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Abstract Employing the covariant phase space formalism, we discuss black hole thermodynamics in four-dimensional scalar-tensor Einstein-Gauss-Bonnet gravity. We argue that logarithmic corrections to Wald entropy previously reported in this theory do not appear, due to the symmetry of the theory under constant shifts of the scalar field. Instead, we obtain the standard Bekenstein entropy of general relativity. Then, to satisfy the first law of black hole mechanics, the Hawking temperature must be modified. It has been proposed that such temperature modifications occur generically in scalar-tensor theories, due to different propagation speeds of gravitons and photons. We show that the temperature modifications also emerge in the Euclidean canonical ensemble approach to black hole thermodynamics. Notably, the boundary terms of the type we consider here can be considered in any scalar-tensor gravitational theories. Hence, we illustrate that adding a suitable boundary term to action may drastically affect black hole thermodynamics, changing both the entropy and the temperature.

Abstract Employing the novel exact dictionary between the laws of extended black hole thermodynamics and the laws of the dual CFT, we study the extended thermodynamics for CFT states that are dual to neutral singly-spinning asymptotically AdS black holes in d bulk spacetime dimensions. On the field theory side we include two independent pairs of thermodynamic conjugate variables: the central charge-chemical potential term and the pressure-volume term. In this setting we uncover various phase transitions and critical behaviour in the CFT, focusing on three different thermodynamic ensembles. Namely, for fixed angular momentum and central charge, we show there is a Van der Waals-like criticality for d = 4, 5 and reentrant phase transitions for d ≥ 6. At fixed angular velocity and central charge, there is a first-order (de)confinement phase transition in all dimensions d ≥ 3. Finally, at fixed angular momentum and chemical potential we find a plethora of zero-order phase transitions and unstable phases in both d = 4 and d = 6.

Abstract We study the holographic dual of the extended thermodynamics of spherically symmetric, charged AdS black holes in the context of the AdS/CFT correspondence. The gravitational thermodynamics of AdS black holes can be extended by allowing for variations of the cosmological constant and Newton’s constant. In the dual CFT this corresponds to including the central charge C and its chemical potential μ as a new pair of conjugate thermodynamic variables, in addition to the standard pairs: temperature vs. entropy (T, S), electric potential vs. charge ( Φ ~ , Q ~ $$ \overset{\sim }{\Phi},\overset{\sim }{Q} $$ ) and field theory pressure vs. volume (p, V $$ \mathcal{V} $$ ). For the (grand) canonical ensembles at fixed ( Q ~ , V , C $$ \overset{\sim }{Q},\mathcal{V},C $$ ), ( Φ ~ , V , C $$ \overset{\sim }{\Phi},\mathcal{V},C $$ ) and ( Q ~ , V , μ $$ \overset{\sim }{Q},\mathcal{V},\mu $$ ), based on the holographic dictionary, we argue the CFT description of charged AdS black holes contains either critical phenomena or interesting phase behaviour. In the fixed ( Q ~ , V , μ $$ \overset{\sim }{Q},\mathcal{V},\mu $$ ) we find a new zeroth-order phase transition between a high- and low-entropy phase at some μ-dependent temperature. Finally, we point out there is no critical behaviour in the fixed p ensembles, i.e. there is no p − V $$ \mathcal{V} $$ criticality, and hence the CFT state dual to a classical charged black hole cannot be a Van der Waals fluid. Whether or not this phase structure is supported by CFT computations remains an interesting open question.

