Smarr Formula and an Extended First Law for Lovelock Gravity

We study properties of static, asymptotically AdS black holes in Lovelock gravity. Our main result is a Smarr formula that gives the mass in terms of geometrical quantities together with the parameters of the Lovelock theory. As in Einstein gravity, the Smarr formula follows from applying the first law to an infinitesimal change in the overall length scale. However, because the Lovelock couplings are dimensionful, we must first prove an extension of the first law that includes their variations. Key ingredients in this construction are the Killing-Lovelock potentials associated with each of the the higher curvature Lovelock interactions. Geometric expressions are obtained for the new thermodynamic potentials conjugate to variation of the Lovelock couplings.Comment: 20 pages; v2 - references added; v3 - includes important corrections to result

Chemical Potential in the First Law for Holographic Entanglement Entropy

Entanglement entropy in conformal field theories is known to satisfy a first law. For spherical entangling surfaces, this has been shown to follow via the AdS/CFT correspondence and the holographic prescription for entanglement entropy from the bulk first law for Killing horizons. The bulk first law can be extended to include variations in the cosmological constant $\Lambda$, which we established in earlier work. Here we show that this implies an extension of the boundary first law to include varying the number of degrees of freedom of the boundary CFT. The thermodynamic potential conjugate to $\Lambda$ in the bulk is called the thermodynamic volume and has a simple geometric formula. In the boundary first law it plays the role of a chemical potential. For the bulk minimal surface $\Sigma$ corresponding to a boundary sphere, the thermodynamic volume is found to be proportional to the area of $\Sigma$, in agreement with the variation of the known result for entanglement entropy of spheres. The dependence of the CFT chemical potential on the entanglement entropy and number of degrees of freedom is similar to how the thermodynamic chemical potential of an ideal gas depends on entropy and particle number.Comment: 18 pages; v2 - reference adde

Birkhoff's Theorem in Higher Derivative Theories of Gravity

In this paper we present a class of higher derivative theories of gravity which admit Birkhoff's theorem. In particular, we explicitly show that in this class of theories, although generically the field equations are of fourth order, under spherical (plane or hyperbolic) symmetry, all the field equations reduce to second order and have exactly the same or similar structure to those of Lovelock theories, depending on the spacetime dimensions and the order of the Lagrangian.Comment: 7 pages, no figures. v1: This version received an Honorable Mention from the Gravity Research Foundation - 2011 Awards for Essays on Gravitation. v2: Expanded version. To appear in CQ

Hairy black holes sourced by a conformally coupled scalar field in D dimensions

There exist well-known no-hair theorems forbidding the existence of hairy black hole solutions in general relativity coupled to a scalar conformal field theory in asymptotically flat space. Even in the presence of cosmological constant, where no-hair theorems can usually be circumvented and black holes with conformal scalar hair were shown to exist in dimensions three and four, no-go results were reported for D>4. In this paper we prove that these obstructions can be evaded and we answer in the affirmative a question that remained open: Whether hairy black holes do exist in general relativity sourced by a conformally coupled scalar field in arbitrary dimensions. We find the analytic black hole solution in arbitrary dimension D>4, which exhibits a backreacting scalar hair that is regular everywhere outside and on the horizon. The metric asymptotes to (Anti-)de Sitter spacetime at large distance and admits spherical horizon as well as horizon of a different topology. We also find analytic solutions when higher-curvature corrections O(R^n) of arbitrary order n are included in the gravity action.Comment: 5 pages, no figures. V2: minor changes. Published versio

Birkhoff's Theorem in Higher Derivative Theories of Gravity II: Asymptotically Lifshitz Black Holes

As a continuation of a previous work, here we examine the admittance of Birkhoff's theorem in a class of higher derivative theories of gravity. This class is contained in a larger class of theories which are characterized by the property that the trace of the field equations are of second order in the metric. The action representing these theories are given by a sum of higher curvature terms. Moreover the terms of a fixed order k in the curvature are constructed by taking a complete contraction of k conformal tensors. The general spherically (hyperbolic or plane) symmetric solution is then given by a static asymptotically Lifshitz black hole with the dynamical exponent equal to the spacetime dimensions. However, theories which are homogeneous in the curvature (i.e., of fixed order k) possess additional symmetry which manifests as an arbitrary conformal factor in the general solution. So, these theories are analyzed separately and have been further divided into two classes depending on the order and the spacetime dimensions.Comment: 10 pages, no figures. v2: minor corrections. Rejected by CQG. v3: Final version, to appear in PRD with the title "Birkhoff's Theorem in Higher Derivative Theories of Gravity II

Extended First Law for Entanglement Entropy in Lovelock Gravity

The first law for the holographic entanglement entropy of spheres in a boundary CFT (Conformal Field Theory) with a bulk Lovelock dual is extended to include variations of the bulk Lovelock coupling constants. Such variations in the bulk correspond to perturbations within a family of boundary CFTs. The new contribution to the first law is found to be the product of the variation δa\u27\u3eδaδa of the “A”-type trace anomaly coefficient for even dimensional CFTs, or more generally its extension δa*\u27\u3eδa∗δa* to include odd dimensional boundaries, times the ratio S/a*\u27\u3eS/a∗S/a* . Since a*\u27\u3ea∗a* is a measure of the number of degrees of freedom N per unit volume of the boundary CFT, this new term has the form μδN\u27\u3eμδNμδN , where the chemical potential μ is given by the entanglement entropy per degree of freedom