Tangherlini Black Hole Research Articles

Bogomol’nyi-like equations in gravity theories

Using the Bogomol’nyi–Prasad–Sommerfield Lagrangian method, we show that gravity theory coupled to matter in various dimensions may possess Bogomol’nyi-like equations, which are first-order differential equations, satisfying the Einstein equations and the Euler–Lagrange equations of classical fields (U(1) gauge and scalar fields). In particular we consider static and spherically symmetric solutions by taking proper ansatzes and then we find an effective Lagrangian density that can reproduce the Einstein equations and the Euler–Lagrange equations of the classical fields. We consider the BPS Lagrangian density to be linear function of first-order derivative of all the fields. From these two Lagrangian desities we are able to obtain the Bogomol’nyi-like equations whose some of solutions are well-known such as Schwarzschild, Reissner–Nordström, Tangherlini black holes, and the recent black holes with scalar hair in three dimensions (Phys. Rev. D 107, 124047). Using these Bogomol’nyi-like equations, we are also able to find new solutions for scalar hair black holes in three and four dimensional spacetime. Furthermore we show that the Bogomol’nyi–Prasad–Sommerfield Lagrangian method can provide a simple alternative proof of black holes uniqueness theorems in any dimension.

Greybody factors for higher-dimensional non-commutative geometry inspired black holes

Greybody factors are computed for massless fields of spin 0, 1/2, 1, and 2 emitted from higher-dimensional non-commutative geometry inspired black holes. Short-range potentials are used with path-ordered matrix exponentials to numerically calculate transmission coefficients. The resulting absorption cross sections and emission spectra are computed on the brane and compared with the higher-dimensional Schwarzschild–Tangherlini black hole. A non-commutative black hole at its maximum temperature in seven extra dimensions will radiate a particle flux and power of 0.72–0.81 and 0.75–0.81, respectively, times lower than a Schwarzschild–Tangherlini black hole of the same temperature. A non-commutative black hole at its maximum temperature in seven extra dimensions will radiate a particle flux and power of 0.64–0.72 and 0.60–0.64, respectively, times lower than a Schwarzschild–Tangherlini black hole of the same mass.

Nonassociative black holes in R-flux deformed phase spaces and relativistic models of Perelman thermodynamics

This paper explores new classes of black hole (BH) solutions in nonassociative and noncommutative gravity, focusing on features that generalize to higher dimensions. The theories we study are modelled on (co) tangent Lorentz bundles with a star product structure determined by R-flux deformations in string theory. For the nonassociative vacuum Einstein equations we consider both real and complex effective sources. In order to analyze the nonassociative vacuum Einstein equations we develop the anholonomic frame and connection deformation methods, which allows one to decoupled and solve these equations. The metric coefficients can depend on both space-time coordinates and energy-momentum. By imposing conditions on the integration functions and effective sources we find physically important, exact solutions: (1) 6-d Tangherlini BHs, which are star product and R-flux distorted to 8-d black ellipsoids (BEs) and BHs; (2) nonassocitative space-time and co-fiber space double BH and/or BE configurations generalizing Schwarzschild-de Sitter metrics. We also investigate the concept of Bekenstein-Hawking entropy and find it applicable only for very special classes of nonassociative BHs with conventional horizons and (anti) de Sitter configurations. Finally, we show how analogs of the relativistic Perelman W-entropy and related geometric thermodynamic variables can be defined and computed for general classes of off-diagonal solutions with nonassociative R-flux deformations.

Absorption and scattering of a high dimensional non-commutative black hole

In this work, we investigate the scattering of massless plane scalar waves by the high dimensional non-commutative Schwarzschild–Tangherlini black hole. We use the partial wave approach to determine the scattering and absorption cross sections in the incident wavelength range. Our numerical results demonstrate that the bigger the non-commutative parameter, the smaller the maximum value of the related partial absorption cross section, however the tendency is slightly. We also discovered that when the non-commutative parameter is weak, the absorption cross section of the high dimensional black hole oscillates in the low frequency zone. The total absorption cross section oscillates around the geometrical optical limit in the high frequency range, and the scattering characteristics of black holes with various parameters are visibly different. The influence on the differential scattering cross section is particularly pronounced at large angles.

