We report on some properties of a quantum black hole obtained recently. The correction to the Newtonian gravitational potential is proportional to a coupling <math altimg='si1.svg'><mi>&#945;</mi></math>, which is the only free parameter of the theory. We constrain the coupling using the perihelion advance, we compute the quasi-normal modes for scalar (both massless and massive) and electromagnetic perturbations. We find that all modes computed here are complex numbers characterized by a positive real part and a negative imaginary part, while both parts increase with the mass of the test scalar field. Also thermodynamics properties are investigated from the classical and topological point of view. In this regard, the quantum black hole exhibits the same behavior as the classical Reissner-Nordstr&#246;m space-time, that is, it presents stable/unstable branches in the Gibbs potential, one generating point and a topological charge <math altimg='si29.svg'><mrow><mi>W</mi><mo linebreak='goodbreak'>=</mo><mn>0</mn></mrow></math>.

Chile

Grigoris Panotopoulos

Francisco Tello-Ortiz

Quantum black holes: Perihelion advance, quasi normal modes and classical/topological thermodynamics

This article studies the thermodynamics phase transitions and critical phenomena of an FLRW cosmological model under truncated Kaniadakis's statistics. The EoS is derived from the corrected Friedmann field equations and the thermodynamics unified first law. To approach thermodynamics, we use an approximation for <math><mi>K</mi><msub><mrow><mi>S</mi></mrow><mrow><mtext>AH</mtext></mrow></msub><mo>≪</mo><mn>1</mn></math>, on the cosmological equation, and also check the validity of our approximation. Then for different values from Kaniadakis parameter K, order <math><mi>O</mi><mo>(</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>37</mn></mrow></msup><mo>)</mo></math> for the relic-abundance, <math><mi>O</mi><mo>(</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>84</mn></mrow></msup><mo>)</mo></math> for $^{7}$Li-abundance and <math><mi>O</mi><mo>(</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>125</mn></mrow></msup><mo>)</mo></math> in the recombination era, the EoS reveals non-trivial critical points where a first-order phase transition occurs a sort of a van der Waals fluid. Interestingly, the numerical values of the critical exponents are the same as those of the van der Waals system. Besides, to obtain more insights into the thermodynamics description, the so-called Ruppeiner's geometry is studied through the normalized scalar curvature, disclosing this invariant zone where the system undergoes repulsive/attractive interactions. Near the critical point, this curvature provides again the same critical exponent and universal constant value as for van der Waals fluid. Despite the similarity, both systems are quite different because the present one considers a relativistic entropy (Kaniadakis entropy) while the Van der Waals gas responds to a classical entropy (Maxwell-Boltzmann).

Joaquín Housset

Joel F. Saavedra

Cosmological FLRW phase transitions and micro-structure under Kaniadakis statistics

In this article, using gravitational decoupling by means of minimal geometric deformation approach, we obtain a new spherically symmetric and static black hole solution. To progress, we close the system by assuming that the average pressure of the  $$\theta $$  <math> <mi>θ</mi> </math>  -sector is vanishing. Also, we tackle the problem regarding how, for a given minimally deformed black hole solution, one can connect it to a wormhole space–time.

Spain

Ángel Rincón

A. Alvarez

India

Saibal Ray

Gravitationally decoupled non-Schwarzschild black holes and wormhole space–times

A Correction to this paper has been published: 10.1140/epjc/s10052-021-09179-5

Oman

S. Maurya

Pedro Bargueño

Erratum to: Minimally deformed wormholes

In this article, we develop a theoretical framework to study compact stars in Einstein gravity with the Gauss–Bonnet (GB) combination of quadratic curvature terms. We mainly analyzed the dependence of the physical properties of these compact stars on the Gauss–Bonnet coupling strength. This work is motivated by the relations that appear in the framework of the minimal geometric deformation approach to gravitational decoupling (MGD-decoupling), we establish an exact anisotropic version of the interior solution in Einstein–Gauss–Bonnet gravity. In fact, we specify a particular form for gravitational potentials in the MGD approach that helps us to determine the decoupling sector completely and ensure regularity in interior space-time. The interior solutions have been (smoothly) joined with the Boulware–Deser exterior solution for 5D space-time. In particular, two different solutions have been reported which comply with the physically acceptable criteria: one is the mimic constraint for the pressure and the other approach is the mimic constraint for density. We present our solution both analytically and graphically in detail.