Abstract The Lense-Thirring spacetime describes a 4-dimensional slowly rotating approximate solution of vacuum Einstein equations valid to a linear order in rotation parameter. It is fully characterized by a single metric function of the corresponding static (Schwarzschild) solution. In this paper, we introduce a generalization of the Lense-Thirring spacetimes to the higher-dimensional multiply-spinning case, with an ansatz that is not necessarily fully characterized by a single (static) metric function. This generalization lets us study slowly rotating spacetimes in various higher curvature gravities as well as in the presence of non-trivial matter. Moreover, the ansatz can be recast in Painlevé-Gullstrand form (and thence is manifestly regular on the horizon) and admits a tower of exact rank-2 and higher rank Killing tensors that rapidly grows with the number of dimensions. In particular, we construct slowly multiply-spinning solutions in Lovelock gravity and notably show that in four dimensions Einstein gravity is the only non-trivial theory amongst all up to quartic curvature gravities that admits a Lense-Thirring solution characterized by a single metric function.

Abstract Gravitational shockwaves are simple exact solutions of Einstein equations representing the fields of ultrarelativistic sources and idealized gravitational waves (shocks). Historically, much work has focused on shockwaves in the context of possible black hole formation in high energy particle collisions, yet they remain at the forefront of research even today. Representing hard modes in the bulk, shocks give rise to the gravitational memory effect at the classical level and implant supertranslation (BMS) hair onto a classical spacetime at the quantum level. The aim of this paper is to further our understanding of the ‘information content’ of such supertranslations. Namely, we show that, contrary to the several claims in the literature, a gravitational shockwave does leave a quantum imprint on the vacuum state of a test quantum field and that this imprint is accessible to local observers carrying Unruh-DeWitt (UDW) detectors in this spacetime.

Theories of non-linear electrodynamics naturally describe deviations from Maxwell's theory in the strong field regime. Among these, of special interest is the recently discovered ModMax electrodynamics, which is a unique 1-parametric generalization of Maxwell's theory that possesses the conformal invariance as well as the electromagnetic duality. In this paper we construct the asymptotically AdS accelerated black holes in this theory and study their thermodynamics – providing thus a first example of accelerated solutions coupled to non-linear electrodynamics. Our study opens a window towards studying radiative spacetimes in non-linear electromagnetic regime as well as raises new challenges for their corresponding holographic interpretation.

Abstract We comment on the recently introduced Gauss-Bonnet gravity in four dimensions. We argue that it does not make sense to consider this theory to be defined by a set of D → 4 solutions of the higher-dimensional Gauss-Bonnet gravity. We show that a well-defined D → 4 limit of Gauss-Bonnet Gravity is obtained generalizing a method employed by Mann and Ross to obtain a limit of the Einstein gravity in D = 2 dimensions. This is a scalar-tensor theory of the Horndeski type obtained by dimensional reduction methods. By considering simple spacetimes beyond spherical symmetry (Taub-NUT spaces) we show that the naive limit of the higher-dimensional theory to D = 4 is not well defined and contrast the resultant metrics with the actual solutions of the new theory.

Abstract We extend the work on thermodynamics of Lorentzian NUTty solutions, by including simultaneously the effect of rotation and electromagnetic charges. Due to the fact that Misner strings carry electric charge, and similar to the non-rotating case, we observe an interesting interplay between the horizon and asymptotic charges. Namely, upon employing the Euclidean action calculation we derive two alternative full cohomogeneity first laws, one that includes the variations of the horizon electric charge and the asymptotic magnetic charge and another that involves the horizon magnetic charge and the asymptotic electric charge. When one of the horizon charges vanishes, we obtain the corresponding ‘electric’ and ‘magnetic’ first laws that are connected by the electromagnetic duality. We also briefly study the free energy of the corresponding system.

Abstract The superradiant instability modes of ultralight massive vector bosons are studied for weakly charged rotating black holes in Einstein-Maxwell gravity (the Kerr- Newman solution) and low-energy heterotic string theory (the Kerr-Sen black hole). We show that in both these cases, the corresponding massive vector (Proca) equations can be fully separated, exploiting the hidden symmetry present in these spacetimes. The resultant ordinary differential equations are solved numerically to find the most unstable modes of the Proca field in the two backgrounds and compared to the vacuum (Kerr black hole) case.

