Angular momentum and Killing potentials

Transcription

1 Angular momentum and Killing potentials E. N. Glass a) Physics Department, University of Michigan, Ann Arbor, Michigan 4809 Received 6 April 995; accepted for publication September 995 When the Penrose Goldberg PG superpotential is used to compute the angular momentum of an axial symmetry, the Killing potential Q for that symmetry is needed. Killing potentials used in the PG superpotential must satisfy Penrose s equation. It is proved for the Schwarzschild and Kerr solutions that the Penrose equation does not admit a Q at finite r and therefore the PG superpotential can only be used to compute angular momentum asymptotically. 996 American Institute of Physics. S I. INTRODUCTION In this work computing angular momentum with the use of Killing potentials is studied for the Schwarzschild and Kerr solutions. Killing potentials are bivectors Q whose divergence yields a Killing vector. Both solutions have explicit rotational Killing symmetries, spherical for Schwarzschild and axial for Kerr, and we have obtained an axial Killing potential Q for both solutions. We expected to use that Q in the Penrose Goldberg PG superpotential to compute angular momentum in the same way that Q t has been previously used to compute mass 2 and found, to our surprise, that this was not possible. Killing potentials used in the PG superpotential must satisfy Penrose s equation 3 P : Q Q ) g Q ] ; 0 such that Q is a Killing vector. Penrose showed that ten independent Q exist in Minkowski space, but there can be no solutions in a general space time which has no Killing symmetries. For Penrose s quasi-local mass integral we exhibit, in the following section, a Killing potential for the Kerr spacetime which satisfies and yields a quasi-local Kerr mass. Unfortunately, one cannot use the PG superpotential to compute quasi-local angular momentum and so this work has a negative result. It is proved for the Schwarzschild and Kerr solutions that the Penrose equation does not admit a Q at finite r and thus the PG superpotential cannot be used to compute angular momentum at finite r. A Newman Penrose null tetrad for the Kerr solution is given in Appendix A together with the details of an anti-self-dual bivector basis. Bivector components of the Penrose equation are presented in Appendix B. The conformal Penrose equation is given in Appendix C. Sign conventions used here are 2A ;[ ] A R, and R R. II. KILLING POTENTIALS For Killing vector k there is an antisymmetric Killing potential Q such that k 3 Q. It is the Killing potential which is the core of the PG superpotential for computing conserved Noether quantities such as mass and angular momentum. The PG superpotential is U g 2G Q, 2 a Permanent address: Physics Department, University of Windsor, Windsor, Ontario N9B 3P4, Canada /96/37()/42/9/$6.00 J. Math. Phys. 37 (), January American Institute of Physics 42

2 422 E. N. Glass: Angular momentum and Killing potentials where G is the negative right and left dual of the Riemann tensor. In order for U gg k, it is necessary that the Killing potential Q satisfy the Penrose equation. The Kerr solution has two Killing vectors, stationary k (t) and axial k, and the metric, in Boyer Lindquist coordinates, is given by g Kerr dx dx dt 2 / dr 2 2a sin 2 dtd d 2 sin 2 2 a 2 sin 2 d 2, 3 where R r ia cos, RR, r 2 a 2 2mr, and 2mr/. The Killing potential for is k (t) Q t 2 RM R M. 4 Here M is an anti-self-dual bivector, M* im, given in terms of Newman Penrose null vectors in Appendix A. One-third the divergence of Eq. 4 yields the stationary Killing vector k t n /2 l ia sin /& R m Rm. 5 Direct substitution of Q (t) in Eq. verifies that Q (t) satisfies the Penrose equation. One can now use the stationary Killing potential with the PG superpotential to compute the mass 2 of the Kerr source: M S 2 6 gc S 2 Q t ds 6 where S 2 is a closed t const, r const two-surface. The result is m for any r beyond the outer event horizon. An axial Killing potential for the Kerr solution is given by Q ar sin 2 2 r 2 3a 2 cos 2, Q Q M Q 2 V c.c., ir sin Q 2 &R r2 3a 2 cos 2, 7 and one-third the divergence of Q yields the axial Killing vector k a sin 2 n 2 l i r2 a 2 sin R m Rm. 8 & When the Kerr rotation parameter is set to zero, one obtains the Schwarzschild results Q ir2 sin & V c.c., 9 k ir sin & m c.c. 0 Neither the Q for Kerr nor the Q for Schwarzschild satisfy the Penrose equation.

