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. *

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, this 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 long-standing 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.

Previous work on dynamical black hole instability is further elucidated within the Hamilton–Jacobi method for horizon tunneling and the reconstruction of the classical action by means of the null expansion method. Everything is based on two natural requirements, namely that the tunneling rate is an observable and therefore it must be based on invariantly defined quantities, and that coordinate systems which do not cover the horizon should not be admitted. These simple observations can help to clarify some ambiguities, like the doubling of the temperature occurring in the static case when using singular coordinates and the role, if any, of the temporal contribution of the action to the emission rate. The formalism is also applied to FRW cosmological models, where it is observed that it predicts the positivity of the temperature naturally, without further assumptions on the sign of energy.

The instability against emission of massless particles by the trapping horizon of an evolving black hole is analyzed with the use of the Hamilton-Jacobi method. The method automatically selects one special expression for the surface gravity of a changing horizon. Indeed, the strength of the horizon singularity turns out to be governed by the surface gravity as was defined a decade ago by Hayward using Kodama's theory of spherically symmetric gravitational fields. The theory also applies to point masses embedded in an expanding universe, were the surface gravity is still related to KodamaHayward theory. As a bonus of the tunneling method, we gain the insight that the surface gravity still defines a temperature parameter as long as the evolution is sufficiently slow that the black hole pass through a sequence of quasi-equilibrium states.

The instability against emission of fermionic particles by the trapping horizon of an evolving black hole is analyzed using the Hamilton-Jacobi tunneling method. This method automatically selects one special expression for the surface gravity of a changing horizon. The results also apply to point masses embedded in an expanding universe. As a bonus of the tunneling method, we gain the insight that the surface gravity still defines a temperature parameter as long as the evolution is sufficiently slow that the black hole pass through a sequence of quasi-equilibrium states, and that black holes should be semi-classically unstable even in a hypothetical world without bosonic fields.

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 ΛCDM 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.

Previous work on dynamical black hole instability is further elucidated within the Hamilton-Jacobi method for horizon tunnelling and the reconstruction of the classical action by means of the null-expansion method and making use of a coordinate invariance approach.

the netherlands), are experimentally characterized. Their behaviour is compared against simulations performed in anSYS Workbench™, confirming good accuracy of the predictive method. Furthermore, the electromechanical multiphysical behaviour of the Flc eh is also analysed in Workbench, by adding a layer with piezoelectric conversion properties in the simulation. The measured and simulated data reported in this paper confirm that the MeMS converter exhibits multiple resonant modes in the frequency range below 1 khz, where most of the environmental vibration energy is scattered, and extracted power levels of 0.2 μW can be achieved as well, in closed-loop conditions. Further developments of this work are expected to fully prove the high-performance of the Flc concept, and are going to be addressed by the authors of this work in the on-going activities.