Abstract We study a critical limit in which asymptotically-AdS black holes develop maximal conical deficits and their horizons become non-compact. When applied to stationary rotating black holes this limit coincides with the “ultraspinning limit” and yields the Superentropic black holes whose entropy was derived recently and found to exceed the maximal possible bound imposed by the Reverse Isoperimetric Inequality [1, 2]. To gain more insight into this peculiar result, we study this limit in the context of accelerated AdS black holes that have unequal deficits along the polar axes, hence the maximal deficit need not appear on both poles simultaneously. Surprisingly, we find that in the presence of acceleration, the critical limit becomes smooth, and is obtained simply by taking various upper bounds in the parameter space that we elucidate. The Critical black holes thus obtained have many common features with Superentropic black holes, but are manifestly not superentropic. This raises a concern as to whether Superentropic black holes actually are superentropic.1 We argue that this may not be so and that the original conclusion is likely attributed to the degeneracy of the resulting first law.

We investigate the sensitivity of an Unruh–DeWitt detector to the global features of a deficit angle that are otherwise classically inaccessible. Specifically, we consider a detector placed inside an infinite thin hollow cylinder whose spacetime is everywhere flat but outside of which the spacetime has a deficit angle and study its response to a scalar field to which it couples. We find that the response of the detector is sensitive to the deficit angle, despite the fact that it does not interact with the cylinder.

We construct a novel charged Taub-NUT spacetime, providing a first non-trivial example of a self-gravitating solution to the recently proposed ModMax theory [1,2], the most general (1-parametric) theory of non-linear electrodynamics that is characterized by both the conformal symmetry and the SO(2) duality-rotation invariance. The spacetime features non-trivial magnetic fields, in their presence the non-linearity of the field becomes apparent and the solution is distinguished from that of the Maxwell theory. The thermodynamics of the new solution, the charged black hole limit of vanishing NUT parameter, and the self-dual solutions are also briefly discussed.

Many black hole spacetimes with a 3-form field exhibit a hidden symmetry encoded in a torsion generalization of the principal Killing–Yano tensor. This tensor determines basic properties of such black holes while also underlying the separability of the Hamilton–Jacobi, Klein–Gordon, and (torsion-modified) Dirac field equations in their background. As a specific example, we consider the Chong–Cvetič–Lü–Pope black hole of D=5 minimal gauged supergravity and show that the torsion-modified vector field equations can also be separated, with the principal tensor playing a key role in the separability ansatz. For comparison, separability of the Proca field in higher-dimensional Kerr–NUT–AdS spacetimes (including new explicit formulae in odd dimensions) is also presented.

Abstract Picking up the threads on a recent proposal [1], we show how to formulate consistent thermodynamics of Lorentzian Taub-NUT spacetimes in the presence of electric and magnetic charges. Namely, with an entropy identified with a quarter of the horizon area and no Misner time periodicity condition imposed, we show that a new pair of conjugate quantities can be introduced so that the NUT parameter can be independently varied and the corresponding first law and Smarr relation can be consistently formulated. Moreover, we show that (contrary to the statements in the literature) the electric and magnetic parameters need not be proportional to one another and a full cohomogeneity first law including variations of both charges can be written down, provided one charge is considered on the horizon and the other at infinity. The corresponding phase transitions are also briefly discussed.

Abstract We show how to obtain a consistent thermodynamic description of accelerating asymptotically AdS black holes, extending our previous results by including charge and rotation. We find that the key ingredient of consistent thermodynamics is to ensure that the system is not over-constrained by including the possibility of varying the ‘string’ tensions that are responsible for the acceleration of the black hole, yielding a first law of full cohomogeneity. The first law assumes the standard form, with the entropy given by one quarter of the horizon area and other quantities identified by standard methods. In particular we compute the mass in two independent ways: through a Euclidean action calculation and by the method of conformal completion. The ambiguity in the choice of the normalization of the timelike Killing vector can be fixed by explicit coordinate transformation (in the case of rotation) to the standard AdS form or by holographic methods (in the case of charge). This resolves a long-standing problem of formulating the thermodynamics of accelerating black holes, opening the way to detailed studies of their phase behaviour.

