On the basis of the state parameter of de Sitter space-time satisfying the first law of thermodynamics,we can derive some effective thermodynamic quantities.When the temperature of the black hole horizon is equal to that of the cosmological horizon, we think that the effective temperature of the space-time should have the same value. Using this condition, we obtain a differential equation of the entropy of the de Sitter black hole in the higherdimensional de Rham, Gabadadze and Tolley (dRGT) massive gravity. Solving the differential equation, we obtain the corrected entropy and effective thermodynamic quantities of the de Sitter black hole. The results show that for multiparameter black holes, the entropy satisfied differential equation is invariable with different independent state parameters. Therefore, the entropy of higher-dimensional dS black holes in dRGT massive gravity is only a function of the position of the black hole horizon, and is independent of other state parameters. It is consistent with the corresponding entropy of the black hole horizon and the cosmological horizon. The thermodynamic quantities of self-consistent de Sitter spacetime are given theoretically, and the equivalent thermodynamic quantities have the second-order phase transformation similar to AdS black hole, but unlike AdS black hole, the equivalent temperature of de Sitter space-time has a maximum value. By satisfying the requirement of thermodynamic equilibrium and stability of space-time, the conditions for the existence of dS black holes in the universe are obtained.
In this paper, we study the thermodynamics and geothermodynamics of spherical black hole solutions in dRGT massive gravity in a new extended phase space. Inspired by the work of Kastor et al. [1], by interpreting the graviton mass as a thermodynamical variable, we propose a first law of thermodynamics which include a mass term and establish a new Smarr Formula. Then, we perform a thermodynamical analysis to reveal the existence of a critical behavior for black holes in dRGT massive gravity with two different critical points through canonical and grand canonical ensembles. To consolidate these …
In this paper, the holographic p-wave superfluid model with charged complex vector field is studied in dRGT massive gravity beyond the probe limit. The stability of p-wave and p+ip solutions are compared in the grand canonical ensemble. The p-wave solution always get lower value of grand potential than the p+ip solution, showing that the holographic system still favors an anisotropic (p-wave) solution even with considering a massive gravity theory in bulk. In the holographic superconductor models with dRGT massive gravity in bulk, a key sailing symmetry is found to be violated by fixing the re…
An interesting proposal has recently been made to extend massive gravity models beyond dRGT by a disformal transformation of the metric. In this Letter we want to note that it can be viewed as a mimetic extension of dRGT gravity which enormously simplifies the Hamiltonian analysis. In particular, pure gravity sector is equivalent to the usual dRGT gravity coupled to a constrained scalar field. And we also give some comments about possible matter couplings.
In the context of the dRGT massive gravity coupled with nonlinear electrodynamics, we present new dRGT black hole solutions. Together with the thermodynamical properties of the solutions, we study the greybody factor of the corresponding black hole solution. To this end, we compute the rigorous bound on the greybody factor for the obtained dRGT black holes. The obtained results are graphically represented for different values of the theory's physical parameters. Our analysis shows that the charged dRGT black holes of nonlinear electrodynamics evaporate quicker than the charged dRGT black holes…
We explore the quasi-stationary profile of massive charged scalar field in a class of charged black hole in dRGT massive gravity. We discuss how the linear term in the metric which is a unique character of the dRGT massive gravity affects structure of the spacetime. Numerical calculations of the quasinormal modes are performed for the charged scalar field in the dRGT black hole background. For asymptotically de Sitter (dS) black hole, an improved asymptotic iteration method is used to obtain the associated quasinormal frequencies. The unstable modes are found for $\ell=0$ case and their corres…
