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In this second part of our two-series on extracting the Hawking temperatureof dynamical black holes, we focus into spacetimes that are conformaltransformations of static spacetimes. Our previous investigation builds uponthe Unruh-Hawking analogy, which relates the spacetime of a uniformlyaccelerating observer to the near-horizon region of a black hole, to obtain theHawking temperature. However, in this work, we explicitly compute theBogoliubov coefficients associated with incoming and outgoing modes, which notonly yields the temperature but also thermal spectrum of particles emitted by ablack hole. For illustration, we take the simplest nontrivial example of thelinear Vaidya spacetime, which is conformal to the static metric and using thisproperty, we analytically solve the massless scalar field in its background.This allows the explicit computations of the Bogoliubov coefficients to studythe particle production in this spacetime. We also derive an expression for thetotal mass of such dynamical spacetimes using the conformal Killing vector. Wethen perform differential variations of the mass formula to determine whetherthe laws of dynamical black hole mechanics correspond to the laws ofthermodynamics.

The Hawking temperature of dynamical black holes via conformal transformations