Classical holographic relations and alternative boundary conditions for Lovelock gravity

In this paper, we combine Lovelock gravity with gravity's rainbow to construct Lovelock gravity's rainbow. Considering the Lovelock gravity's rainbow coupled to linear and also nonlinear electromagnetic gauge fields, we present two new classes of topological black hole solutions. We compute conserved and thermodynamic quantities of these black holes (such as temperature, entropy, electric potential, charge and mass) and show that these quantities satisfy the first law of thermodynamics. In order to study the thermal stability in canonical ensemble, we calculate the heat capacity and determinant of the Hessian matrix and show in what regions there are thermally stable phases for black holes. Also, we discuss the dependence of thermodynamic behavior and thermal stability of black holes on rainbow functions. Finally, we investigate the critical behavior of black holes in the extended phase space and study their interesting properties.

Brief statements are given on recent results in the following areas: 1) The derivation of a variational (maximum) principle for black hole entropy, and an outline of the physically reasonable properties of the related solution for the density of states in the microcanonical equilibrium ensemble (with Jeffrey Melmed, University of Maine). 2) Analysis of topological contributions to black hole entropy in Lovelock gravity, and the corresponding thermodynamic identity (with Jonathan Z. Simon, University of Maryland). 3) Stability analysis for a shell of matter surrounding a black hole in microcanonical (thermal) equilibrium (with Gerald Horwitz, Hebrew University, Jerusalem)—results from this work can be compared with an earlier analysis of the purely mechanical stability of the shell, and with known properties of both the dynamical and thermal stability for the black hole without a shell. 4) The study of a simple ‘singularity'-free model of gravitational collapse, and an examination of the relationship between the ensuing preservation of quantum coherence and the usual perception that mixed states should be associated with the late time emission of Hawking radiation from a classically formed black hole (with Gerard ’t Hooft and Chris Stephens, University of Utrecht).

Conserved vectors are divergencies of superpotentials. In field theory on curved backgrounds, they are useful in calculating global ‘charges’ in arbitrary coordinates and local conserved quantities for small perturbations with specific gauge conditions. Superpotentials are, however, ill–defined. A new criterion of Julia and Silva selects uniquely for Dirichlet boundary conditions the ‘KBL superpotential’ as proposed by Katz, Bičák and Lynden–Bell, which has remarkable properties.Here, we show that a Belinfante–type addition to the KBL superpotential in general relativity gives an expression that is independent of boundary conditions defined by a variational principle. The modified superpotential has the same global properties as the KBL one, except for angular momentum at null infinity, and it does not differ from the KBL superpotential in the linearized theory of gravitation.As an illustration in linearized theory on curved backgrounds, we calculate conserved quantities for small perturbations on a Friedmann–Robertson–Walker spacetime associated with conformal Killing vectors. Our unifying view relates a number of applications in cosmology found in the literature. Globally conserved quantities have simple physical interpretations in the ‘uniform Hubble expansion’ gauge.

Lovelock theory is a natural extension of the Einstein theory of general relativity to higher dimensions in which the first and second orders correspond, respectively, to general relativity and Einstein-Gauss-Bonnet gravity. We present exact black hole solutions of $D\geq 4$-dimensional spacetime for first-, second-, and third-order Lovelock gravities in a string cloud background. Further, we compute the mass, temperature, and entropy of black hole solutions for the higher-dimensional general relativity and Einstein-Gauss-Bonnet theories and also perform thermodynamic stability of black holes. It turns out that the presence of the Gauss-Bonnet term and/or background string cloud completely changes the black hole thermodynamics. Interestingly, the entropy of a black hole is unaffected due to a background string cloud. We rediscover several known spherically symmetric black hole solutions in the appropriate limits.

Previous explorations of the Asymptotic Safety scenario in Quantum Einstein Gravity (QEG) by means of the effective average action and its associated functional renormalization group (RG) equation assumed spacetime manifolds which have no boundaries. Here we take a first step towards a generalization for non-trivial boundaries, restricting ourselves to action functionals which are at most of second order in the derivatives acting on the metric. We analyze two examples of truncated actions with running boundary terms: full fledged QEG within the single-metric Einstein-Hilbert (EH) truncation, augmented by a scale dependent Gibbons-Hawking (GH) surface term, and a bi-metric truncation for gravity coupled to scalar matter fields. The latter contains 17 running couplings, related to both bulk and boundary terms, whose beta-functions are computed in the induced gravity approximation. We find that the bulk and the boundary Newton constant, pertaining to the EH and GH term, respectively, show opposite RG running; proposing a scale dependent variant of the ADM mass we argue that the running of both couplings is consistent with gravitational anti-screening. We describe a simple device for counting the number of field modes integrated out between the infrared cutoff scale and the ultraviolet. This method makes it manifest that, in an asymptotically safe theory, there are effectively no field modes integrated out while the RG trajectory stays in the scaling regime of the underlying fixed point. As an application, we investigate how the semiclassical theory of Black Hole Thermodynamics gets modified by quantum gravity effects and compare the new picture to older work on `RG-improved black holes' which incorporated the running of the bulk Newton constant only. We find, for instance, that the black hole's entropy vanishes and its specific heat capacity turns positive at Planckian scales.