Higher-dimensional polytropic and modified Chaplygin black holes: Thermodynamics and heat engines

In this paper, we have assumed that the thermodynamic pressure may be generated by the negative cosmological constant and the thermodynamic parameters of the n-dimensional asymptotically Anti-de Sitter (AdS)-Tangherlini black holes are identical with the polytropic gas and modified Chaplygin gas separately. The black hole mass, volume, entropy, and temperature have been written due to the assigned gases’ thermodynamic system. We found the solutions of Einstein’s field equations of n-dimensional black hole for polytropic gas and modified Chaplygin gas. The null energy condition, weak energy condition, strong energy condition, and dominant energy condition have been examined for the fluid sources in n-dimensional black hole system due to both the gases. We found that the four energy conditions completely depend on the model parameters and the dimensions of the spacetime. We have studied the heat engine phenomena for both the black holes and analyzed the work done with the efficiencies due to the Carnot and other cycles.

Gravitational Capture Cross-Section of Particles by Schwarzschild-Tangherlini Black Holes

We study the capture cross-section of massless (photon) and massive test particles by the Schwarzschild–Tangherlini black hole, which is a solution of pure general relativity in higher dimensional spacetime with R×SD−2 topology. It is shown that an extra dimension weakens the gravitational attraction of a black hole, and consequently, radii of all the characteristic circular orbits, such as the radius of a photonsphere decrease in the higher dimensions. Furthermore, it is shown that in higher dimensions, there are no stable and bounded circular orbits. The critical impact parameters and capture cross-sections of photons and massive particles are calculated for several higher dimensions and it is shown that they also decrease with increasing dimension. Moreover, we calculate the capture cross-section of relativistic and non-relativistic test particles in the higher dimensions.

Fuzzy and discrete black hole models* *The first author was partially supported by CONACyT (México).

Using quantum Riemannian geometry, we solve for a Ricci = 0 static spherically-symmetric solution in 4D, with the S 2 at each t, r a noncommutative fuzzy sphere, finding a dimension jump with solutions having the time and radial form of a classical 5D Tangherlini black hole. Thus, even a small amount of angular noncommutativity leads to radically different radial behaviour, modifying the Laplacian and the weak gravity limit. We likewise provide a version of a 3D black hole with the S 1 at each t, r now a discrete circle , with the time and radial form of the inside of a classical 4D Schwarzschild black hole far from the horizon. We study the Laplacian and the classical limit . We also study the 3D FLRW model on with S 2 an expanding fuzzy sphere and find that the Friedmann equation for the expansion is the classical 4D one for a closed Universe.

Sparsity of Hawking radiation in D + 1 space-time dimensions for massless and massive particles

Hawking radiation from an evaporating black hole has often been compared to black body radiation. However, this comparison misses an important feature of Hawking radiation: its low density of states. This can be captured in an easy to calculate, heuristic, and semi-analytic measure called ‘sparsity’. In this note we shall present both the concept of sparsities and its application to D + 1-dimensional Tangherlini black holes and their evaporation. In particular, we shall also publish for the first time sparsity expressions taking into account in closed form effects of non-zero particle mass. We will also see how this comparatively simple method reproduces results of (massless) Hawking radiation in higher dimensions and how different spins contribute to the total radiation in this context.

Higher dimensional black hole scalarization

In the simplest scalar-tensor theories, wherein the scalar field is non-minimally coupled to the Ricci scalar, spontaneous scalarization of electrovacuum black holes (BHs) does not occur. This ceases to be true in higher dimensional spacetimes, d > 4. We consider the scalarization of the higher dimensional Reissner-Nordström BHs in scalar-tensor models and provide results on the zero modes for different d, together with an explicit construction of the scalarized BHs in d = 5, discussing some of their properties. We also observe that a conformal transformation into the Einstein frame maps this model into an Einstein-Maxwel- scalar model, wherein the non-minimal coupling occurs between the scalar field and the Maxwell invariant (rather than the Ricci scalar), thus relating the occurence of scalarization in the two models. Next, we consider the spontaneous scalarization of the Schwarzschild- Tangherlini BH in extended-scalar-tensor-Lovelock gravity in even dimensions. In these models, the scalar field is non-minimally coupled to the (d/2)th Euler density, in d spacetime dimensions. We construct explicitly examples in d = 6, 8, showing the properties of the four dimensional case are qualitatively generic, but with quantitative differences. We compare these higher d scalarized BHs with the hairy BHs in shift-symmetric Horndeski theory, for the same d, which we also construct.

