Killing tensors and a new geometric duality

NUCLEAR PHYSICS B ELSEVIER

Nuclear Physics B 472 (1996) 427-446

Killing tensors and a new geometric duality R . H . R i e t d i j k a,1, J.W. v a n H o l t e n b a University of Durham, Durham, UK b NIKHEE Amsterdam, The Netherlands

Received 23 November 1995; accepted 16 April 1996

Abstract

We present a theorem describing a dual relation between the local geometry of a space admitting a symmetric second-rank Killing tensor, and the local geometry of a space with a metric specified by this Killing tensor. The relation can be generalized to spinning spaces, but only at the expense of introducing torsion. This introduces new supersymmetfies in their geometry. Interesting examples in four dimensions include the Kerr-Newman metric of spinning black holes and self-dual TaubNUT.

1. Introduction

The importance of symmetries in the description of physical systems can hardly be overestimated. In the case of dynamical systems in particular, continuous symmetries determine the structure of the algebra of observables by Noether's theorem, giving rise to constants of motion in classical mechanics and quantum numbers labeling stationary states in quantum theory. In a geometrical setting, symmetries are connected with isometries associated with Killing vectors and, more generally, Killing tensors on the configuration space of the system. An example is the motion of a point particle in a space with isometries [ 1 ], which is a physicist's way of studying the geodesic structure of a manifold. Contact with the algebraic approach is made through Lie-derivatives and their commutators. In [ 1,2] such studies were extended to spinning space-times described by supersymmetric 1Address since 1 September 1995: Kon. Shell Expl. and Prod. Lab., PO. Box 60, 2280 AB Rijswijk, The Netherlands. 0550-3213/96/$15.00 Copyright (~) 1996 Elsevier Science B.V. All rights reserved PH S 0 5 5 0 - 3 2 1 3 ( 9 6 ) 0 0 2 0 6 - 4

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extensions of the geodesic motion, and in [3] it was shown that this can give rise to interesting new types of supersymmetry as well. This paper concerns spaces on which there exists a symmetric second-rank Killing tensor, including such interesting cases as the four-dimensional Kerr-Newman spacetime describing spinning and/or charged black holes, and four-dimensional Taub-NUT which describes a gravitational instanton [4] and appears as a low-energy effective action for monopole scattering [5-7]. These second-rank Killing tensors correspond to constants of motion which are quadratic in the momenta and play an important role in the complete solution of the problem of geodesic motion in these spaces [8]. Similar results concerning the monopole with spin and 3-D fermion systems have been presented in [9-12]. The main aim of the paper is to present and illustrate a theorem concerning the reciprocal relation between two local geometries described by metrics which are Killing tensors with respect to one another. In Section 2 the basic theorem is presented. It is illustrated in Section 3 with the four-dimensional examples mentioned above. In Section 4 we extend the discussion to the motion of particles with spin and charge in curved space, including torsion and electro-magnetic background fields. This generalizes results obtained by Tanimoto [ 13]. In Section 5 the relation between Killing tensors and new supersymmetries, representing a certain square root of the associated constants of motion, is explained. The basic ingredient is the existence of so-called Killing-Yano tensors [ 3 ]. The duality theorem is then generalized to include geometries with KillingYano tensors, and it is shown that in general this requires the introduction of torsion. Finally the explicit expressions for the Killing-Yano tensors and the associated torsion tensors for the examples of Kerr-Newman and Taub-NUT are given.

2. Dual geometries Suppose a space (of either Euclidean or Lorentzian signature) with metric admits a second-rank Killing tensor field Ku~ (x), K(v~;a) = O.

g~(x) (1)

Here the semicolon denotes a Riemannian covariant derivative, and the parentheses denote complete symmetrization over the component indices. The equation of motion of a particle on a geodesic is derived from the action

S = f dr ( ½gu,,Jct*.icv).

