arXiv subjects

M. M. Akbar

Publications and source records attributed to M. M. Akbar.

13 recordsLinked to original sources

arXiv2025

Spacetime Structure of a Regular Accelerating Black Hole Pair in General Relativity

We revisit the one-parameter generalization of the C-metric derived by Ernst, which solves the vacuum Einstein equations. Resolving conflicting claims in the literature, we determine the correct value of the parameter that ensures the regularity of the metric on the axis. This "regularized C-metric" describes a pair of accelerating black holes without the line source present in the original C-metric. Additionally, this generalization changes the Petrov type from D to I. We use the Gauss-Bonnet theorem to analyze the nodal singularities, the line source, and their relation to the horizon topology. Both the black hole and acceleration horizons are found to be embeddable in $\mathrm{E}^3$. We examine various geometric and asymptotic properties in detail using several coordinate systems and construct the corresponding 2D and 3D conformal diagrams. This process is more involved than for the original C-metric due to the presence of the exponential factors. These exponential factors also introduce curvature singularities at infinity, which obstructs asymptotic flatness. Contrary to Bonnor's expectation, we demonstrate why Bondi's algorithm for obtaining the standard Bondi form fails for the C-metric, despite its asymptotic flatness. We also show that Ernst's solution-generating prescription in boost-rotation symmetric coordinates is a symmetry of the wave equation.

gr-qc↗

arXiv2025

Near-Horizon Symmetries of Local Black Holes in General Relativity

We analyze the near-horizon symmetries of static, axisymmetric, four-dimensional black holes with spherical and toroidal horizon topologies in vacuum general relativity. These black hole solutions, collectively referred to as local/distorted black holes, are known in closed form and are not asymptotically flat. Building on earlier works in the literature that primarily focused on black holes with spherical topology, we compute the algebra of the Killing vector fields that preserve the asymptotic structure near the horizons and the algebra of the associated Noether-Wald charges under the boundary conditions that produce the spin-$s$ BMS$_d$ and the Heisenberg-like algebras. We show that a similar analysis extends to all local axisymmetric black holes. The toroidal topology of the holes changes the algebras considerably. For example, one obtains two copies of spin-$s$ BMS$_3$ instead of spin-$s$ BMS$_4$. We also revisit the thermodynamics of black holes under these boundary conditions. While previous studies suggested that spin-$s$ BMS$_d$ preserves the first law of thermodynamics for isolated horizons ($κ= \text{const.}$), our analysis indicates that this is not generally the case when the spin parameter $s$ is nonzero. A nonzero $s$ can be seen as introducing a conical singularity (in the Euclidean quantum gravity sense) or a Hamiltonian that causes soft hairs to contribute to the energy. This leads us to interpret the spin-$s$ BMS$_d$ boundary condition as arising in the context of dynamical black holes.

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arXiv2025

Higher-Dimensional Vacuum Einstein Equations: Symmetry, New Solutions, and Ricci Solitons

We show that the system of vacuum Einstein equations (i.e., Ricci-flat metrics) with two hypersurface-orthogonal, commuting Killing vector fields in $d \ge 5$ dimensions is invariant under the action of a one-parameter Lie group, and the group action on any metric can be expressed in a closed, universal form. This enables the generation of a one-parameter family of solutions from any given ``seed" solution of the system without solving additional equations, as well as one-parameter families of local steady Ricci solitons. This extends the Lie point symmetry in four dimensions, found earlier for axisymmetric static vacuum systems, and provides the first example of solution generation in higher-dimensional vacuum Einstein equations that can be realized purely algebraically.

gr-qc↗

arXiv2017

Spherical Spacelike Geometries in Static Spherically Symmetric Spacetimes: Generalized Painlevè-Gullstrand Coordinates, Foliation, and Embedding

