A unified treatment of null and spatial infinity in general relativity. I. Universal structure, asymptotic symmetries, and conserved quantities at spatial infinity
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10.One does not expect asymptotic flatness in one regime, by itself, to imply asymptotic flatness in the other. Thus, for example, given a space—time which is asymptotically flat in both regimes, one can redistribute matter such that some neighborhood of l continues to remain empty but such that every neighborhood of spatial infinity contains some matter, thereby destroying the asymptotic flatness in the spatial regime but not in the null. The converse situation can be arranged by using zero rest‐mass fields with appropriate strength and support.
11.B. G. Schmidt and J. Stewart; private communication.
12.The usual strategy to obtain a boosted Cauchy surface from the given one is to move along normal geodesics using a lapse function which resembles the boost lapse function in Minkowski space (see, e.g., Ref. 8). However, since this function diverges “like r,” there is no a priori guarantee that geodesics will not start crossing before one can evolve by a sufficient amount. Other strategies run into similar problems. Although these can be avoided in special cases—e.g., in stationary space—times—the issue is unresolved in the general context.
13.Consider, for example, the statement that the ADM 4‐momentum is invariant under time evolution. Since the very definition of this quantity rests crucially on asymptotic flatness of initial data sets, the statement of invariance has nontrivial content only if one is guaranteed the preservation of asymptotic conditions under evolution.
14.R. Penrose, in Battelle Rencontres, Lectures in Mathematical Physics, edited by C. DeWitt and J. A. Wheeler (Benjamin, New York, 1968).
15.See Ref. 4, or, e.g., S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space—time (Cambridge U.P., London, 1973).
16.For example, in Schwarzschild space—time, the scalar at spatial infinity. Hence, in terms of the rescaled metric diverges at 1°.
17.We thank David Deutch for suggesting this terminology.
18.One normally decomposes Maxwell field into electric and magnetic parts relative to spacelike 3‐surfaces. K, on the other hand, is timelike. Perhaps a better terminology would have been asymptotic pseudoelectric and asymptotic pseudomagnetic fields. The same remark holds for the decomposition of the Weyl tensor.
19.That is, the limit is finite but depends on the direction of approach to Λ. For a precise definition, see Ref. 8 or Appendix A.
20.A. Ashtekar, Lectures on Asymptotics at the University of Chicago.
20.Lecture notes are being prepared for the Springer Series by B. Xanthopoulos.
21.For definitions and detailed discussion, see Appendix A.
22.A somewhat different but more natural definition will appear in Ref. 20; l will be introduced in terms of rather than in terms of l.
23.In the case of Minkowski space, the Einstein cylinder can serve as see, e.g., Ref. 15.
24.More generally, arbitrary matter sources for which the stress‐energy tensor field (with this index structure) on admits a regular direction‐dependent limit at are permissible in the analysis of spatial infinity.
25.Let be the physical metric and and be any two rescaled metrics at Then, and Consistency of these conditions requires that ω be at and
26.In this case the resulting analysis would be essentially equivalent to that in Ref. 1 provided we require, in addition, that admits a regular direction‐dependent limit at (In Ref. 1 the same additional condition is required although it is formulated in terms of electric and magnetic parts of the Weyl tensor relative to a Cauchy surface.)
27.Throughout, we use the term geodesics for curves whose tangent vector satisfies rather than We do not use affine parametrization (w.r.t. the physical metric) because the parameter is ill behaved at
28.Because the rescaled metric is at it follows that —and hence —admits a regular direction‐dependent limit at (see Appendix A). Using this fact and the l’Hopital’s rule, it follows that exists as a direction‐dependent tensor at
29.Note that, given any two curves with the same tangent vector at their tangential components of acceleration at agree w.r.t. one metric in the conformal class if and only if they agree w.r.t. any other metric. Hence, the equivalence relation is well defined.
30.For example, no point of Spi is left invariant by the group action, and, the only group element whose action leaves Spi invariant is the identity.
31.See, e.g., R. L. Bishop and R. J. Crittenden, Geometry of Manifolds (Academic, New York and London, 1964), page 41.
32.Note that, since is covariant and contravariant, these fields do not give rise to a (nondegenerate) metric on Spi.
33.The resulting construction of Spi is the most natural one for discussing physical fields. It will be described in Ref. 20.
34.Thus, the situation is the same as at null infinity; the BMS group can also be characterized as the subgroup of the diffeomorphism group of l which leaves all the universal structure invariant. See, e.g., the sixth paper in Ref. 5.
35.Note, however, that there is an important difference between the BMS group and G: While the center of the BMS group is the identity, the center of G is the one‐parameter family of supertranslations corresponding to the structure group of Spi.
36.This restriction may be weakened to allow charges whose current vector has the property that without affecting the final results.
37.Note that, on K, does not imply that Indeed, by the first of these equations, it follows that there exists a scalar V with and the second then reduces only to Since this last equation is hyperbolic, it admits lots of (regular) solutions.
38.Note that, although the 2‐forms and are both closed (i.e., curl‐free), the integrals are not necessarily zero: Since K admits 2‐sphere cross sections which cannot be contracted continuously to a point, closed 2‐forms on K need not be exact.
39.More precisely, they are invariant under the action of the group G of asymptotic symmetries.
40.See last three papers in Ref. 5 or Ref. 1.
41.The hyperboloid K admits ten—the maximum possible number of—conformal Killing vectors. Furthermore, the Killing form on the Lie algebra is nondegenerate. Hence, one can divide the ten‐dimensional vector space underlying the Lie algebra unambiguously into two complementary subspaces; a six‐dimensional subspace of Killing vectors and a four‐dimensional subspace of “pure” conformal Killing vectors. The former satisfy while the latter, (The “pure” conformal Killing vectors are gradients of functions on K which correspond to translations.)
42.See, e.g., the analysis of Kerr space—times in Appendix C.
43.Although it has been occasionally argued that the differentiability requirements in Ref. 4 are too strong. See, e.g., W. E. Couch and R. J. Torrence, J. Math. Phys. 13, 69 (1973);
43.W. H. Press and J. M. Bardeen, J. Math. Phys. 14, 7 (1973).
43.Stephaney Novak has investigated in detail the case when the rescaled metric fails to be on l (private communication).
44.See, e.g., P. Sommers, “Geometry of Space‐like Infinity,” preprint. It would not be surprising if Sommers’ and the present approach turn out to be completely equivalent. Indeed, Sommers’ hyperboloid “Psi” can, in essence, be constructed by associating, with each equivalence class of geodesics in the physical space—time which are tangential to each other at a boundary point to the given space—time.
45.R. P. Kerr, Phys. Rev. Lett. 11, 237 (1963);
45.R. H. Boyer and R. W. Lindquist, J. Math. Phys. 8, 265 (1967).
46.The choice of these coordinates makes the conformally rescaled metric (rather than ) at We thank Bernd Schmidt and Martin Walker for making available to us their calculations which led to this choice.
47.J. M. Bardeen and W. H. Press, the second paper in Ref. 43.
48.A. I. Janis and E. T. Newman, J. Math. Phys. 6, 902 (1965).
49.That is, there do not appear to exist charts in which the rescaled metric connection admits direction‐dependent limits as one approaches along l.
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