$$$$-Dimensional charged black holes with scalar hair in Einstein–Power–Maxwell Theory

In \((2+1)\)-dimensional AdS spacetime, we obtain new exact black hole solutions, including two different models (power parameter \(k=1\) and \(k\ne 1\)), in the Einstein–Power–Maxwell (EPM) theory with nonminimally coupled scalar field. For the charged hairy black hole with \(k\ne 1\), we find that the solution contains a curvature singularity at the origin and is nonconformally flat. The horizon structures are identified, which indicates the physically acceptable lower bound of mass in according to the existence of black hole solutions. Later, the null geodesic equations for photon around this charged hairy black hole are also discussed in detail.     

Particle collisions near a three-dimensional warped AdS black hole

In this paper we consider the warped \(\hbox {AdS}_{3}\) black hole solution of topologically massive gravity with a negative cosmological constant, and we study the possibility that it acts as a particle accelerator by analyzing the energy in the center of mass (CM) frame of two colliding particles in the vicinity of its horizon, which is known as the Bañnados, Silk and West (BSW) process. Mainly, we show that the critical angular momentum \((L_c)\) of the particle decreases when the warping parameter(\(\nu \)) increases. Also, we show that despite the particle with \(L_c\) being able to exist for certain values of the conserved energy outside the horizon, it will never reach the event horizon; therefore, the black hole cannot act as a particle accelerator with arbitrarily high CM energy on the event horizon. However, such a particle could also exist inside the outer horizon, with the BSW process being possible on the inner horizon. On the other hand, for the extremal warped \(\hbox {AdS}_{3}\) black hole, the particle with \(L_c\) and energy E could exist outside the event horizon and, the CM energy blows up on the event horizon if its conserved energy fulfills the condition \(E^{2}>\frac{(\nu ^{2}+3)l^{2}}{3(\nu ^{2}-1)}\), with the BSW process being possible.     

Reentrant phase transitions of higher-dimensional AdS black holes in dRGT massive gravity

We study the P–V criticality and phase transition in the extended phase space of anti-de Sitter (AdS) black holes in higher-dimensional de Rham, Gabadadze and Tolley (dRGT) massive gravity, treating the cosmological constant as pressure and the corresponding conjugate quantity is interpreted as thermodynamic volume. Besides the usual small/large black hole phase transitions, the interesting thermodynamic phenomena of reentrant phase transitions (RPTs) are observed for black holes in all \(d\ge 6\)-dimensional spacetime when the coupling coefficients \(c_i m^2\) of massive potential satisfy some certain conditions.     

Geometrothermodynamics of Van der Waals black hole

We study the geometrothermodynamics of a special asymptotically AdS black hole, i.e. Van der Waals \(\left( VdW\right) \) black hole, in the extended phase space where the negative cosmological constant \(\Lambda \) can be regarded as thermodynamic pressure. Analysing some special conditions of this black hole with geometrothermodynamical method, we find a good correlation with ordinary cases according to the state equation.     

Thermodynamic variables of first-order entropy corrected Lovelock-AdS black holes: $$P{-}V$$ criticality analysis

We investigate the effect of thermal fluctuations on the thermodynamics of a Lovelock-AdS black hole. Taking the first order logarithmic correction term in entropy we analyze the thermodynamic potentials like Helmholtz free energy, enthalpy and Gibbs free energy. We find that all the thermodynamic potentials are decreasing functions of correction coefficient \(\alpha \). We also examined this correction coefficient must be positive by analysing \(P{-}V\) diagram. Further we study the \(P{-}V\) criticality and stability and find that presence of logarithmic correction in it is necessary to have critical points and stable phases. When \(P{-}V\) criticality appears, we calculate the critical volume \(V_c\), critical pressure \(P_c\) and critical temperature \(T_c\) using different equations and show that there is no critical point for this black hole without thermal fluctuations. We also study the geometrothermodynamics of this kind of black holes. The Ricci scalar of the Ruppeiner metric is graphically analysed.     