(2)

The corresponding world-line Hamiltonian is constructed in terms of the inverse (contravariant) metric g*'~,

H = ½g*Z*'Pl*Pv, and the elementary Poisson brackets are

(3)

R.H. Rietdijk, J.W. van Holten/Nuclear Physics B 472 (1996) 427-446

(4)

{x~',p~} = 6~.

The equation of motion for a phase space function F ( x , p ) Poisson brackets with the Hamiltonian,

429

can be computed from the

dF

dr

{F,/-/}.

(5)

From the contravariant components K u~ of the Killing tensor one can construct a constant of motion K, n n K = 1/¢qzu 2"" ruy~"

(6)

Its bracket with the Hamiltonian vanishes precisely because of the condition (1). The constant of motion K generates symmetry transformations on the phase space linear in momentum, {X ~, K } = KU~p~.

(7)

The infinitesimal transformation with parameter ot can also be written in terms of the velocity as ~(ot) x u = ceK~± ~.

(8)

The formal similarity between the constants of motion H and K, and the symmetrical nature of the condition implying the existence of the Killing tensor {H, K} = 0,

(9)

amount to a reciprocal relation between two different models: the model with Hamiltonian H and constant of motion K, and a model with constant of motion H and Hamiltonian K. In the second model H generates a symmetry in the phase space of the system. Thus the relation between the two models has a geometrical interpretation: it implies that if K u~ are the contravariant components of a Killing tensor with respect to the inverse metric g~U, then gU~ must represent a Killing tensor with respect to the inverse metric defined by K u~. A more direct proof of this result can be given purely in terms of geometrical quantities. Given the metric gu~ and the Killing tensor Ku~ satisfying Eq. (1), we identify the contravariant components of K with a new contravariant metric ~u~, ~

---- K u~ = gUaKa~gK~.

(10)

If these components form a nonsingular (d × d)-dimensional matrix 2 , we denote its inverse by guv, ~ a ga~ = 6~. 2 A brief discussion of singular Killing tensors is presented in the appendix.

(11)

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Now we can interpret ~,~ as the metric of another space. Define the associated Riemann-Christoffel connection p u a as usual through the metric postulate /3ag~,~ = 0.

(12)

Then it follows from Eq. (1) that /) is antisymmetric by 0 d definition (93) and furthermore f~a/)~ satisfies the Killing-Yano equation for space I. Hence spaces I and [ are dual to each other and e~a1)~" must satisfy the Killing-Yano equation in space [. The torsion in space [ is found to be defined by •

O

A(i)

¢(t)# ¢(/)v ¢(t)x X

abc = a a

a b

J c

"~vK,

(98)

with At,,~ having only one nonvanishing component, 2r2m 2 sin 0 (r -+- m) 2 "

,40~¢, =

(99)

Notice that ,4~,~K is the same for all spaces [. The conclusion is that there is only one space dual to Taub-NUT space• The spaces [ have the same metric and the same torsion and thus represent the same geometry• Using the expression for the torsion we have checked explicitly that e~a1)~" satisfies the Killing-Yano equation for space [. The corresponding antisymmetric three-tensors (space [, Killing-Yano tensor e~at)~) are defined by :(t)u :(l)x z cabc ( = iJ¢(/)~z a , d lb )d c '-'l~VX,

(10o)

with ?r0~ =

2 r m ( r + 2m) (2r 2 + 4 r m + 5m 2) sin0 3(r q- m) 4

(101)

being the only nonvanishing component of ?~,~, which is the same for all representations [ of dual Taub-NUT space. The duality between Taub-NUT space and dual Taub-NUT space is not complete. The described procedure gives only one Killing-Yano tensor for dual Taub-NWr space. The f~t)~ are not solutions of the Killing-Yano equation for space ] if J ~ I. And although it is easily checked that fa~J)~' satisfies the Killing-Yano equation for space I, also for I 4: J (the covariant derivatives are the same in all spaces 1) there is no duality between the spaces I and Jr because Jt - la ) ~ r ~J Ja > is not antisymmetric for I :~ J. Hence also the e~a/)~' will not automatically be Killing-Yano tensors for space J, J 4= 1. We have checked this explicitly and found that this is indeed not the case. Finally we remark that in space J both ea(1)~, and fa(s)" are Killing-Yano tensors which implies, as was shown in [ 3 ], that in that space the symmetric contraction, (1)~ ea

f~( J ) v

_(l)v¢(J)~ + ea Ja

'