We study thermodynamics and critical behaviors of higher-dimensional Lovelock black holes with non-maximally symmetric horizons in the canonical ensemble of extended phase space. The effects from non-constancy of the horizon of the black hole via appearing two chargelike parameters in thermodynamic quantities of third-order Lovelock black holes are investigated. We find that Ricci flat black holes with nonconstant curvature horizon show critical behavior. This is an interesting feature that is not seen for any kind of black hole in Einstein or Lovelock gravity in the literature. We examine how various interesting thermodynamic phenomena such as standard first-order small-large black hole phase transition, a reentrant phase transition, or zeroth order phase transition happens for Ricci flat, spherical, or hyperbolic black holes with nonconstant curvature horizon depending on the values of Lovelock coefficient and chargelike parameters. While for a spherical black hole of third order Lovelock gravity with constant curvature horizon phase transition is observed only for $7\leq d \leq11$, for our solution criticality and phase transition exist in every dimension. With a proper choice of the free parameters, a large-small-large black hole phase transition occurs. This process is accompanied by a finite jump of the Gibbs free energy referred to as a zeroth-order phase transition. For the case $\kappa=-1$ a novel behavior is found for which three critical points could exist.

In topological terms, we compute the spectral flow of an arbitrary family of self-adjoint Dirac-type operators with classical (local) boundary conditions on a compact Riemannian manifold with boundary under the assumption that the initial and terminal operators of the family are conjugate by an automorphism of the bundle in which the operators act. We use this result to study conditions for the existence of a nonzero spectral flow of a family of self-adjoint Dirac-type operators with local boundary conditions in a two-dimensional domain with a nontrivial topology and discuss possible physical realizations of a nonzero spectral flow.

In this paper we show how to translate into tensorial language the Chern-Weil theorem for the Lorentz symmetry, which equates the difference of the Euler densities of two manifolds to the exterior derivative of a transgression form. For doing so we need to introduce an auxiliary, hybrid, manifold whose geometry we construct explicitely. This allows us to find the vector density, constructed out of spacetime quantities only, whose divergence is the exterior derivative of the transgression form. As a consequence we can show how the Einstein-Hilbert, Gauss-Bonnet and, in general, the Euler scalar densities can be written as the divergences of genuine vector densities in the critical dimensions $D=2,4$, etc. As Lovelock gravity is a dimensional continuation of Euler densities, these results are of relevance for Gauss-Bonnet and, in general, Lovelock gravity. Indeed, these vectors which can be called generalized Katz vectors ensure, in particular, a well-posed Dirichlet variational principle.

Thermodynamics plays an important role in gravitational theories. It is a principle that is independent of gravitational dynamics, and there is still no rigorous proof to show that it is consistent with the dynamical principle. We consider a self-gravitating perfect fluid system with the general diffeomorphism-covariant purely gravitational theory. Based on the Noether charge method proposed by Iyer and Wald, considering static off/on-shell variational configurations, which satisfy the gravitational constraint equation, we rigorously prove that the extrema of the total entropy of a perfect fluid inside a compact region for a fixed total particle number demands that the static configuration is an on-shell solution after we introduce some appropriate boundary conditions, i.e., it also satisfies the spatial gravitational equations. This means that the entropy principle of the fluid stores the same information as the gravitational equation in a static configuration. Our proof is universal and holds for any diffeomorphism-covariant purely gravitational theories, such as Einstein gravity, gravity, Lovelock gravity, f(Gauss-Bonnet) gravity and Einstein-Weyl gravity. Our result indicates the consistency between ordinary thermodynamics and gravitational dynamics.

In this chapter, we discuss the results obtained in [38] with the use of the relative index superposition principle for the spectral flow of families of self-adjoint Dirac type operators with classical elliptic boundary conditions on a compact Riemannian manifold with boundary. In the two-dimensional case, such operators arise, for example, when describing electron states in graphene [39] or in topological insulators [33,68], and their spectral flow has an important physical meaning, being related to the creation of electron–hole (or, more generally, particle–antiparticle) pairs.