3 E. N. Glass: Angular momentum and Killing potentials 423 III. NO AXIAL PENROSE SOLUTION We will show for the Schwarzschild solution and the Kerr solution that the Penrose equation does not allow an axial Killing potential at finite r. Penrose s equation, 3 A (A W BC) 0 for symmetric spinor W BC equivalent to the antisymmetric Killing potential Q, was used in linearized theory where Penrose 4 showed existence of ten independent Killing potentials, one for each Minkowski Killing vector. In Goldberg s generalization to a fully curved metric there is no discussion of the existence of solutions of the Penrose equation at finite r. We know that a solution exists for Q (t). It is given in Eq. 4 for the Kerr solution with anti-self-dual components where Q 0 0, Q 2R, Q 2 0, Q Q 0 U Q M Q 2 V c.c. We also know that Penrose obtained asymptotic results for angular momentum J. For axial symmetry k at the conformal boundary he found J 0 for Schwarzschild s solution and J ma for Kerr s, so it is reasonable to expect a Q for use in the PG superpotential at finite r. The argument presented below assumes that Q exists, goes through a long set of equations which are the components of the bivector form of Penrose s equation given in Appendix B, and ends with no possible Q. To integrate the equations it is assumed that Q 0, Q, and Q 2 are independent of t and, i.e., it is assumed that L Q 0 where is a Killing vector that commutes with the Kerr k (t) and k. If this assumption is false, then L Q ( ) X. Penrose s equation with Q 3k can be written as Q Q ] 3k ]. 2 Since the Lie and covariant derivatives commute, the nonzero bivector X must satisfy X X ]. 3 The Kerr and Schwarzschild solutions do not admit a nonzero X at finite r. We investigate the existence of Q for the Schwarzschild solution since the equations are simpler with the Kerr rotation parameter set to zero but the argument can be extended in a straightforward manner to the Kerr solution. The null tetrad and spin coefficients given in Appendix A are used. Penrose s equation B4 has n component L 0 0 r Q 0, 4 with solution Q 0 h( ); h an arbitrary function. The m component is M 0 0 &r Q 0 cot Q 0, 5 with solution Q 0 f (r) sin, f arbitrary. The two separate solutions require Q 0 c 0 sin, c 0 const. 6 Equation B2 has l component N 2 0 r r 2m r Q 2 2mQ 2, 7 with solution Q 2 ( 2m/r)h( ). The m component is

4 424 E. N. Glass: Angular momentum and Killing potentials B 2 0 &r Q 2 cot Q 2, 8 with solution Q 2 f (r) sin. The two solutions for Q 2 require Q 2 c 2 2m/r sin, c 2 const. 9 The n component of B2 is L 2 2B 0, r Q 2 2 r Q 2 & r Q Using Q 2 from 9 we find Q c 2 & 3m/r cos f r. 2 We now have functional forms for Q 0, Q, and Q 2. The Q components are further restricted by using the m component of B2 : &r Q 2 cot Q 2 M 2 2N 0, 2m r r Q r Q 0. Using Q 2 from 9 and Q from 2 we obtain the equation 22 c 2 6&m r 2 cos r f r f No solution is possible unless one chooses c 2 0. Then Q c r. The Q components are now The l component of B4 is Q 0 c 0 sin, Q c r, Q N 0 2M 0, 2 2m r r Q 0 m r 2 Q 0 r 2m r Q 0 & r Q 0. Substituting 24 requires c 0 0. Comparing 24 and one can now see that the only solution possible is the one for Q (t) given above. We have proved that, for the Schwarzschild and Kerr solutions, only the timelike Killing vector k (t) can have a Killing potential that satisfies the Penrose equation at finite r. 25 IV. NULL INFINITY We proceed to solve the Penrose equation at the boundary of Schwarzschild space time. The Schwarzschild solution is given in outgoing null coordinates as g dx dx 2m/r du 2 2 dudr r 2 d 2 sin 2 d We use the null tetrad

