Black hole chemistry: thermodynamics with Lambda
We review 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. In this approach, the mass of the black hole is no longer regarded as internal energy, rather it is identified with the chemical enthalpy. This leads to an extended dictionary for black hole thermodynamic quantities, in particular a notion of thermodynamic volume emerges for a given black hole spacetime. This volume is conjectured to satisfy the reverse isoperimetric inequality—an inequality imposing a bound on the amount of entropy black hole can carry for a fixed thermodynamic volume. New thermodynamic phase transitions naturally emerge from these identifications. Namely, we show that black holes can be understood from the viewpoint of chemistry, in terms of concepts such as Van der Waals fluids, reentrant phase transitions, and triple points. We also review the recent attempts at extending the AdS/CFT dictionary in this setting, discuss the connections with horizon thermodynamics, applications to Lifshitz spacetimes, and outline possible future directions in this field.
a,b]Robert B. Mann,
c]and Mae Teo
\affiliation[a]Department of Physics and Astronomy, University of Waterloo,
Waterloo, Ontario N2L 3G1, Canada \affiliation[b]Perimeter Institute for Theoretical Physics,
31 Caroline St. N.,Waterloo, Ontario N2L 2Y5, Canada \affiliation[c]Stanford Institute for Theoretical Physics, Stanford University,
Stanford, CA 94305, USA \emailAdddkubiznak@perimeterinstitute.ca \emailAddrbmann@uwaterloo.ca \emailAddmaehwee@stanford.edu \keywordsBlack Holes, Thermodynamics, Volume, AdS/CFT correspondence
Over the past four decades a preponderance of evidence has accumulated suggesting a fundamental relationship between gravitation, thermodynamics, and quantum theory. This evidence is rooted in our understanding of black holes and their relationship to quantum physics, and developed into the sub-discipline of black hole thermodynamics.
This subject was originally quite counter-intuitive . Classically black holes were nature’s ultimate sponges, absorbing all matter and emitting nothing. Superficially they had neither temperature nor entropy, and were characterized by only a few basic parameters: mass, angular momentum, and charge (if any) . However the advent of quantum field theory in curved spacetime changed all of this, leading to the famous results that the area of a black hole corresponds to its entropy  and its surface gravity corresponds to its temperature . All theoretical evidence indicated that black holes radiate heat, analogous to black body radiation, and the subject of black hole thermodynamics was born.
Black hole thermodynamics stimulated a whole new set of techniques for analyzing the behavior of black holes and gave rise to some deep insights concerning the relationship between gravity and quantum physics. Further investigation led to one of the most perplexing conundrums in physics, namely that the process of black hole radiation leads to a loss of information that is incompatible with the basic foundations of quantum physics [5, 6] that have yet to be resolved [7, 8, 9]. Black hole entropy turned out to be the Noether charge associated with diffeomorphism symmetry . The laws of gravitation were posited to be intimately connected with the laws of thermodynamics [11, 12]. The introduction of a negative cosmological constant implied that black holes could exhibit phase behaviour , and somewhat later led to the holographic deployment of black holes as systems dual to those in conformal field theories [14, 15, 16], quantum chromodynamics , and condensed matter physics [18, 19]. Deep connections were discovered between the quantum information concept of entanglement entropy  and the ‘architecture of spacetime’ ; the linearized Einstein equations were later shown to follow from the first law of entanglement entropy . Geometric approaches to the thermodynamics of black holes were summarized in .
Somewhat more recently reconsideration of the role of the cosmological constant, , has led to the realization that black hole thermodynamics is a much richer subject than previously thought. It led to the introduction of pressure, and with it a concept of volume for a black hole. New phase behaviour, analogous to that seen in gels and polymers, was found to be present. Triple points for black holes, analogous to those in water, were discovered. Black holes could further be understood as heat engines. In general black holes were found quite analogous to Van der Waals fluids, and in general exhibited the diverse behavior of different substances we encounter in everyday life. This burgeoning subfield was hence given the name black hole chemistry [24, 25].