(lO2)

defines a second-rank Killing tensor. For 1 v~ J these tensors are nontrivial and are defined precisely by the conserved quantities K (i) given in (24) for the scalar particle. We conclude that these second-rank Killing tensors are of a different type than the ones

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given by (19) and (22). Although they do define a dual bosonic space, they do not define a dual spinning space, since they are not the square of a Killing-Yano tensor.

Acknowledgements RHR wishes to thank the Engineering and Physical Sciences Research Council for post-doctoral support, and Koos de Vos for very useful and pleasant conversations.

Appendix A. Killing vectors and zero modes The Taub-NUT Killing tensor is nonsingular for all values of the parameter m for which the metric is well defined (m ~ 0). Thus it always has an inverse and a well-defined dual metric. In contrast, in the limit a ~ 0 the Killing tensor of the KerrNewman space-time has two zero-modes. This case happens to be a direct generalization of the magnetic monopole discussed in [ 11 ]: in both examples the Killing tensor is conformal to the standard metric on the 2-sphere ds~s2 ) = dO 2 + sin 2 0 d~b2.

(A.1)

This reflects the spherical symmetry present in both the monopole system and in the Kerr-Newman solution for a = 0 (i.e. the Reissner-Nordstrom black hole). The existence of a Killing tensor in these and other spaces can be understood more generally as a result of the presence of Killing vectors. Suppose we are given a ddimensional manifold with a metric g ~ ( x ) and n Killing vector fields u~r)(x), r = l~...,n,

u(r) (r) = 0. v;~ _ "1- U/z;v

(A.2)

This manifold admits ½n2 special Killing tensors of the form Kix(r,s) u(r)U(s) , (r), (s)~ , = ½ (~--/x --v + u/.~ " v 1 .

(A.3)

With r = s these Killing tensors have rank one, and are singular for any d > 1. For r 4: s they are at most of rank two, and are always singular for d > 2. However, one may take linear combinations with constant coefficients to form new Killing tensors, and if there are sufficiently many of them (n > d) which are linearly independent globally, then there may exist combinations K[c] = ~

Crs u (~) ® u (s),

(A.4)

r,$

with constant coefficients Crs, which are actually nonsingular, and thus have an inverse and define a regular dual metric. The 2-sphere provides a simple example. In spherical coordinates with the standard metric (A.1) there are three Killing vectors corresponding to rotations about three independent axes. Their contravariant components are

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u (1) ~' = (cos ¢, cot 0 sin ¢) , u (2) ~ = ( - sin ¢, cot 0 cos ¢ ) , u (3) ~ = (0, 1).

(A.5)

Each of these Killing vectors defines a singular Killing tensor of rank one, constructed as a quadratic form with itself. Taking a linear combination of these three Killing tensors one can obtain a nonsingular Killing tensor. In particular, one can reobtain the metric of the two-sphere itself by taking Crs = ~rs, 3

E

3

u(r) tZu(r) v = g ~

r=l

~

E

,(r),,,~(r) = g ~ . -~

(A.6)

r=l

This particular nonsingular combination is favored, because it is invariant under the three-dimensional rotations of the Killing vectors among themselves, the unit tensor with components 8rs being invariant under SO(3). Of course this is not the case for the singular individual terms corresponding to fixed r. The key result from the studies of Killing-Yano tensors [ 18,14,3] is that a quite similar construction exists also for Killing tensors like those of the generic Kerr-Newman and Taub-NUT spaces: they can be expressed as a quadratic form of Killing-Yano tensors, which may be viewed alternatively as a set of d generalized Killing vectors f a, a = 1 ..... d, transforming among themselves as a local Lorentz vector,

g =E

Tab fa ® fb.

