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

Abstract

Using minimalist assumptions we develop a natural functional decomposition for the spacetime metric, and explicit tractable formulae for the surface gravities, in arbitrary stationary circular (PT symmetric) axisymmetric spacetimes. We relate rigidity results, (the existence of a Killing horizon), and the zeroth law to the absence of curvature singularities at the would-be horizons. These observations are of interest to both observational astrophysicists (modelling the cold, dark, heavy objects at the center of most spiral galaxies), and to the analogue spacetime community, (wherein the presence of naked singularities is not necessarily deprecated, and the occurrence of non-Killing horizons is relatively common).

A trick for calculating surface gravities of Killing horizons

“…The surface gravity matches the remarkably simple formula given by [33,35]. Furthermore, the issue for the sign of κ appears again.…”

“…where the subscript H means to take value on the horizon. Baines and Visser [35] has carefully compared the inaffinity with other different perspectives of surface gravities and ensures consistency. Back to the Schwarzchild spacetime, the advanced EF coordinates make the direct calculation more trackable.…”

Photon Rings and Shadow Size for General Axisymmetric and Stationary Integrable Spacetimes

“…In Appendix D, we further demonstrate that it is either identical to or encompasses a number of other proposed families of stationary axisymmetric metrics from the literature. These include those proposed by Baines & Visser (2023b) and Carson & Yagi (2020) (after applying the observational constraints discussed near Equation (50) of Johannsen 2013), and the regular black hole spacetime in Azreg-Aïnou (2014). For each of these metric families, the mapping of their free functions to N(r, a), B(r, a), F(r, a), and f (r, a) is summarized at the bottom of Table 1.…”