Surface gravity in dynamical spherically symmetric spacetimes

TL;DR

This paper proposes a new, practical definition of surface gravity at the apparent horizon of dynamical spherically symmetric spacetimes, linking it to observable redshift and matter fields, with applications to various black hole solutions.

Contribution

It introduces a unique foliation-based definition of surface gravity that aligns with static cases and provides a simple observational method for its determination.

Findings

Surface gravity expressed as an integral of matter fields along null lines.

A practical method for observers to measure surface gravity via redshift.

Formulas for surface gravity in multiple dynamical black hole solutions.

Abstract

A definition of surface gravity at the apparent horizon of dynamical spherically symmetric spacetimes is proposed. It is based on a unique foliation by ingoing null hypersurfaces. The function parametrizing the hypersurfaces can be interpreted as the phase of a light wave uniformly emitted by some far-away static observer. The definition gives back the accepted value of surface gravity in the static case by virtue of its nonlocal character. Although the definition is motivated by the behavior of outgoing null rays, it turns out that there is a simple connection between the generalized surface gravity, the acceleration of any radially moving observer, and the observed frequency change of the infalling light signal. In particular, this gives a practical and simple method of how any geodesic observer can determine surface gravity by measuring only the redshift of the infalling light wave. The surface gravity can be expressed as an integral of matter field quantities along an ingoing null line, which shows that it is a continuous function along the apparent horizon. A formula for the area change of the apparent horizon is presented, and the possibility of thermodynamical interpretation is discussed. Finally, concrete expressions of surface gravity are given for a number of four-dimensional and two-dimensional dynamical black hole solutions.