Schwarzschild-de Sitter Solution Research Articles

Repulsive gravity in regular black holes

We evaluate the effects of repulsive gravity using first order geometric invariants, i.e. the Ricci scalar and the eigenvalues of the Riemann curvature tensor, for three regular black holes, namely the Bardeen, Hayward, and Dymnikova spacetimes. To examine the repulsive effects, we calculate their respective onsets and regions of repulsive gravity. Afterwards, we compare the repulsive regions obtained from these metrics among themselves and then with the predictions got from the Reissner–Nordström and Schwarzschild–de Sitter. A notable characteristic, observed in all these metrics, is that the repulsive regions appear to be unaffected by the mass that generates the regular black hole. This property emerges due to the invariants employed in our analysis, which do not change sign through linear combinations of the mass and the free coefficients of the metrics. As a result, gravity can change sign independently of the specific values acquired by the mass. This conclusion suggests a potential incompleteness of regular solutions, particularly in terms of their repulsive effects. To further highlight this finding, we numerically compute, for the Reissner–Nordström and Schwarzschild–de Sitter solutions, the values of mass, M, that emulate the repulsive effects found in the Bardeen and Hayward spacetimes. These selected values of M provide evidence that regular black holes do not incorporate repulsive effects by means of the masses used to generate the solutions themselves. Implications and physical consequences of these results are then discussed in detail.

Quasi-local masses and cosmological coupling of black holes and mimickers

Motivated by the recent heated debate on whether the masses of local objects, such as compact stars or black holes (BHs), may be affected by the large-scale, cosmological dynamics, we analyze the conditions under which, in a general relativity framework, such a coupling small/large scales is allowed. We shed light on some controversial arguments, which have been used to rule out the latter possibility. We find that the cosmological coupling occurs whenever the energy of the central objects is quantified by the quasi-local Misner-Sharp mass (MS). Conversely, the decoupling occurs whenever the MS mass is fully equivalent to the (nonlocal) Arnowitt-Deser-Misner (ADM) mass. Consequently, for singular BHs embedded in cosmological backgrounds, like the Schwarzschild-de Sitter or McVittie solutions, we show that there is no cosmological coupling, confirming previous results in the literature. Furthermore, we show that nonsingular compact objects couple to the cosmological background, as quantified by their MS mass. We conclude that observational evidence of cosmological coupling of astrophysical BHs would be the smoking gun of their nonsingular nature.

Uniqueness of the extremal Schwarzschild de Sitter spacetime

We prove that any analytic vacuum spacetime with a positive cosmological constant in four and higher dimensions, that contains a static extremal Killing horizon with a maximally symmetric compact cross-section, must be locally isometric to either the extremal Schwarzschild de Sitter solution or its near-horizon geometry (the Nariai solution). In four-dimensions, this implies these solutions are the only analytic vacuum spacetimes that contain a static extremal horizon with compact cross-sections (up to identifications). We also consider the analogous uniqueness problem for the four-dimensional extremal hyperbolic Schwarzschild anti-de Sitter solution and show that it reduces to a spectral problem for the laplacian on compact hyperbolic surfaces, if a cohomological obstruction to the uniqueness of infinitesimal transverse deformations of the horizon is absent.

Gravitational Light Bending in Weyl Gravity and Schwarzschild–de Sitter Spacetime

The topic of gravitational lensing in the Mannheim–Kazanas solution of Weyl conformal gravity and the Schwarzschild–de Sitter solution in general relativity has featured in numerous publications. These two solutions represent a spherical massive object (lens) embedded in a cosmological background. In both cases, the interest lies in the possible effect of the background non-asymptotically flat spacetime on the geometry of the local light curves, particularly the observed deflection angle of light near the massive object. The main discussion involves possible contributions to the bending angle formula from the cosmological constant Λ in the Schwarzschild–de Sitter solution and the linear term γr in the Mannheim–Kazanas metric. These effects from the background geometry, and whether they are significant enough to be important for gravitational lensing, seem to depend on the methodology used to calculate the bending angle. In this paper, we review these techniques and comment on some of the obtained results, particularly those cases that contain unphysical terms in the bending angle formula.