This is the subject of this topical review. While more focussed overviews of this subject have appeared [26, 27, 24, 28, 29, 25], our aim here is to be comprehensive, reviewing the subject from its historical roots to its modern developments.
We begin in Sec. 2 by reviewing the laws of black hole mechanics and their relationship to black hole thermodynamics. After discussing early attempts at incorporating into the thermodynamic laws, we shall investigate how a proper treatment suggests that this quantity should be interpreted as thermodynamic pressure, thereby completing the parallel between the laws of black hole mechanics and thermodynamics. The conjugate quantity, volume, naturally emerges and its properties are discussed in Sec. 3. Once these basic notions of pressure and volume for black hole systems are defined and understood it becomes possible to analyze the extended thermodynamic phase space, one that includes these variables along with the more established quantities of temperature, entropy, potential, charge, angular velocity, angular momentum, and energy. It is in this context that the rich panoply of chemical behavior of black holes is manifest, a subject we review in Sec. 4. We then turn to more recent developments that endeavour to understand black hole chemistry from a holographic viewpoint in Sec. 5. We then consider in Sec. 6 how the concept of thermodynamic pressure can be applied and re-interpreted in more general situations, such as cosmological expanding spacetimes, or Lifshitz spacetimes that are conjectured to be dual to certain condensed matter systems. We relegate supplementary technical material into appendices and conclude our review with a discussion of what has been accomplished in this subject and what remains to be done.
By introducing as a thermodynamic variable, black hole thermodynamics has been given a new life. Let us begin exploring.
2 Thermodynamics with Lambda
2.1 A brief review of standard black hole thermodynamics
Since black holes are classical solutions to Einstein’s equations there is no a-priori reason to expect them to exhibit thermodynamic behaviour. The first indication linking these two subjects came from Hawking’s area theorem , which states that the area of the event horizon of a black hole can
The surface gravity is constant over the event horizon of a stationary black hole.
For a rotating charged black hole with a mass , an angular momentum , and a charge ,
where is its surface gravity, its angular velocity, and its electric potential.
Hawking’s area theorem: , i.e. the area of a black hole’s event horizon can never decrease.
It is impossible to reduce the surface gravity to zero in a finite number of steps.
The surface gravity is defined in the presence of a Killing horizon via
for a suitably normalized Killing vector that generates the horizon. For a static black hole, such as the Schwarzschild black hole, the surface gravity is the force exerted at infinity that is required to keep an object of unit mass at the horizon.
If we only consider black holes classically, these laws are merely a formal analogy between black hole mechanics and thermodynamics, where comparison to the first law of ordinary thermodynamics
is made with playing the role of temperature and event horizon area playing the role of entropy. In fact classical black holes have zero temperature. They never emit anything; a classical black hole immersed in a radiation bath at any finite temperature will always absorb the radiation.
By taking quantum effects into account, Hawking discovered  that black holes do emit radiation with a blackbody spectrum at a characteristic temperature
inserting the factors of Boltzmann’s constant , the speed of light , and Planck’s constant , the latter quantity underscoring the intrinsically quantum-mechanical nature of black hole temperature. This discovery, verified by many subsequent derivations, led to a paradigm shift: black holes are actual physical thermodynamic systems that have temperature and entropy; they are no longer systems that are simply described by a convenient analogy with thermodynamics.
With the relation (4) between and established, we can compare the term in the first law of thermodynamics with the term for black holes to infer that the entropy is directly related to the area by
for a black hole of mass , charge , and angular momentum , upon setting
a convention that implies that has units of [length]. Keeping this in mind, we shall henceforth suppress the explicit appearance of these quantities, restoring them on an as-needed basis.
The black hole mass is identified with the energy of the system, , and the chemical potential and electromagnetic work terms in (3) play roles analogous to the rotational and electromagnetic work terms in (6).
The thermodynamic variables in (6) are related by a useful Smarr–Gibbs–Duhem relation, which (in four dimensions) reads
expressing a relationship between the extensive and intensive thermodynamic variables.
2.2 History of variable
One of the noteworthy features of the first law (6) is the omission of a pressure-volume term
. This quantity is commonplace in everyday thermodynamics, but there is no obvious notion
of pressure or volume associated with a black hole.