Recently discovered models of ghost-free massive gravity and bigravity are characterized by a non-trivial potential that gives rise to a rich vacuum structure. We review maximally symmetric vacua of the de Rham-Gabadadze-Tolley (dRGT) massive gravity and of the Hassan-Rosen (HR) bigravity, and discuss their perturbative stability. In particular, we discuss perturbations about self-accelerating vacua in HR bigravity, and argue that, analogously to what was found in the case of dRGT gravity, some of them contain strongly coupled modes. We then show that it is impossible to construct regular inst…
It is commonly believed that the dRGT theory is the unique way to describe a massive spin-2 field without ghosts. While dRGT is arguably the most elegant massive gravity theory, it seems that it may not be the unique ghost-free one if one relaxes the assumption of locality. In this work, we derive a new massive gravity theory by using a disformal transformation of the metric tensor in the dRGT action. Its decoupling limit lives inside the class of beyond-Horndeski Lagrangians as long as the transformation of the metric remains purely disformal. This proves the absence of ghosts in this decoupl…
We study the $P-V$ criticality and phase transition in the extended phase space of anti-de Sitter (AdS) black holes in higher-dimensional de Rham, Gabadadze and Tolley (dRGT) massive gravity, treating the cosmological constant as pressure and the corresponding conjugate quantity is interpreted as thermodynamic volume. Besides the usual small/large black hole phase transitions, the interesting thermodynamic phenomena of reentrant phase transitions (RPTs) are observed for black holes in all $d\geq6$-dimensional spacetime when the coupling coefficients $c_i m^2$ of massive potential satisfy some …
The greybody factor of the massive Dirac field around the black hole in the dRGT massive gravity theory is investigated using the rigorous bound and the WKB approximation methods. In both methods, the greybody factor significantly depends on the shape of the potential. If the potential is smaller, there is more probability for the Dirac field to transmit through the black hole, therefore, the greybody factor is higher. Moreover, there exists a critical mass of the Dirac field such that the greybody factor is maximum. By comparing the results from these two methods, we argue that it is useful t…
The Bardeen black hole solution is the first spherically symmetric regular black hole based on the Sakharov and Gliner proposal which is a modification of the Schwarzschild black hole. We present the Bardeen black hole solution in the presence of the de Rham, Gabadaadze, and Tolly (dRGT) massive gravity, which is regular everywhere in the presence of a nonlinear source. The obtained solution reduces to the Bardeen black hole in the absence of a massive gravity parameter, and the Schwarzschild black hole when magnetic charge $g = 0$. We investigated the thermodynamic quantities, that is, mass (…
It is known that the no-boundary proposal in the traditional Einstein gravity does not prefer inflation, that is, the probability of realizing a large number of e-folds is exponentially suppressed. This situation may be changed drastically in a class of nonlinear massive gravity theories recently proposed by deRham, Gabadadze and Tolley, called dRGT massive gravity. We show that the contribution from the massive gravity sector can enhance the probability of a large number of e-folds substantially for a sufficiently large mass parameter mg comparable to the Hubble parameter during inflation, sa…
In this work, we study the thin-shell wormholes in dRGT massive gravity. In order to joint two bulks of the spacetime geometry, we first derive junction conditions of the dRGT spacetime. This results the dynamics of the spherical thin-shell wormholes in the dRGT theory. We show that the massive graviton correction term of the dRGT theory in the Einstein equation is represented in terms of the effective anisotropic pressure fluid. However, if there is only this correction term, without invoking exotic fluids, we find that the thin-shell wormholes can not be stabilized. We then examine the stabi…
The recent Event Horizon Telescope (EHT) observations of the M87* have led to a surge of interest in studying the shadow of black holes. Besides, investigation of time evolution and lifetime of black holes helps us to veto/restrict some theoretical models in gravitating systems. Motivated by such exciting properties, we study optical features of black holes, such as the shadow geometrical shape and the energy emission rate in modified gravity. We consider a charged AdS black hole in dRGT massive gravity and look for criteria to restrict the free parameters of the theory. The main goal of this …