On the stability of scale-invariant black holes

Quadratic scale-invariant gravity non minimally coupled to a scalar field provides a competitive model for inflation, characterized by the transition from an unstable to a stable fixed point, both characterized by constant scalar field configurations. We provide a complementary analysis of the same model in the static, spherically symmetric setting, obtaining two Schwarzschild-de Sitter solutions, which corresponds to the two fixed points existing in the cosmological scenario. The stability of such solutions is thoroughly investigated from two different perspectives. First, we study the system at the classical level by the analysis of linear perturbations. In particular, we provide both analytical and numerical results for the late-time behavior of the perturbations, proving the stable and unstable character of the two solutions. Then we perform a semi-classical, non-linear analysis based on the Euclidean path integral formulation. By studying the difference between the Euclidean on-shell actions evaluated on both solutions, we prove that the unstable one has a meta-stable character and is spontaneously decaying into the stable fixed point which is always favoured.

Black holes in dS3

In three-dimensional de Sitter space classical black holes do not exist, and the Schwarzschild-de Sitter solution instead describes a conical defect with a single cosmological horizon. We argue that the quantum backreaction of conformal fields can generate a black hole horizon, leading to a three-dimensional quantum de Sitter black hole. Its size can be as large as the cosmological horizon in a Nariai-type limit. We show explicitly how these solutions arise using braneworld holography, but also compare to a non-holographic, perturbative analysis of backreaction due to conformally coupled scalar fields in conical de Sitter space. We analyze the thermodynamics of this quantum black hole, revealing it behaves similarly to its classical four-dimensional counterpart, where the generalized entropy replaces the classical Bekenstein-Hawking entropy. We compute entropy deficits due to nucleating the three-dimensional black hole and revisit arguments for a possible matrix model description of dS spacetimes. Finally, we comment on the holographic dual description for dS spacetimes as seen from the braneworld perspective.

Effective field theory of black hole perturbations with timelike scalar profile: formulation

We formulate the Effective Field Theory (EFT) of perturbations within scalar-tensor theories on an inhomogeneous background. The EFT is constructed while keeping a background of a scalar field to be timelike, which spontaneously breaks the time diffeomorphism. We find a set of consistency relations that are imposed by the invariance of the EFT under the 3d spatial diffeomorphism. This EFT can be generically applied to any inhomogeneous background metric as long as the scalar profile is everywhere timelike. For completeness, we report a dictionary between our EFT parameters to those of Horndeski theories. Finally, we compute background equations for a class of spherically symmetric, static black hole backgrounds, including a stealth Schwarzschild-de Sitter solution.

Effective models of nonsingular quantum black holes

We investigate how the resolution of the singularity problem for the Schwarzschild black hole could be related to the presence of quantum gravity effects at horizon scales. Motivated by the analogy with the cosmological Schwarzschild-de Sitter solution, we construct a broad class of nonsingular, static, asymptotically flat black-hole solutions with a de Sitter (dS) core, sourced by an anisotropic fluid, which effectively encodes the quantum corrections. The latter are parametrized by a single length-scale $\ensuremath{\ell}$, which has a dual interpretation as an effective ``quantum hair'' and as the length-scale resolving the classical singularity. Depending on the value of $\ensuremath{\ell}$, these solutions can have two horizons, be extremal (when the two horizons merge) or be horizonless exotic stars. We also investigate the thermodynamic behavior of our black-hole solutions and propose a generalization of the area law in order to account for their entropy. We find a second-order phase transition near extremality, when $\ensuremath{\ell}$ is of order of the classical Schwarzschild radius ${R}_{\mathrm{S}}$. Black holes with $\ensuremath{\ell}\ensuremath{\sim}{R}_{\mathrm{S}}$ are thermodynamically preferred with respect to those with $\ensuremath{\ell}\ensuremath{\ll}{R}_{\mathrm{S}}$, supporting the relevance of quantum corrections at horizon scales. We also find that the extremal configuration is a zero-temperature, zero-entropy state with its near-horizon geometry factorizing as ${\mathrm{AdS}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathrm{S}}^{2}$, signalizing the possible relevance of these models for the information paradox. Finally, we show that the presence of quantum corrections with $\ensuremath{\ell}\ensuremath{\sim}{R}_{\mathrm{S}}$ have observable phenomenological signatures in the photon orbits and in the quasinormal modes (QNMs) spectrum. In particular, in the near-extremal regime, the imaginary part of the QNMs spectrum scales with the temperature as ${c}_{1}/\ensuremath{\ell}+{c}_{2}\ensuremath{\ell}{T}_{\mathrm{H}}^{2}$, while it goes to zero linearly in the near-horizon limit. Our general findings are confirmed by revisiting two already known models, which are particular cases of our general class of models, namely the Hayward and Gaussian-core black holes.