Trapped photons in Schwarzschild–Tangherlini spacetimes

We study the set of trapped photons outside the event horizon of Schwarzschild–Tangherlini black holes in dimension . We prove that the only trapped photons are those with constant Boyer–Lindquist coordinate radius. We then identify the set of trapped photons as a submanifold of the tangent bundle of the spacetime, and prove that this submanifold has topology , as suggested by Cederbaum–Jahns.

Compatibility complexes of overdetermined PDEs of finite type, with applications to the Killing equation

In linearized gravity, two linearized metrics are considered gauge-equivalent, , when they differ by the image of the Killing operator, . A universal (or complete) compatibility operator for K is a differential operator K1 such that and any other operator annihilating K must factor through K1. The components of K1 can be interpreted as a complete (or generating) set of local gauge-invariant observables in linearized gravity. By appealing to known results in the formal theory of overdetermined PDEs and basic notions from homological algebra, we solve the problem of constructing the Killing compatibility operator K1 on an arbitrary background geometry, as well as of extending it to a full compatibility complex Ki (), meaning that for each Ki the operator Ki+1 is its universal compatibility operator. Our solution is practical enough that we apply it explicitly in two examples, giving the first construction of full compatibility complexes for the Killing operator on these geometries. The first example consists of the cosmological FLRW spacetimes, in any dimension. The second consists of a generalization of the Schwarzschild–Tangherlini black hole spacetimes, also in any dimension. The generalization allows an arbitrary cosmological constant and the replacement of spherical symmetry by planar or pseudo-spherical symmetry.

Study of semiclassical instability of the Schwarzschild AdS black hole in the large D limit

We analyze the semiclassical stability of the Schwarzschild AdS black hole in the Euclidean partition function approach. We perform this computation in the large D limit and focus on scalar perturbations. We obtain the equations for non-spherically symmetric scalar perturbations in a simple form. For a class of perturbations stability is demonstrated by the S-deformation method. For some other classes we rule out unstable modes of . We also analyze the spherically symmetric perturbations and demonstrate the appearance of an unstable mode for small black holes in the large D limit. We obtain an expression for the eigenvalue corresponding to the unstable mode to next to leading order in a 1/D expansion. This result agrees with a previously obtained numerical bound on this eigenvalue. For cosmological constant zero, our answer matches a previous result obtained for the corresponding eigenvalue for the D dimensional Schwarzschild–Tangherlini black hole to next to leading order in a 1/D expansion.

Shadow of Schwarzschild–Tangherlini black holes

We study the shadow cast by the higher-dimensional Schwarzschild–Tangherlini black hole, and analytically calculate the influence of extra dimensions on the shadow of a black hole. A black hole casts a shadow as an optical appearance because of its strong gravitational field which is known to be a dark zone covered by a circle for a Schwarzschild black hole. We demonstrate that the null geodesic equation can be integrated by Hamilton–Jacobi approach, which enable us to investigate the shadow cast by the higher-dimensional Schwarzschild–Tangherlini black holes. Interestingly, it turns out that, for fixed values of the mass parameter, the shadow in higher-dimensional spacetimes are smaller when compared with four-dimensional Schwarzschild black hole. Further, the shadows of higher-dimensional Schwarzschild–Tangherlini black holes are concentric circles with radius of the circle decreases with increase in D. We visualize the photon regions and the shadows in various dimensions for different values of the parameters, and the energy emission rates are is also investigated. Our results, in the limit D=4, reduced exactly to vis-à-vis Schwarzschild black hole case.

Slowly damped quasinormal modes of the massive Dirac field in d -dimensional Tangherlini spacetime

We consider quasinormal modes of the massive Dirac field in the background of a Schwarzschild- Tangherlini black hole. Different dimensions of the spacetime are considered, from d = 4 to d = 9. The quasinormal modes are calculated using two independent methods: WKB and continued fraction. We obtain the spectrum of quasinormal modes for different values of the overtone number and angular quantum number. An analytical approximation of the spectrum valid in the case of large values of the angular quantum number and mass is calculated. Although we don't find unstable modes in the spectrum, we show that for large values of the mass, the quasinormal modes can become very slowly damped, giving rise to quasistationary perturbations.

