Sub-Planckian black holes and the Generalized Uncertainty Principle

The Black Hole Uncertainty Principle correspondence suggests that there could exist black holes with mass beneath the Planck scale but radius of order the Compton scale rather than Schwarzschild scale. We present a modified, self-dual Schwarzschild-like metric that reproduces desirable aspects of a variety of disparate models in the sub-Planckian limit, while remaining Schwarzschild in the large mass limit. The self-dual nature of this solution under $M \leftrightarrow M^{-1}$ naturally implies a Generalized Uncertainty Principle with the linear form $\Delta x \sim \frac{1}{\Delta p} + \Delta p$. We also demonstrate a natural dimensional reduction feature, in that the gravitational radius and thermodynamics of sub-Planckian objects resemble that of $(1+1)$-D gravity. The temperature of sub-Planckian black holes scales as $M$ rather than $M^{-1}$ but the evaporation of those smaller than $10^{-36}$g is suppressed by the cosmic background radiation. This suggests that relics of this mass could provide the dark matter.Comment: 12 pages, 9 figures, version published in J. High En. Phy

Holographic Renormalization for Asymptotically Lifshitz Spacetimes

A variational formulation is given for a theory of gravity coupled to a massive vector in four dimensions, with Asymptotically Lifshitz boundary conditions on the fields. For theories with critical exponent z=2 we obtain a well-defined variational principle by explicitly constructing two actions with local boundary counterterms. As part of our analysis we obtain solutions of these theories on a neighborhood of spatial infinity, study the asymptotic symmetries, and consider different definitions of the boundary stress tensor and associated charges. A constraint on the boundary data for the fields figures prominently in one of our formulations, and in that case the only suitable definition of the boundary stress tensor is due to Hollands, Ishibashi, and Marolf. Their definition naturally emerges from our requirement of finiteness of the action under Hamilton-Jacobi variations of the fields. A second, more general variational principle also allows the Brown-York definition of a boundary stress tensor.Comment: 34 pages, Added Reference

Thermodynamic instability of doubly spinning black objects

We investigate the thermodynamic stability of neutral black objects with (at least) two angular momenta. We use the quasilocal formalism to compute the grand canonical potential and show that the doubly spinning black ring is thermodynamically unstable. We consider the thermodynamic instabilities of ultra-spinning black objects and point out a subtle relation between the microcanonical and grand canonical ensembles. We also find the location of the black string/membrane phases of doubly spinning black objects.Comment: 25 pages, 7 figures v2: matches the published versio

Pathologies in Asymptotically Lifshitz Spacetimes

There has been significant interest in the last several years in studying possible gravitational duals, known as Lifshitz spacetimes, to anisotropically scaling field theories by adding matter to distort the asymptotics of an AdS spacetime. We point out that putative ground state for the most heavily studied example of such a spacetime, that with a flat spatial section, suffers from a naked singularity and further point out this singularity is not resolvable by any known stringy effect. We review the reasons one might worry that asymptotically Lifshitz spacetimes are unstable and employ the initial data problem to study the stability of such systems. Rather surprisingly this question, and even the initial value problem itself, for these spacetimes turns out to generically not be well-posed. A generic normalizable state will evolve in such a way to violate Lifshitz asymptotics in finite time. Conversely, enforcing the desired asymptotics at all times puts strong restrictions not just on the metric and fields in the asymptotic region but in the deep interior as well. Generically, even perturbations of the matter field of compact support are not compatible with the desired asymptotics.Comment: 36 pages, 1 figure, v2: Enhanced discussion of singularity, including relationship to Gubser's conjecture and singularity in RG flow solution, plus minor clarification

Extended phase space thermodynamics for charged and rotating black holes and Born-Infeld vacuum polarization

We investigate the critical behaviour of charged and rotating AdS black holes in d spacetime dimensions, including effects from non-linear electrodynamics via the Born-Infeld action, in an extended phase space in which the cosmological constant is interpreted as thermodynamic pressure. For Reissner-Nordstrom black holes we find that the analogy with the Van der Walls liquid-gas system holds in any dimension greater than three, and that the critical exponents coincide with those of the Van der Waals system. We find that neutral slowly rotating black holes in four space-time dimensions also have the same qualitative behaviour. However charged and rotating black holes in three spacetime dimensions do not exhibit critical phenomena. For Born-Infeld black holes we define a new thermodynamic quantity B conjugate to the Born-Infeld parameter b that we call Born-Infeld vacuum polarization. We demonstrate that this quantity is required for consistency of both the first law of thermodynamics and the corresponding Smarr relation.Comment: 23 pages, 32 figures, v2: minor changes, upgraded reference