Static and spherically symmetric general relativity solutions in minimal theory of bigravity

We investigate static and spherically symmetric solutions in the Minimal Theory of Bigravity (MTBG). First, we show that a pair of Schwarzschild-de Sitter spacetimes with different cosmological constants and black hole masses written in the spatially-flat Gullstrand-Painlev\'e (GP) coordinates is a solution in the self-accelerating branch of MTBG, while it cannot be a solution in the normal branch. We then illustrate how Schwarzschild-de Sitter solutions can become compatible with the normal branch when using different coordinates. We also confirm that the self-accelerating branch of MTBG admits static and spherically symmetric general relativity solutions with matter written in the spatially-flat coordinates, including neutron stars with arbitrary matter equations of state. Finally, we show that in the self-accelerating branch nontrivial solutions are given by the Schwarzschild-de Sitter metrics written in nonstandard coordinates.

Scalar lumps with two horizons

We study generalisations of the Schwarzschild-de Sitter solution in the presence of a scalar field with a potential barrier. These static, spherically symmetric solutions have two horizons, in between which the scalar interpolates at least once across the potential barrier, thus developing a lump. In part, we recover solutions discussed earlier in the literature and for those we clarify their properties. But we also find a new class of solutions in which the scalar lump curves the spacetime sufficiently strongly so as to change the nature of the erstwhile cosmological horizon into an additional trapped horizon, resulting in a scalar lump surrounded by two black holes. These new solutions appear in a wide range of the parameter space of the potential. We also discuss (challenges for) the application of all of these solutions to black hole seeded vacuum decay.

A Way to Quantum Gravity

We present a simple way to approach the hard problem of quantization of the gravitational field in four-dimensional space-time, due to non-linearity of the Einstein equations. The difficulty may be overcome when the cosmological constant is non-null. Treating the cosmological contribution as the energy-momentum of vacuum, and representing the metric tensor onto the tetrad of its eigenvectors, the corresponding energy-momentum and, consequently, the Hamiltonian are easily quantized assuming a correspondence rule according to which the eigenvectors are replaced by creation and annihilation operators for the gravitational field. So the geometric Einstein tensor, which is opposite in sign respect to the vacuum energy-momentum (plus the possible known matter one), is also quantized. Physical examples provided by Schwarzschild-De Sitter, Robertson-Walker-De Sitter and Kerr-De Sitter solutions are examined.

Effect of the cosmological parameters on gravitational waves: general analysis

Some time ago it was pointed out that the presence of cosmological components could affect the propagation of gravitational waves (GW) beyond the usual cosmological redshift and that such effects might be observable in pulsar timing arrays (PTA). These analyses were done at leading order in the Hubble constant H 0, which is proportional to and (ρ i being the various cosmological fluid densities). In this work, we study in detail the propagation of metric perturbations on a Schwarzschild–de Sitter (SdS) background, close to the place where GW are produced, and obtain solutions that incorporate corrections linear in ρ i and Λ. At the next-to-leading order the corrections do not appear in the form of H 0 thus lifting the degeneracy among the various cosmological components. We also determine the leading corrections proportional to the mass of the final object; they are very small for the distances considered in PTA but may be of relevance in other cases. When transformed into comoving coordinates, the ones used in cosmological measurements, this SdS solution does satisfy the perturbation equations in a Friedmann–Lemaître–Robertson–Walker metric up to and including terms. This analysis is then extended to the other cosmological fluids, allowing us to consider GW sources in the Gpc range. Finally, we investigate the influence of these corrections in PTA observations.

The Study of the Energy Conditions of the Universe and Unique Solutions of the Einstein’s Field Equations in the Lights of the Theory of General Relativity

