Abstract
Abstract We propose new concepts, a dynamically transversely trapping surface (DTTS) and a marginally DTTS, as indicators for a strong gravity region. A DTTS is defined as a two-dimensional closed surface on a spacelike hypersurface such that photons emitted from arbitrary points on it in transverse directions are acceleratedly contracted in time, and a marginally DTTS is reduced to the photon sphere in spherically symmetric cases. (Marginally) DTTSs have a close analogy with (marginally) trapped surfaces in many aspects. After preparing the method of solving for a marginally DTTS in the time-symmetric initial data and the momentarily stationary axisymmetric initial data, some examples of marginally DTTSs are numerically constructed for systems of two black holes in the Brill–Lindquist initial data and in the Majumdar–Papapetrou spacetimes. Furthermore, the area of a DTTS is proved to satisfy the Penrose-like inequality $A_0\le 4\pi (3GM)^2$, under some assumptions. Differences and connections between a DTTS and the other two concepts proposed by us previously, a loosely trapped surface [Prog. Theor. Exp. Phys. 2017, 033E01 (2017)] and a static/stationary transversely trapping surface [Prog. Theor. Exp. Phys. 2017, 063E01 (2017)], are also discussed. A (marginally) DTTS provides us with a theoretical tool to significantly advance our understanding of strong gravity fields. Also, since DTTSs are located outside the event horizon, they could possibly be related with future observations of strong gravity regions in dynamical evolutions.
Highlights
There are two characteristic positions in a black hole spacetime
Taking a Schwarzschild spacetime as an example, one is the horizon r = 2GM that determines the black hole region, where r is the circumferential radius, G is the Newtonian constant of gravitation, and M is the Arnowitt–Deser–Misner (ADM) mass that represents the total gravitational energy evaluated at spatial infinity
Ref. [24], clarifying the relation of loosely trapped surface (LTS) to the behavior of photons was left as a remaining problem, and we show here the fact that a dynamically transversely trapping surface (DTTS) is an LTS at the same time under certain conditions
Summary
There are two characteristic positions in a black hole spacetime. Taking a Schwarzschild spacetime as an example, one is the horizon r = 2GM that determines the black hole region, where r is the circumferential radius, G is the Newtonian constant of gravitation, and M is the Arnowitt–Deser–Misner (ADM) mass that represents the total gravitational energy evaluated at spatial infinity. Extended concepts of r = 2GM like the event and apparent horizons, if properly defined, provide us with tools to greatly advance our understanding of the properties of spacetimes This fact motivates us to consider extended concepts of the photon sphere r = 3GM. We study extended concepts of a photon sphere to characterize a strong gravity region outside a black hole. Shortcomings of our previous works are that in the case of an LTS, the relation to the behavior of photons cannot be read from the definition directly, and in the case of a static/stationary TTS, its straightforward generalization to dynamical cases does not necessarily represent a strong gravity region to a photon surface.
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