The third law of black hole dynamics states that it is impossible, through any classical perturbation of a stationary configuration, to reduce the surface gravity of a black hole to zero. In this work, we examine the validity of this law for static, spherically symmetric charged black holes in the Lovelock theory of gravity. By studying infinitesimal variations in mass and charge, we derive a set of inequalities that constrain these variations. Our analysis shows that as the surface gravity approaches zero ( <italic>&#954;</italic>&#8239;&#8594;&#8239;0), the range of admissible perturbations gradually diminishes, thereby forbidding the attainment of extremality through any finite classical process. The saturation of the inequality is interpreted as the emergence of a dynamical barrier near extremality, which prevents further evolution toward the extremal configuration.

India

Jyotirmoy De

Chiranjeeb Singha

South Africa

Naresh Dadhich

The third law of black hole dynamics in Lovelock gravity

In General Relativity, there exist two objects of limiting compactness, one with a null boundary defining the horizon of a black hole and the other with a timelike boundary defining a Buchdahl star. The two are characterized by gravitational energy equal to or half the mass. Since non-gravitational mass-energy is the source of gravitational energy, these configurations suggest an underlying energetic balance. In this letter, we demonstrate, in a simple and general manner, that the corresponding equilibrium state always represents a minima of an appropriate quasi-local energy functional, thereby establishing energetic (variational) stability within the quasi-local configuration space.

Soumya Chakrabarti

On the stability of the objects of limiting compactness: Black hole and Buchdahl star

By resolving the Riemann curvature into electric and magnetic parts, Einstein’s equation can accordingly be written in terms of electric (active and passive) and magnetic parts. The electrogravity duality is defined by the interchange of active and passive parts. It turns out that in static and stationary spacetimes, there is a subset of the equations (that identifies the effective vacuum equation) that is sufficient to yield the vacuum solution. In spherically symmetric spacetime, the electrograv dual of the effective equation solves to give the Schwarzschild black hole with a global monopole. Interestingly, this is not so for axial symmetry, where the Kerr vacuum solution turns out to be electrograv self-dual. However, in the asymptotic limit where the effect of rotation dies out, the situation reverts to the static case, admitting a global monopole. This is also what follows when we apply the Newman-Janis transformation to the static black hole with a global monopole.

On electrogravity duality and black hole with global monopole

Buchdahl star is the most compact object without an event horizon and is an excellent candidate for a black hole mimicker. Unlike black holes, rotating Buchdahl star can be over-extremal with respect to the black hole, sustaining a larger spin. We show that it can also develop an ergosphere above the threshold spin  $$\beta &gt; 1/\sqrt{2}$$  <math> <mrow> <mi>β</mi> <mo>&gt;</mo> <mn>1</mn> <mo>/</mo> <msqrt> <mn>2</mn> </msqrt> </mrow> </math>  , which allows extraction of its rotational energy. Electromagnetic field around Buchdahl star is also expected to differ from that of black hole in both strength and topology. In this paper, we explore the energetics of Buchdahl star focusing on the magnetic Penrose process in the two magnetic field configurations, i.e., uniform and dipole. Below the threshold spin, Buchdahl star is expected to be quiet, while above the threshold it can be much more efficient than the black hole if a dipolar magnetic field is developed on its surface.

China

Uzbekistan

Azerbaijan

Sanjar Shaymatov

Czechia

Germany

Arman Tursunov

Energetics of Buchdahl stars and the magnetic Penrose process

It was shown long back in [1] that a non-extremal black hole cannot be converted into an extremal one by test particle adiabatic accretion. The Buchdahl star is the most compact object without horizon and is defined by <math><mi>Φ</mi><mo>(</mo><mi>R</mi><mo>)</mo><mo>=</mo><mn>4</mn><mo>/</mo><mn>9</mn></math>, while black hole by <math><mi>Φ</mi><mo>(</mo><mi>R</mi><mo>)</mo><mo>=</mo><mn>1</mn><mo>/</mo><mn>2</mn></math>. Here <math><mi>Φ</mi><mo>(</mo><mi>R</mi><mo>)</mo></math> is the gravitational potential experienced by a particle, radially falling for static and axially for the rotating object. In this short note we examine the question of extremalization of the Buchdhal star and show that the same result holds good as for the black hole. That is, a non-extremal Buchdahl star cannot be extremalized by test particle accretion. Further since extremal limit for BS is &gt;1, it could facilitate formation of extremal black holes by neutral and spinless accretion. That is perhaps the only way they could be formed.

Like black holes, Buchdahl stars cannot be extremalized

Buchdahl sphere, the limiting stable isotropic stellar structure without exotic matter, plays a very important role in our understanding of how compact an astrophysical object can be. Here, we show certain universal properties associated with the Buchdahl sphere, in the sense that these properties will not change with the inclusion of electric charge in the stellar structure, or, will hold good in the pure Lovelock theories of gravity as well. Using these universalities, we have proposed a Buchdahl limit for a slowly-rotating stellar configuration, for the first time. Finally, the universality of the Buchdahl sphere in terms of the gravitational and non-gravitational field energies, as well as for the photon sphere have also been discussed.

Sumanta Chakraborty

Universality of the Buchdahl sphere

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.

Strong cosmic censorship conjecture for a charged BTZ black hole

We obtain an exact  $$\Lambda $$  <math> <mi>Λ</mi> </math>  -vacuum solution in the pure Gauss–Bonnet gravity with NUT charge in six dimension, with horizon having the product topology of  $$S^{(2)} \times S^{(2)}$$  <math> <mrow> <msup> <mi>S</mi> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </msup> <mo>×</mo> <msup> <mi>S</mi> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </msup> </mrow> </math>  . We discuss its horizon and singularity structure, and consequently arrive at parameter windows for its physical viability. It turns out that for the curvatures to remain function of r alone for NUT black hole spacetime, horizon topology has to be product of  $$S^{(2)}$$  <math> <msup> <mi>S</mi> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </msup> </math>   spheres. This is true for Einstein as well as for pure Gauss–Bonnet gravity, and perhaps would hold good for higher order pure Lovelock as well.

Sajal Mukherjee

Pure Gauss–Bonnet NUT black hole solution: I

It is known that NUT solution has many interesting features and pathologies like being non-singular and having closed timelike curves. It turns out that in higher dimensions horizon topology cannot be spherical but it has instead to be product of 2-spheres so as to retain radial symmetry of spacetime. In this letter we wish to present a new solution of pure Gauss–Bonnet  $$\Lambda $$  <math> <mi>Λ</mi> </math>  -vacuum equation describing a black hole with NUT charge. It has three interesting cases: (a) black hole with both event and cosmological horizons with singularity being hidden behind the former, (b) a regular spacetime free of both horizon and singularity, and (c) black hole with event horizon without singularity and cosmological horizon. Singularity here is always non-centric at  $$r \ne 0$$  <math> <mrow> <mi>r</mi> <mo>≠</mo> <mn>0</mn> </mrow> </math>  .

Pure Gauss–Bonnet NUT black hole with and without non-central singularity

We wish to construct a model for charged star as a generalization of the uniform density Schwarzschild interior solution. We employ the Vaidya and Tikekar ansatz (Astrophys Astron 3:325, 1982) for one of the metric potentials and electric field is chosen in such a way that when it is switched off the metric reduces to the Schwarzschild. This relates charge distribution to the Vaidya–Tikekar parameter, k, indicating deviation from sphericity of three dimensional space when embedded into four dimensional Euclidean space. The model is examined against all the physical conditions required for a relativistic charged fluid sphere as an interior to a charged star. We also obtain and discuss charged analogue of the Buchdahl compactness bound.