In the last few years a new perspective has emerged that incorporates these notions
into black hole thermodynamics. The basic idea is that pressure can be associated with a negative cosmological constant , a form of energy whose (positive) pressure is equal in magnitude to its (negative) energy density
An asymptotically anti de Sitter (AdS) black hole in spacetime dimensions is a solution to the Einstein equations
where is often parameterized by the AdS radius according to
and is the matter stress-energy tensor (that vanishes sufficiently quickly as we approach the asymptotic region).
For example, let us consider a vacuum () static spherically symmetric black hole solution, generalizing the asymptotically flat higher-dimensional Schwarzschild–Tangherlini solution . The metric reads
where the metric function
is the metric on a compact -dimensional space of constant curvature with sign , with being the -sphere, being a torus, and being a compact hyperbolic space [38, 39]. The object is the metric of a -sphere. The parameter is related to the black hole mass. The appearance of the parameter with its three distinct values is specific to AdS spacetimes; appropriate identifications can render any constant section of the spacetime compact . The presence of in the metric function ensures the proper AdS asymptotic behavior.
Concentrating on the spherical case, sufficiently large (as compared to the AdS radius ) black holes (11) have positive specific heat (unlike their asymptotically flat counterparts) and can be in stable equilibrium at a fixed temperature (with AdS space acting like a gravitational box). They can also undergo a phase transition to pure radiation depending on the temperature . In the context of the AdS/CFT correspondence, this transition, known as the Hawking–Page transition, was later understood as a confinement/deconfinement phase transition in the boundary Conformal Field Theory (CFT) ; we shall further discuss this transition in Sec. 4.
The notion that itself might be a dynamical variable was proposed by Teitelboim and Brown
[41, 42], and the corresponding thermodynamic term was formally incorporated into the first law somewhat later , though no interpretation of the conjugate variable was considered.
The idea of associating
with pressure was subsequently explored from several perspectives [44, 45], but its proper association along with the notion of a conjugate black hole volume
was achieved once the laws of black hole mechanics were generalized to include
whose derivation is reproduced in App. A. Here
is interpreted as thermodynamic pressure, and the quantity , given by
is its conjugate thermodynamic volume [48, 49], whose interpretation we postpone until Sec. 3. The quantities and are the conserved charges respectively associated with the time-translation and rotational Killing vectors of the spacetime. As before, the area of the black hole event horizon is related to the entropy according to and temperature with its surface gravity.
The interpretation of as thermodynamic pressure naturally follows from the realization that induces a positive vacuum pressure in spacetime. Comparing (14) with (3) we see that in the presence of the cosmological constant, the mass has no longer meaning of internal energy. Rather, can be interpreted as a gravitational version of chemical enthalpy , which is the total energy of a system including both its internal energy and the energy required to displace the vacuum energy of its environment:
with the two quantities related by standard Legendre transformation. In other words, is the total energy required to “create a black hole and place it in a cosmological (negative ) environment”.
Hence by permitting to be a variable quantity we recover the familiar pressure-volume term from chemical thermodynamics. Extending to cases with multiple rotations and charges, the generalized first law of black hole thermodynamics reads [47, 49, 50, 27]
where is the largest integer less than or equal to and represents an upper bound on possible number of independent rotations in dimensions . (higher-dimensional rotating black hole solutions with were constructed in [52, 53, 54] and are reviewed in App. B.) Here the are the conjugate (gauge independent) potentials for the electric (and magnetic) charges, , allowing for both: a non-trivial potential on the horizon and at infinity . Similarly, , where the quantities allow for the possibility of a rotating frame at infinity . The thermodynamic volume may therefore be interpreted as the change in the mass under variations in , with the black hole entropy, angular momenta, and charges held fixed.
A “practical reason” for including the pressure volume term is connected with the Smarr formula, which now reads
generalizing the relation (8) to AdS spacetimes in -dimensions. Note the presence of the crucial term; we shall demonstrate the need for this term in an example below.
upon taking the derivative with respect to . Taking the mass to be a homogeneous function , and noting that the scaling dimensions of and are , and are both , and is we have
Employing now the first law (18), together with identification of with and with , we for example have and so on, and so (21) yields (19). Note that the inclusion of the term is required for (19) to hold.