We consider the D→3 limit of Gauss–Bonnet gravity. We find two distinct but similar versions of the theory and obtain black hole solutions for each. For one theory the solution is an interesting generalization of the BTZ black hole that does not have constant curvature but whose thermodynamics is identical. The other theory admits a solution that is asymptotically AdS but does not approach the BTZ black hole in the limit of small Gauss–Bonnet coupling. We also discuss the distinction between our solutions and those obtained by taking a D→3 limit of solutions to D-dimensional Einstein Gauss–Bonnet gravity. We find that these latter metrics are not solutions of the theories we consider except for particular constraints on the parameters.

In this paper we explicitly demonstrate separability of the Maxwell equations in a wide class of higher-dimensional metrics which include the Kerr–NUT–(A)dS solution as a special case. Namely, we prove such separability for the most general metric admitting the principal tensor (a non-degenerate closed conformal Killing–Yano 2-form). To this purpose we use a special ansatz for the electromagnetic potential, which we represent as a product of a (rank 2) polarization tensor with the gradient of a potential function, generalizing the ansatz recently proposed by Lunin. We show that for a special choice of the polarization tensor written in terms of the principal tensor, both the Lorenz gauge condition and the Maxwell equations reduce to a composition of mutually commuting operators acting on the potential function. A solution to both these equations can be written in terms of an eigenfunction of these commuting operators. When incorporating a multiplicative separation ansatz, it turns out that the eigenvalue equations reduce to a set of separated ordinary differential equations with the eigenvalues playing a role of separability constants. The remaining ambiguity in the separated equations is related to an identification of D−2 polarizations of the electromagnetic field. We thus obtained a sufficiently rich set of solutions for the Maxwell equations in these spacetimes.

Abstract The Jacobi equation for geodesic deviation describes finite size effects due to the gravitational tidal forces. In this paper we show how one can integrate the Jacobi equation in any spacetime admitting completely integrable geodesics. Namely, by linearizing the geodesic equation and its conserved charges, we arrive at the invariant Wronskians for the Jacobi system that are linear in the ‘deviation momenta’ and thus yield a system of first-order differential equations that can be integrated. The procedure is illustrated on an example of a rotating black hole spacetime described by the Kerr geometry and its higher-dimensional generalizations. A number of related topics, including the phase space formulation of the theory and the derivation of the covariant Hamiltonian for the Jacobi system are also discussed.

Abstract We investigate a holographic version of Maxwell’s equal area law analogous to that for the phase transition in the black hole temperature/black hole entropy plane of a charged AdS black hole. We consider proposed area laws for both the black hole temperature/holographic entanglement entropy plane and the black hole temperature/2- point correlation function plane. Despite recent claims to the contrary, we demonstrate numerically that neither proposal is valid. We argue that there is no physical reason to expect such a construction in these planes.

Abstract The study of higher-dimensional black holes is a subject which has recently attracted vast interest. Perhaps one of the most surprising discoveries is a realization that the properties of higher-dimensional black holes with the spherical horizon topology and described by the Kerr–NUT–(A)dS metrics are very similar to the properties of the well known four-dimensional Kerr metric. This remarkable result stems from the existence of a single object called the principal tensor. In our review we discuss explicit and hidden symmetries of higher-dimensional Kerr–NUT–(A)dS black hole spacetimes. We start with discussion of the Killing and Killing–Yano objects representing explicit and hidden symmetries. We demonstrate that the principal tensor can be used as a “seed object” which generates all these symmetries. It determines the form of the geometry, as well as guarantees its remarkable properties, such as special algebraic type of the spacetime, complete integrability of geodesic motion, and separability of the Hamilton–Jacobi, Klein–Gordon, and Dirac equations. The review also contains a discussion of different applications of the developed formalism and its possible generalizations.