Classical defects in higher-dimensional Einstein gravity coupled to nonlinear $$\sigma $$-models

We construct solutions of higher-dimensional Einstein gravity coupled to nonlinear \(\sigma \)-model with cosmological constant. The \(\sigma \)-model can be perceived as exterior configuration of a spontaneously-broken \(SO(D-1)\) global higher-codimensional “monopole”. Here we allow the kinetic term of the \(\sigma \)-model to be noncanonical; in particular we specifically study a quadratic-power-law type. This is some possible higher-dimensional generalization of the Bariola–Vilenkin (BV) solutions with k-global monopole studied recently. The solutions can be perceived as the exterior solution of a black hole swallowing up noncanonical global defects. Even in the absence of comological constant its surrounding spacetime is asymptotically non-flat; it suffers from deficit solid angle. We discuss the corresponding horizons. For \(\Lambda >0\) in 4d there can exist three extremal conditions (the cold, ultracold, and Nariai black holes), while in higher-than-four dimensions the extremal black hole is only Nariai. For \(\Lambda <0\) we only have black hole solutions with one horizon, save for the 4d case where there can exist two horizons. We give constraints on the mass and the symmetry-breaking scale for the existence of all the extremal cases. In addition, we also obtain factorized solutions, whose topology is the direct product of two-dimensional spaces of constant curvature (\(M_2\), \(dS_2\), or \(AdS_2\)) with (D-2)-sphere. We study all possible factorized channels.     

Holographic screening length in a hot plasma of two sphere

We study the screening length \(L_{\mathrm{max}}\) of a moving quark–antiquark pair in a hot plasma, which lives in a two sphere, \(S^2\), using the AdS/CFT correspondence in which the corresponding background metric is the four-dimensional Schwarzschild–AdS black hole. The geodesic of both ends of the string at the boundary, interpreted as the quark–antiquark pair, is given by a stationary motion in the equatorial plane by which the separation length L of both ends of the string is parallel to the angular velocity \(\omega \). The screening length and total energy H of the quark–antiquark pair are computed numerically and show that the plots are bounded from below by some functions related to the momentum transfer \(P_c\) of the drag force configuration. We compare the result by computing the screening length in the reference frame of the moving quark–antiquark pair, in which the background metrics are “Boost-AdS” and Kerr–AdS black holes. Comparing both black holes, we argue that the mass parameters \(M_{\mathrm{Sch}}\) of the Schwarzschild–AdS black hole and \(M_{\mathrm{Kerr}}\) of the Kerr–AdS black hole are related at high temperature by \(M_{\mathrm{Kerr}}=M_{\mathrm{Sch}}(1-a^2l^2)^{3/2}\), where a is the angular momentum parameter and l is the AdS curvature.     

Dynamical analysis and cosmological viability of varying G and $$\Lambda $$ cosmology

The cosmological viability of varying \(G\left( t\right) \) and \(\Lambda \left( t\right) \) cosmology is discussed by determining the cosmological eras provided by the theory. Such a study is performed with the determination of the critical points while stability analysis is performed. The application of renormalization group in the ADM formalism of general relativity provides a modified second-order theory of gravity where varying \(G\left( t\right) \) plays the role of a minimally coupled field, different from that of scalar–tensor theories, while \(\Lambda \left( t\right) =\Lambda \left( G\left( t\right) \right) \) is a potential term. We find that the theory provides two de Sitter phases and a tracking solution. In the presence of matter source, two new critical points are introduced, where the matter source contributes to the universe. One of those points describes the \(\Lambda \)CDM cosmology and in order for the solution at the point to be cosmologically viable, it has to be unstable. Moreover, the second point, where matter exists, describes a universe where the dark energy parameter for the equation of state has a different value from that of the cosmological constant.     

