dS Black Hole Research Articles

Thermodynamics and phase transition of topological dS black holes with a nonlinear source

We discuss black hole solutions of Einstein-Λ gravity in the presence of nonlinear electrodynamics in dS spacetime. Considering the correlation of the thermodynamic quantities respectively corresponding to the black hole horizon and cosmological horizon of dS spacetime and taking the region between the two horizons as a thermodynamic system, we derive effective thermodynamic quantities of the system according to the first law of thermodynamics, and investigate the thermodynamic properties of the system under the influence of nonlinearity parameter α. It is shown that nonlinearity parameter α influences the position of the black hole horizon and the critical state of the system, and along with electric charge has an effect on the phase structure of the system, which is obvious, especially as the effective temperature is below the critical temperature. The critical phase transition is proved to be second-order equilibrium phase transition by using the Gibbs free energy criterion and Ehrenfest equations.

Quasinormal modes for integer and half-integer spins within the large angular momentum limit

While independent observations have been made regarding the behaviour of effective quasinormal mode (QNM) potentials within the large angular momentum limit, we demonstrate analytically here that a uniform expression emerges for non-rotating, higher-dimensional, and spherically-symmetric black holes (BHs) in this regime for fields of integer and half-integer spin in asymptotically flat and dS BH contexts; a second uniform expression arises for these QNM potentials in AdS BH spacetimes. We then proceed with a numerical analysis based on the multipolar expansion method recently proposed by Dolan and Ottewill to determine the behaviour of quasinormal frequencies (QNF) for varying BH parameters in the eikonal limit. We perform a complete study of Dolan and Ottewill's method for perturbations of spin $s \in \{0,1/2,1,3/2,2 \}$ in 4D Schwarzschild, Reissner-Nordstr{\"o}m, and Schwarzschild de Sitter spacetimes, clarifying expressions and presenting expansions and results to higher orders $(\mathcal{O}(L^{-6}))$ than many of those presented in the literature $(\sim \mathcal{O}(L^{-2}))$. We find good agreement with known results of QNFs for low-lying modes; in the large-$\ell$ regime, our results are highly consistent with those of Konoplya's 6th-order WKB method. We confirm a universality in the trends of physical features recorded in the literature for the low-lying QNFs (that the real part grows indefinitely, the imaginary tends to a constant as $\ell \rightarrow \infty$, etc.) as we approach large values of $\ell$ within these spacetimes, and explore the consequent interplay between BH parameters and QNFs in the eikonal limit.

Geodesic motion in the spacetime of a SU(2)-Colored (A)dS black hole in conformal gravity

In this paper, we study the geodesic motion in the spacetime of a SU(2)-colored (A)dS black hole in conformal gravity, and also we investigate spacetime features, such as light spheres and horizons. Moreover, we derive the analytical solutions for the equation of motion of test particles and light rays using Weierstrass elliptic and Kleinian \(\sigma \) functions. Depending on the particle energy levels and angular momentums, we classify the solutions of the geodesic equations. Furthermore, several examples of possible types of orbits illustrate the results.

Yang\u2013Mills black holes in quasitopological gravity

In this paper, we formulate two new classes of black hole solutions in higher curvature quartic quasitopological gravity with nonabelian Yang–Mills theory. At first step, we consider the SO(n) and SO(n-1,1) semisimple gauge groups. We obtain the analytic quartic quasitopological Yang–Mills black hole solutions. Real solutions are only accessible for the positive value of the redefined quartic quasitopological gravity coefficient, mu _{4}. These solutions have a finite value and an essential singularity at the origin, r=0 for space dimension higher than 8. We also probe the thermodynamic and critical behavior of the quasitopological Yang–Mills black hole. The obtained solutions may be thermally stable only in the canonical ensemble. They may also show a first order phase transition from a small to a large black hole. In the second step, we obtain the pure quasitopological Yang–Mills black hole solutions. For the positive cosmological constant and the space dimensions greater than eight, the pure quasitopological Yang–Mills solutions have the ability to produce both the asymptotically AdS and dS black holes for respectively the negative and positive constant curvatures, k=-1 and k=+1. This is unlike the quasitopological Yang–Mills theory which can lead to just the asymptotically dS solutions for Lambda >0. The pure quasitopological Yang–Mills black hole is not thermally stable.