By regarding the Einstein equations as equation(s) of state, we demonstrate that a full cohomogeneity horizon first law can be derived in horizon thermodynamics. In this approach both the entropy and the free energy are derived concepts, while the standard (degenerate) horizon first law is recovered by a Legendre projection from the more general one we derive. These results readily generalize to higher curvature gravities where they naturally reproduce a formula for the entropy without introducing Noether charges. Our results thus establish a way of how to formulate consistent black hole thermodynamics without conserved charges.

Abstract Recently we have shown [1] how to formulate a thermodynamic first law for a single (charged) accelerated black hole in AdS space by fixing the conical deficit angles present in the spacetime. Here we show how to generalise this result, formulating thermodynamics for black holes with varying conical deficits. We derive a new potential for the varying tension defects: the thermodynamic length, both for accelerating and static black holes. We discuss possible physical processes in which the tension of a string ending on a black hole might vary, and also map out the thermodynamic phase space of accelerating black holes and explore their critical phenomena.

We address a long-standing problem of describing the thermodynamics of rotating Taub–NUT solutions. The obtained first law is of full cohomogeneity and allows for asymmetric distributions of Misner strings as well as their potential variable strengths—encoded in the gravitational Misner charges. Notably, the angular momentum is no longer given by the Noether charge over the sphere at infinity and picks up non-trivial contributions from Misner strings.

We apply the hyperboloid membrane limit to the general Kerr-AdS metrics and their recently studied super-entropic cousins and obtain a new class of rotating black holes, for which the rotational parameters in multiple directions attain their maximal value—equal to the AdS radius. These new solutions have a potential application in the description of holographic fluids with vorticity. They also possess interesting thermodynamic properties: we show that—despite the absence of Misner strings—the Bekenstein–Hawking entropy/area law is still violated, raising a question about the origin of this violation.

We find an explicit solution of the source free Maxwell equations in a generalized Kerr–NUT–(A)dS spacetime in all dimensions. This solution is obtained as a linear combination of the closed conformal Killing–Yano tensor hab, which is present in such a spacetime, and a derivative of the primary Killing vector, associated with hab. For the vanishing cosmological constant the obtained solution reduces to the Wald's electromagnetic field generated from the primary Killing vector.

In this review we summarize, expand, and set in context recent developments on the thermodynamics of black holes in extended phase space, where the cosmological constant is interpreted as thermodynamic pressure and treated as a thermodynamic variable in its own right. We specifically consider the thermodynamics of higher-dimensional rotating asymptotically flat and AdS black holes and black rings in a canonical (fixed angular momentum) ensemble. We plot the associated thermodynamic potential—the Gibbs free energy—and study its behavior to uncover possible thermodynamic phase transitions in these black hole spacetimes. We show that the multiply-rotating Kerr-AdS black holes exhibit a rich set of interesting thermodynamic phenomena analogous to the “every day thermodynamics” of simple substances, such as reentrant phase transitions of multicomponent liquids, multiple first-order solid/liquid/gas phase transitions, and liquid/gas phase transitions of the van derWaals type. Furthermore, the reentrant phase transitions also occur for multiply-spinning asymptotically flat Myers–Perry black holes. These phenomena do not require a variable cosmological constant, though they are more naturally understood in the context of the extended phase space. The thermodynamic volume, a quantity conjugate to the thermodynamic pressure, is studied for AdS black rings and demonstrated to satisfy the reverse isoperimetric inequality; this provides a first example of calculation confirming the validity of isoperimetric inequality conjecture for a black hole with non-spherical horizon topology. The equation of state P = P(V,T) is studied for various black holes both numerically and analytically—in the ultraspinning and slow rotation regimes.

In the context of extended phase space, where the negative cosmological constant is treated as a thermodynamic pressure in the first law of black hole thermodynamics, we find an asymptotically AdS metric whose thermodynamics matches exactly that of the Van der Waals fluid. We show that as a solution of Einstein's equations, the corresponding stress energy tensor obeys (at least for certain range of metric parameters) all three weak, strong, and dominant energy conditions.