Angular momentum conservation for dynamical black holes

Abstract

Angular momentum can be defined by rearranging the Komar surface integral in terms of a twist form, encoding the twisting around of space-time due to a rotating mass, and an axial vector. If the axial vector is a coordinate vector and has vanishing transverse divergence, it can be uniquely specified under certain generic conditions. Along a trapping horizon, a conservation law expresses the rate of change of angular momentum of a general black hole in terms of angular momentum densities of matter and gravitational radiation. This identifies the transverse-normal block of an effective gravitational-radiation energy tensor, whose normal-normal block was recently identified in a corresponding energy conservation law. Angular momentum and energy are dual respectively to the axial vector and a previously identified vector, the conservation equations taking the same form. Including charge conservation, the three conserved quantities yield definitions of an effective energy, electric potential, angular velocity and surface gravity, satisfying a dynamical version of the so-called first law of black-hole mechanics. A corresponding zeroth law holds for null trapping horizons, resolving an ambiguity in taking the null limit.

Light cone structure near null infinity of the Kerr metric

“…Using the NP equations and the explicit expressions of the spin coefficients given in (24), it may be deduced that the above angular momentum expressions are all special cases of the the linkage expression [18] (or equivalently the Komar integral) written in terms of the NP tetrad chosen here. From (26), we also see that the angular momentum is a measure of nonintegrability of the timelike two plane spanned by the null vectors l a , n a (see also [19] in this connection).…”

“…By further stipulating that asymptotically the angular part of m a should take the form 1 2 p r @~i csc~@' , we may choose 0 to be zero and conclude from (19) that…”

Approximate Killing vectors onS2

“…It has been noted [7,8] that an approximate Killing vector should be divergence free D i ξ i = 0, in part because the angular momenta computed using such an approximate Killing vector possess a certain gauge invariance. Approximate Killing vectors constructed using the KT method are not guaranteed to be divergenceless, although at least in certain cases [4] this can be enforced a posteriori.…”

Energy and Angular Momentum in Strong Gravitating Systems

“…The angular momentum surface density j is defined by j = ψ A ̟ A , where ψ is a vector tangent to S satisfying i ψ ϑ 0 | S = 0 and i ψ ϑ 1 | S = 0, with the boundary conditions [5,6,7,8,10,25],…”

“…Traditional description of black holes in terms of event horizons is inadequate for the expected observational data. For non-stationary spacetimes, quasi-local notions of trapped and marginal surfaces have now been found to be more useful within the framework of isolated, and dynamical trapping horizons [2,3,4,5,6,7,8]. These frameworks enable one to significantly extend the laws of black hole mechanics to the dynamical regime with the associated notions of energy, angular momentum and their fluxes, and have been applied to several problems in mathematical general relativity, numerical relativity, and quantum gravity [9].…”