The Hawking temperature of dynamical black holes via conformal transformations

Abstract

In this second part of our two-series on extracting the Hawking temperature of dynamical black holes, we focus into spacetimes that are conformal transformations of static spacetimes. Our previous investigation builds upon the Unruh-Hawking analogy, which relates the spacetime of a uniformly accelerating observer to the near-horizon region of a black hole, to obtain the Hawking temperature. However, in this work, we explicitly compute the Bogoliubov coefficients associated with incoming and outgoing modes, which not only yields the temperature but also thermal spectrum of particles emitted by a black hole. For illustration, we take the simplest nontrivial example of the linear Vaidya spacetime, which is conformal to the static metric and using this property, we analytically solve the massless scalar field in its background. This allows the explicit computations of the Bogoliubov coefficients to study the particle production in this spacetime. We also derive an expression for the total mass of such dynamical spacetimes using the conformal Killing vector. We then perform differential variations of the mass formula to determine whether the laws of dynamical black hole mechanics correspond to the laws of thermodynamics.

19 pages, 2 figures, Published Version; Comments Welcome!

References in corpus (12)

• Gravitational Test Beyond the First Post-Newtonian Order with the Shadow of the M87 Black Hole
• Observation of Incipient Black Holes and the Information Loss Problem
• Hawking-like radiation from evolving black holes and compact horizonless objects
• A gravito-electromagnetic analogy based on tidal tensors
• Black-hole microstate spectroscopy: ringdown, quasinormal modes, and echoes
• Surfaces away from horizons are not thermodynamic
• Black holes as spherically-symmetric horizon-bound objects
• Elementary Integral Series for Heun Functions. With an Application to Black-Hole Perturbation Theory
• Surface gravity from tidal acceleration
• Trapped region in Kerr-Vaidya space-time
• On thermodynamics of compact objects
• The Hawking temperature of dynamical black holes via Rindler transformations