Black holes and other spherical solutions in quadratic gravity with a cosmological constant

We study static spherically symmetric solutions to the vacuum field equations of quadratic gravity in the presence of a cosmological constant $\Lambda$. Motivated by the trace no-hair theorem, we assume the Ricci scalar to be constant throughout a spacetime. Furthermore, we employ the conformal-to-Kundt metric ansatz that is valid for all static spherically symmetric spacetimes and leads to a considerable simplification of the field equations. We arrive at a set of two ordinary differential equations and study its solutions using the Frobenius-like approach of (infinite) power series expansions. While the indicial equations considerably restrict the set of possible leading powers, careful analysis of higher-order terms is necessary to establish the existence of the corresponding classes of solutions. We thus obtain various non-Einstein generalizations of the Schwarzschild, (anti-)de Sitter [or (A)dS for short], Nariai, and Pleba\'{n}ski-Hacyan spacetimes. Interestingly, some classes of solutions allow for an arbitrary value of $\Lambda$, while other classes admit only discrete values of $\Lambda$. For most of these classes, we give recurrent formulas for all series coefficients. We determine which classes contain the Schwarzschild-(A)dS black hole as a special case and briefly discuss the physical interpretation of the spacetimes. In the discussion of physical properties, we naturally focus on the generalization of the Schwarzschild-(A)dS black hole, namely the Schwarzschild-Bach-(A)dS black hole, which possesses one additional Bach parameter. We also study its basic thermodynamical properties and observable effects on test particles caused by the presence of the Bach tensor. This work is a considerable extension of our letter [Phys. Rev. Lett., 121, 231104, 2018].

Gravitational redshift/blueshift of light emitted by geodesic test particles, frame-dragging and pericentre-shift effects, in the Kerr\u2013Newman\u2013de Sitter and Kerr\u2013Newman black hole geometries

We investigate the redshift and blueshift of light emitted by timelike geodesic particles in orbits around a Kerr–Newman–(anti) de Sitter (KN(a)dS) black hole. Specifically we compute the redshift and blueshift of photons that are emitted by geodesic massive particles and travel along null geodesics towards a distant observer-located at a finite distance from the KN(a)dS black hole. For this purpose we use the killing-vector formalism and the associated first integrals-constants of motion. We consider in detail stable timelike equatorial circular orbits of stars and express their corresponding redshift/blueshift in terms of the metric physical black hole parameters (angular momentum per unit mass, mass, electric charge and the cosmological constant) and the orbital radii of both the emitter star and the distant observer. These radii are linked through the constants of motion along the null geodesics followed by the photons since their emission until their detection and as a result we get closed form analytic expressions for the orbital radius of the observer in terms of the emitter radius, and the black hole parameters. In addition, we compute exact analytic expressions for the frame dragging of timelike spherical orbits in the KN(a)dS spacetime in terms of multivariable generalised hypergeometric functions of Lauricella and Appell. We apply our exact solutions of timelike non-spherical polar KN geodesics for the computation of frame-dragging, pericentre-shift, orbital period for the orbits of S2 and S14 stars within the 1^{prime prime } of SgrA*. We solve the conditions for timelike spherical orbits in KN(a)dS and KN spacetimes. We present new, elegant compact forms for the parameters of these orbits. Last but not least we derive a very elegant and novel exact formula for the periapsis advance for a test particle in a non-spherical polar orbit in KNdS black hole spacetime in terms of Jacobi’s elliptic function sn and Lauricella’s hypergeometric function F_D.

Phase transition and entropic force of de Sitter black hole in massive gravity

It is well known that de Sitter(dS) black holes generally have a black hole horizon and a cosmological horizon, both of which have Hawking radiation. But the radiation temperature of the two horizons is generally different, so dS black holes do not meet the requirements of thermal equilibrium stability, which brings certain difficulties to the study of the thermodynamic characteristics of black holes. In this paper, dS black hole is regarded as a thermodynamic system, and the effective thermodynamic quantities of the system are obtained. The influence of various state parameters on the effective thermodynamic quantities in the massive gravity space-time is discussed. The condition of the phase transition of the de Sitter black hole in massive gravity space-time is given. We consider that the total entropy of the dS black hole is the sum of the corresponding entropy of the two horizons plus an extra term from the correlation of the two horizons. By comparing the entropic force of interaction between black hole horizon and the cosmological horizon with Lennard-Jones force between two particles, we find that the change rule of entropic force between the two system is surprisingly the same. The research will help us to explore the real reason of accelerating expansion of the universe.