Objectives: To study the energy conditions of the universe and to find unique solutions of the Einstein’s field equations. Methods: A mathematical formulation developed to study the energy conditions of the universe and a new way of unique solutions of the famous Einstein’s field equations with appropriate theoretical and mathematical analysis of the theory of general relativity. Findings: With the reference to the power series expansion of the mass function ˆM(u; r) developed by Wang and Wu (1999), we have derived two new ways to solve Einstein’s field equations by introducing new values of n as n=2 and n=-2. Novelty: With the reference to the power series expansion of the mass function ˆM(u; r), we have found new ways to solve Einstein’s field equations with n=2 and n=-2. And we have derived a total of four new solutions of the Einstein’s field equations. And the solutions are very innovative and different from (i) Schwarzschild solution and (ii) de Sitter solution. The solutions own (i) the first solution with the line element in the equation (11) describes a stationary solution. It has coordinate singularity at r = (2m)􀀀1 . (ii) The second solution with the line element in the equation (14) will be reduced to those Schwarzschild black holes when m=0 with singularities at r=2M. Also, it will be that dark energy when M=0 with singularities at r = (2m)􀀀1 . (iii) The third solution with the line element in the equation (17) describes a stationary solution. It has coordinate singularity at r = (2m)13 (iv) The fourth solution with the line element in the equation (20) describes a stationary solution. It has coordinate singularity at r = (2m)13. Keywords: The Mass Function; Schwarzschild Solution; Einstein’s Field Equation; The Theory of General Relativity; Space - time Curvature

Study of gravastars in Rastall gravity

Gravastars have been considered as a feasible alternative to black holes in the past couple of decades.Stable models of gravastar have been studied in many of the alternative gravity theories besides standard GeneralRelativity (GR). The Rastall theory of gravity is a popular alternative to GR, specially in the cosmological andastrophysical context. Here, we propose a stellar model under the Rastall gravity following Mazur-Mottola's [1,2]conjecture. The gravastar consists of three regions, viz., (I) Interior region, (II) Intermediate shell region,and (III) Exterior region. The pressure within the interior core region is assumed with a constant negativematter-energy density which provides a repulsive force over the entire thin shell region. The shell is assumed tobe made up of fluid of ultrarelativistic plasma which follows the Zel'dovich's conjecture of stiff fluid [3,4].It is also assumed that the pressure is proportional to the matter-energy density according to Zel'dovich's conjecture,which cancel the repulsive force exerted by the interior region. The exterior region is completely vacuum which isdescribed by the Schwarzschild-de Sitter solution. Under all these specifications we obtain a set of exact andsingularity-free solutions of the gravastar model presenting several physically valid features within the frameworkof Rastall gravity. The physical properties of the shell region namely, the energy density, proper length, totalenergy and entropy are explored. The stability of the gravastar model is investigated using the surface redshiftagainst the shell thickness and maximizing the entropy of the shell within the framework of Rastall gravity.

Spherically symmetric configuration in f(Q) gravity

General relativity can be formulated equivalently with a non-Riemannian geometry that associates with an affine connection of nonzero nonmetricity $Q$ but vanishing curvature $R$ and torsion $T$. Modification based on this description of gravity generates the $f(Q)$ gravity. In this work we explore the application of $f(Q)$ gravity to the spherically symmetric configurations. We discuss the gauge fixing and connections in this setting. We demonstrate the effects of $f(Q)$ by considering the external and internal solutions of compact stars. The external background solutions for any regular form of $f(Q)$ coincide with the corresponding solutions in general relativity, i.e., the Schwarzschild-de Sitter solution and the Reissner-Nordstr\"om-de Sitter solution with an electromagnetic field. For internal structure, with a simple model $f(Q)=Q+\alpha Q^2$ and a polytropic equation of state, we find that a negative modification ($\alpha<0$) provides support to more stellar masses while a positive one ($\alpha>0$) reduces the amount of matter of the star.

Riemannian manifolds dual to static spacetimes

We establish a one-to-one correspondence between static spacetimes and Riemannian manifolds that maps causal geodesics to geodesics, as suggested by L. C. Epstein. We then explore constant curvature spacetimes - such as the de Sitter and the anti-de Sitter universes - and find that they map to constant curvature Riemannian manifolds, namely the Euclidean space, the sphere and the hyperbolic space. By imposing the conditions required to map to the sphere, we obtain the spherically symmetric metrics for which there is radial oscillatory motion with a period independent of the amplitude. We then consider the case of a perfect fluid and an Einstein cluster and determine the pressure and density profiles required to find this type of motion. Finally, we give examples of surfaces corresponding to certain types of motion for metrics that do not exhibit constant curvature, such as the Schwarzschild, Schwarzschild de Sitter and Schwarzschild anti-de Sitter solutions, and even for a simplified model of a wormhole.

