensor bracket operation yields a new symmetrictensor whose associated scalar function is [F , G]p. Inthe se t of constants of motion, those which have thesimple form Kab cp aP b' . Pc thereby constitute asubset which is closed under the Poisson bracket operation.

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Another important property of constants of motionfollows immediately from these considerations. I f K isa constant of motion, then the geodesic spray is Liedragged along the gradient vector field Ka:(3.7)

This follows from the fact that, since n all gil is a gradient and Or nail] = 0, then gn all = O. Using thiS, to gether with \h e hypothesis that gao cf\. == g K = 0, oneobtains0= os(gK) = gOIlK = g(naIlKa) = nasgKa. (3.8)Because nas is of maximal rank, this implies gKa =- K g a = O. In this waY,a constant of motion K movesgeodesics onto geodesics.In a four-dimensi onal space, suppose three independentcommuting constants of motion, K, L,M, are known. OnT*M, since gao aK, Ka o ag , Kaocl-, etc., all vanish, thevector fields ga, Ka, La, Ma are tangent to the 4-surfaces of constantg,K,L,M. By the argument of thepreceding paragraph,ga, Ka, La, Ma al l commute asvector fields and, therefore, ma y serve as a basis onthese 4-surfaces. Functions k, 1, m may be chosen suchthat Ka o 0.= % k , Lcxo a = 0/01, Mcxo a = %m , andsuch that the points of a lifted geodesic are given byspecifying constant values for g,K, L,M,k , l, m.4. CHARGED PARTICLE ORBITSTh e discussion of the preceding section generalizes tothe case of charged particles mOving in a Maxwell field.In four-dimensional space-times with physical Maxwellfield, the charged particle orbits are determined by th eLorentz equation of motion:

(4.1)where q is the charge of the particle and Fa b is th e Maxwell field. The orbits satisfying the Lorentz equationfo r a fixed value of q constitute a Single congruence ofcurves when lifted to T*M. The vector field tangent tothe congruence is given by

(4.2)In this equaUon,g is the energy function constructedfrom the metric as before, and n as[q] is the symplectic structure appropriate to the charge q [cf. AppendixA]. Since al l the properties of the canonical Hamiltonianstructure which were used in the discussion of geodesics are also properties of n as[q], it follows that constants of motion fo r particles of charge q possess th esame algebraic structure.The relationship of Killing tensors to constants ofmotion fo r charged particle orbits is slightly more involved than in the case of geodesics. Suppose a functionK on T*M is of the form

(4.3)o 1 2for symmetric tensor fields K ,Ka,Kab,'" on M. When

the geodesic spray gCX [in the form given by Eq. (A2) ofAppendix A] acts on K, term by term, each resultingterm is of a different degree in P" and must thereforevanish by itself i f K is to be a constant of motion fo ral l geodesics. The individual terms then show that eacho 1 2of the tensors K, K", K" b, . . , must be a Killing tensor.When gcx[q] [in th e form given by Eq. (A3) of Appendix A]



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789 Paul Sommers: On Killing tensors and constants of motion

is applied to K, however, the result vanishes for arbitrary Pa if and only if,8 fo r each value of n,n (n +)v(aKb",c) + (n + 1)qFm (a K b",c)m=: 0, (4,4)

Two special cases, which occur in the charged Kerrmetric, deserve mention, Suppose K is linear in Pa sothat o 1K = K + Kap a' (4,5)The conditions that g '1q]oaK = : 0, in this case, are

V (aKb) = : 0, (4,6)and

(4.7))Therefore Ka must be a Killing vector, and, uSing theMaxwell equation O[aFbc] = 0, the Eq. (4. 7) is seen tobe equivalent 9 to the condition

(4.8)The two Killing vectors of the charged Kerr metric inthis way yield coostants of motion fo r particles of arbitrary charge.The Kerr spacetime also admits a quadratic constantof motion fo r charged particles,2.10

(4.9)2where Kab is the irreducible Killing tensor.3 The condi-tions (4.4) in this case yield the two relations

