  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.