Stable, thin wall, negative mass bubbles in de Sitter space-time

Negative mass makes perfect physical sense as long as the dominant energy condition is satisfied by the corresponding energy-momentum tensor. Heretofore, only {\it configurations} of negative mass had been found \cite{Belletete:2013nqa,Mbarek:2014ppa}, the analysis did not address stability or dynamics. In this paper, we analyze both of these criteria. We demonstrate the existence of {\it stable}, static, negative mass bubbles in an asymptotically de Sitter space-time. The bubbles are solutions of the Einstein equations and correspond to an interior region of space-time containing a specific mass distribution, separated by a thin wall from the exact, negative mass Schwarzschild-de Sitter space-time in the exterior. We apply the Israel junction conditions at the wall. For the case of an interior corresponding simply to de Sitter space-time with a different cosmological constant from the outside space-time, separated by a thin wall with energy density that is independent of the radius, we find static but unstable solutions which satisfy the dominant energy condition everywhere. The bubbles can collapse through spherically symmetric configurations to the exact, singular, negative mass Schwarzschild-de Sitter solution. Interestingly, this provides a counter-example of the cosmic censorship hypothesis. Alternatively, the junction conditions can be used to give rise to an interior mass distribution that depends on the potential for the radius of the wall. We show that for no choice of the potential, for positive energy density on the wall that is independent of the radius, can we get a solution that is non-singular at the origin. However, if we allow the energy density on the wall to depend on the radius of the bubble, we can find {\it stable}, static, non-singular solutions of negative mass which everywhere satisfy the dominant energy condition.

Gravitational deflection angle of light: Definition by an observer and its application to an asymptotically nonflat spacetime

The gravitational deflection angle of light for an observer and source at finite distance from a lens object has been studied by Ishihara et al. [Phys. Rev. D, 94, 084015 (2016)], based on the Gauss-Bonnet theorem with using the optical metric. Their approach to finite-distance cases is limited within an asymptotically flat spacetime. By making several assumptions, we give an interpretation of their definition from the observer's viewpoint: The observer assumes the direction of a hypothetical light emission at the observer position and makes a comparison between the fiducial emission direction and the direction along the real light ray. The angle between the two directions at the observer location can be interpreted as the deflection angle by Ishihara et al. The present interpretation does not require the asymptotic flatness. Motivated by this, we avoid such asymptotic regions to discuss another integral form of the deflection angle of light. This form makes it clear that the proposed deflection angle can be used not only for asymptotically flat spacetimes but also for asymptotically nonflat ones. We examine the proposed deflection angle in two models for the latter case; Kottler (Schwarzschild-de Sitter) solution in general relativity and a spherical solution in Weyl conformal gravity. Effects of finite distance on the light deflection in Weyl conformal gravity result in an extra term in the deflection angle, which may be marginally observable in a certain parameter region. On the other hand, those in Kottler spacetime are beyond reach of the current technology.

Study of gravastars under [formula omitted] gravity

In the present paper, we propose a stellar model under the f(T) gravity following the conjecture of Mazur-Mottola [1,2] known in literature as gravastar, a viable alternative to the black hole. This gravastar has three different regions, viz., (A) Interior core region, (B) Intermediate thin shell, and (C) Exterior spherical region. It is assumed that in the interior region the fluid pressure is equal to a negative matter-energy density providing a constant repulsive force over the spherical thin shell. This shell at the intermediate region is assumed to be formed by a fluid of ultrarelativistic plasma and the pressure, which is directly proportional to the matter-energy density according to Zel'dovich's conjecture of stiff fluid [Zel'dovich (1972) [3]], does nullify the repulsive force exerted by the interior core region for a stable configuration. On the other hand, the exterior spherical region can be described by the exterior Schwarzschild-de Sitter solution. With all these specifications we have found out a set of exact and singularity-free solutions of the gravastar which presents several physically interesting as well as valid features within the framework of alternative gravity.

Existence and stability of circular orbits in time-dependent spherically symmetric spacetimes

For a general spherically-symmetric four-dimensional metric the notion of “circularity” of a family of equatorial geodesic trajectories is defined in geometrical terms. The main object turns out to be the angular-momentum function J obeying a consistency condition involving the mean extrinsic curvature of the submanifold containing the geodesics. The analysis of linear stability is reduced to a simple dynamical system. For static metrics the existence of such geodesics is given when $$J^2 > 0$$ , and $$(J^2)^{\prime } > 0$$ for stability. The formalism is then applied to the Schwarzschild–de Sitter solution, both in its static and in its time-dependent cosmological version, as well to the Kerr–de Sitter solution. In addition we present an approximate solution to a cosmological metric containing a massive source and solving the Einstein field equation for a massless scalar.