Abstract:
It is known now that a typical gravitational collapse in general relativity, evolving from regular initial data and under physically reasonable conditions would end in either a black hole or a naked singularity final state. An important question that needs to be answered in this connection is, whether the analogues of the laws of thermodynamics, as formulated for relativistic horizons are respected by the dynamical spacetimes for collapse that end in the formation of a naked singularity. We investigate here the thermodynamical behaviour of the dynamical horizons that form in spherically symmetric gravitational collapse and we show that the first and second laws of black hole thermodynamics, as extended to dynamical spacetimes in a suitable manner, are not violated whether the collapse ends in a black hole or a naked singularity. We then make a distinction between the naked singularities that result from gravitational collapse, and those that exist in solutions of Einstein equations in vacuum axially symmetric and stationary spacetimes, and discuss their connection with thermodynamics in view of the cosmic censorship conjecture and the validity of the third law of black hole mechanics.

Abstract:
We study numerically gravitational collapse of a spherically symmetric instanton particle in five dimensions. We show that the late stages of the process are characterized by a nearly constant ``free energy'', the value of which matches (within numerical uncertainties) the value obtained from standard black-hole thermodynamics. This suggests a purely classical interpretation of the free energy of a black hole.

Abstract:
We show that it is possible to obtain a picture of equilibrium thermodynamics on the apparent horizon in the expanding cosmological background for a wide class of modified gravity theories with the Lagrangian density $f(R, \phi, X)$, where $R$ is the Ricci scalar and $X$ is the kinetic energy of a scalar field $\phi$. This comes from a suitable definition of an energy momentum tensor of the "dark" component that respects to a local energy conservation in the Jordan frame. In this framework the horizon entropy $S$ corresponding to equilibrium thermodynamics is equal to a quarter of the horizon area $A$ in units of gravitational constant $G$, as in Einstein gravity. For a flat cosmological background with a decreasing Hubble parameter, $S$ globally increases with time, as it happens for viable $f(R)$ inflation and dark energy models. We also show that the equilibrium description in terms of the horizon entropy $S$ is convenient because it takes into account the contribution of both the horizon entropy $\hat{S}$ in non-equilibrium thermodynamics and an entropy production term.

Abstract:
By studying the Euclidean partition function on a cone, we argue that pure and mixed gravitational anomalies generate a "Casimir momentum" which manifests itself as parity violating coefficients in the hydrodynamic stress tensor and charge current. The coefficients generated by these anomalies enter at a lower order in the hydrodynamic gradient expansion than would be naively expected. In 1+1 dimensions, the gravitational anomaly affects coefficients at zeroth order in the gradient expansion. The mixed anomaly in 3+1 dimensions controls the value of coefficients at first order in the gradient expansion.

Abstract:
The concept of gravitational pressure is naturally defined in the context of the teleparallel equivalent of general relativity. Together with the definition of gravitational energy, we investigate the thermodynamics of rotating black holes in the teleparallel framework. We obtain the value of the gravitational pressure over the external event horizon of the Kerr black hole, and write an expression for the thermodynamic relation $TdS =dE + pdV$, where the variations refer to the Penrose process for the Kerr black hole. We employ only the notions of gravitational energy and pressure that arise in teleparallel gravity, and do not make any consideration of the area or the variation of the area of the event horizon. However, our results are qualitatively similar to the standard expression of the literature.

Abstract:
It is shown that linearized gravitational radiation confined in a cavity can achieve thermal equilibrium if the mean density of the radiation and the size of the cavity satisfy certain constraints.

Abstract:
We examine the dynamics of the gravitational collapse in a 4-dim Lorentzian brane embedded in a 5-dim bulk with an extra timelike dimension. By considering the collapse of pure dust on the brane we derive a bouncing FLRW interior solution and match it with a corrected Schwarzschild exterior geometry. In the physical domain considered for the parameters of the solution, the analytical extension is built, exhibiting an exterior event horizon and a Cauchy horizon, analogous to the Reissner-Nordstr\"om solution. For such an exterior geometry we examine the effects of the bulk-brane corrections in the Hawking radiation. In this scenario the model extends Bekenstein's black hole geometrical thermodynamics for quasi-extremal configurations, with an extra work term in the laws associated with variations of the brane tension. We also propose a simple statistical mechanics model for the entropy of the bouncing collapsed matter by quantizing its fluctuations and constructing the associated partition function. This entropy differs from the geometrical entropy by an additive constant proportional to the area of the extremal black hole and satisfies an analogous first law of thermodynamics. A possible connection between both entropies is discussed.

Abstract:
We explore the relationship between the first law of thermodynamics and gravitational field equation at a static, spherically symmetric black hole horizon in Ho\v{r}ava-Lifshtiz theory with/without detailed balance. It turns out that as in the cases of Einstein gravity and Lovelock gravity, the gravitational field equation can be cast to a form of the first law of thermodynamics at the black hole horizon. This way we obtain the expressions for entropy and mass in terms of black hole horizon, consistent with those from other approaches. We also define a generalized Misner-Sharp energy for static, spherically symmetric spacetimes in Ho\v{r}ava-Lifshtiz theory. The generalized Misner-Sharp energy is conserved in the case without matter field, and its variation gives the first law of black hole thermodynamics at black hole horizon.

Abstract:
Hamiltonian dynamics of gravitational field contained in a spacetime region with boundary $S$ being a null-like hypersurface (a wave front) is discussed. Complete Hamiltonian formula for the dynamics (with no surface integrals neglected) is derived. A quasi-local proof of the first law of black holes thermodynamics is obtained as a consequence, in case when $S$ is a non-expanding horizon. The zeroth law and Penrose inequalities are discussed from this point of view.

Abstract:
We define, by an integral of geometric quantities over a spherical shell of arbitrary radius, an invariant gravitational entropy. This definition relies on defining a gravitational energy and pressure, and it reduces at the horizon of both black branes and black holes to Wald's Noether charge entropy. We support the thermodynamic interpretation of the proposed entropy by showing that, for some cases, the field theory duals of the entropy, energy and pressure are the same as the corresponding quantities in the field theory. In this context, the Einstein equations are equivalent to the field theory thermodynamic relation TdS=dE+PdV supplemented by an equation of state.