5 E. N. Glass: Angular momentum and Killing potentials 425 l dx du, n dx 2 2m/r du dr, m dx r/& d i sin d, and spin coefficients given in Eq. A2 with Kerr rotation parameter a 0. The general equations for a conformal map are given in Appendix C. We choose /r z. OnI, where z 0, the metric is ĝ dx dx 2 dudz d 2 sin 2 d Here the conformal Bondi frame is lˆ dx du, nˆ dx dz, mˆ dx /& d i sin d, with nonzero spin coefficients ˆ cot 2& ˆ. The Penrose equation comprises eight complex equations B2 B4 for Qˆ 0, Qˆ, and Qˆ 2. Three establish finite values for the Qs on the boundary: z Qˆ 0 0, z Qˆ 2 ˆ 2 ˆ Qˆ 0 0, z Qˆ 2 2 ˆ Qˆ 0, where Dˆ z, ˆ u, and on I ( ˆ 2s ˆ ) Z for a spin weight s scalar we use the original definition 5 of edth with spin weight opposite to the helicity of outgoing radiation. Inthe following a zero superscript denotes independence of z, and (Qˆ 0 0,Qˆ 0,Qˆ 2 0 ) have spin weights,0,. The remaining five equations on I are The solutions are u Qˆ 2 0 0, 28a Z p Qˆ 2 0 0, 28b Z p Qˆ u Qˆ 0 0, 28c ZQˆ 0 0 0, 28d 2ZQˆ 0 u Qˆ e Qˆ 2 0 k m Y m, Qˆ 0 2u ZQˆ 2 0 f,, Qˆ u 2 Z 2 Qˆ 2 0 2uZf c m Y m, 29 where k m and c m are complex constants. Here we can go beyond Goldberg and integrate 28e since the Schwarzschild null surfaces are shear-free. The asymptotic Killing vectors are kˆ u Qˆ 0 c.c., kˆ c.c. Qˆ 0 2, kˆ Qˆ The supertranslations of the BMS group have a full function s worth of freedom in Qˆ 0 but at the Schwarzschild boundary f, is restricted to four parameters for ordinary translations and Z f 0. The solution of the Penrose equation for Q (t) is contained above. The nonzero anti-self-dual component of Eq. 4 is Q r/2 or Qˆ 2. This solution coincides with the values k m 0, c m 0, and f, 2. Now lets take the asymptotic solutions found above in 29 and 30 and use them to construct a Killing potential Q. Thus our candidate has the form

6 426 E. N. Glass: Angular momentum and Killing potentials Q r 2 Q 0 2 V c.c. 3 For the Schwarzschild solution we compute the divergence: 3 Q rq 0 2 m r & cot Q 0 2 l c.c. 32 Equating with k ir sin & m c.c. yields i sin Q 0 2 &. The l term in 32 vanishes when the complex conjugate is added. We have constructed the Killing potential which was already given above as Eq. 9. The anti-self-dual components are Q 0 0, Q 0, Q 2 i/& r 2 sin. 33 Of the twelve terms entering the Penrose equation defined in Appendix B, four are nonzero for the components of Eq. 33 : L 2 i&r sin, N 2 i/& 3m r sin, M 2 ir cos, B i/& r sin. Although Q 2 has the r 2 dependence that one expects for an asymptotic solution and the angular dependence dictated by k, the components of Eq. B2, particularly N 2 0, show directly that this Killing potential fails to satisfy the Penrose equation. V. CONCLUSION To find a Killing potential one can write the divergence equation relating k and Q as a three-form relation, one-third the exterior derivative of dual Q equal to the dual of k dx, 3d*Q * k dx, and then integrate if possible. We have seen that not just any Killing potential can be used in the PG superpotential but only one which satisfies Penrose s equation. Although a Q whose divergence yielded the axial Killing vector was presented for the Kerr solution, it could not be used to compute quasi-local angular momentum although asymptotically it yields ma. It has been shown that a Q cannot be found for either the Kerr or Schwarzschild solutions which will satisfy the Penrose equation in curved space and so the PG superpotential cannot be used to compute quasilocal angular momentum. Some interesting questions remain. What are the complete integrability conditions for the Penrose equation? What is the physical reason that no quasi-local Killing potential for rotational symmetry can satisfy the Penrose equation?