In this note we propose a topological action for a Poincare times diffeomorphism invariant gauge theory. We show that there is Higgs phase where the gauge symmetry is spontaneous broken to a diagonal Lorentz subgroup and gives the Einstein-Hilbert action plus the dRGT potential terms. In this vacuum, there are five (three from Goldstone modes) propagating degrees of freedom which form polarizations of a massive spin 2 particle, an extra healthy heavy scalar (Higgs) mode and no Boulware-Deser ghost mode. We further show that the action can be derived in a limit from a topological de Sitter inva…
We study relativistic stars in the simplest model of the de Rham-Gabadadze-Tolley massive gravity which describes the massive graviton without ghost propagating mode. We consider the hydrostatic equilibrium, and obtain the modified Tolman-Oppenheimer-Volkoff equation and the constraint equation coming from the potential terms in the gravitational action. We give analytical and numerical results for quark and neutron stars and discuss the deviations compared with General Relativity and $F(R)$ gravity. It is shown that theory under investigation leads to small deviation from the General Relativi…
In dRGT massive gravity, to get the equations of motion, the square root tensor is assumed to be invertible in the variation of the action. However, this condition can not be fulfilled when the reference metric is degenerate. This implies that the resulting equations of motion might be different from the case where the reference metric has full rank. In this paper, by generalizing the Moore-Penrose inverse to the symmetric tensor on Lorentz manifolds, we get the equations of motion of the theory with degenerate reference metric. It is found that the equations of motion are a little bit differe…
We study the background cosmology of two extensions of dRGT massive gravity. The first is variable mass massive gravity, where the fixed graviton mass of dRGT is replaced by the expectation value of a scalar field. We ask whether self-inflation can be driven by the self-accelerated branch of this theory, and we find that, while such solutions can exist for a short period, they cannot be sustained for a cosmologically useful time. Furthermore, we demonstrate that there generally exist future curvature singularities of the "big brake" form in cosmological solutions to these theories. The second …
The Schwarzschild-de Sitter solution in the Einstein theory with a positive cosmological constant $Λ=m^2/α$ becomes an exact solution to the dRGT non-linear massive gravity theory with the mass parameter $m$ when the theory parameters $α$ and $β$ satisfy the relation $β=α^2$. We study the perturbative behaviour of this black hole solution in the non-linear dRGT theory with $β=α^2$. We find that the linear perturbation equations become identical to those for the vacuum Einstein theory when they are expressed in terms of the gauge-invariant variables. This implies that this black hole is stable …
We study the Coleman-de Luccia (CDL) instanton characterizing the tunneling from a false vacuum to the true vacuum in a semi-classical way in dRGT (deRham-Gabadadze-Tolley) massive gravity theory, and evaluate the dependence of the tunneling rate on the model parameters. It is found that provided with the same physical Hubble parameters for the true vacuum $H_{\rm T}$ and the false vacuum $H_{\rm F}$ as in General Relativity (GR), the thin-wall approximation method implies the same tunneling rate as GR. However, deviations of tunneling rate from GR arise when one goes beyond the thin-wall appr…
I explain in a simple and compact form the origin of the apparent loss of predictability inside the dRGT non-linear formulation of massive gravity. This apparent pathology was first reported by Kodama and the author when the stability of the Schwarzschild de-Sitter (S-dS) black-hole in dRGT was analyzed. If we study the motion of a massive test particle around the S-dS solution, we find that the total energy is not conserved in the usual sense. The conserved quantity associated with time appears as a combination of the total energy and a velocity-dependent term. If the equations of motion are …
The linear theory of massive spin-two field in arbitrary curved background is investigated. In flat spacetime, the Fierz-Pauli model is well-known as the unique linear theory describing the massive spin-two field. On the other hand, in order to construct the massive spin-two theory in fixed curved background with arbitrary metric, infinite series of nonminimal coupling terms are necessary. In [Nucl. Phys. B 584 (2000) 615], Buchbinder et al. have derived the condition for the ghost-freeness and they have solved the condition in small curvature approximation. In the leading order approximation,…
