We review recent progress in massive gravity. We start by showing how different theories of massive gravity emerge from a higher-dimensional theory of general relativity, leading to the Dvali-Gabadadze-Porrati model, cascading gravity and ghost-free massive gravity. We then explore their theoretical and phenomenological consistency, proving the absence of Boulware-Deser ghosts and reviewing the Vainshtein mechanism and the cosmological solutions in these models. Finally we present alternative and related models of massive gravity such as new massive gravity, Lorentz-violating massive gravity and non-local massive gravity.

Massive Gravity

We show that the linearization of all exact solutions of classical chiral gravity around the ${\mathrm{AdS}}_{3}$ vacuum have positive energy. Nonchiral and negative-energy solutions of the linearized equations are infrared divergent at second order, and so are removed from the spectrum. In other words, chirality is confined and the equations of motion have linearization instabilities. We prove that the only stationary, axially symmetric solutions of chiral gravity are BTZ black holes, which have positive energy. It is further shown that classical log gravity---the theory with logarithmically relaxed boundary conditions---has finite asymptotic symmetry generators but is not chiral and hence may be dual at the quantum level to a logarithmic conformal field theories (CFT). Moreover we show that log gravity contains chiral gravity within it as a decoupled charge superselection sector. We formally evaluate the Euclidean sum over geometries of chiral gravity and show that it gives precisely the holomorphic extremal CFT partition function. The modular invariance and integrality of the expansion coefficients of this partition function are consistent with the existence of an exact quantum theory of chiral gravity. We argue that the problem of quantizing chiral gravity is the holographic dual of the problem of constructing an extremal CFT, while quantizing log gravity is dual to the problem of constructing a logarithmic extremal CFT.

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Chiral gravity, log gravity, and extremal CFT

The Rise of Gauge Theory of Gravity Up to 1961: From Special to General Relativity Theory Analyzing General Relativity Theory A Fresh Start by Yang - Mills and Utiyama Poincare Gauge Theory: Einstein - Cartan( - Sciama - Kibble) Theory as a Viable Gravitational Theory General Structure of Poincare Gauge Theory (Including Quadratic Lagrangians) Translational Gauge Theory Fallacies About Torsion Extending the Gauge Group of Gravity: Poincare Group Plus Scale Transformations: Weyl - Cartan Gauge Theory of Gravity From the Poincare to the Affine Group: Metric-Affine Gravity Conformal Gauge Theory of Gravity (Anti-)de Sitter Gauge Theory of Gravity From the Square Root of Translations to the Super-Poincare Group Specific Subjects of Metric-Affine Gravity and Poincare Gauge Theory: Hamiltonian Structure Equations of Motion for Matter Cosmological Models Exact Solutions Poincare Gauge Theory in Three Dimensions Dislocations and Torsion The Yang Episode: A Historical Case Study.

Gauge Theories Of Gravitation: A Reader With Commentaries

We present a new cubic theory of gravity in five dimensions which has second-order traced field equations, analogous to BHT new massive gravity in three dimensions. Moreover, for static spherically symmetric spacetimes all the field equations are of second order, and the theory admits a new asymptotically locally flat black hole. Furthermore, we prove the uniqueness of this solution, study its thermodynamical properties and show the existence of a C-function for the theory following the arguments of Anber and Kastor (2008 J. High Energy Phys. JHEP05(2008)061 (arXiv:0802.1290 [hep-th])) in pure Lovelock theories. Finally, we include the Einstein–Gauss–Bonnet and cosmological terms and find new asymptotically AdS black holes at the point where the three maximally symmetric solutions of the theory coincide. These black holes may also possess a Cauchy horizon.

A new cubic theory of gravity in five dimensions: black hole, Birkhoff's theorem and C-function

In this article we review recent progress on the holographic modelling of field theories with Lifshitz symmetry We focus in particular on the holographic dictionary for Lifshitz backgrounds - the relationship between bulk fields and boundary operators, operator correlation functions and underlying geometrical structure The holographic dictionary is essential in identifying the universality class of strongly coupled Lifshitz theories described by gravitational models

Lifshitz holography

We show that three-dimensional massive gravity admits Lifshitz metrics with generic values of the dynamical exponent $z$ as exact solutions. At the point $z=3$, exact black hole solutions that are asymptotically Lifshitz arise. These spacetimes are three-dimensional analogues of those that were recently proposed as gravity duals for anisotropic scale invariant fixed points.

Lifshitz black hole in three dimensions

Fil: Ayon Beato, Eloy Centro de Investigacion y de Estudios Avanzados del Instituto Politecnico Nacional; Mexico

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Analytic Lifshitz black holes in higher dimensions

We study exact solutions to cosmological topologically massive gravity coupled to topologically massive electrodynamics at special values of the coupling constants. For the particular case of the so-called chiral point l{mu}{sub G}=1, vacuum solutions (with vanishing gauge field) are exhibited. These correspond to a one-parameter deformation of general relativity solutions, and are continuously connected to the extremal Banados-Teitelboim-Zanelli black hole with bare constants J=-lM. At the chiral point this extremal Banados-Teitelboim-Zanelli black hole turns out to be massless, and thus it can be regarded as a kind of ground state. Although the solution is not asymptotically AdS{sub 3} in the sense of Brown-Henneaux boundary conditions, it does obey the weakened asymptotic recently proposed by Grumiller and Johansson. Consequently, we discuss the holographic computation of the conserved charges in terms of the stress tensor in the boundary. For the case where the coupling constants satisfy the relation l{mu}{sub G}=1+2l{mu}{sub E}, electrically charged analogues to these solutions exist. These solutions are asymptotically AdS{sub 3} in the strongest sense, and correspond to a logarithmic branch of self-dual solutions previously discussed in the literature.

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Asymptotically AdS 3 solutions to topologically massive gravity at special values of the coupling constants

We revisit the description of the space of asymptotically AdS3 solutions of pure gravity in three dimensions with a negative cosmological constant as a collection of coadjoint orbits of the Virasoro group. Each orbit corresponds to a set of metrics related by diffeomorphisms which do not approach the identity fast enough at the boundary. Orbits contain more than a single element and this fact manifests the global degrees of freedom of AdS3 gravity, being each element of an orbit what we call boundary graviton. We show how this setup allows to learn features about the classical phase space that otherwise would be quite difficult. Most important are the proof of energy bounds and the characterization of boundary gravitons unrelated to BTZs and AdS3. In addition, it makes manifest the underlying mathematical structure of the space of solutions close to infinity. Notably, because of the existence of a symplectic form in each orbit, being this related with the usual Dirac bracket of the asymptotic charges, this approach is a natural starting point for the quantization of different sectors of AdS3 gravity. We finally discuss previous attempts to quantize coadjoint orbits of the Virasoro group and how this is relevant for the formulation of AdS3 quantum gravity.

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Alan Garbarz

Papers

Lifshitz black hole in three dimensions

Analytic Lifshitz black holes in higher dimensions

Asymptotically AdS 3 solutions to topologically massive gravity at special values of the coupling constants

Classification of Boundary Gravitons in AdS$_3$ Gravity

Quantization of BMS3 orbits: A perturbative approach