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We show that the Hawking--Bekenstein entropy formula is modified by a factor of $8/3$ if one also considers a work term in the 1st law of thermodynamics by a pressure stemming from the Hawking radiation. We give an intuitive definition for the corresponding thermodynamical volume by the implicit definition $\epsilon=Mc^2/V$, which is the average energy density of the Hawking radiation. This volume scales as $V \sim M^5$, agreeing with other suggestions. As a result the corresponding Smarr relation describes an extensive entropy and a stable effective equation of state $S(E,V)\sim E^{3/4}V^{1/4}$. These results pertain for charged and rotating Kerr-Newman black holes., Comment: 6 pages, 1 figure; accepted for publication in Phys. Lett. B

The grand challenges of contemporary fundamental physics---dark matter, dark energy, vacuum energy, inflation and early universe cosmology, singularities and the hierarchy problem---all involve gravity as a key component. And of all gravitational phenomena, black holes stand out in their elegant simplicity, while harbouring some of the most remarkable predictions of General Relativity: event horizons, singularities and ergoregions. The hitherto invisible landscape of the gravitational Universe is being unveiled before our eyes: the historical direct detection of gravitational waves by the LIGO-Virgo collaboration marks the dawn of a new era of scientific exploration. Gravitational-wave astronomy will allow us to test models of black hole formation, growth and evolution, as well as models of gravitational-wave generation and propagation. It will provide evidence for event horizons and ergoregions, test the theory of General Relativity itself, and may reveal the existence of new fundamental fields. The synthesis of these results has the potential to radically reshape our understanding of the cosmos and of the laws of Nature. The purpose of this work is to present a concise, yet comprehensive overview of the state of the art in the relevant fields of research, summarize important open problems, and lay out a roadmap for future progress., Comment: White Paper for the COST action "Gravitational Waves, Black Holes, and Fundamental Physics", 272 pages, 12 figures; v4: updated references and author list. Overall improvements and corrections. To appear in Classical and Quantum Gravity

Regarding the volume as independent thermodynamic variable we point out that black hole horizons can hide positive heat capacity and specific heat. Such horizons are mechanically marginal, but thermally stable. In the absence of a canonical volume definition, we consider various suggestions scaling differently with the horizon radius. Assuming Euler-homogeneity of the entropy, besides the Hawking temperature, a pressure and a corresponding work term render the equation of state at the horizon thermally stable for any meaningful volume concept that scales larger than the horizon area. When considering also a Stefan--Boltzmann radiation like equation of state at the horizon, only one possible solution emerges: the Christodoulou--Rovelli volume, scaling as $V\sim R^5$, with an entropy $S = \frac{8}{3}S_{BH}$., Comment: 5 pages, no figures, to be published in Phys. Lett. B

Thermodynamics of rotating black holes described by the R\'enyi formula as equilibrium and zeroth law compatible entropy function is investigated. We show that similarly to the standard Boltzmann approach, isolated Kerr black holes are stable with respect to axisymmetric perturbations in the R\'enyi model. On the other hand, when the black holes are surrounded by a bath of thermal radiation, slowly rotating black holes can also be in stable equilibrium with the heat bath at a fixed temperature, in contrast to the Boltzmann description. For the question of possible phase transitions in the system, we show that a Hawking-Page transition and a first order small black hole/large black hole transition occur, analogous to the picture of rotating black holes in AdS space. These results confirm the similarity between the R\'enyi-asymptotically flat and Boltzmann-AdS approaches to black hole thermodynamics in the rotating case as well. We derive the relations between the thermodynamic parameters based on this correspondence., Comment: 29 pages, 20 figures

Thermodynamic stability of black holes, described by the R\'enyi formula as equilibrium compatible entropy function, is investigated. It is shown that within this approach, asymptotically flat, Schwarzschild black holes can be in stable equilibrium with thermal radiation at a fixed temperature. This implies that the canonical ensemble exists just like in anti-de Sitter space, and nonextensive effects can stabilize the black holes in a very similar way as it is done by the gravitational potential of an anti-de Sitter space. Furthermore, it is also shown that a Hawking-Page-like black hole phase transition occurs at a critical temperature which depends on the $q$-parameter of the R\'enyi formula., Comment: 6 pages, 6 figures, accepted for publication in Phys. Lett. B