Anirudh Pradhan

South Africa

Ayan Banerjee

Riju Nag

Minimally deformed anisotropic stars by gravitational decoupling in Einstein–Gauss–Bonnet gravity

In the present article, the solution of the Einstein–Maxwell field equations in the presence of a massive scalar field under the Brans-Dicke (BD) gravity is obtained via embedding approach, which describes a charged anisotropic strange star model. The interior spacetime is described by a spherically symmetric static metric of embedding class I. This reduces the problem to a single-generating function of the metric potential which is chosen by appealing to physics based on regularity at each interior point of the stellar interior. The resulting model is subjected to rigorous physical checks based on stability, causality and regularity for particular object PSR J1903+327. We also show that our solutions describe compact objects such as PSR J1903+327; Cen X-3; EXO 1785-248 and LMC X-4 to an excellent approximation. Novel results of our investigation reveal that the scalar field leads to higher surface charge densities which in turn affects the compactness and upper and lower values imposed by the modified Buchdahl limit for charged stars. Our results also show that the electric field and scalar field which originate from entirely different sources couple to alter physical characteristics such as mass-radius relation and surface redshift of compact objects. This superposition of the electric and scalar fields is enhanced by an increase in the BD coupling constant,  $$\omega _{BD}$$  <math> <msub> <mi>ω</mi> <mrow> <mi>BD</mi> </mrow> </msub> </math>  .

Ksh. Singh

M. Govender

Morocco

Abdelghani Errehymy

Anisotropic stars via embedding approach in Brans–Dicke gravity

A spherically symmetric solution of the field equations of the Hořava–Lifshitz gravity–gauge vector interaction theory is obtained and analyzed. It describes a charged throat. The solution exists provided a restriction on the relation between the mass and charge is satisfied. The restriction reduces to the Reissner–Nordström one in the limit in which the coupling constants tend to the relativistic values of General Relativity. We introduce the correct charts to describe the solution across the entire manifold, including the throat connecting an asymptotic Minkowski space-time with a singular 3+1 dimensional manifold. The solution external to the throat on the asymptotically flat side tends to the Reissner–Nordström space-time at the limit when the coupling parameter, associated with the term in the low energy Hamiltonian that manifestly breaks the relativistic symmetry, tends to zero. Also, when the electric charge is taken to be zero the solution becomes the spherically symmetric and static solution of the Hořava–Lifshitz gravity.

Alvaro Restuccia

Charged throats in the Hořava–Lifshitz theory

This work is devoted to the study of wormhole solutions in the framework of gravitational decoupling by means of the minimal geometric deformation scheme. As an example, to analyze how this methodology works in this scenario, we have minimally deformed the well-known Morris–Thorne model. The decoupler function f(r) and the  $$\theta $$  <math> <mi>θ</mi> </math>  -sector are determined considering the following approaches: (i) the most general linear equation of state relating the  $$\theta _{\mu \nu }$$  <math> <msub> <mi>θ</mi> <mrow> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </math>   components is imposed and (ii) the generalized pseudo-isothermal dark matter density profile is mimicked by the temporal component of the  $$\theta $$  <math> <mi>θ</mi> </math>  -sector. It is found that the first approach leads to a non-asymptotically flat space-time with an unbounded mass function. To address this issue we have matched both the wormhole and the Schwarzschild vacuum solutions, via a thin-shell at the junction surface. Using the second approach, it can be seen that, on one hand, the solution for  $$\gamma =1$$  <math> <mrow> <mi>γ</mi> <mo>=</mo> <mn>1</mn> </mrow> </math>   does not give place to a bounded mass and it presents a topological defect at large distances; on the other hand, the wormhole manifold is asymptotically flat in the  $$\gamma =2$$  <math> <mrow> <mi>γ</mi> <mo>=</mo> <mn>2</mn> </mrow> </math>   case. In order to satisfy the flare-out condition, we have found restrictions on the value of the  $$\alpha $$  <math> <mi>α</mi> </math>   parameter, which is related with the amount of exotic matter distribution. Finally, the averaged weak energy condition has been analyzed by using the volume integral quantifier.