Phase transitions and entropy force of charged de Sitter black holes with cloud of string and quintessence

In this paper, we investigate the combined effects of the cloud of strings and quintessence on the thermodynamics of a Reissner–Nordström–de Sitter black hole. Based on the equivalent thermodynamic quantities considering the correlation between the black hole horizon and the cosmological horizon, we extensively discuss the phase transitions of the spacetime. Our analysis proves that similar to the case in AdS spacetime, second-order phase transitions could take place under certain conditions, with the absence of first-order phase transition in the charged de Sitter (dS) black holes with cloud of string and quintessence. The effects of different thermodynamic quantities on the phase transitions are also quantitatively discussed, which provides a new approach to study the thermodynamic qualities of unstable dS spacetime. Focusing on the entropy force generated by the interaction between the black hole horizon and the cosmological horizon, as well as the Lennard–Jones force between two particles, our results demonstrate the strong degeneracy between the entropy force of the two horizons and the ratio of the horizon positions, which follows the surprisingly similar law given the relation between the Lennard–Jones force and the ratio of two particle positions. Therefore, the study of the entropy force between two horizons is not only beneficial to the deep exploration of the three modes of cosmic evolution, but also helpful to understand the correlation between the microstates of particles in black holes and those in ordinary thermodynamic systems.

Horizon thermodynamics in theory * * Supported in part by Hebei Provincial Natural Science Foundation of China (A2014201068)

We investigate whether the new horizon first law still holds in theory. For this complicated theory, we first determine the entropy of a black hole by using the Wald method, and then derive the energy of the black hole by using the new horizon first law, the degenerate Legendre transformation, and the gravitational field equations. For application, we consider the quadratic-curvature gravity, and first calculate the entropy and energy of a static spherically symmetric black hole, which are in agreement with the results obtained in the literature for a Schwarzschild-(A)dS black hole.

Effective thermodynamics and critical phenomena of rotating regular-de Sitter black holes

We consider the horizon structure of a rotating regular-de Sitter (dS) black hole, which has an additional parameter k because of nonlinear electrodynamics (NED) apart from its mass (M) and angular momentum (a). The rotating regular de Sitter black holes, like Kerr dS black holes, admit a cosmological horizon (rc) beside the inner Cauchy (r−) and outer event (rh) horizons. Considering the relation between rh and rc, we analyze the effective thermodynamic quantities associated with rotating regular dS black holes. The expressions for the effective temperature and the effective pressure are obtained, and plotted for various values of k. The second order phase transition for thermodynamic stability is marked by the divergence of the heat capacity of constant pressure CP, the volume expansion coefficient α, and the isothermal compressibility κT at multiple critical points xc = rh/rc with the stable (unstable) branch with positive (negative) heat capacity. It turns out that at a critical point, the heat capacity of constant pressure, the isothermal compressibility, and the volume expansion coefficient of the rotating regular dS black holes exhibit an infinite peak suggesting a second order phase transition. Our results, in the absence of a charge from the NED (k = 0), vis-à-vis go over to that of Kerr dS black holes.

Gravitational phase transition mediated by thermalon in Einstein-Gauss-Bonnet-Maxwell-Kalb-Ramond gravity

In this work, we study the possible existence of gravitational phase transition from AdS to dS asymptotic geometries in Einstein-Gauss-Bonnet gravity by adding the Maxwell one-form field (Aμ) and the Kalb-Ramond two-form field (Bμν) as impurity substitutions. The phase transitions proceed via the bubble nucleation of spherical thin-shells described by different branches of the solutions which host a dS black hole in the interior and asymptotic thermal AdS state in the exterior. We analyze the phase diagrams of the free energy and temperature to demonstrate the existence of the phase transitions in the grand canonical ensemble (fixed electrical potential). The phase transitions of having the one-form and two-form charges are possible in which the critical temperature is lower than that of the neutral case. Comparing results with existing literature, more importantly, our analyses show that the critical temperature and the Gauss-Bonnet coupling λ of the phase transitions get decreased by adding more types of the charges.

Thermodynamic properties of higher-dimensional dS black holes in dRGT massive gravity

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 higher-dimensional 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 multi-parameter 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 space-time 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.