7 E. N. Glass: Angular momentum and Killing potentials 427 ACKNOWLEDGMENTS We thank Bob Geroch for questioning the existence of Killing potentials which satisfy the Penrose equation and David Garfinkle for many stimulating discussions. This work was partially supported by a NSERC of Canada grant. APPENDIX A: NULL TETRAD AND BIVECTORS A Newman Penrose tetrad (l,n,m,m ) for the Kerr metric 3 with l and n as principal null vectors is chosen as l r2 a 2 t r a, n 2 r2 a 2 t r a, A m &R ia sin t i sin, where R r ia cos, RR, and r 2 a 2 2mr. The nonzero spin coefficients and Weyl tensor component are, R 2 R, r m 2, ia sin ia sin,, & &R 2 cot 2&R,, 2 m R 3. A2 A basis of anti-self-dual bivectors is given by U 2m n ], M 2l n ] 2m m ], V 2l m ]. A3 Their inner products are U V Ū V 2, M M M M 4, and all others zero. As a basis, they satisfy the completeness relation 2 g i U V V U 2M M, A4 where g g g g g, and 2 is the dual tensor. It is useful to list their covariant derivatives: U 2U a M b, a n l m m, M 2U c 2V b, b n l m m, A5 V 2V a M c, c n l m m. APPENDIX B: THE PENROSE EQUATION Equation, which a Killing potential must satisfy in order to be valid for use in the PG superpotential, can be written in terms of anti self-dual bivectors with the definition Q Q 0 U Q M Q 2 V c.c. B

8 428 E. N. Glass: Angular momentum and Killing potentials Substituting the bivector expansion into provides equations for the components Q 0,Q,Q 2, which can be most simply written with the use of twelve terms: L 0 D 2 Q 0 2 Q, L DQ Q 2 Q 0, L 2 D 2 Q 2 2 Q, N 0 2 Q 0 2 Q, N Q Q 2 Q 0, N 2 2 Q 2 2 Q, M 0 2 Q 0 2 Q, M Q Q 2 Q 0, M 2 2 Q 2 2 Q, B 0 2 Q 0 2 Q, B Q Q 2 Q 0, B 2 2 Q 2 2 Q. Here D l, n, and m. The Penrose equation has the following U, M, and V components, respectively: l 3N 2 n L 2 2B m 3B 2 m M 2 2N 0, l M 2 2N n B 0 2L m 2B L 2 m 2M N 0 0, l N 0 2M n 3L 0 m B 0 2L m 3M 0 0. B2 B3 B4 If Q is to be a Killing potential for k, then its divergence must satisfy 3k l N M 2 n B 0 L m B L 2 m N 0 M c.c. B5 APPENDIX C: THE CONFORMAL PENROSE EQUATION For asymptotically simple space times with future null infinity I we follow Penrose and Rindler 6 case iv to conformally map from the physical metric g to the unphysical metric ĝ : ĝ 2 g C with the spinor basis mapping as ô A o A, î A ı A. Here 0 defines the future null boundary with a null vector tangent to the generators of I. It follows from the map of the spinor basis that the tetrad derivatives transform as The spin coefficients conformally map as Dˆ 2 D, ˆ, ˆ. C2 ˆ 3, ˆ 2, ˆ 2, ˆ, ˆ, ˆ 2 3 D, ˆ 2, ˆ 2, ˆ, ˆ, ˆ, ˆ 2. Since the Killing potential obeys the conformal transformation Qˆ Q, it s anti-self-dual bivector components map as

9 E. N. Glass: Angular momentum and Killing potentials 429 Qˆ 0 Q 0, Qˆ Q, Qˆ 2 2 Q 2. C3 The twelve terms in Appendix B which comprise the components of the Penrose equation map as Lˆ 0 2 L 0, Lˆ L Q 0 2 Q 2 D, Lˆ 2 L 2 2Q 2Q 2 D, Nˆ 0 N 0 2Q 0 2Q, Nˆ N Q Q 2, Nˆ 2 2 N 2, Mˆ 0 N 0, Mˆ M Q 0 Q, Mˆ 2 M 2 2Q 2Q 2, Bˆ 0 B 0 2Q 0 2 2Q 2 D, Bˆ B Q Q 2 D, Bˆ 2 B 2. Finally, by direct substitution of the twelve terms above into Eqs. B2 B4 we find the anti-selfdual components of the Penrose equation conformally transform as This result is confirmed by the conformal maps Bˆ 2 B2, Bˆ 3 B3, Bˆ 4 2 B4. C4 Pˆ 3 P C5 and Û 3 U, Mˆ 2 M, Vˆ V. C6 J. N. Goldberg, Phys. Rev. D 4, E. N. Glass and M. G. Naber, J. Math. Phys. 35, R. Penrose, Proc. R. Soc. London Ser. A 38, R. Penrose and W. Rindler, Spinors and Space Time Cambridge U.P., Cambridge, 986, Vol. 2, Eq E. T. Newman and R. Penrose, J. Math. Phys. 7, R. Penrose and W. Rindler, Spinors and Space Time Cambridge U.P., Cambridge, 986, Vol., p. 358.