(A.7)

a

The generalization referred to is that the equation for the Killing-Yano tensor,

O~fv a + Ovf• = 0,

(A.8)

differs from the Killing equation (A.2) only in that it includes the spin connection for the local Lorentz group in the covariant derivative. In this sense it is a gauged version of the Killing equation: the f a are components of a Killing vector modulo a local Lorentz transformation. Note also that in Eq. (A.7) for K only the complete sum is locally SO(d)-invariant, as it is constructed in terms of the local flat-space metric ~ob. Hence only this Killing tensor defines a locally Lorentz-invariant constant of motion. In the singular Reissner-Nordstrom limit a = 0 of the Kerr-Newman geometry, two of the generalized Killing vectors (83) defining the Killing-Yano tensor vanish identically. Like for the monopole, the Killing-Yano tensor then is an antisymmetric matrix of rank two. At the same time, in both cases the symmetric Killing tensors are of the special type discussed above: modulo an r-dependent factor they are given by the same quadratic form in the rotational Killing vectors (A.1), (A.6), corresponding to a twosphere embedded in a 3- or 4-dimensional space, respectively. Thus these Killing tensors can be viewed either as a quadratic form of three Killing vectors, or a quadratic form of two Killing-Yano tensors. Finally we comment on how to extend our construction of dual metrics to include singular Killing tensors. Some basic ideas have already been presented in the context of

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the example in Ref. [ 11 ]. In the general case we proceed as follows: expand the Killing tensor, which is a symmetric d x d matrix, in eigenvectors, d

,~iei @ e i,

K = ~

(A.9)

i=l

with ,~i the eigenvalues (which can be zero), and el,i a complete orthonormal set of eigenvectors (including zero modes). Note that in general these eigenvectors are not Killing vectors, although this might happen in special cases. Note also, that the values of the components Kjz~ depend on the space-time point, and so does the expansion (A.9): both the eigenvectors and the eigenvalues may be functions of the position, and hence of the coordinates x i'. This is in contrast to the expansion (A.4) in Killing vectors, where the coefficients Crs are taken to be constant. In terms of the eigenmode expansion (A.9), we can now define a generalized inverse of the Killing tensor as ~ - 1 = ~-"' 1 ei ® ei '

(A.10)

~7" ai where the primed sum is over nonzero modes only. Alternatively, one may regularize the zero modes of K to finite nonzero eigenvalues, take the inverse and then disregard any dependence on the regulator modes, for example by taking the limit in which the regulators tend to infinity. This procedure is like gauge-fixing to obtain a propagator for gauge fields with a singular classical action, and our definition of the inverse (A.10) corresponds in that terminology to the propagator in the Landau gauge. With this definition it follows that K . k -1 = H--= 1 - ~-'~e (°~ ® e (°),

(A.11)

where e (°) is a collective notation for the zero modes. Clearly H is the projection operator on the space of nonzero modes. In general the covariant components of this projection operator define a metric on a curved hypersurface. In the case of the monopole and the Reissner-Nordstrom black hole we know this is the projection on a two-sphere. Returning to the generic case, on the space of nonzero modes the inverse and the dual metric defined in terms of the Killing tensor K can now be constructed as before, using the metric projected w i t h / / o n the hypersurface as the starting point. Our metric duality is thus seen to be well defined in the space of nonzero modes. Note, however, that in the special examples of the monopole and the Reissner-Nordstrom black hole the projected Killing tensor actually coincides with the metric on the two-sphere. Therefore, in these examples there is no separate nontrivial dual metric. This confirms the conclusion in Ref. [ 11 ] that the Killing tensor and the new supercharge become the ordinary Hamiltonian and supercharge for the motion of a particle restricted to a spherical surface. In other cases, however, genuine nontrivial metric duality may be found in the hypersurface of nonzero modes.

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