Rotating black holes with equal-magnitude angular momenta in d=5 Einstein-Gauss-Bonnet theory

We construct rotating black hole solutions in Einstein-Gauss-Bonnet theory in five spacetime dimensions. These black holes are asymptotically flat, and possess a regular horizon of spherical topology and two equal-magnitude angular momenta associated with two distinct planes of rotation. The action and global charges of the solutions are obtained by using the quasilocal formalism with boundary counterterms generalized for the case of Einstein-Gauss-Bonnet theory. We discuss the general properties of these black holes and study their dependence on the Gauss-Bonnet coupling constant $\alpha$. We argue that most of the properties of the configurations are not affected by the higher derivative terms. For fixed $\alpha$ the set of black hole solutions terminates at an extremal black hole with a regular horizon, where the Hawking temperature vanishes and the angular momenta attain their extremal values. The domain of existence of regular black hole solutions is studied. The near horizon geometry of the extremal solutions is determined by employing the entropy function formalism.Comment: 25 pages, 7 figure

Lovelock-Lifshitz Black Holes

In this paper, we investigate the existence of Lifshitz solutions in Lovelock gravity, both in vacuum and in the presence of a massive vector field. We show that the Lovelock terms can support the Lifshitz solution provided the constants of the theory are suitably chosen. We obtain an exact black hole solution with Lifshitz asymptotics of any scaling parameter $z$ in both Gauss-Bonnet and in pure 3rd order Lovelock gravity. If matter is added in the form of a massive vector field, we also show that Lifshitz solutions in Lovelock gravity exist; these can be regarded as corrections to Einstein gravity coupled to this form of matter. For this form of matter we numerically obtain a broad range of charged black hole solutions with Lifshitz asymptotics, for either sign of the cosmological constant. We find that these asymptotic Lifshitz solutions are more sensitive to corrections induced by Lovelock gravity than are their asymptotic AdS counterparts. We also consider the thermodynamics of the black hole solutions and show that the temperature of large black holes with curved horizons is proportional to $r_0^z$ where $z$ is the critical exponent; this relationship holds for black branes of any size. As is the case for asymptotic AdS black holes, we find that an extreme black hole exists only for the case of horizons with negative curvature. We also find that these Lovelock-Lifshitz black holes have no unstable phase, in contrast to the Lovelock-AdS case. We also present a class of rotating Lovelock-Lifshitz black holes with Ricci-flat horizons.Comment: 26 pages, 10 figures, a few references added, typo fixed and some comments have been adde

Shaping black holes with free fields

Starting from a metric Ansatz permitting a weak version of Birkhoff's theorem we find static black hole solutions including matter in the form of free scalar and p-form fields, with and without a cosmological constant \Lambda. Single p-form matter fields permit multiple possibilities, including dyonic solutions, self-dual instantons and metrics with Einstein-Kaelher horizons. The inclusion of multiple p-forms on the other hand, arranged in a homogeneous fashion with respect to the horizon geometry, permits the construction of higher dimensional dyonic p-form black holes and four dimensional axionic black holes with flat horizons, when \Lambda<0. It is found that axionic fields regularize black hole solutions in the sense, for example, of permitting regular -- rather than singular -- small mass Reissner-Nordstrom type black holes. Their cosmic string and Vaidya versions are also obtained.Comment: 38 pages. v2: minor changes, published versio

Gauss-Bonnet Black Holes and Heavy Fermion Metals

We consider charged black holes in Einstein-Gauss-Bonnet Gravity with Lifshitz boundary conditions. We find that this class of models can reproduce the anomalous specific heat of condensed matter systems exhibiting non-Fermi-liquid behaviour at low temperatures. We find that the temperature dependence of the Sommerfeld ratio is sensitive to the choice of Gauss-Bonnet coupling parameter for a given value of the Lifshitz scaling parameter. We propose that this class of models is dual to a class of models of non-Fermi-liquid systems proposed by Castro-Neto et.al.Comment: 17 pages, 6 figures, pdfLatex; small corrections to figure 10 in this versio