Dimensional General Relativity Research Articles

The mass gap in five dimensional Einstein–Gauss–Bonnet black holes: a geometrical explanation

It is well known that, unlike in higher dimensional general relativity (GR), we cannot have a black hole with an arbitrarily small mass in five dimensional Einstein–Gauss–Bonnet gravity. When we study the dynamical black hole formation via the radiation collapse in the radiating Boulware–Deser spacetime in five dimensions, the central zero mass singularity is weak, conical and naked, and the horizon forms only when a finite amount of matter, that depends on the coupling constant of the Gauss–Bonnet term, falls into the central singularity. To understand this phenomenon transparently and geometrically, we study the radiating Boulware–Deser spacetime in five dimensions using a 1+1+3 spacetime decomposition, for the first time. We find that the geometric and thermodynamic quantities can be expressed in terms of the gravitational mass and the Gauss–Bonnet (GB) parameter and separate each of them into their Gauss–Bonnet and matter parts. Drawing comparisons with five dimensional GR at every step, we explicitly show how the mass gap arises for a general mass function M(v) and what functions for M(v) make certain geometrical quantities well defined at the central singularity. We show in the case of self-similar radiation collapse in the modified theory, the central singularity is not a sink for timelike geodesics and is extendable. This clearly demonstrates how the GB invariant affects the nature of the final state of a continual collapse in this modified theory.

Big-bang limit of 2 + 1 gravity and Thurston boundary of Teichmüller space

We study the asymptotic behavior of the solution curves of the dynamics of spacetimes of the topological type Σp×R, p > 1, where Σp is a closed Riemann surface of genus p, in the regime of 2 + 1 dimensional classical general relativity. The configuration space of the gauge fixed dynamics is identified with the Teichmüller space (TΣp≈R6p−6) of Σp. Utilizing the properties of the Dirichlet energy of certain harmonic maps, estimates derived from the associated elliptic equations in conjunction with a few standard results of the theory of the compact Riemann surfaces, we prove that every non-trivial solution curve runs off the edge of the Teichmüller space at the limit of the big bang singularity and approaches the space of projective measured laminations/foliations (PMLPMF), the Thurston boundary of the Teichmüller space.

Radiating stars and Riccati equations in higher dimensions

The objective of this study is to investigate spherically symmetric radiating stars undergoing gravitational collapse, in higher dimensional general relativity, inclusive of acceleration, expansion, shear, an electromagnetic field and a cosmological constant. Methods that can be used to obtain exact solutions to the boundary condition with/without a linear equation state are studied. Two distinct approaches are investigated. In the first approach, the boundary condition is expressed as a Riccati equation in terms of one of the dependent variables, and restrictions are placed to obtain new exact solutions. In the second approach, transformations that map the boundary condition into a new Riccati equation are investigated. The resulting new transformed equation is solved, by placing restrictions on the coefficients, to obtain new exact models. Special properties of the transformation are shown when appropriate restrictions on the parameters of the transformation are placed. This allows the order of the boundary condition to be reduced from a second order partial differential equation into a first order partial differential equation. The versatility of the transformation on other equations is exhibited when new solutions to the system of equations consisting of both the boundary condition and equation of state are obtained. When the dimension is set to four, some known solutions are recovered. It is shown that horizons can be identified by using a special case of the transformation. Our results elucidates the importance of the use of transformations that map the coordinates of differential equations into new and different coordinate systems.

R2 corrections to the black string instability and the boosted black string

We consider a perturbative Gauss-Bonnet term supplementing the Einstein-Hilbert action, and evaluate its effect on the spectrum of the scalar mode that triggers the Gregory-Laflamme instability of black strings in five dimensional General Relativity. After studying some properties of the static black string, we provide the correction to the Lichnerowicz operator up to $\mathcal{O}(\alpha^2)$. For the scalar mode of the gravitational perturbation, we find a master variable and study its spectrum, providing an analysis of the regime of validity of our scheme. We show that the instability persists under the inclusion of the $R^2$-correction, and that the critical wavelength increases with the value of $\alpha/r_+^2<<1$, providing new evidence of the phenomenon which was already observed in other approaches. We also construct the boosted black strings and compute the correction to the mass, the momentum and the tension due to the higher curvature term. The presence of the dimensionful coupling $\alpha$ spoils the validity of the Smarr relation, which is the gravitational version of the Euler relation which must hold for every homogeneous, thermodynamic system. We show that this identity can be restored by working in an extended thermodynamic setup that includes variations of the Gauss-Bonnet coupling.