Massive gravity theory introduced by de Rham, Gabadadze, Tolley (dRGT) is restricted by several uniqueness theorems that protect the form of the potential and kinetic terms, as well as the matter coupling. These restrictions arise from the requirement that the degrees of freedom match the expectation from Poincaré representations of a spin--2 field. Any modification beyond the dRGT form is known to invalidate a constraint that the theory enjoys and revive a dangerous sixth mode. One loophole is to exploit the effective nature of the theory by pushing the sixth mode beyond the strong coupling s…
It is known that de Rham-Gabadadze-Tolley (dRGT) massive gravity does not permit a homogeneous and isotropic universe with flat or spherical spatial metrics. We demonstrate that a singular reference metric solves this problem in an economic and straightforward way. In the dRGT massive gravity with a singular reference metric, there are sound homogeneous and isotropic cosmological solutions. We investigate cosmologies with the static and dynamical singular reference metrics, respectively. The term like dark energy appears naturally and the universe accelerates itself in some late time evolution…
In the Hamiltonian language we provide a study of flat-space cosmology in bigravity and massive gravity constructed mostly with de Rham, Gabadadze, Tolley (dRGT) potential. It is demonstrated that the Hamiltonian methods are powerful not only in proving the absence of the Boulware-Deser ghost, but also in solving other problems. The purpose of this work is to give an introduction both to the Hamiltonian formalism and to the cosmology of bigravity. We sketch three roads to the Hamiltonian of bigravity with the dRGT potential: the metric, the tetrad and the minisuperspace approaches.
We investigate gravitational lensing and particle motions around non-asymptotically flat black hole spacetime in non-linear, ghost-free massive gravity theory, called de Rham-Gabadadze-Tolley (dRGT) massive gravity. Deflection angle formulae are derived in terms of perihelion parameter. The deflection angle can be positive, zero or even negative with various perihelion distance. The negative angle reveals repulsive behaviour of gravity from a linear term $γ$ in the dRGT black hole solution. We also find an analytically approximated formula of deflection angle in two regimes: large and small $γ…
In this paper we study the possibility of assigning a geometric structure to the Lie groups. It is shown the Poincaré and Weyl groups have geometrical structure of the Riemann-Cartan and Weyl space-time respectively. The geometric approach to these groups can be carried out by considering the most general (non)metricity conditions, or equivalently, tetrad postulates which we show that can be written in terms of the group's gauge fields. By focusing on the conformal group we apply this procedure to show that a nontrivial 3-metrics geometry may be extracted from the group's Maurer-Cartan structu…
We investigate primordial gravitational waves and curvature perturbations in de Rham-Gabadadze-Tolley (dRGT) bimetric gravity. We evaluate the power-spectra in the leading order in slow roll. Taking into account the decay of massive graviton, we find that the action up to the second order reduces to the Einstein theory with a non-minimally coupled scalar field, which is simplified to a minimally coupled model by conformal transformation. We also find that the tensor to scalar ratio for large field inflation with power law potential is larger than the general relativity counterpart for any choi…
We investigate the static, spherically symmetric black hole solutions in the quasi-dilaton model and its generalizations, which are scalar extended dRGT massive gravity with a shift symmetry. We show that, unlike generic scalar extended massive gravity models, these theories do not admit static, spherically symmetric black hole solutions until the theory parameters in the dRGT potential is fine-tuned. When fine-tuned, the geometry of the static, spherically symmetric black hole is necessarily that of general relativity and the quasi-dilaton field is constant across the spacetime. The fine-tuni…