Minimally deformed wormholes

This work is devoted to the study of relativistic anisotropic compact objects. To obtain this class of solutions of the Einstein field equations, we have developed a general scheme to generate the metric of the space–time describing the interior of the compact structure. This approach is based on the class I space–time and the extended gravitational decoupling by means of an extended geometric deformation (EGD). The class I condition provides a differential equation relating both metric potential  $$\nu $$  <math> <mi>ν</mi> </math>   and  $$\lambda $$  <math> <mi>λ</mi> </math>  , whilst the EGD translates the metric potentials to  $$ \nu =\xi +\beta \,h(r)$$  <math> <mrow> <mi>ν</mi> <mo>=</mo> <mi>ξ</mi> <mo>+</mo> <mi>β</mi> <mspace width="0.166667em"></mspace> <mi>h</mi> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> </math>   and  $$ \lambda =-\ln [\mu +\beta \,f(r)]$$  <math> <mrow> <mi>λ</mi> <mo>=</mo> <mo>-</mo> <mo>ln</mo> <mo>[</mo> <mi>μ</mi> <mo>+</mo> <mi>β</mi> <mspace width="0.166667em"></mspace> <mi>f</mi> <mo>(</mo> <mi>r</mi> <mo>)</mo> <mo>]</mo> </mrow> </math>  , where h(r) and f(r) are the deformation functions and  $$\beta $$  <math> <mi>β</mi> </math>   a dimensionless constant. In this case the pair  $$\{\xi ,\mu \}$$  <math> <mrow> <mo>{</mo> <mi>ξ</mi> <mo>,</mo> <mi>μ</mi> <mo>}</mo> </mrow> </math>   represents the seed solution satisfying the class I condition without any deformation. Once the deformed metric potentials are inserted into the class I, the main task is to obtain h(r) or f(r). So, in this case a particular ansatz for h(r) is considered in conjunction with  $$\beta =0.5$$  <math> <mrow> <mi>β</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </math>   to get f(r). In order to check feasibility of our model, we have performed a thoroughly physical, mathematical and graphical analysis.

M. Jasim

An EGD model in the background of embedding class I space–time

A spherically symmetric wormhole family of solutions, with null red-shift, in the context of f(R)-gravity is presented. The model depends on two parameters: m and $$\beta $$ <math><mi>β</mi></math>  and meets all requirements to be an asymptotically and traversable wormhole. To solve the field equations, an EoS is imposed: $$p_{\perp }=-\rho $$ <math><mrow><msub><mi>p</mi><mo>⊥</mo></msub><mo>=</mo><mo>-</mo><mi>ρ</mi></mrow></math> . It is found that for $$m=1$$ <math><mrow><mi>m</mi><mo>=</mo><mn>1</mn></mrow></math>  the solution satisfies the null energy condition, although $$F(R)&lt;0$$ <math><mrow><mi>F</mi><mo>(</mo><mi>R</mi><mo>)</mo><mo>&lt;</mo><mn>0</mn></mrow></math>  everywhere. For $$m=0$$ <math><mrow><mi>m</mi><mo>=</mo><mn>0</mn></mrow></math> , the model satisfies the null energy condition away from the throat, where the function F(R) is everywhere positive and together with dF(R)/dR vanish at the throat of the wormhole. This fact is beyond the scope of the non-existence theorem. Furthermore, the cosmological viability of the model, to address the late – time accelerated epoch, is analyzed on the background of a flat FLRW space-time. The model satisfies consistency of local gravity tests, stability under cosmological perturbations, ghosts free and stability of the de Sitter point.