It is well known that static spherically symmetric spacetimes can admit foliations by flat spacelike hypersurfaces, which are best described in terms of the Painlevè--Gullstrand coordinates. The uniqueness and existence of such foliations were addressed earlier. In this paper, we prove, purely geometrically, that any possible foliation of a static spherically symmetric spacetime by an arbitrary codimension-one spherical spacelike geometry, up to time translation and rotation, is unique, and we find the algebraic condition under which it exists. This leads us to what can be considered as the most natural generalization of the Painlevè--Gullstrand coordinate system for static spherically symmetric metrics, which, in turn, makes it easy to derive generic conclusions on foliation and to study specific cases as well as to easily reproduce previously obtained generalizations as special cases. In particular, we note that the existence of foliation by flat hypersurfaces guarantees the existence of foliation by hypersurfaces whose Ricci curvature tensor is everywhere non-positive (constant negative curvature is a special case). The study of uniqueness and the existence concurrently solves the question of embeddability of a spherical spacelike geometry in one-dimensional higher static spherically symmetric spacetimes, and this produces known and new results geometrically, without having to go through the momentum and Hamiltonian constraints.

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arXiv2017

Embedding FLRW Geometries in Pseudo-Euclidean and Anti-de Sitter Spaces

Contrary to the general consensus in the literature that Friedmann--Lemaître--Robertson--Walker (FLRW) geometries are of embedding class one (i.e.,\ embeddable in one higher dimensional pseudo-Euclidean spaces), we show that the most general $k=0$ and $k=-1$ FLRW geometries are of embedding class two, and their corresponding pseudo-Euclidean spaces have strictly one and two negative eigenvalues, respectively. These are particular results that follow from the new perspective on FLRW embedding that we develop in this paper, namely that these embeddings are equivalent to unit-speed parametrized curves in two or three dimensions. A careful analysis of appropriate tensor fields then gives identical results and further explains why the class-two geometries remained hidden. However, the signatures of the embedding spaces, as well as the explicit embedding formulae, follow only from the curve picture. This also streamlines the comparatively difficult $k=0$ class and provides new explicit embedding formulae for it and reproduces known embedding formulae for the $k=1,-1$ classes. Embedding into anti-de Sitter space in one higher dimension can likewise be done by constructing associated curves. In particular, we find that all $k=1$ and mildly restricted subclasses of $k=0, -1$ geometries are embeddable in anti-de Sitter space in one higher dimension.

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arXiv2015

Static Axisymmetric Einstein Equations in Vacuum: Symmetry, New Solutions and Ricci Solitons

An explicit one-parameter Lie point symmetry of the four-dimensional vacuum Einstein equations with two commuting hypersurface-orthogonal Killing vector fields is presented. The parameter takes values over all of the real line and the action of the group can be effected algebraically on any solution of the system. This enables one to construct particular one-parameter extended families of axisymmetric static solutions and cylindrical gravitational wave solutions from old ones, in a simpler way than most solution-generation techniques, including the prescription given by Ernst for this system. As examples, we obtain the families that generalize the Schwarzschild solution and the $C$-metric. These in effect superpose a Levi-Civita cylindrical solution on the seeds. Exploiting a correspondence between static solutions of Einstein's equations and Ricci solitons (self-similar solutions of the Ricci flow), this also enables us to construct new steady Ricci solitons.

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arXiv2012

Pseudo-Riemannian Ricci-flat and Flat Warped Geometries and New Coordinates for the Minkowski metric

It is well-known that the Einstein condition on warpedgeometries requires the fibres to be necessarily Einstein. However, exact warped solutions have often been obtained using one- and two-dimensional bases. In this paper, keeping the dimensions and signatures of the base and the fibre independently arbitrary, we obtain all Ricci-flat warped metrics with flat base in closed form and show that the number of free parameters is one less than the dimension of the base. Without any assumptions on the base and fibre geometry, we then show that a warped geometry is flat, i.e, has vanishing Riemann curvature, only if its base is flat and its fibre is maximally symmetric, i.e. of constant curvature. Applying this result systematically all possible warped forms of the Euclidean, Minkowski, and flat metrics of arbitrary signature can be obtained in closed form up to disjoint diffemorphisms of the base and fiber metrics. In particular, we obtained four new time-dependent forms of the Minkowski metric in four dimensions in addition to reproducing all of its known warped forms.

gr-qc↗

arXiv2004

Schwarzschild-anti de Sitter within an Isothermal Cavity: Thermodynamics, Phase Transitions and the Dirichlet Problem

