Angular momentum, fluxes, and angular correlations

A variety of definitions of angular momentum at null infinity have been proposed.  Much of this work is rather formal, and one would like to understand the physical significances of the different proposals, and especially what distinguishes them.

One longstanding debate has been whether the emitted angular radiation should be derivable from a flux density (three-form) on null infinity.  I will show here that for this to occur the emitted angular momentum cannot depend on correlations in the shear (or news) in different asymptotic directions.  This, for example, will be the case for the Dray-Streubel angular momentum [1], and the Ashtekar-Streubel fluxes [2].

By contrast, I will show that the twistorial angular momentum [3] does involve such correlations, and moreover they occur for physically plausible reasons.  These come about because the proposal gives corrections to the center of mass, which automatically compensate for supertranslational gauge effects.  

To understand the main idea, suppose there are a number systems S_j, each of which makes a contribution whose Newtonian part contains a term r_j x p_j.  When we include general-relativistic, supertranslation-compensating, effects according to twistor theory, each S_j gives a correction to all the r_k's.  There will then be cross-terms, involving corrections to the center of mass of S_j correlated with momentum from S_j.  That these correlations depend on the shear and news in different asymptotic directions is due to the nonlocality of the center-of-mass corrections, which come from an angular potential for the shear.

References

1. T. Dray and M. Streubel, Classical and Quantum Gravity 1, 15 (1984).

2. A. Ashtekar and M. Streubel, Proc. Roy. Soc. Lond. A376, 585 (1981).

3. A. D. Helfer, Phys. Rev. D 106, 084044 (2022).