Strong cosmic censorship conjecture for a charged BTZ black hole

The strong cosmic censorship conjecture, whose validation asserts the deterministic nature of general relativity, has been studied for charged BTZ black holes in three dimensional general relativity, as well as for Nth order pure Lovelock gravity in d=2N+1 spacetime dimensions. Through both analytical and numerical routes, we have computed the ratio of the imaginary part of the quasi-normal mode frequencies with the surface gravity at the Cauchy horizon. The lowest of which corresponds to the key parameter associated with violation of strong cosmic censorship conjecture. Our results demonstrate that this parameter is always less than the critical value (1/2), thereby respecting the strong cosmic censorship conjecture. This is in complete contrast to the four or, higher dimensional black holes, as well as for rotating BTZ black hole, where the violation of strong cosmic censorship conjecture exists. Implications and possible connection with the stability of the photon orbits have been discussed.

Understanding gravitational entropy of black holes: A new proposal via curvature invariants

Partly motivated by the arrow of time problem in cosmology and the Weyl curvature hypothesis formulated by Roger Penrose, previous works in the literature have proposed - among other possibilities - the square of the Weyl curvature, as being the underlying entropy density function of black hole entropy, but the proposal suffers from a few drawbacks. In this work, we propose a new entropy density function also based solely on the Weyl curvature, but adopting some other combinations of curvature invariants. As an improvement we find that our method works for all static black hole solutions in four and five dimensional general relativity regardless of whether they are empty space solutions or not. It should also be possible to generalize our method to higher dimensions. This allows us to discuss the physical interpretation of black hole entropy, which remains somewhat mysterious. Extending to modified theories of gravity, our work also suggests that gravitational entropy in some theories is a manifestation of different physical effects since we need to choose different combinations of curvature quantities

Bound on Photon Circular Orbits in General Relativity and Beyond

The existence of a photon circular orbit can tell us a lot about the nature of the underlying spacetime, since it plays a pivotal role in the understanding of the characteristic signatures of compact objects, namely the quasi-normal modes and shadow radius. For this purpose, determination of the location of the photon circular orbit is of utmost importance. In this work, we derive bounds on the location of the photon circular orbit around compact objects within the purview of general relativity and beyond. As we have explicitly demonstrated, contrary to the earlier results in the context of general relativity, the bound on the location of the photon circular orbit is not necessarily an upper bound. Depending on the matter content, it is possible to arrive at a lower bound as well. This has interesting implications for the quasi-normal modes and shadow radius, the two key observables related to the strong field tests of gravity. Besides discussing the bound for higher dimensional general relativity, we have also considered how the bound on the photon circular orbits gets modified in the braneworld scenario, for pure Lovelock and general Lovelock theories of gravity. Implications of these results for compact objects were also discussed.

Binary black hole system at equilibrium

An exact and analytical solution of four dimensional vacuum General Relativity representing a system of two static black holes at equilibrium is presented. The metric is completely regular outside the event horizons, both from curvature and conical singularities. The balance between the two Schwarzschild sources is granted by an external gravitational field, without the need of extra matter fields besides gravity, nor strings or struts. The geometry of the solution is analysed. The Smarr law, the first and the second law of black hole thermodynamics are discussed.

Warping wormholes with dust: a metric construction of the Python\u2019s Lunch

We show how wormholes in three spacetime dimensions can be customizably warped using pressureless matter. In particular, we exhibit a large new class of solutions in (2 + 1)-dimensional general relativity with energy-momentum tensor describing a negative cosmological constant and positive-energy dust. From this class of solutions, we construct wormhole geometries and study their geometric and holographic properties, including Ryu- Takayanagi surfaces, entanglement wedge cross sections, mutual information, and outer entropy. Finally, we construct a Python’s Lunch geometry: a wormhole in asymptotically anti-de Sitter space with a local maximum in size near its middle.