Charged (A)dS black hole solutions in conformal teleparallel equivalent of general relativity

We continue our study in 4-dimension to derive non-charged and charged (Anti)-de Sitter black hole solutions in conformal teleparallel equivalent of general relativity. The non-charged and charged equations of motion are applied to a spherically symmetric tetrad and the non-linear differential equations are derived. It is shown that the output solutions of the two cases are identical to those obtained in teleparallel equivalent theory to general relativity. As a result, it is found that in the conformal teleparallel equivalent theory to general relativity, the scalar field cannot influence on the manifold of spherical symmetry, i.e. the scalar field must equal 1 in order to have a well known asymptote spacetime.

Black hole nonmodal linear stability: Even perturbations in the Reissner-Nordström case

This paper is a companion of [Phys. Rev. D 95, 124041 (2017)] in which, following a program on black hole nonmodal linear stability initiated in Phys. Rev. Lett. 112 (2014) 191101, odd perturbations of the Einstein-Maxwell equations around a Reissner-Nordstr\"om (A)dS black hole were analyzed. Here we complete the proof of the nonmodal linear stability of this spacetime by analyzing the even sector of the linear perturbations. We show that all the gauge invariant information in the metric and Maxwell field even perturbations is encoded in two spacetime scalars: ${\mathcal S}$, which is a gauge invariant combination of $\delta (C_{\alpha \beta \gamma \epsilon}C^{\alpha \beta \gamma \epsilon})$ and $\delta (C_{\alpha \beta \gamma \delta} F_{\alpha \beta} F^{\gamma \delta})$, and ${\mathcal T}$, a gauge invariant combination of $\delta ( \nabla _\mu F_{\alpha \beta} \nabla^\mu F^{\alpha \beta })$ and $\delta ( \nabla_{\mu} C_{\alpha \beta \gamma \delta} \nabla^{\mu}C^{\alpha \beta \gamma \delta})$. Here $C_{\alpha \beta \gamma \delta}$ is the Weyl tensor, $F_{\alpha \beta}$ the Maxwell field and $\delta$ means first order variation. We prove that $\mathcal{S}$ and $\mathcal{T}$ are are in one-one correspondence with gauge classes of even linear perturbations, and that the linearized Einstein-Maxwell equations imply that these scalar fields are pointwise bounded on the outer static region.

Thermodynamics of charged A(dS) black holes with quadratically-extended electrodynamics

In this paper, the Einstein-nonlinear electromagnetic theory has been studied in the four-dimensional de Sitter (dS) and anti-de Sitter (AdS) spacetimes. A new class of nonlinearly charged A(dS) black hole solution, in the presence of quadratically-extended Maxwell field, has been obtained and the thermodynamical properties have been analyzed. The black hole entropy, temperature and electric potential have been calculated, making use of the geometric methods. Through a Smarr mass formula the black hole mass has been written as a function of the complete set of extensive parameters (i.e. charge and entropy). It has been found that the intensive parameters (i.e. electric potential and temperature) obtained from the mass formula coincide with their values obtained from geometric approaches. It confirms the validity of the first law of black hole thermodynamics. A black hole stability analysis has been performed from the canonical ensemble approach, regarding the black hole heat capacity with the black hole charge as a constant. The points of type one and type two phase transitions, and the ranges at which the new charged A(dS) black holes are locally stable have been determined. Also, for the new charged black holes to be physically reasonable, their charge has been restricted to some specific ranges.

Master equations and quasinormal modes of spin- 3/2 fields in Schwarzschild (A)dS black hole spacetimes

In this work we consider spin-3/2 fields in Schwarzschild (A)dS black hole spacetimes. As this spacetime is different from the Ricci-flat cases, it is necessary to modify the covariant derivative to the supercovariant derivative in order to maintain the gauge symmetry, as noted in our earlier works, where this is done here by including terms related to the cosmological constant. Together with the eigenmodes for spin-3/2 fields on an $n$-sphere, we derive the master radial equations, which have effective potentials that in general include an explicitly imaginary part and energy dependence. We found that for the asymptotically AdS cases, the explicit imaginary dependence automatically disappears, because of the negative cosmological constant. We take this case as an example and obtain the quasi-normal modes by using the Horowitz-Hubeny method [6].

