36 pages, 7 figures. Invited contribution to the Focus Issue on "Black holes and fundamental fields" to appear in Classical and Quantum Gravity; v2: typos corrected, minor revision, added references, matches published version in CQG
We review the black hole solutions of the ghost-free massive gravity theory and its bimetric extension and outline the main results on the stability of these solutions against small perturbations. Massive (bi)-gravity accommodates exact black hole solutions, analogous to those of General Relativity. In addition to these solutions, hairy black holes -- solutions with no correspondent in General Relativity -- have been found numerically, whose existence is a natural consequence of the absence of Birkhoff's theorem in these theories. The existence of extra propagating degrees of freedom, makes the stability properties of these black holes richer and more complex than those of General Relativity. In particular, the bi-Schwarzschild black hole exhibits an unstable spherically symmetric mode, while the bi-Kerr geometry is also generically unstable, both against the spherical mode and against superradiant instabilities. If astrophysical black holes are described by these solutions, the superradiant instability of the Kerr solution imposes stringent bounds on the graviton mass.
This is a heuristic introduction to massive gravity based on an analogy with perfect fluids. I will argue that massive gravity can be thought of as Einstein gravity in the presence of a medium with unusual properties.
The linearized equations of `New Massive Gravity' propagate a parity doublet of massive spin-2 modes in 3D Minkowski spacetime, but a different non-linear extension is made possible by `third-way' consistency. There is a `Chern-Simons-like' action, as for other 3D massive gravity models, but the new theory is `exotic': its action is parity odd. This `Exotic Massive Gravity' is the next-to-simplest case in an infinite sequence of third-way consistent 3D gravity theories, the simplest being the `Minimal Massive Gravity' alternative to `Topologically Massive Gravity'.
We discuss locally Weyl (scale) covariant generalisations of gravitational theories using Riemann-Cartan-Weyl space-times in arbitrary dimensions. We demonstrate the procedure of Weyl gauging on two examples in particular: General relativity, and topologically massive gravity in three dimensions.
We construct a generalized massive gravity by combining quadratic curvature gravity with the Chern-Simons term in four dimensions. This may be a candidate for the parity-odd tricritical gravity theory. Considering the AdS$_4$ vacuum solution, we derive the linearized Einstein equation, which is not similar to that of the three dimensional (3D) generalized massive gravity. When a perturbed metric tensor is chosen to be the Kerr-Schild form, the linearized equation reduces to a single massive scalar equation. At the tricritical points where two masses are equal to -1 and 2, we obtain a log-squar…
We present locally scale (Weyl) covariant generalisation of Minimal Massive Gravity theory using the language of exterior differential forms on Riemann-Cartan-Weyl space-times. The theory is expressed by a locally scale invariant action and locally scale covariant field equations are found by a first order variational formalism.
Constraint analysis of the three-dimensional massive gravity, the so- called new massive gravity, is studied in the Palatini formalism. We show that amongst 6 components of the metric, 2 are dynamical, which is compatible with the existence of one vector massive graviton in the linearized theory (Fierz-Pauli theory).
An auxiliary metric (reference metric) is inevitable in massive gravity theory. In the scenario of gauge/gravity duality, a singular reference metric corresponds to momentum dissipations, which describes the electric and heat conductivity for normal conductors. We demonstrate in detail that the massive gravity with singular reference metric is ghost-free.
We consider a recently proposed non-minimal matter coupling in massive gravity and multi-gravity. We argue that, when formulated in terms of unconstrained vielbeins, this matter coupling contains the primary constraints necessary to remove the Boulware-Deser ghost to all orders away from the decoupling limit.
In a recent paper in EPJC March 2016, Faria has shown that quantum massive conformal gravity is renormalizable but has ghost states. We comment this paper on the aspect of renormalizability.
We study the generalized massive gravity in three dimensional flat spacetime. A massive logarithmic mode is propagating in the flat spacetime at the critical point where two masses degenerate. Furthermore, we discuss the logarithmic extension of the Galilean conformal algebra (GCA) which may arise from the exotic and standard rank-2 logarithmic conformal field theory (LCFT) on the boundary of AdS3 spacetime.
We investigate the new massive conformal gravity which is not invariant under conformal transformations, in comparison to the massive conformal gravity. We find five polarization modes of gravitational waves propagating on the Minkowski spacetimes. The stability of Minkowski spacetimes is guaranteed if the mass squared is not negative and the linearized Ricci tenor is employed to describe a massive spin-2 graviton. However, the small Schwarzschild black hole is unstable against the $s$-mode massive graviton perturbations.