We obtain the fully covariant linearized field equations for the metric perturbation in the de Rham-Gabadadze-Tolley (dRGT) ghost free massive gravities. For a subset of these theories, we show that the non dynamical metric that appears in the dRGT setup can be completely eliminated leading to the theory of a massive graviton moving in a single metric. This has a mass term which contains non trivial contributions of the space-time curvature. We show further how 5 covariant constraints can be obtained including one which leads to the tracelessness of the graviton on flat space-time and removes …
We study the cosmological perturbations of the recently proposed extension of non-linear massive gravity with a scalar field. The added scalar field ensures a new symmetry on the field space of the theory. The theory has the property of having a flat dS solution, in contrast to the standard dRGT massive gravity. The tensor part is the same as that of the standard dRGT and shows gravitational waves with a time dependent mass which modifies the dispersion relation. We obtain the curvature perturbation of the model on superhorizon scales for a specific choice of $ω=0$ and find that the theory doe…
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 extend the four-dimensional de Rham-Gabadadze-Tolley (dRGT) massive gravity model to a general scalar massive-tensor theory in arbitrary dimensions, coupling a dRGT massive graviton to multiple scalars and allowing for generic kinetic and mass matrix mixing between the massive graviton and the scalars, and derive its Hamiltonian formulation and associated constraint system. When passing to the Hamiltonian formulation, two different sectors arise: a general sector and a special sector. Although obtained via different ways, there are two second class constraints in either of the two sectors, …
We propose a new theory of massive gravity with only two propagating degrees of freedom. After defining the theory in the unitary gauge in the vielbein language, we shall perform a Hamiltonian analysis to count the number of physical degrees of freedom, and then study some phenomenologies. While the homogeneous and isotropic background cosmology and the tensor linear perturbations around it are described by exactly the same equations as those in the de Rham-Gabadadze-Tolley (dRGT) massive gravity, the scalar and vector gravitational degrees of freedom are absent in the new theory at the fully …
The canonical structure of the massive gravity in the first order moving frame formalism is studied. We work in the simplified context of translation invariant fields, with mass terms given by general non-derivative interactions, invariant under the diagonal Lorentz group, depending on the moving frame as well as a fixed reference frame. We prove that the only mass terms which give 5 propagating degrees of freedom are the dRGT mass terms, namely those which are linear in the lapse. We also complete the Hamiltonian analysis with the dynamical evolution of the system.
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.
We present a detailed study of the Vaidya solution and its generalization in de Rham-Gabadadze-Tolley (dRGT) theory. Since the diffeomorphism invariance can be restored with the Stückelberg fields $ϕ^a$ introduced, there is a new invariant $I^{ab}=g^{μν}\partial_μϕ^a\partial_νϕ^b$ in the massive gravity, which adds to the ones usually encountered in general relativity. There is no conventional Vaidya solution if we choose unitary gauge. In this paper, we obtain three types of self-consistent ansatz with some nonunitary gauge, and find accordingly the Vaidya, generalized Vaidya and furry Vaidya…
Motivated by recent developments of black hole thermodynamics in de Rham, Gabadadze and Tolley(dRGT) massive gravity, we study the critical behaviors of four-dimensional topological Anti-de Sitter(AdS) black holes in the presence of Born-Infeld nonlinear electrodynamics by treating the cosmological constant as pressure and the corresponding conjugate quantity is interpreted as thermodynamic volume. It shows that besides the Van der Waals-like SBH/LBH phase transitions appears, the so-called reentrant phase transitions (RPTs) are also observed when the coupling coefficients $c_i m^2$ of massive…
We compare the standard Higgs mechanism corresponding to the scalar field, with the dynamical origin of the graviton mass inside the scenario of the dRGT theory of massive gravity. We demonstrate that the effective mass perceived locally by different observers, depends on the way how they define the local time with respect to the preferred notion of time defined by the Stückelberg function $T_0(r,t)$.
