Recently, definitions of total 4‐momentum and angular momentum of isolated gravitating systems have been introduced in terms of the asymptotic behavior of the Weyl curvature (of the underlying space–time) at spatial infinity. Given a space–time equipped with isometries, on the other hand, one can also construct conserved quantities using the presence of the Killing fields. Thus, for example, for stationary space–times, the Komar integral can be used to define the total mass, and, the asymptotic value of the twist of the Killing field, to introduce the dipole angular momentum moment. Similarly, for axisymmetric space–times, one can obtain the (’’z‐component’’ of the) total angular momentum in terms of the Komar integral. It is shown that, in spite of their apparently distinct origin, in the presence of isometries, quantities defined at spatial infinity reduce to the ones constructed from Killing fields. This agreement reflects one of the many subtle aspects of Einstein’s (vacuum) equation.

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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).
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These conserved quantities arise only after Einstein’s vacuum equation is imposed asymptotically, i.e., only after the stress—energy tensor T̂ab (with this index structure) is required to admit a regular direction dependent limit at i0.
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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.
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Note that unlike the definitions of angular momentum at null infinity, the definition at spatial infinity is free of supertranslation ambiguities.
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For details, See Ref. 9.
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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 i0; one cannot, e.g., use the conformal Killing transport equations in a straightforward way.
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See, e.g., A. Lichnerowicz, Théories Relativistes de la Gravitation et de I’électromagnétisme (Masson, Paris, 1955), or,
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Choose an orthonormal basis at i0 such that Ka points along the z axis. Consider, e.g., the cross section of D defined by t⋅η = 0 where ta is the timelike vector in the basis. On this 2‐sphere, K̃a = −sinθ(∂/∂θ) and e = f(θ)(sinθ)−1. Since e is smooth, it follows that f(θ) = 0, whence e = 0.
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Our convention is such that ∫SdS = 1.
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See, e.g., R. Geroch in Asymptotic Structure of Space—time, edited by P. Esposito and L. Witten (Plenum, New York, 1977), p. 96.
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Hansen’s dipole moment S̃a is defined by S̃a = 12lim→Λ grad. (Ω̃−1/2λ̂ω̂), where Λ is the point at infinity on the three‐manifold of orbits of t̂a, Ω̃ the conformal factor on this three‐manifold, ω̂ the “twist potential” (ω̂a = gradω̂) and grad, stands for “gradient.” A simple calculation yields,
S̃aVa = 12 ∫ S2λ−1Va(ωa−12ωm⋅ηm)dS =  ∫ S2(ωm⋅ηm)(V⋅η)dS,
where S2 is the 2‐sphere of unit vectors η at Λ, ωa is the limiting value of Ω−12ωa and Va is an arbitrary fixed vector at Λ.
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At spatial infinity, the presence of a rotational Killing field in the physical space—time implies that Bab must vanish on D and thus reduces the Spi group to the Poineare. Is there any hope of a similar reduction at null infinity?
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© 1979 American Institute of Physics.
1979
American Institute of Physics
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