Recently a new extended nonlinear massive gravity model has been proposed which includes the $F(R)$ modifications to dRGT model.We follow the $F(R)$ nonlinear massive gravity and study its implications on cosmological evolutions. We derive the critical points of the cosmic system and study the corresponding kinetics by performing the phase-plane analysis.
Recently in (Eur. Phys. J. C 76:341, 2016) [arXiv:1604.08706], Myung has suggested that the renormalizability of massive conformal gravity is meaningless unless the massive ghost states of the theory are stable. Here we show that massive conformal gravity can be renormalizable having unstable ghost states.
We perform the Hamiltonian analysis of 1+1 dimensional non-linear massive gravity studied in arXiv:1107.3820. We find the constraint structure of given theory and perform the counting of the physical degrees of freedom.
In this paper, we investigate the fluid/gravity correspondence in the framework of massive Einstein gravity. Treating the gravitational mass terms as an effective energy-momentum tensor and utilizing the Petrov-like boundary condition on a timelike hypersurface, we find that the perturbation effects of massive gravity in bulk can be completely governed by the incompressible Navier-Stokes equation living on the cutoff surface under the near horizon and nonrelativistic limits. Furthermore, we have concisely computed the ratio of dynamical viscosity to entropy density for two massive Einstein gra…
A proof that new massive gravity - the massive 3D gravity model proposed by Bergshoeff, Hohm and Townsend (BHT) - is the only unitary system at the tree level that can be constructed by augmenting planar gravity through the curvature-squared terms, is presented. Two interesting gravitational properties of the BHT model, namely, time dilation and time delay, which have no counterpart in the usual Einstein 3D gravity, are analyzed as well.
One of the fundamental open questions in cosmology is whether we can regard the universe evolution without singularity like a Big Bang or a Big Rip. This challenging subject stimulates one to regard a nonsingular universe in the far past with an arbitrarily large vacuum energy. Considering the high energy regime in the cosmic history, it is believed that Einstein gravity should be corrected to an effective energy dependent theory which could be acquired by gravity's rainbow. On the other hand, employing massive gravity provided us with solutions to some of the long standing fundamental problem…
A ghost free massive deformation of unimodular gravity (UG), in the spirit of {\em mimetic massive gravity}, is shown to exist. This construction avoids the no-go theorem for a Fierz-Pauli type of mass term in UG by giving up on Lorentz invariance. In our framework, the mimetic degree of freedom vanishes on-shell.
We define conserved gravitational charges in -cosmologically extended- topologically massive gravity, exhibit them in surface integral form about their de-Sitter or flat vacua and verify their correctness in terms of two basic types of solution.
We present a simplified formulation of massive gravity where the Higgs fields have quadratic kinetic term. This new formulation allows us to prove in a very explicit way that all massive gravity theories considered so far inevitably have Boulware-Deser ghost in non-trivial fluctuations of background metric.
We consider the massless limit of Higgs gravity, where the graviton becomes massive when the scalar fields acquire expectation values. We determine the Vainshtein scale and prove that massive gravity smoothly goes to General Relativity below this scale. We find that the Vainshtein scale depends on the particular action of scalar fields used to give mass to the graviton.
We study the null geodesics in a static circularly symmetric (SCS) black hole spacetime which is a solution in the $(2+1)$-dimensional massive gravity proposed by Bergshoeff, Hohm and Townsend (BHT massive gravity). We obtain analytic solutions for the null geodesic equation in the SCS black hole background and find the explicit form of deflection angles. We see that for various values of the impact parameter, the deflection angle can be positive, negative or even zero in this black hole spacetime. The negative deflection angle indicates the repulsive behavior of the gravity which comes from t…
In this paper, we study the collapsing scenario for the k-essence emergent Vaidya spacetime in the context of massive gravity's rainbow. For this study, we consider that the background metric is Vaidya spacetime in massive gravity's rainbow. We show that the k-essence emergent gravity metric resembles closely to the new type of generalized Vaidya massive gravity metric with the rainbow deformations for null fluid collapse where we consider the k-essence scalar field as a function solely of the advanced or the retarded time. The k-essence emergent Vaidya massive gravity rainbow mass function is…
We compute the tree-level scattering amplitude between two covariantly conserved sources in generic Cosmological Topologically Massive Gravity augmented with a Fierz-Pauli term that has three massive degrees of freedom. We consider the Chiral Gravity limit in the anti-de Sitter space as well as the limit of Flat-Space Chiral Gravity. We show that Chiral Gravity cannot be unitarily deformed with a Fierz-Pauli mass. We calculate the non-relativistic potential energy between two point-like spinning sources. In addition to the expected mass-mass and spin-spin interactions, there are mass-spin inte…
The "complexity = action" duality states that the quantum complexity is equal to the action of the stationary AdS black holes within the Wheeler-DeWitt patch at late time approximation. We compute the action growth rates of the neutral and charged black holes in massive gravity and the neutral, charged and Kerr-Newman black holes in $f(R)$ gravity to test this conjecture. Besides, we investigate the effects of the massive graviton terms, higher derivative terms and the topology of the black hole horizon on the complexity growth rate.