Critical gravity from four dimensional scale invariant gravity

We show that a critical condition exists in four dimensional scale invariant gravity given by the pure quadratic action \U0001d6fd CμvσρCμνσρ+ \U0001d6fc R2 where {C}_{nu sigma rho}^{mu } is the Weyl tensor, R is the Ricci scalar and \U0001d6fd and \U0001d6fc are dimensionless parameters. The critical condition in a dS or AdS background is \U0001d6fd = 6\U0001d6fc. This leads to critical gravity where the massive spin two physical ghost becomes a massless spin two graviton. In contrast to the original work on critical gravity, no Einstein gravity with a cosmological constant is added explicitly to the higher-derivative action. The critical condition is obtained in two independent ways. In the first case, we show the equivalence between the initial action and an action containing Einstein gravity, a cosmological constant, a massless scalar field plus Weyl squared gravity. The scale invariance is spontaneously broken. The linearized Einstein-Weyl equations about adS or AdS background yield the critical condition \U0001d6fd = 6\U0001d6fc. In the second case, we work directly with the original quadratic action. After a suitable field redefinition, where the metric perturbation is traceless and transverse, we obtain linearized equations about a dS or AdS background that yield the critical condition \U0001d6fd = 6\U0001d6fc. As in the first case, we also obtain a propagating massless scalar field. Substituting \U0001d6fd = 6\U0001d6fc into the energy and entropy formula for the Schwarzschild and Kerr AdS or dS black hole in higher-derivative gravity yields zero, the same value obtained in the original work on critical gravity. We discuss the role of boundary conditions in relaxing the \U0001d6fd = 6\U0001d6fc condition.

Entropy of higher-dimensional topological dS black holes with nonlinear source

On the basis of the first law of black hole thermodynamics, we propose the concept of effective temperature of de Sitter (dS) black holes and conjecture that the effective temperature should be the temperature of the dS black holes when the Hawking radiation temperatures of the black hole horizon and the cosmological horizon are equal. Choosing different independent variables, we can find a differential equation satisfied by the entropy of the dS black hole. It is shown that the differential equation of entropy is independent of the choice of independent variables. From the differential equation, we get the entropy of dS black hole and other effective thermodynamic quantities. We also discuss the influence of several parameters on the effective thermodynamic quantities.

Thermodynamic description and quasinormal modes of adS black holes in Born-lnfeld massive gravity with a non-abelian hair

We construct a new class of asymptotically (a)dS black hole solutions of Einstein-Yang-Mills massive gravity in the presence of Born-Infeld nonlinear electrody­ namics. The obtained solutions possess a Coulomb electric charge, massive term and a non-abelian hair as well. We calculate the conserved and thermodynamic quantities, and investigate the validity of the first law of thermodynamics. Also, we investigate thermal stability conditions by using the sign of heat capacity through canonical ensemble. Next, we consider the cosmological constant as a thermodynamical pressure and study the van der Waals like phase transition of black holes in the extended phase space thermodynamics. Our results indicate the existence of a phase transition which is affected by the parameters of theory. Finally, we consider a massless scalar perturbation in the background of asymptotically adS solutions and calculate the quasinormal modes by employing the pseu­ dospectral method. The imaginary part of quasinormal frequencies is the time scale of a thermal state (in the conformal field theory) for the approach to thermal equilibrium.

Entropy spectrum of charged BTZ black holes in massive gravity’s rainbow

Regarding the significant interests in massive gravity and combining it with gravity's rainbow and also BTZ black holes, we apply the formalism introduced by Jiang and Han in order to investigate the quantization of the entropy of black holes. We show that the entropy of BTZ black holes in massive gravity's rainbow is quantized with equally spaced spectra and it depends on the black holes' properties including massive parameters, electrical charge, the cosmological constant and also rainbow functions. In addition, we show that quantization of the entropy results into the appearance of novel properties for this quantity such as; the existence of divergencies, non-zero entropy in vanishing horizon radius and possibility of tracing out the effects of black holes' properties. Such properties are absent in the non-quantized version of these black holes' entropy. Furthermore, we investigate the effects of quantization on the thermodynamical behavior of the solutions. We confirm that due to quantization, novel phase transitions points are introduced and stable solutions are limited to only dS black holes (AdS and asymptotically flat solutions are unstable).