Summary This work addresses the development of some alternative hybrid and mixed variational formulations for the geometrically-exact three-dimensional first-order-shear shell boundary value problem [1,2]. In the framework of the complementaryenergy-based formulations, a Legendre transformation is used to introduce the complementary energy density in the variational statements as a function of the cross-sectional resultants only. The corresponding variational principles are shown to feature stationarity within the framework of the boundary-value-problem. The main features of the principles are highlighted, giving special attention to their relationships from both theoretical and numerical point of view. Variational principles constitute the core of the development of numerical methods in Solid Mechanics. The utility of such principles is two-fold: first, they provide a very convenient method for the derivation of the governing equations and natural boundary conditions of the boundary value problem and, second, they provide the mathematical foundation required to produce consistent numerical approximations. In this second role, the variational methods have been most useful in Computational Solid Mechanics [3]. Many different variational principles can be constructed depending on the equations enforced in the weak form. Perhaps the most remarkable variational principle is the Principle of Stationary Total Potential Energy, which states that, among all kinematically admissible displacement fields, those that satisfy the equilibrium conditions in the domain and at the boundary, lead to a stationary value of the total potential energy functional. In this paper, after the presentation of many mixed variational principles for the nonlinear analysis of shells, we derive a hybrid form of the Two-field Principle of Total Complementary Potential Energy, which is used on a totally consistent derivation of equilibrated co-rotational finite elements [4].

Study of thermal boundary conditions for a plain journal bearing

The desire for new mostly fluent “soft spaces” in architecture cannot be overseen any more. Therefore new tools and approaches are tried out to create architecture with special spatial qualities. In combination with common building-technologies and considered as architectural elements, spatially curved membrane structures and especially anticlastic Minimal Surfaces offer one possible approach to this topic. This paper presents the result of the research on Minimal Surfaces of membrane corner solutions representing one part of a comprehensive basic research in this field. In connection with the exploration of corner solutions, the meaning and potential of analog and digital approaches became obvious. Depending on the potential of the selected tool, the desired results or the purpose of the research one of these strategies will be preferred. Further on this paper partly reveals new correlations between Minimal Surface and boundary conditions and so far unknown rules in its selforganizing processes. Case studies document new capabilities in designing and creating space in architecture. Latest approaches are dealing with alternative boundary conditions, case studies and with software implementation in terms of scripting found rules.

If gravity is an emergent phenomenon, as suggested by several recent results, then the structure of the action principle for gravity should encode this fact. With this motivation we study several features of the Einstein-Hilbert action and establish direct connections with horizon thermodynamics. We begin by introducing the concept of holographically conjugate variables (HCVs) in terms of which the surface term in the action has a specific relationship with the bulk term. In addition to g_{ab} and its conjugate momentum \sqrt{-g} M^{cab}, this procedure allows us to (re)discover and motivate strongly the use of f^{ab}=\sqrt{-g}g^{ab} and its conjugate momentum N^c_{ab}. The gravitational action can then be interpreted as a momentum space action for these variables. We also show that many expressions in classical gravity simplify considerably in this approach. For example, the field equations can be written in a form analogous to Hamilton's equations for a suitable Hamiltonian if we use these variables. More importantly, the variation of the surface term, evaluated on any null surface which acts a local Rindler horizon can be given a direct thermodynamic interpretation. The term involving the variation of the dynamical variable leads to T\delta S while the term involving the variation of the conjugate momentum leads to S\delta T. We have found this correspondence only for the choice of variables (g_{ab}, \sqrt{-g} M^{cab}) or (f^{ab}, N^c_{ab}). We use this result to provide a direct thermodynamical interpretation of the boundary condition in the action principle, when it is formulated in a spacetime region bounded by the null surfaces. We analyse these features from several different perspectives and provide a detailed description, which offers insights about the nature of classical gravity and emergent paradigm.

We discuss the criteria that must be satisfied by a well-posed variational principle. We clarify the role of Gibbons-Hawking-York type boundary terms in the actions of higher derivative models of gravity, such as $F(R)$ gravity, and argue that the correct boundary terms are the naive ones obtained through the correspondence with scalar-tensor theory, despite the fact that variations of normal derivatives of the metric must be fixed on the boundary. We show in the case of $F(R)$ gravity that these boundary terms reproduce the correct Arnowitt-Deser-Misner energy in the Hamiltonian formalism, and the correct entropy for black holes in the semiclassical approximation.