Critical behavior and phase transition of dilaton black holes with nonlinear electrodynamics

In this paper, we take into account the dilaton black hole solutions of Einstein gravity in the presence of logarithmic and exponential forms of nonlinear electrodynamics. First of all, we consider the cosmological constant and nonlinear parameter as thermodynamic quantities which can vary. We obtain thermodynamic quantities of the system such as pressure, temperature and Gibbs free energy in an extended phase space. We complete the analogy of the nonlinear dilaton black holes with the Van der Waals liquid–gas system. We work in the canonical ensemble and hence we treat the charge of the black hole as an external fixed parameter. Moreover, we calculate the critical values of temperature, volume and pressure and show that they depend on the dilaton coupling constant as well as on the nonlinear parameter. We also investigate the critical exponents and find that they are universal and independent of the dilaton and nonlinear parameters, which is an expected result. Finally, we explore the phase transition of nonlinear dilaton black holes by studying the Gibbs free energy of the system. We find that in the case of \(T>T_c\), we have no phase transition. When \(T=T_c\), the system admits a second-order phase transition, while for \(T=T_\mathrm{f}<T_c\) the system experiences a first-order transition. Interestingly, for \(T_\mathrm{f}<T<T_c\) we observe a zeroth-order phase transition in the presence of a dilaton field. This novel zeroth-order phase transition occurs due to a finite jump in the Gibbs free energy which is generated by the dilaton–electromagnetic coupling constant, \(\alpha \), for a certain range of pressure.     

Dynamical system approach to scalar–vector–tensor cosmology

Using scalar–vector–tensor Brans Dicke (VBD) gravity (Ghaffarnejad in Gen Relativ Gravit 40:2229, 2008; Gen Relativ Gravit 41:2941, 2009) in presence of self interaction BD potential \(V(\phi )\) and perfect fluid matter field action we solve corresponding field equations via dynamical system approach for flat Friedmann Robertson Walker metric (FRW). We obtained three type critical points for \(\Lambda CDM\) vacuum de Sitter era where stability of our solutions are depended to choose particular values of BD parameter \(\omega \). One of these fixed points is supported by a constant potential which is stable for \(\omega <0\) and behaves as saddle (quasi stable) for \(\omega \ge 0\). Two other ones are supported by a linear potential \(V(\phi )\sim \phi \) which one of them is stable for \(\omega =0.27647\). For a fixed value of \(\omega \) there is at least 2 out of 3 critical points reaching to a unique critical point. Namely for \(\omega =-0.16856(-0.56038)\) the second (third) critical point become unique with the first critical point. In dust and radiation eras we obtained one critical point which never become unique fixed point. In the latter case coordinates of fixed points are also depended to \(\omega \). To determine stability of our solutions we calculate eigenvalues of Jacobi matrix of 4D phase space dynamical field equations for de Sitter, dust and radiation eras. We should point also potentials which support dust and radiation eras must be similar to \(V(\phi )\sim \phi ^{-\frac{1}{2}}\) and \(V(\phi )\sim \phi ^{-1}\) respectively. In short our study predicts that radiation and dust eras of our VBD–FRW cosmology transmit to stable de Sitter state via non-constant potential (effective variable cosmological parameter) by choosing \(\omega =0.27647\).     

<Emphasis Type="Italic">P</Emphasis>–<Emphasis Type="Italic">v</Emphasis> criticality in the extended phase space of a noncommutative geometry inspired Reissner–Nordström black hole in AdS space-time

The P–v criticality and phase transition in the extended phase space of a noncommutative geometry inspired Reissner–Nordström (RN) black hole in Anti-de Sitter (AdS) space-time are studied, where the cosmological constant appears as a dynamical pressure and its conjugate quantity is thermodynamic volume of the black hole. It is found that the P–v criticality and the small black hole/large black hole phase transition appear for the noncommutative RN-AdS black hole. Numerical calculations indicate that the noncommutative parameter affects the phase transition as well as the critical temperature, horizon radius, pressure and ratio. The critical ratio is no longer universal, which is different from the result in the van de Waals liquid–gas system. The nature of phase transition at the critical point is also discussed. Especially, for the noncommutative geometry inspired RN-AdS black hole, a new thermodynamic quantity \(\varPsi \) conjugate to the noncommutative parameter \(\theta \) has to be defined further, which is required for consistency of both the first law of thermodynamics and the corresponding Smarr relation.     