The thermodynamics of Schwarzschild black holes within an isothermal cavity and the associated Euclidean Dirichlet boundary-value problem are studied for four and higher dimensions in anti-de Sitter (AdS) space. For such boundary conditions classically there always exists a unique hot AdS solution and two or no Schwarzschild-AdS black-hole solutions depending on whether or not the temperature of the cavity-wall is above a minimum value, the latter being a function of the radius of the cavity. Assuming the standard area-law of black-hole entropy, it was known that larger and smaller holes have positive and negative specific heats and hence are locally thermodynamically stable and unstable respectively. In this paper we present the first derivation of this by showing that the standard area law of black-hole entropy holds in the semi-classical approximation of the Euclidean path integral for such boundary conditions. We further show that for wall-temperatures above a critical value a phase transition takes hot AdS to the larger Schwarzschild-AdS within the cavity. The larger hole thus can be globally thermodynamically stable above this temperature. The phase transition can occur for a cavity of arbitrary radius above a (corresponding) critical temperature. In the infinite cavity limit this picture reduces to that considered by Hawking and Page. The case of five dimensions is found to be rather special since exact analytic expressions can be obtained for the masses of the two holes as functions of cavity radius and temperature thus solving exactly the Euclidean Dirichlet problem. This makes it possible to compute the on-shell Euclidean action as functions of them from which other quantities of interest can be evaluated exactly.

hep-th↗

arXiv2004

$\bbbc P^2$ and $\bbbc P^{1}$ Sigma Models in Supergravity: Bianchi type IX Instantons and Cosmologies

We find instanton/cosmological solutions with biaxial Bianchi-IX symmetry, involving non-trivial spatial dependence of the $\bbbc P^{1}$- and $\bbbc P^{2}$-sigma-models coupled to gravity. Such manifolds arise in N=1, $d=4$ supergravity with supermatter actions and hence the solutions can be embedded in supergravity. There is a natural way in which the standard coordinates of these manifolds can be mapped into the four-dimensional physical space. Due to its special symmetry, we start with $\bbbc P^{2}$ with its corresponding scalar Ansatz; this further requires the spacetime to be $SU(2) \times U(1)$-invariant. The problem then reduces to a set of ordinary differential equations whose analytical properties and solutions are discussed. Among the solutions there is a surprising, special-family of exact solutions which owe their existence to the non-trivial topology of $\bbbc P^{2}$ and are in 1-1 correspondence with matter-free Bianchi-IX metrics. These solutions can also be found by coupling $\bbbc P^{1}$ to gravity. The regularity of these Euclidean solutions is discussed -- the only possibility is bolt-type regularity. The Lorentzian solutions with similar scalar Ansatz are all obtainable from the Euclidean solutions by Wick rotation.

hep-th↗

arXiv2004

Entropy Corrections for Schwarzschild and Reissner-Nordström Black Holes

Schwarzschild black hole being thermodynamically unstable, corrections to its entropy due to small thermal fluctuations cannot be computed. However, a thermodynamically stable Schwarzschild solution can be obtained within a cavity of any finite radius by immersing it in an isothermal bath. For these boundary conditions, classically there are either two black hole solutions or no solution. In the former case, the larger mass solution has a positive specific heat and hence is locally thermodynamically stable. We find that the entropy of this black hole, including first order fluctuation corrections is given by: ${\cal S} = S_{BH} - \ln[\f{3}{R} (S_{BH}/4\p)^{1/2} -2]^{-1} + (1/2) \ln(4\p)$, where $S_{BH}=A/4$ is its Bekenstein-Hawking entropy and $R$ is the radius of the cavity. We extend our results to four dimensional Reissner-Nordström black holes, for which the corresponding expression is: ${\cal S} = S_{BH} - \f{1}{2} \ln [ {(S_{BH}/\p R^2) ({3S_{BH}}/{\p R^2} - 2\sqrt{S_{BH}/{\p R^2 -\a^2}}) \le(\sqrt{S_{BH}/{\p R^2}} - \a^2 \ri)}/ {\le({S_{BH}}/{\p R^2} -\a^2 \ri)^2} ]^{-1} +(1/2)\ln(4\p).$ Finally, we generalise the stability analysis to Reissner-Nordström black holes in arbitrary spacetime dimensions, and compute their leading order entropy corrections. In contrast to previously studied examples, we find that the entropy corrections in these cases have a different character.

