Abstract:
Much attention has been recently devoted to modified theories of gravity in the attempt to efficiently describe both early inflation and late-time acceleration of our universe without referring to the cosmological constant or other ad hoc kinds of fluids. The simplest models overcome General Relativity simply by replacing $R$ with $F(R)$ in the Einstein--Hilbert action. Unfortunately, such models typically lack most of the beautiful solutions discovered in Einstein's gravity. Nonetheless, in $F(R)$ gravity, it has been possible to get at least few black holes, but still we do not know any empty wormhole-like spacetime solution. The present paper aims to explain why it is so hard to get such solutions (given that they exist). Few solutions are derived in the simplest cases while only an implicit form has been obtained in the non-trivial case.

Abstract:
The aim of this work is to review the tunnelling method as an alternative description of the quantum radiation from black holes and cosmological horizons. The method is first formulated and discussed for the case of stationary black holes, then a foundation is provided in terms of analytic continuation throughout complex space-time. The two principal implementations of the tunnelling approach, which are the null geodesic method and the Hamilton-Jacobi method, are shown to be equivalent in the stationary case. The Hamilton-Jacobi method is then extended to cover spherically symmetric dynamical black holes, cosmological horizons and naked singularities. Prospects and achievements are discussed in the conclusions.

Abstract:
Studying the behaviour of a quantum field in a classical, curved, spacetime is an extraordinary task which nobody is able to take on at present time. Independently by the fact that such problem is not likely to be solved soon, still we possess the instruments to perform exact predictions in special, highly symmetric, conditions. Aim of the present contribution is to show how it is possible to extract quantitative information about a variety of physical phenomena in very general situations by virtue of the so-called Hamilton-Jacobi method. In particular, we shall prove the agreement of such semi-classical method with exact results of quantum field theoretic calculations.

Abstract:
In the paper, the temperature associated with a dynamical spherically symmetric black hole or with a cosmological horizon is investigated from the point of view of a point-like detector. First, we briefly review the Hamilton-Jacobi tunneling method for a generic dynamical spherically symmetric space-time, and present two applications of the tunneling method. Then, we apply a well-known relativistic quantum theoretical technique, namely the Unruh-DeWitt detector formalism for a conformally coupled scalar field in a generic FRW space-time. As an application, for the generic static black hole case and the FRW de Sitter case, making use of peculiar Kodama observer trajectories, the tunneling semiclassical results are fully recovered, automatically corrected by Tolman factors. Some remarks on the temperature of FRW universe are presented. For more general spaces interpolating de Sitter space with the Einstein-de Sitter universe a second set of poles is present, whose exact role remains to be clarified, plus an extra fluctuating term describing the way equilibrium is reached, similarly to de Sitter space. The simple thermal interpretation found for de Sitter space is lost and forces, at a same time, a different quantum interpretation of the horizon surface gravity for the cosmological FRW models.

Abstract:
We give an interpretation of the temperature in de Sitter universe in terms of a dynamical Unruh effect associated with the Hubble sphere. As with the quantum noise perceived by a uniformly accelerated observer in static space-times, observers endowed with a proper motion can in principle detect the effect. In particular, we study a "Kodama observer" as a two-field Unruh detector for which we show the effect is approximately thermal. We also estimate the back-reaction of the emitted radiation and find trajectories associated with the Kodama vector fields are stable.

Abstract:
The arguments of the above article [arXiv:0910.3934] do not apply to the papers which it criticizes, and contain several key errors, including a fundamental misunderstanding about the equivalence principle.

Abstract:
A local Hawking temperature is derived for any future outer trapping horizon in spherical symmetry, using a Hamilton-Jacobi variant of the Parikh-Wilczek tunneling method. It is given by a dynamical surface gravity as defined geometrically. The operational meaning of the temperature is that Kodama observers just outside the horizon measure an invariantly redshifted temperature, diverging at the horizon itself. In static, asymptotically flat cases, the Hawking temperature as usually defined by the Killing vector agrees in standard cases, but generally differs by a relative redshift factor between the horizon and infinity, being the temperature measured by static observers at infinity. Likewise, the geometrical surface gravity reduces to the Newtonian surface gravity in the Newtonian limit, while the Killing definition instead reflects measurements at infinity. This may resolve a longstanding puzzle concerning the Hawking temperature for the extremal limit of the charged stringy black hole, namely that it is the local temperature which vanishes. In general, this confirms the quasi-stationary picture of black-hole evaporation in early stages. However, the geometrical surface gravity is generally not the surface gravity of a static black hole with the same parameters.

Abstract:
A local Hawking temperature was recently derived for any future outer trapping horizon in spherical symmetry, using a Hamilton-Jacobi tunneling method, and is given by a dynamical surface gravity as defined geometrically. Descriptions are given of the operational meaning of the temperature, in terms of what observers measure, and its relation to the usual Hawking temperature for static black holes. Implications for the final fate of an evaporating black hole are discussed.

Abstract:
In the present paper, Unruh--DeWitt detectors are used in order to investigate the issue of temperature associated with a spherically symmetric dynamical space-times. Firstly, we review the semi-classical tunneling method, then we introduce the Unruh--DeWitt detector approach. We show that for the generic static black hole case and the FRW de Sitter case, making use of peculiar Kodama trajectories, semiclassical and quantum field theoretic techniques give the same standard and well known thermal interpretation, with an associated temperature, corrected by appropriate Tolman factors. For a FRW space-time interpolating de Sitter space with the Einstein--de Sitter universe (that is a more realistic situation in the frame of $\Lambda$CDM cosmologies), we show that the detector response splits into a de Sitter contribution plus a fluctuating term containing no trace of Boltzmann-like factors, but rather describing the way thermal equilibrium is reached in the late time limit. As a consequence, and unlike the case of black holes, the identification of the dynamical surface gravity of a cosmological trapping horizon as an effective temperature parameter seems lost, at least for our co-moving simplified detectors. The possibility remains that a detector performing a proper motion along a Kodama trajectory may register something more, in which case the horizon surface gravity would be associated more likely to vacuum correlations than to particle creation.