Horizon structure of rotating Bardeen black hole and particle acceleration

We investigate the horizon structure and ergosphere in a rotating Bardeen regular black hole, which has an additional parameter (g) due to the magnetic charge, apart from the mass (M) and the rotation parameter (a). Interestingly, for each value of the parameter g, there exists a critical rotation parameter (\(a=a_{E}\)), which corresponds to an extremal black hole with degenerate horizons, while for \(a<a_{E}\) it describes a non-extremal black hole with two horizons, and no black hole for \(a>a_{E}\). We find that the extremal value \(a_E\) is also influenced by the parameter g, and so is the ergosphere. While the value of \(a_E\) remarkably decreases when compared with the Kerr black hole, the ergosphere becomes thicker with the increase in g. We also study the collision of two equal mass particles near the horizon of this black hole, and explicitly show the effect of the parameter g. The center-of-mass energy (\(E_\mathrm{CM}\)) not only depend on the rotation parameter a, but also on the parameter g. It is demonstrated that the \(E_\mathrm{CM}\) could be arbitrarily high in the extremal cases when one of the colliding particles has a critical angular momentum, thereby suggesting that the rotating Bardeen regular black hole can act as a particle accelerator.     

Rainbow black hole thermodynamics and the generalized uncertainty principle

We study the phase transition of rainbow inspired higher dimensional Schwarzschild black hole incorporating the effects of the generalized uncertainty principle. First, we obtain the relation between the mass and Hawking temperature of the rainbow inspired black hole taking into account the effects of the modified dispersion relation and the generalized uncertainty principle. The heat capacity is then computed from this relation which reveals that there are remnants. The entropy of the black hole is next obtained in \(3+1\) and \(4+1\)-dimensions and is found to have logarithmic corrections only in \(3+1\)-dimensions. We further investigate the local temperature, free energy and stability of the black hole in an isothermal cavity. From the analysis of the free energy, we find that there are two Hawking–Page type phase transitions in \(3+1\) and \(4+1\)-dimensions if we take into account the generalized uncertainty principle. However, in the absence of the generalized uncertainty principle, only one Hawking–Page type phase transition exists in spacetime dimensions greater than four.     

Absorption and radiation of nonminimally coupled scalar field from charged BTZ black hole

In this paper we investigate the absorption and radiation of nonminimally coupled scalar field from the charged BTZ black hole. We find the analytical expressions for the reflection coefficient, the absorption cross section and the decay rate in strong coupling case. We find that the reflection coefficient is directly governed by Hawking temperature \(T_{H}\), scalar wave frequency \(\omega \), Bekenstein–Hawking entropy \(S_{BH}\), angular momentum m and coupling constant \(\xi \).     

The generalization of charged AdS black hole specific volume and number density

In this paper, by proposing a generalized specific volume, we restudy the P–V criticality of charged AdS black holes in the extended phase space. The results show that most of the previous conclusions can be generalized without change, but the ratio \({\tilde{\rho }}_c\) should be \(3 {\tilde{\alpha }}/16\) in general case. Further research on the thermodynamical phase transition of black hole leads us to a natural interpretation of our assumption, and more black hole properties can be generalized. Finally, we study the number density for charged AdS black hole in higher dimensions, the results show the necessity of our assumption.     

Stable exponential cosmological solutions with two factor spaces in the Einstein–Gauss–Bonnet model with a $$\Lambda $$-term

We study D-dimensional Einstein–Gauss–Bonnet gravitational model including the Gauss–Bonnet term and the cosmological term \(\Lambda \). We find a class of solutions with exponential time dependence of two scale factors, governed by two Hubble-like parameters \(H >0\) and h, corresponding to factor spaces of dimensions \(m >2\) and \(l > 2\), respectively. These solutions contain a fine-tuned \(\Lambda = \Lambda (x, m, l, \alpha )\), which depends upon the ratio \(h/H = x\), dimensions of factor spaces m and l, and the ratio \(\alpha = \alpha _2/\alpha _1\) of two constants (\(\alpha _2\) and \(\alpha _1\)) of the model. The master equation \(\Lambda (x, m, l,\alpha ) = \Lambda \) is equivalent to a polynomial equation of either fourth or third order and may be solved in radicals. The explicit solution for \(m = l\) is presented in “Appendix”. Imposing certain restrictions on x, we prove the stability of the solutions in a class of cosmological solutions with diagonal metrics. We also consider a subclass of solutions with small enough variation of the effective gravitational constant G and show the stability of all solutions from this subclass.     