(4. 10)and

(4.11)The first equation is the Killing equation. Exploitingthe antisymmetry of F ab' the second relation is easilyverified by writing the Killing tensor of the Kerr metricin the form 10

(4. 12)APPENDIX A: HAMILTONIAN STRUCTURES AN D

LIFTED ORBITS ON PMThe canonical Hamiltonian structure on the cotangentbundle, T*M, may be defined invariantly by means ofcertain bundle projection maps. Let a be a vector ata point of T*M, so a may be regarded as a point inTT*M, the tangent bundle to T*M. The bundle projectionh from TT*M to T*M maps into the point of T*M towhich it is attached. This point may be regarded as acovector p a at the point of M under the fiber to which

a is attached. At the same time, the projection 11 fromT*M to M determines a differential map 1f * fromTT*M to TM which maps into a contravariant vectora at the point of M under the fiber to which a isattached. The scalar ap a is naturally determined inthis way from a. Doing this for each vector a on

T*M defines a canonical1-form (J a on T*M with values(J a = a' The curl of (J a is the canonical Hamiltonian symplectic structure: 11(A1)

The s)lmplectic structures 0 C


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790 Paul Sommers: On Killing tensors and constants of motion

(3) An affine collineation is a vector field a satisfying t rg c = O. Using the identity

(B3)it is seen that V ( b V c ~ a ) == 0 i f is an affine collineation.In that case V b) is a Killing tensor.A projective collineation, 7] a, satisfies

for some scalar field 1/>. The same identity (B3) thenshows that the tensor

satisfies the Killing equation when 7]a is a projectivecollineation. 13

(B4)

(B5)

(4) In four-dimensional space-time, each vacuum solution of the Einstein field equation for which the Weyltensor is of type {2, 2} admits a trace-free conformalKilling tensor3 Pab

which satisfies the equation(B6)

Except in the cases of the C-metric and it s rotating/feneralization due to Kinnersley,14 the divergence3vmPcm is the gradient of a scalar a,15 Then the tensorKab:== P ab - ag ab (B7)

is a Killing tensor. The result holds equally well forthe charged versions of these metrics.

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ACKNOWLEDGMENTS1 thank R. Geroch, L. Hughston, R. Matzner, J. Rosenman,A. SChild, L. Smarr, M. Walker, and J. York for helpfuldiscussions. I thank The University of Texas at Dallasand Max Planck Institute fo r their hospitality.

*This work supported in part by National Science Foundation GrantGP-34639X.I In this paper "geodesic" will mean "affinely parametrized geodesic."2B. Carter, Phys. Rev. 174, 1559 (1968).3M. Walker andR. Penrose, Commun. Math. Phys. 18,265 (1970).4A. Nijenhuis, Idag. Math. 17, 390 (1955).sR. Geroch, J. Math. Phys. 11, 1955 (1970).6Abstract indices for tensors on T* M will be chosen from the Greekalphabet to distinguish tensors on T *M from tensors on M.7Since the energy function is a function on T*M, it should rigorouslybe given by g: (l/2)(gab o1T )PaPb where 1T is the bundle projection. Theprojection map will be omitted from such expressions for the sake ofcompact equations since there is little danger of confusion.8The relationship (4.4) was deduced previously by L. Hughston (privatecommunication) using a different approach.9M. Walker, Ph.D. thesis, Birkbeck College, 1969.1OL. Hughston, R. Penrose, P. Sommers, and M. Walker, Commun.Math. Phys. 27, 303 (1972).llR. Bishop and S. Goldberg, Tensor Analysis on Manifolds (MacMillan,New York, 1968).12J. Schouten, Ricci Calculus (Springer-Verlag, Berlin, 1954).13W. Davis, M. Moss, and J. York, Nuovo Cimento B 65, 19 (1970).14W. Kinnersley, 1. Math. Phys. 10,1195 (1969).I'L. Hughston and P. Sommers, "Spacet imes with Killing tensors",Commun. Math. Phys. (to appear).

On Killing Tensors and Constants of Motion - Sommers P.

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