Exact inner metric and microscopic state of AdS3-Schwarzschild BHs

Through full solvability of 2+1 dimensional general relativity we derive out exact dynamic inner metric of collapsing stars with inhomogeneous initial mass distribution but joining with outside Anti-deSitt-Schwarzschild black holes smoothly. We prove analytically by standard quantum mechanics that the log-number of such solutions, or microscopic states of the system is proportional to the perimeter of the outside black holes. Key formulas for generalizing to 3+1D Schwarzschild black holes are also presented. Our result provides a bulk space viewpoint to questions on what the microscopic degrees of freedom are and who their carriers are in various holographic and/or asymptotic symmetry methods to black hole entropies. It may also shed light for singularity theorem and cosmic censorship related researches.

Attempts at vacuum counterexamples to cosmic censorship in AdS

We consider vacuum solutions of four dimensional general relativity with Λ < 0. We numerically construct stationary solutions that asymptotically approach a boundary metric with differential rotation. Smooth solutions only exist up to a critical rotation. We thus argue that increasing the differential rotation by a finite amount will cause the curvature to grow without bound. This holds for both zero and nonzero temperature, and both compact and noncompact boundaries. However, the boundary metric always develops an ergoregion before reaching the critical rotation, which probably means that the energy is unbounded from below for these counterexamples to cosmic censorship.

Extended naked conical singularity in radiation collapse in Einstein-Gauss-Bonnet gravity

In this paper, we investigate the continual gravitational collapse of a spherically symmetric radiation shell in five-dimensional Einstein-Gauss-Bonnet gravity. We show that the final fate of such a collapse is an extended and weak curvature naked conical singularity at the centre, which then subsequently becomes covered by an apparent horizon. This process is entirely different from the five dimensional general relativity counterpart, where a strong curvature singularity develops at the centre. Since the singularity in the case of Einstein-Gauss-Bonnet gravity is sufficiently weak, we argue that the spacetime can be extended through it, which gives us an elegant way of constructing regular black holes in higher dimensions without violating any energy conditions. We also extend our study to spacetimes with null and string fluids, which are the counterpart of generalized Vaidya spacetimes in general relativity. We show that similar end states are also possible in those cases. Higher-dimensional spacetimes are then considered.

Lichnerowicz modes and black hole families in Ricci quadratic gravity

A new branch of black hole solutions occurs along with the standard Schwarzschild branch in $n$-dimensional extensions of general relativity including terms quadratic in the Ricci tensor. The standard and new branches cross at a point determined by a static negative-eigenvalue eigenfunction of the Lichnerowicz operator, analogous to the Gross-Perry-Yaffe eigenfunction for the Schwarzschild solution in standard $n=4$ dimensional general relativity. This static eigenfunction has two r\^oles: both as a perturbation away from Schwarzschild along the new black-hole branch and also as a threshold unstable mode lying at the edge of a domain of Gregory-Laflamme-type instability of the Schwarzschild solution for small-radius black holes. A thermodynamic analogy with the Gubser and Mitra conjecture on the relation between quantum thermodynamic and classical dynamical instabilities leads to a suggestion that there may be a switch of stability properties between the old and new black-hole branches for small black holes with radii below the branch crossing point.

The Cartan algorithm in five dimensions

In this paper, we introduce an algorithm to determine the equivalence of five dimensional spacetimes, which generalizes the Karlhede algorithm for four dimensional general relativity. As an alternative to the Petrov type classification, we employ the alignment classification to algebraically classify the Weyl tensor. To illustrate the algorithm, we discuss three examples: the singly rotating Myers-Perry solution, the Kerr (Anti-) de Sitter solution, and the rotating black ring solution. We briefly discuss some applications of the Cartan algorithm in five dimensions.

Reduced Quantum General Relativity in Higher Dimensions

The higher dimensional Quantum General Relativity of a Riemannian manifold being an embedded space in a space-time being a Lorentzian manifold is investigated. The model of quantum geometrodynamics, based on the Wheeler-DeWitt equation reduced to a first order functional quantum evolution supplemented through an additional eigenequation for the scalar curvature, is formulated. Furthermore, making use of the objective quantum gravity and global one-dimensional conjecture, the general wave function beyond the Feynman path integral technique is derived. The resulting quantum gravity model creates the opportunity of potentially new theoretical and phenomenological applications for astrophysics, cosmology, and physics.