Absorption of scalars by extremal black holes in string theory

We show that the low frequency absorption cross section of minimally coupled test massless scalar fields by extremal spherically symmetric black holes in d dimensions is equal to the horizon area, even in the presence of string-theoretical \(\alpha '\) corrections. Classically one has the relation \(\sigma = 4 GS\) between that absorption cross section and the black hole entropy. By comparing in each case the values of the horizon area and Wald’s entropy, we discuss the validity of such relation in the presence of higher derivative corrections for extremal black holes in many different contexts: in the presence of electric and magnetic charges; for nonsupersymmetric and supersymmetric black holes; in \(d=4\) and \(d=5\) dimensions. The examples we consider seem to indicate that this relation is not verified in the presence of \(\alpha '\) corrections in general, although being valid in some specific cases (electrically charged maximally supersymmetric black holes in \(d=5\)). We argue that the relation \(\sigma = 4 GS\) should in general be valid for the absorption cross section of scalar fields which, although being independent from the black hole solution, have their origin from string theory, and therefore are not minimally coupled.     

Critical Two-Point Function for Long-Range <Emphasis Type="Italic">O</Emphasis>(<Emphasis Type="Italic">n</Emphasis>) Models Below the Upper Critical Dimension

We consider the n-component \(|\varphi |^4\) lattice spin model (\(n \ge 1\)) and the weakly self-avoiding walk (\(n=0\)) on \(\mathbb Z^d\), in dimensions \(d=1,2,3\). We study long-range models based on the fractional Laplacian, with spin-spin interactions or walk step probabilities decaying with distance r as \(r^{-(d+\alpha )}\) with \(\alpha \in (0,2)\). The upper critical dimension is \(d_c=2\alpha \). For \(\varepsilon >0\), and \(\alpha = \frac{1}{2} (d+\varepsilon )\), the dimension \(d=d_c-\varepsilon \) is below the upper critical dimension. For small \(\varepsilon \), weak coupling, and all integers \(n \ge 0\), we prove that the two-point function at the critical point decays with distance as \(r^{-(d-\alpha )}\). This “sticking” of the critical exponent at its mean-field value was first predicted in the physics literature in 1972. Our proof is based on a rigorous renormalisation group method. The treatment of observables differs from that used in recent work on the nearest-neighbour 4-dimensional case, via our use of a cluster expansion.     

Black holes in the generalized Proca theory

We investigate static and spherically symmetric black hole solutions in the generalized Proca theory which corresponds to the generalization of the shift-symmetric scalar–tensor Horndeski theory to the vector–tensor theory. Any solution obtained in this paper possesses a constant spacetime norm of the vector field, \(X:=-\frac{1}{2}g^{\mu \nu }A_\mu A_\nu =X_0=\mathrm{constant}\). The solutions in the theory with generalized quartic coupling \(G_4(X)\) generalize the stealth Schwarzschild and the Schwarzschild- (anti-) de Sitter solutions obtained in the theory with the nonminimal coupling to the Einstein tensor \(G^{\mu \nu } A_\mu A_\nu \). While in the vector–tensor theory with the coupling \(G^{\mu \nu }A_\mu A_\nu \) the electric charge does not explicitly affect the spacetime geometry, in more general cases with nonzero \(G_{4XX}(X_0)\ne 0\) this property does not hold in general. The solutions in the theory with generalized cubic coupling \(G_3(X)\) are given by the Schwarzschild- (anti-) de Sitter spacetime, where the dependence on \(G_3(X)\) does not appear in the metric function.     

Abelian cosmic string in the extended Starobinsky model of gravity

We analyze numerically the behaviour of the solutions corresponding to an Abelian cosmic string taking into account an extension of the Starobinsky model, where the action of general relativity is replaced by \(f(R) = R - 2\Lambda + \eta R^2 + \rho R^m\), with \(m > 2\). As an interesting result, we find that the angular deficit which characterizes the cosmic string decreases as the parameters \(\eta \) and \(\rho \) increase. We also find that the cosmic horizon due to the presence of a cosmological constant is affected in such a way that it can grows or shrinks, depending on the vacuum expectation value of the scalar field and on the value of the cosmological constant.