Thermodynamics with conformal Killing vector in the charged Vaidya metric

TL;DR

This paper explores the conformal symmetry of charged Vaidya spacetime, identifying conformal Killing horizons and linking their surface gravity to Hawking temperature, revealing new insights into dynamical black hole thermodynamics.

Contribution

It introduces a conformal Killing vector in charged Vaidya spacetime, classifies horizons, and connects conformal horizons to static spacetime properties, including Hawking temperature.

Findings

Conformal Killing horizons map to static spacetime horizons under conformal transformation.

The conformal factor is not unique and can depend on the ratio of radius to mass.

Hawking temperature remains invariant under conformal transformations of the horizons.

Abstract

We investigate the charged Vaidya spacetime with conformal symmetry by classifying the horizons and finding its connection to Hawking temperature. We find a conformal Killing vector whose existence requires the mass and electric charge functions to be proportional, as well as linear in time. Solving the Killing equations for the conformally transformed metric from the linear charged Vaidya metric yields the required form of the conformal factor. From the vanishing of the norm of the conformal Killing vector, we find three conformal Killing horizons which, under the transformation, are mapped to the Killing horizons of the associated static spacetime, if the spherical symmetry is maintained. We find that the conformal factor is not uniquely determined, but can take any function of the ratio of the radial coordinate to the dynamical mass. As an example, we illustrate a static spacetime with our choice of the conformal factor and explicitly show that the surface gravity of the conformal Killing horizons, which is conformally invariant, yield the expected Hawking temperature in the static spacetime. This static black hole spacetime contains a cosmological horizon, but it is not asymptotically de Sitter. We also investigate the case when the mass parameter is equal to the constant electric charge. While in this case the standard pair of horizons, the loci of the time component of the metric, degenerate, the conformal Killing horizons do not degenerate. This therefore leads to a non-zero Hawking temperature in the associated static spacetime.