Buchdahl’s inequality in five dimensional Gauss–Bonnet gravity

The Buchdahl limit for static spherically symmetric isotropic stars is generalised to the case of five dimensional Gauss-Bonnet gravity. Our result depends on the sign of the Gauss-Bonnet coupling constant $\alpha$. When $\alpha>0$, we find, unlike in general relativity, that the bound is dependent on the stellar structure, in particular the central energy density. We find that stable stellar structures can exist arbitrarily close to the event horizon. Thus stable stars can exist with extra mass in this theory compared to five dimensional general relativity. For $\alpha<0$ it is found that the Buchdahl bound is more restrictive than the general relativistic case.

Wormhole Dynamics

Wormholes are theoretical products in general relativity, and are popular tools in science fictions. We know numerically the four-dimensional Ellis wormhole solution (the so- called Morris-Thorne's traversable wormhole) is unstable against an input of scalar-pulse from one side. We investigate this feature for higher-dimensional versions, both in n-dimensional general relativity and in Gauss-Bonnet gravity. We derived Ellis-type wormhole solution in n- dimensional general relativity, and found existence of unstable modes in its linear perturbation analysis. We also evolved it numerically in dual-null coordinate system, and confirmed its instability. The wormhole throat will change into black-hole horizon for the input of (relatively) positive energy, while it will change into inflationary expansion for (relatively) negative energy input. If we add Gauss-Bonnet terms (higher curvature correction terms in gravity), then wormhole tends to expand (or change to black-hole) if the coupling constant α is positive (negative).

Locally Inertial Approximations of Balance Laws Arising in (1+1)-Dimensional General Relativity

An elementary model of (1+1)-dimensional general relativity, known as “${\cal R}={\cal T}$” and mainly developed by Mann and coworkers in the early 1990s, is set up in various contexts. Its formulation, mostly in isothermal coordinates, is derived and a relativistic Euler system of self-gravitating gas coupled to a Liouville equation for the metric's conformal factor is deduced. First, external field approximations are carried out: both a Klein--Gordon equation is studied along with its corresponding density, and a Dirac one inside a hydrostatic gravitational field induced by a static, piecewise constant mass repartition. Finally, the coupled Euler--Liouville system is simulated, by means of a locally inertial Godunov scheme: the gravitational collapse of a static random initial distribution of density is displayed. Well-balanced discretizations rely on the treatment of source terms at each interface of the computational grid, hence the metric remains flat in every computational cell.

Cloud of strings for radiating black holes in Lovelock gravity

We present exact spherically symmetric null dust solutions in the third order Lovelock gravity with a string cloud background in arbitrary $N$ dimensions,. This represents radiating black holes and generalizes the well known Vaidya solution to Lovelock gravity with a string cloud in the background. We also discuss the energy conditions and horizon structures, and explicitly bring out the effect of the string clouds on the horizon structure of black hole solutions for the higher dimensional general relativity and Einstein-Gauss-Bonnet theories. It turns out that the presence of the coupling constant of the Gauss-Bonnet terms and/or background string clouds completely changes the structure of the horizon and this may lead to a naked singularity. We recover known spherically symmetric radiating models as well as static black holes in the appropriate limits.

Deconstructing dimensions and massive gravity

We show that the ghost-free models of massive gravity and their multi-graviton extensions follow from considering higher dimensional General Relativity in Einstein–Cartan form on a discrete extra dimension, according to the dimensional deconstruction paradigm. We show that dimensional deconstruction is equivalent to a truncation of the Kaluza–Klein tower at the nonlinear level. Higher dimensional gravity is not recovered from a lower dimensional multi-graviton theory in the limit of a continuous extra dimension (infinite Kaluza–Klein tower) due to the appearance of a low strong coupling scale that depends on IR physics. This strong coupling scale, which is associated with the mass of the lowest Kaluza–Klein mode, controls the onset of the Vainshtein mechanism and is crucial to the theoretical and observational viability of the truncated theory.