Thickness corrections to static, axisymmetric Dirac-Nambu-Goto branes embedded into spherically symmetric black hole spacetimes with arbitrary number of dimensions are studied. First, by applying a perturbative approximation, it is found that the thick solutions deviate significantly in their analytic properties from the thin ones near the axis of the system, and perturbative approaches around the thin configurations can not provide regular thick solutions above a certain dimension. For the general case, a non-perturbative, numerical approach is applied and regular solutions are obtained for arbitrary brane and bulk dimensions. As a special case, it has been found that 2-dimensional branes are exceptional, as they share their analytic properties with the thin branes rather than the thick solutions of all other dimensions., Comment: 4 pages, 2 figures, Proceedings of the Spanish Relativity Meeting in Portugal ERE2012

Stationary, D-dimensional test branes, interacting with N-dimensional Myers-Perry bulk black holes, are investigated in arbitrary brane and bulk dimensions. The branes are asymptotically flat and axisymmetric around the rotation axis of the black hole with a single angular momentum. They are also spherically symmetric in all other dimensions allowing a total of O(1)xO(D-2) group of symmetry. It is shown that even though this setup is the most natural extension of the spherical symmetric problem to the simplest rotating case in higher dimensions, the obtained solutions are not compatible with the spherical solutions in the sense that the latter ones are not recovered in the non-rotating limit. The brane configurations are qualitatively different from the spherical problem, except in the special case of a 3-dimensional brane. Furthermore, a quasi-static phase transition between the topologically different solutions cannot be studied here, due to the lack of a general, stationary, equatorial solution., Comment: 8 pages, 4 figures, to be published in Phys. Rev. D

By regarding the Hawking-Bekenstein entropy of Schwarzschild black hole horizons as a non-extensive Tsallis entropy, its formal logarithm, the R\'enyi entropy, is considered. The resulting temperature - horizon-radius relation has the same form as the one obtained from a 3+1-dimensional black hole in anti-de Sitter space using the original entropy formula. In both cases the temperature has a minimum. A semi-classical estimate of the horizon radius at this minimum leads to a Bekenstein bound for the q-parameter in the R\'enyi entropy of micro black holes, (q >= 1 + 2/pi^2), which is surprisingly close to fitted q-parameters of cosmic ray spectra and power-law distribution of quarks coalescing to hadrons in high energy accelerator experiments., Comment: 5 pages, 1 figure, to be published in Phys. Lett. B. arXiv admin note: text overlap with arXiv:1011.3442

We consider curvature corrections to static, axisymmetric Dirac-Nambu-Goto membranes embedded into a spherically symmetric black hole spacetime with arbitrary number of dimensions. Since the next to leading order corrections in the effective brane action are quadratic in the brane thickness l, we adopt a linear perturbation approach in l^2. The perturbations are general in the sense that they are not restricted to the Rindler zone nor to the near-critical solutions of the unperturbed system. As a result, an unexpected asymmetry in the perturbed system is found. In configurations, where the brane does not cross the black hole horizon, the perturbative approach does not lead to regular solutions if the number of the brane's spacetime dimensions D>3. This condition, however, does not hold for the horizon crossing solutions. Consequently we argue that the presented perturbative approach breaks down for subcritical type solutions near the axis of the system for D>3. Nevertheless, we can discuss topology-changing phase transitions in cases when D=2 or 3, i.e. when the brane is a 1-dimensional string or a 2-dimensional sheet, respectively. For the general case, a different, non-perturbative approach should be sought. Based on the energy properties of those branes that are quasi-statically evolved from the equatorial configuration, we illustrate the results of the phase transition in the case of a D=3 brane. It is found that small thickness perturbations do not modify the order of the transition, i.e. it remains first order just as in the case of vanishing thickness., Comment: 17 pages, 8 figures; several corrections made, substantial new material added, the title has been changed