On the holographic complexity dual to the bulk action, we investigate the action growth for a shock wave geometry in a massive gravity theory within the Wheeler-De Witt (WDW) patch at the late time limit. For a global shock wave, the graviton mass does not affect the action growth in the bulk, i.e. the complexity on the boundary, showing that the action growth (complexity) is the same for both the Einstein gravity and the massive gravity. Nevertheless, for a local shock wave that depends on transverse coordinates, the action growth (complexity) is proportional to the butterfly velocity for the…
We introduce the Brans-Dick de Rham-Gabadadze-Tolley massive gravity theory which is the new extension of nonlinear massive gravity. We demonstrate a detailed study of the cosmological properties of this theory of gravity, and we show the transformation of the Jordan frame to the Einstein frame. We obtain the cosmological background equations and show the analyses of self-accelerating solutions for explaining the accelerated expansion of the Universe. In the following, we analyze the background perturbations, which consist of tensor, vector, and scalar perturbations within the framework of the…
We analyze the classical stability of Schwarzschild black hole in massive conformal gravity which was recently proposed for another massive gravity model. This model in the Jordan frame is conformally equivalent to the Einstein-Weyl gravity in the Einstein frame. The coupled linearized Einstein equation is decomposed into the traceless and trace equation when one chooses $6m^2φ=δR$. Solving the traceless equation exhibits unstable modes featuring the Gregory-Laflamme $s$-mode instability of five-dimensional black string, while we find no unstable modes when solving the trace equation. It is sh…
We study spherically-symmetric solutions in Massive Gravity generated by matter sources with polytropic equation of state. We concentrate in the non-perturbative regime where the mass term non-linearities are important, and present the main features of the solutions.
In this work, a correspondence between black hole solutions of conformal and massive theories of gravity is found. It is seen that this correspondence imposes some constraints on parameters of these theories. What is more, a relation between the mass of black holes and the parameters of massive gravity is found. Indeed, the acceptable ranges of massive gravity parameters ($c_{1}$\ and $c_{2}$) are found. It is shown that by considering the positive mass of black holes, some ranges of $c_{1}$\ and $c_{2}$ are acceptable.
In this paper, we generalize the study of the model of holographic superconductor in the excited states to the framework of massive gravity. By taking into account the effect of massive graviton, we numerically present a family of solutions of the holographic superconductor in excited states, and find that the critical temperatures can be higher due to the effect of massive graviton comparing with the superconductor in Einstein gravity. We also investigate the condensates and conductivities in the ground state and the excited states with studying various parameters that determine the framework…
We investigate the possibility of a new massive gravity theory with derivative interactions as an extension of de Rham-Gabadadze-Tolley massive gravity. We find the most general Lagrangian of derivative interactions using Riemann tensor whose cutoff energy scale is $Λ_3$, which is consistent with de Rham-Gabadadze-Tolley massive gravity. Surprisingly, this infinite number of derivative interactions can be resummed with the same method in de Rham-Gabadadze-Tolley massive gravity, and remaining interactions contain only two parameters. We show that the equations of motion for scalar and tensor m…
I review how "classical SUGRA" embeddability establishes positive energy E for D=3 topologically massive gravity (TMG), with or without a cosmological term, a procedure familiar from D=4 Einstein gravity (GR). It also provides explicit expressions for E. In contrast to GR, E is not manifestly positive, due to the peculiar two-term nature of TMG.
The existence of a ghost free theory of massive gravity begs for an interpretation as a Higgs phase of General Relativity. We revisit the study of massive gravity as a Higgs phase. Absent a compelling microphysical model of spontaneous symmetry breaking in gravity, we approach this problem from the viewpoint of nonlinear realizations. We employ the coset construction to search for the most restrictive symmetry breaking pattern whose low energy theory will both admit the de Rham-Gabadadze-Tolley (dRGT) potentials and nonlinearly realize every symmetry of General Relativity, thereby providing a …
Using the metric formalism, we study the derivative mixings of spin-2 fields in massive bi-Gravity. Necessary (but not sufficient) criteria are given for such mixings to be ghost free. Examples satisfying those criteria are studied and it is shown that in the decoupling limit they host a ghost.