hep-th↗

arXiv2003

Classical Boundary-value Problem in Riemannian Quantum Gravity and Taub-Bolt-anti-de Sitter Geometries

For an $SU(2)\times U(1)$-invariant $S^3$ boundary the classical Dirichlet problem of Riemannian quantum gravity is studied for positive-definite regular solutions of the Einstein equations with a negative cosmological constant within biaxial Bianchi-IX metrics containing bolts, i.e., within the family of Taub-Bolt-anti-de Sitter (Taub-Bolt-AdS) metrics. Such metrics are obtained from the two-parameter Taub-NUT-anti-de Sitter family. The condition of regularity requires them to have only one free parameter ($L$) and constrains $L$ to take values within a narrow range; the other parameter is determined as a double-valued function of $L$ and hence there is a bifurcation within the family. We found that {\it{any}} axially symmetric $S^3$-boundary can be filled in with at least one solution coming from each of these two branches despite the severe limit on the permissible values of $L$. The number of infilling solutions can be one, three or five and they appear or disappear catastrophically in pairs as the values of the two radii of $S^3$ are varied. The solutions occur simultaneously in both branches and hence the total number of independent infillings is two, six or ten. We further showed that when the two radii are of the same order and large the number of solutions is two. In the isotropic limit this holds for small radii as well. These results are to be contrasted with the one-parameter self-dual Taub-NUT-AdS infilling solutions of the same boundary-value problem studied previously.

gr-qc↗

arXiv2003

Ricci-flat Metrics with U(1) Action and the Dirichlet Boundary-value Problem in Riemannian Quantum Gravity and Isoperimetric Inequalities

The Dirichlet boundary-value problem and isoperimetric inequalities for positive definite regular solutions of the vacuum Einstein equations are studied in arbitrary dimensions for the class of metrics with boundaries admitting a U(1) action. We show that in the case of non-trivial bundles Taub-Bolt infillings are double-valued whereas Taub-Nut and Eguchi-Hanson infillings are unique. In the case of trivial bundles, there are two Schwarzschild infillings in arbitrary dimensions. The condition of whether a particular type of filling in is possible can be expressed as a limitation on squashing through a functional dependence on dimension in each case. The case of the Eguchi-Hanson metric is solved in arbitrary dimension. The Taub-Nut and the Taub-Bolt are solved in four dimensions and methods for arbitrary dimension are delineated. For the case of Schwarzschild, analytic formulae for the two infilling black hole masses in arbitrary dimension have been obtained. This should facilitate the study of black hole dynamics/thermodynamics in higher dimensions. We found that all infilling solutions are convex. Thus convexity of the boundary does not guarantee uniqueness of the infilling. Isoperimetric inequalities involving the volume of the boundary and the volume of the infilling solutions are then investigated. In particular, the analogues of Minkowski's celebrated inequality in flat space are found and discussed providing insight into the geometric nature of these Ricci-flat spaces.

hep-th↗

arXiv2002

Classical Boundary-value Problem in Riemannian Quantum Gravity and Self-dual Taub-NUT-(anti)de Sitter Geometries

The classical boundary-value problem of the Einstein field equations is studied with an arbitrary cosmological constant, in the case of a compact ($S^{3}$) boundary given a biaxial Bianchi-IX positive-definite three-metric, specified by two radii $(a,b).$ For the simplest, four-ball, topology of the manifold with this boundary, the regular classical solutions are found within the family of Taub-NUT-(anti)de Sitter metrics with self-dual Weyl curvature. For arbitrary choice of positive radii $(a,b),$ we find that there are three solutions for the infilling geometry of this type. We obtain exact solutions for them and for their Euclidean actions. The case of negative cosmological constant is investigated further. For reasonable squashing of the three-sphere, all three infilling solutions have real-valued actions which possess a ``cusp catastrophe'' structure with a non-self-intersecting ``catastrophe manifold'' implying that the dominant contribution comes from the unique real positive-definite solution on the ball. The positive-definite solution exists even for larger deformations of the three-sphere, as long as a certain inequality between $a$ and $b$ holds. The action of this solution is proportional to $-a^{3}$ for large $a (\sim b)$ and hence larger radii are favoured. The same boundary-value problem with more complicated interior topology containing a ``bolt'' is investigated in a forthcoming paper.

gr-qc↗