The existence of possible massive white dwarfs more than the Chandrasekhar limit ($1.45M_{\odot }$, in which $M_{\odot }$ is mass of the sun) is a challenging topic. In this regard and motivated by the important effect of massive graviton on the structure of white dwarfs we study the white dwarfs in Vegh's massive gravity which is known as one of theories of de Rham, Gabadadze and Tolley (dRGT) like massive gravity. First, we consider the modified Tolman-Oppenheimer-Volkoff equation in this theory of massive gravity and solve it numerically by using the Chandrasekhar's equation of state. Our r…
Massive gravity in three dimensions accepts several different formulations. Recently, the 3-dimensional bigravity dRGT model in first order form, Zwei-Dreibein gravity, was considered by Bergshoeff {\it et al.} and it was argued that the Boulware-Deser mode is killed by extra constraints. We revisit this assertion and conclude that there are sectors on the space of initial conditions, or subsets of the most general such model, where this mode is absent. But, generically, the theory does carry 3 degrees of freedom and thus the Boulware-Deser mode is still active. Our results also sheds light on…
We perform a numerical study to recognize the difference between various massive potentials in the dRGT massive gravity on the holographic thermalization in the AdS and AdS Gauss-Bonnet gravities. The massive potential in $4+1$ dimensions includes three symmetric polynomial terms which we denote them as $a_1$, $a_2$ and $a_3$ terms. We observe, in the case of time evolution of entanglement entropy that there is a critical size of the entangling surface on the boundary below which both signs of $a_1$ and above the critical size $a_3$ are able to reduce the thermal value of entanglement entropy.…
It is well known that the black hole can has temperature and radiate the particles with black body spectrum, i.e. Hawking radiation. Therefore, if the black hole is surrounded by an isolated box, there is a thermal equilibrium between the black hole and radiation gas. A simple case considering the thermal equilibrium between the Schwarzschild black hole and radiation gas in an isolated box has been well investigated previously in detail, i.e. taking the conservation of energy and principle of maximal entropy for the isolated system into account. In this paper, following the above spirit, the e…
We derive analyticity constraints on a nonlinear ghost-free effective theory of a massive spin-2 particle known as pseudo-linear massive gravity, and on a generalized theory of a massive spin-1 particle, both of which provide simple IR completions of Galileon theories. For pseudo-linear massive gravity we find that, unlike dRGT massive gravity, there is no window of parameter space which satisfies the analyticity constraints. For massive vectors which reduce to Galileons in the decoupling limit, we find that no two-derivative actions are compatible with positivity, but that higher derivative a…
This article is devoted to static spherically symmetric black hole solutions of dRGT (de Rham-Gabadadze-Tolley) massive gravity in the presence of cosmological constant. The unitary and non-unitary gauges are used to find the solutions in three, four and five dimensions. We show that there are two general classes of solutions. In one of them, the effect of massive potential is appeared as the effective cosmological constant. By investigating these solutions in different dimensions, we find an expression for effective cosmological constant in arbitrary dimensions.
We present a detailed study of the static spherically symmetric solutions in de Rham-Gabadadze-Tolley (dRGT) theory. Since the diffeomorphism invariance can be restored by introducing the Stückelberg fields $ϕ^a$, there is new invariant $I^{ab}=g^{μν}\partial_μϕ^a\partial_νϕ^b$ in the massive gravity, which adds to the ones usually encountered in general relativity (GR). In the unitary gauge $ϕ^a=x^μδ_μ^a$, any inverse metric $g^{μν}$ that has divergence including the coordinate singularity in GR would exhibit a singularity in the invariant $I^{ab}$. Therefore, there is no conventional Schwarz…
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…
When starting with a static, spherically-symmetric ansatz, there are two types of black hole solutions in dRGT massive gravity: (i) exact Schwarzschild solutions which exhibit no Yukawa suppression at large distances and (ii) solutions in which the dynamical metric and the reference metric are simultaneously diagonal and which inevitably exhibit coordinate-invariant singularities at the horizon. In this work we investigate the possibility of black hole solutions which can accommodate both a non-singular horizon and Yukawa asymptotics. In particular, by adopting a time-dependent ansatz, we deri…
We study perturbations around some cosmological backgrounds in the dRGT theory of massive gravity. We develop a general formalism to calculate the perturbations around any background. We derive the Lagrangian for fluctuations in the small scale limit, and for the open FRW solution we repeat the analysis around the full background. We find that the perturbations display similar properties: the longitudinal modes of the massive graviton are instantaneous at quadratic level, but they acquire a time-kinetic term at cubic order.