We study the conserved charges of supersymmetric solutions in the topologically massive gravity theory for both asymptotically flat and constant curvature geometries.
We study van Dam-Veltman-Zakharov discontinuity in the topologically new massive gravity (TNMG). The reduction from 2 degrees of freedom to one is interpreted as van Dam-Veltman-Zakharov discontinuity appeared when going from anti-de Sitter spacetime to Minkowski spacetime in the linearized TNMG.
We present a 3-dimensional model for massive gravity with masses induced by topological (Chern-Simons) and Proca-like mass terms. Causality and unitarity are discussed at tree-level. Power-counting renormalizability is also contemplated.
We find the Gregory-Laflamme $s$-mode instability of the non-rotating BTZ black hole in new massive gravity. This instability shows that the BTZ black hole could not exist as a stable static solution to the new massive gravity. For non-rotating BTZ black string in four dimensions, however, it is demonstrated that the BTZ black string can be stable against the metric perturbation.
We show, by analyzing its characteristics, that the ghost-free, 5 degree of freedom, Wess--Zumino massive gravity model admits superluminal shock wave solutions and thus is acausal. Ironically, this pathology arises from the very constraint that removes the (sixth) Boulware-Deser ghost mode.
We construct a Weyl-invariant extension of topologically massive gravity which, remarkably, turns out to include topologically massive electrodynamics, with a Proca mass term, conformally coupled to a scalar field. The action has no dimensionful parameters, therefore, the masses are generated via symmetry breaking either radiatively in flat backgrounds or spontaneously in constant curvature backgrounds. The broken phase of the theory, generically, has a single massive spin-2 and a massive spin-1 excitation. Chiral gravity in asymptotically anti-de Sitter spacetimes does not arise as a low ener…
As massive gravity and its extensions offer physically well-defined gravitational theories with a nonzero graviton mass, we present a new extension of the de Rham-Gabadadze-Tolley (dRGT)massive gravity, which is tachyonic massive gravity theory. We firstly introduce the new extension of the dRGT massive gravity, constructed by adding a tachyonic term. We then find the cosmological background equations, and present the analysis of self-accelerating solutions. We examine the tensor perturbations to calculate the dispersion relation of gravitational waves (GWs). In a special case, we consider a c…
We present a new class of solutions describing charged black holes in massive (bi)gravity. For a generic choice of the parameters of the massive gravity action, the solution is the Reissner-Nordstrom-de Sitter metric written in the Eddington-Finkelstein coordinates for both metrics. We also study a special case of the parameters, for which the space of solutions contains an extra symmetry.
We find the excitations and construct the conserved charges ( mass and angular momentum) of the recently found Minimal Massive Gravity (MMG) in 2+1 dimensions in asymptotically Anti-de Sitter (AdS) spacetimes. The field equation of the theory does not come from an action and hence lacks the required Bianchi Identity needed to define conserved charges. But the theory, which also provides a healthy extension of the Topologically Massive Gravity in the bulk and boundary of spacetime, does admit conserved charges for metric that are solutions. Our construction is based on background Killing vector…
We present an exact solution in 3-dimensional topologically massive gravity with negative cosmological constant which dynamically interpolates between a past horizon and a chiral AdS pp-wave. Similarly, upon time reversal, one obtains an AdS pp-wave with a future event horizon
We obtain static, spherically symmetric, and asymptotically flat numerical solutions of massive gravity with a source. Those solutions show, for the first time explicitly, a recovery of the Schwarzschild solution of General Relativity via the so-called Vainshtein mechanism.
Partially massless theory in three dimensions is revisited and its relationship with the self-dual massive gravity is considered. The only mode of the partially massless theory is shown explicitly through an action for a scalar field on (A)dS background. This fact can be generalized to higher dimensions. This degree of freedom is altered when a triadic Chern-Simons is introduced, giving rise to the self-dual massive gravity on (A)dS background. We present another physical system with partially massless symmetry and its connection with topologically massive gravity is discussed.
Motivated by recent developments in black hole thermodynamics, we investigate van der Waals phase transitions of charged black holes in massive gravity. We find that massive gravity theories can exhibit strikingly different thermodynamic behaviour compared to that of Einstein gravity, and that the mass of the graviton can generate a range of new phase transitions for topological black holes that are otherwise forbidden.
