Killing Horizons and Surface Gravities for a Well-Behaved Three-Function Generalization of the Kerr Spacetime

Abstract

Thanks to the recent advent of the event horizon telescope (EHT), we now have the opportunity to test the physical ramifications of the strong-field near-horizon regime for astrophysical black holes. Herein, emphasizing the trade-off between tractability and generality, the authors discuss a particularly powerful three-function distortion of the Kerr spacetime, depending on three arbitrary functions of the radial coordinate r, which on the one hand can be fit to future observational data, and on the other hand is sufficiently general so as to encompass an extremely wide class of theoretical models. In all of these spacetimes, both the timelike Hamilton–Jacobi (geodesic) and massive Klein–Gordon (wave) equations separate, and the spacetime geometry is asymptotically Kerr; hence, these spacetimes are well-suited to modeling real astrophysical black holes. The authors then prove the existence of Killing horizons for this entire class of spacetimes, and give tractable expressions for the angular velocities, areas, and surface gravities of these horizons. We emphasize the validity of rigidity results and zeroth laws for these horizons.

Explicit formulae for surface gravities in stationary circular axi-symmetric spacetimes

“…M V was directly supported by the Marsden Fund, via a grant administered by the Royal Society of New Zealand. This is as expected [41,92].…”

“…In general nothing more can be said, even on horizon. Only after more precisely specifying the spacetime (for example, the specific 3-function distortions of Kerr considered in [41]) can anything more stringent be said concerning horizon areas.…”

“…In recent work [41] we had considered a 3-function extension of Kerr that exhibited some particularly desirable astrophysical properties (Hamilton-Jacobi separability of the geodesic equations, Klein-Gordon separability of the wave equation). We demonstrated that for these 3-function extensions of Kerr all horizons were automatically Killing horizons, and extracted a simple explicit form for the surface gravities.…”

“…We demonstrated that for these 3-function extensions of Kerr all horizons were automatically Killing horizons, and extracted a simple explicit form for the surface gravities. In comparing that formalism to the current framework we first note that for all the 3-function distortions of Kerr [41] we have…”

“…Moving beyond the zeroth order Kerr paradigm can be done either in a 'top down' fashion, (distorting Kerr in some controlled manner [27][28][29][30][31][32][33][34][35][36][37][38][39][40]), or in a 'bottom up' fashion, (starting from minimal symmetry assumptions). In recent work we have investigated a particularly attractive 3-function distortion of Kerr [41] (see also related considerations in [42][43][44]). Herein we shall instead work backwards from minimal symmetry assumptions:…”