We consider the Hawking-Page phase transition between the BTZ black hole of
$M \ge 0$ and the thermal soliton of $M=-1$. In this system, there exists a
mass gap so that there does not seem to exist a continuous thermodynamic phase
transition. We consistently construct the off-shell free energies of the black
hole and the soliton by properly taking into account the conical space. And
then, the continuous off-shell free energy to describe tunneling effect can be
realized through non-equilibrium solitons.Comment: 13 pages, 3 figures, minor clarifications and corrections, accepted
for publication in JHE
Following the recent study on the emergent Friedmann equation from the
expansion of cosmic space for a flat universe, we apply this method to a
nonflat universe, and modify the evolution equation to lead to the Friedmann
equation. In order to maintain the same form with the original evolution
equation, we have to define the time-dependent Plank constant, which shows that
the spatial curvature of $k=0$ and $k=1$ is preferable to $k=-1$ since the
Plank constant of the nonflat open universe is divergent. Finally, we discuss
its physical consequences.Comment: 8 pages, no figure, Version accepted for publication in PR
We revisit the free-fall energy density of scalar fields semi-classically by employing the trace anomaly on a two-dimensional Schwarzschild black hole with respect to various black hole states in order to clarify whether something special at the horizon happens or not. For the Boulware state, the energy density at the horizon is always negative divergent, which is independent of initial free-fall positions. However, in the Unruh state the initial free-fall position is responsible for the energy density at the horizon and there is a critical point to determine the sign of the energy density at the horizon. In particular, a huge negative energy density appears when the freely falling observer is dropped just near the horizon. For the Hartle-Hawking state, it may also be positive or negative depending on the initial free-fall position, but it is always finite. Finally, we discuss physical consequences of these calculations.
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