We study regular spacelike warped black holes in the three dimensional general massive gravity model, which contains both the gravitational Chern-Simons term and the linear combination of curvature squared terms characterizing the new massive gravity besides the Einstein-Hilbert term. The parameters of the metric are found by solving a quartic equation constrained by an inequality that imposes the absence of closed timelike curves. Explicit expressions for the central charges are suggested by exploiting the fact that these black holes are discrete quotients of spacelike warped AdS(3) and a kno…
We perform the Hamiltonian analysis of non-linear massive gravity action studied recently in arXiv:1106.3344 [hep-th]. We show that the Hamiltonian constraint is the second class constraint. As a result the theory possesses an odd number of the second class constraints and hence all non physical degrees of freedom cannot be eliminated.
We show that linear perturbations around the simplest black hole solution of massive bi-gravity theories, the bi-Schwarzschild solution, exhibit an unstable mode featuring the Gregory-Laflamme instability of higher-dimensional black strings. The result is obtained for the massive gravity theory which is free from the Boulware-Deser ghost, as well as for its extension with two dynamical metrics. These results may indicate that static black holes in massive gravity do not exist. For graviton mass of order of Hubble scale, however, the instability timescale is of order of the Hubble time.
We obtain the Misner-Sharp mass in the massive gravity for a four dimensional spacetime with a two dimensional maximally symmetric subspace via the inverse unified first law method. Significantly, the stress energy is conserved in this case with a widely used reference metric. Based on this property we confirm the derived Misner-Sharp mass by the conserved charge method. We find that the existence of the Misner-sharp mass in this case does not lead to extra constraint for the massive gravity, which is notable in modified gravities. In addition, as a special case, we also investigate the Misner…
In this paper, we have analytically investigated $p$-wave holographic superconductor in the framework of massive gravity in the probe limit. We obtained the analytical expressions for the critical temperature, the value of the condensate operator, and the difference of the free energy between the superconductor and normal phases. We studied the behavior of these quantities in the presence of the mass of graviton, which is found to be dependent almost on the sign of the massive gravity couplings. The critical temperature becomes (lower)higher and the condensate value gets (larger)smaller with (…
We study the constraints coming from local causality requirement in various $2+1$ dimensional dynamical theories of gravity. In topologically massive gravity, with a single parity non-invariant massive degree of freedom, and in new massive gravity, with two massive spin-$2$ degrees of freedom, causality and unitarity are compatible with each other and both require the Newton's constant to be negative. In their extensions, such as the Born-Infeld gravity and the minimal massive gravity the situation is similar and quite different from their higher dimensional counterparts, such as quadratic (e.…
We propose and study a new action for three-dimensional massive gravity. This action takes a very simple form when written in terms of connection and triad variables, but the connection can also be integrated out to obtain a triad formulation. The quadratic action for the perturbations around a Minkowski background reproduces the action of self-dual massive gravity, in agreement with the expectation that the theory propagates a massive graviton. We confirm this result at the non-linear level with a Hamiltonian analysis, and show that this new theory does indeed possess a single massive degree …
The solution of topologically massive gravity with cosmological constant is reduced, for space-times with two commuting Killing vectors, to a special-relativistic dynamical problem. This approach is applied to the construction of a class of exact sourceless, horizonless solutions asymptotic to the BTZ extreme black holes.
The paper at hand studies the heat engine provided by black holes in the presence of massive gravity. The main motivation is to investigate the effects of massive gravity on different properties of the heat engine. It will be shown that massive gravity parameters and graviton's mass modify the efficiency of engine on a significant level. Furthermore, it will be shown that it is possible to have the heat engine for non-spherical black holes in massive gravity and we study the effects of topological factor on properties of the heat engine. Surprisingly, it will be shown that the highest efficien…
We study Kundt solutions of topologically massive gravity (TMG) and the new theory of massive gravity (NMG), proposed recently in arXiv:0901.1766. For topologically massive gravity, only the CSI Kundt solutions (i.e., solutions with constant scalar polynomial curvature invariants) were found very recently in arXiv:0906.3559. We find non-CSI explicit solutions of TMG, when $Λ=-μ^2$, and these are the first such solutions. For the new theory of massive gravity, after reducing the field equations to a manageable system of differential equations, the CSI solutions are discussed in detail, with a f…
In general relativity, the instantaneous contributions to the gravitational potentials cancel each other in observable, leaving the theory free of physical instantaneous interactions. In models where these subtle cancellations are spoiled by the presence of fields that break Lorentz invariance, physical instantaneous interactions are possible. Such interactions are studied for a model of Lorentz- violating massive gravity.
I present the stationary Lifshitz black hole solution of three-dimensional New Massive Gravity theory and study its elementary geometric and thermodynamical properties.
