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On conserved quantities in general relativity
1.A. Komar, Phys. Rev. 113, 934 (1959).
2.R. O. Hansen, J. Math. Phys. 15, 46 (1974).
3.A. Ashtekar and R. O. Hansen, J. Math. Phys. 19, 1542 (1978).
4.The 4‐momentum was first introduced by Arnowitt, Deser, and Misner in terms of the asymptotic behavior of the initial data on a Cauchy surface. See, e.g., C. W. Misner, the article in Gravitation, An Introduction to Current Research, edited by L. Witten (Wiley, New York, 1962).
5.H. Bondi, A. W. K. Metzner, and M. J. G. Van der Berg, Proc. R. Soc. (London), Ser. A 269, 21 (1962).
6.These conserved quantities arise only after Einstein’s vacuum equation is imposed asymptotically, i.e., only after the stress—energy tensor (with this index structure) is required to admit a regular direction dependent limit at
7.This situation is to be contrasted with the one in the electromagnetic case: Imposition of Maxwell’s sourcefree equations in the asymptotic region does not restrict the total magnetic charge in any way.
8.Note that unlike the definitions of angular momentum at null infinity, the definition at spatial infinity is free of supertranslation ambiguities.
9.A. Ashtekar, Asymptotic structure of the gravitational field at spatial infinity (to appear in the Einstein birth‐centenary volume, edited by P. Bergmann, J. N. Goldberg, and A. Held. Plenum).
10.For details, See Ref. 9.
11.We do not yet have a proof that every Killing field on a spacetime satisfying Definition 1 must induce a Spi symmetry at spatial infinity. Although one certainly expects the result to be true, the proof may well be quite complicated because of the conformal singularity at one cannot, e.g., use the conformal Killing transport equations in a straightforward way.
12.For details, see, e.g., A. Ashtekar and A. Magnon—Ashtekar, J. Math. Phys. 19, 1567 (1978).
13.See, e.g., A. Lichnerowicz, Théories Relativistes de la Gravitation et de I’électromagnétisme (Masson, Paris, 1955), or, R. Geroch, J. Math. Phys. 12, 918 (1971), Appendix.
14.Choose an orthonormal basis at such that points along the z axis. Consider, e.g., the cross section of D defined by where is the timelike vector in the basis. On this 2‐sphere, and Since e is smooth, it follows that whence
15.Our convention is such that
16.See, e.g., R. Geroch in Asymptotic Structure of Space—time, edited by P. Esposito and L. Witten (Plenum, New York, 1977), p. 96.
17.Hansen’s dipole moment is defined by grad. where Λ is the point at infinity on the three‐manifold of orbits of the conformal factor on this three‐manifold, the “twist potential” and grad, stands for “gradient.” A simple calculation yields, where is the 2‐sphere of unit vectors at is the limiting value of and is an arbitrary fixed vector at Λ.
18.P. D. Sommers, J. Math. Phys. 19, 549 (1978).
19.R. K. Sachs, Proc. R. Soc. (London) Ser. A 270, 103 (1962).
19.R. K. Sachs, Phys. Rev. 128, 2851 (1962);
19.J. Winicour, J. Math. Phys. 9, 861 (1968);
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19.M. Streubel, J. Gen. Rel. Grav. 9, 551 (1978).
20.A. Ashtekar and M. Streubel, “On angular momentum of stationary gravitating systems” (to appear in J. Math. Phys.)
21.At spatial infinity, the presence of a rotational Killing field in the physical space—time implies that must vanish on and thus reduces the Spi group to the Poineare. Is there any hope of a similar reduction at null infinity?
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