Quantum corrections to the thermodynamics of Schwarzschild–Tangherlini black hole and the generalized uncertainty principle

We investigate the thermodynamics of Schwarzschild-Tangherlini black hole in the context of the generalized uncertainty principle. The corrections to the Hawking temperature, entropy and the heat capacity are obtained via the modified Hamilton-Jacobi equation. These modifications show that the GUP changes the evolution of Schwarzschild-Tangherlini black hole. Specially, the GUP effect becomes susceptible when the radius or mass of black hole approach to the order of Planck scale, it stops radiating and leads to black hole remnant. Meanwhile, the Planck scale remnant can be confirmed through the analysis of the heat capacity. Those phenomenons imply that the GUP may give a way to solve the information paradox. Besides, we also investigate the possibilities to observe the black hole at LHC, the results demonstrate that the black hole can not be produced in the recent LHC.

Modified tunneling radiation of the scalar particles and Dirac particles from a five-dimensional black hole

Incorporating the generalized uncertainty principle (GUP) into the tunneling mechanism, we have studied the tunneling radiation of the scalar particles and fermions from the five-dimensional Schwarzschild–Tangherlini black hole. The results showed that the GUP corrected temperatures do not only depend on the mass of ST black hole, but are also affected by the gravity effects correction [Formula: see text]. Besides, the [Formula: see text] slows down the Hawking temperature increasing and causes the existence of remnants in black hole evaporation.

Electrostatic self-force in the field of an (n + 1)-dimensional black hole: Dimensional regularization

The self-energy of a classical charged particle localized at a relatively large distance outside the event horizon of an (n + 1)-dimensional Schwarzschild–Tangherlini black hole for an arbitrary n ≥ 3 is calculated. An expression for the electrostatic Green function is derived in the first two orders of the perturbation theory. Dimensional regularization is proposed to be used to regularize the corresponding formally divergent expression for the self-energy. The derived expression for the renormalized self-energy is compared with the results of other authors.

Throat quantization of the Schwarzschild–Tangherlini(-AdS) black hole

Adopting the throat quantization pioneered by Louko and Mäkelä, we derive the mass and area spectra for the Schwarzschild–Tangherlini black hole and its anti-de Sitter (AdS) generalization in arbitrary dimensions. We find that the system can be quantized exactly in three special cases: the three-dimensional BTZ black hole, toroidal black holes in any dimension, and five-dimensional Schwarzshild–Tangherlini(-AdS) black holes. For the remaining cases the spectra are obtained for large mass using the WKB approximation. For asymptotically flat black holes, the area/entropy has an equally spaced spectrum, as expected from previous work. In the asymptotically AdS case on the other hand, it is the mass spectrum that is equally spaced. Our exact results for the BTZ black hole mass with Dirichlet boundary conditions are consistent with the spectra of the corresponding operators in the dual CFT.

Geometric engineering 5d black holes with rod diagrams

Static solutions of 5-dimensional gravity with two spatial Killing vectors are characterized by their rod structures. In this note we describe how the orbifold singularities and the topologies of the horizons and asymptotic regions can be determined from the corresponding rod diagrams. As an example we introduce the black lens, a static 5-dimensional black hole with a horizon of lens space topology which is asymptotically Minkowski space. The solution is novel in that the asymptotic Minkowski space is not quotiented. However it suffers from a naked singularity. While the conical and orbifold singularities have been removed, two spherical curvature singularities remain. These singularities do not contribute to the ADM mass, and the thermodynamics of the black lens is well behaved, although its entropy is lower than that of a Tangherlini black hole of the same mass.

Ray-tracing in four and higher dimensional black hole spacetimes: An analytical approximation

We study null ray propagation in a spacetime of static Schwarzschild--Tangherlini black holes in an arbitrary number of dimensions. We focus on the bending angle and the retarded time delay for rays emitted in the vicinity of a black hole and propagating to the infinity. We obtain an analytic expression in terms of elementary functions which approximate the bending angle and time delay in these spacetimes with high accuracy. We analyze the relative error of the developed analytic approximations and show that it is quite small in the complete domain of the parameter space for the rays reaching the infinity and for a different number of the spacetime dimensions. Possible applications of the obtained results are briefly discussed.