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From i ° to the 3+1 description of spatial infinity
1.R. Arnowitt, S. Deser, and C. W. Misner, Phys. Rev. 117, 1595 (1960);
1.R. Arnowitt, S. Deser, and C. W. Misner, 118, 1100 (1960); , Phys. Rev.
1.R. Arnowitt, S. Deser, and C. W. Misner, 121, 1566 (1961); , Phys. Rev.
1.R. Arnowitt, S. Deser, and C. W. Misner, 122, 997 (1961); , Phys. Rev.
1.the article in Gravitation, an Introduction to Current Research, edited by L. Witten (Wiley, New York, 1962).
2.A. Ashtekar and R. O. Hansen, J. Math. Phys. 19, 1542 (1978).
3.A. Ashtekar, in General Relativity and Gravitation, edited by A. Held (Plenum, New York, 1980).
4.P. S. Jang, J. Math. Phys. 17, 141 (1976);
4.E. Witten, Commun. Math. Phys. 80, 381 (1981).
5.Actually, it would suffice for our purposes here if the matter stress energy falls off as (for some ) in the physical space‐time.
6.E. T. Newman, L. Tamburino, and T. Unti, J. Math. Phys. 4, 915 (1963);
6.A. Ashtekar and A. Sen, J. Math. Phys. 23, 2168 (1982).
7.A. Ashtekar and A. Magnon‐Ashtekar, J. Math. Phys. 20, 793 (1979).
8.A. Ashtekar, J. Math. Phys. 22, 2885 (1981).
9.R. Penrose, Proc. R. Soc. London Ser. A 381, 53 (1982).
10.W. Shaw, Proc. R. Soc. London Ser. A 390, 191 (1983).
11.S. W. Hawking, Phys. Lett. B 126, 175 (1983).
12.Y. Choquet‐Bruhat and D. Christodoulou, Acta Math. 146, 129 (1981);
12.D. Christodoulou and N. O’Murchadha, Commun. Math. Phys. 80, 271 (1981).
13.R. Geroch, J. Math. Phys. 13, 956 (1972).
14.The boldface indices denote components and take on numerical values 0, 1, 2, 3. Italic indices are “abstract indices” à la Penrose.
15.It is not clear if there are not other types of ambiguities in the various expressions. In the spi framework, such ambiguities would yield inequivalent completions of the physical space‐time, and although the framework would hold for any one completion, the relation between quantities defined using, inequivalent completions may be complicated. For details, see A. Ashtekar, in The Proceedings of the 10th International Conference on General Relativity and Gravitation, edited by B. Bertotti et al. (Reidel, Dordrecht, The Netherlands, 1984);
15.and in The Proceedings of the Oregon Conference on Mass and Asymptotic Behavior of Space‐time, edited by F. Flaherty and J. Isenberg (Springer, Berlin, 1984).
16.Note that our signature is
17.Note that one can also show that lim and use this fact to obtain alternate expressions. This fact is also used in the passage from Eq. (3.4) to (3.12) and (3.13).
18.That an additional assumption is involved in the derivation of (3.11) seems not to have been noticed before. This additional condition can always be satisfied if the additional boundary condition, discussed in Sec. V, for angular momentum to be well‐defined is satisfied. (Also, our expression differs from Geroch’s by a factor of two).
19.In view of Eq. (2.2), Eq. (5.1) is equivalent to the requirement that, the pull‐back, of to the 2‐sphere cross section C of the hyperboloid D defined by should satisfy where is the intrinsic derivative operator on C. Hence, one can demand, in place of Eq. (2.5), on S. This condition removes the spatial‐supertranslation ambiguity associated with Σ.
20. are a set of spherical polar coordinates adapted to The 2‐spheres under consideration are given by
21.In Refs. 2 and 3, is defined to be the limit to of i.e., But since vanishes and since is proportional to with the proportionality factor approaching 1 as one can replace by
22.N. O’Murchadha (private communication).
23.P. Chrusciel (private communication).
24.Obvious generalizations of (5.24), e.g., do not satisfy (5.2).
25.See, e.g., A. Ashtekar and A. Magnon‐Ashtekar, J. Math. Phys. 19, 1567 (1978).
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