Despite the fact that the preceding derivation assumes that the mass is a homogeneous function of
the other thermodynamic variables, the Smarr relation (19) has been demonstrated to have very broad applicability, including
, any dimension,
Before we proceed further, let us illustrate the generalized first law and the Smarr formula for the concrete example of a charged AdS black hole in four dimensions. The metric and the gauge field (characterized by the gauge potential A and field strength ) read
where is given by
and is the metric for the standard element on . The parameter represents the ADM mass of the black hole and its total charge. The outer (event) horizon is located at , determined from . Exploiting this latter relation we have
Taking the variation, we have
upon using (28). It is obvious that without the term, the Smarr formula would not hold.
2.3 Black hole chemistry
This new perspective on black hole thermodynamics, with its different interpretation of black hole mass and the inclusion of as a pressure term [47, 72, 48, 73], has led to a different understanding of known processes and to the discovery of a broad range of new phenomena associated with black holes. Referred to as “Black Hole Chemistry” [24, 25, 29, 56, 74, 75], this approach has led to a new understanding of concepts such as Van der Waals fluids, reentrant phase transitions, triple points, and polymer behavior from a gravitational viewpoint. Both charged and rotating black holes exhibit novel chemical-type phase behaviour that we shall discuss in the sequel.
The first observation is that the thermodynamic correspondence with black hole mechanics is completed  to include the familiar pressure/volume terms:
The dots stand for the work terms. In the black hole case these are , allowing for multiply charged and spinning black hole solutions.
Black hole chemistry has a much broader range of applications than Einstein-AdS gravity. Its concepts generalize to Lovelock gravity and to a broad range of other theories of gravity. We discuss the Lovelock case in the next subsection.
2.4 Lovelock gravity
One of the more fruitful applications of Black Hole Chemistry has been in Lovelock gravity . This refers to a class of gravitational theories whose actions contain terms non-linear in the curvature. Of course infinitely many such theories exist, but Lovelock gravity theories are unique in that they give rise to field equations that are generally covariant and contain at most second order derivatives of the metric. They remain of considerable interest in the context of quantum gravity where it is expected that the Einstein–Hilbert action is only an effective gravitational action valid for small curvature or low energies, and so will be modified by higher-curvature terms.
In spacetime dimensions, the Lagrangian of a Lovelock gravity theory is 
where the are the Lovelock coupling constants. The quantities are the -dimensional Euler densities, given by the contraction of powers of the Riemann tensor
where the ‘generalized Kronecker delta function’ is totally antisymmetric in both sets of indices. The term is defined to be unity and gives the cosmological constant term, , gives the Einstein–Hilbert action, , and corresponds to the quadratic Gauss–Bonnet term. The matter Lagrangian in (30) describes minimal coupling to matter. There is an upper bound of on the sum, which reflects the fact that a given term is a purely topological object in , and vanishes identically for . Only for does this term contribute to the equations of motion. General relativity, whose equations of motion are (9), is recovered upon setting for .
Before proceeding further, we note two special subclasses of Lovelock theories. Introducing the rescaled coupling constants
a special class of theories, called Chern–Simons gravity , arises in odd dimensions for the choice
of Lovelock couplings. Here stands for the AdS radius; it is no longer given by the second equality in (15) but instead is a non-trivial function of the ‘bare’ cosmological constant and the higher-order Lovelock couplings. In this particular case the local Lorentz invariance of the Lovelock action is enhanced to a local (A)dS symmetry. Another special case of Lovelock gravity occurs when
The equations of motion for Lovelock gravity, following from a variational principle using (30), are
where the Einstein-like tensors are given by
and each of them independently satisfies a conservation law .
The same arguments in App. A that use the Hamiltonian formalism can be generalized to the Lovelock case, yielding
for the Smarr formula, using the Euler scaling argument. The Smarr formula (42) for black holes can be expressed in terms of a Noether charge surface integral plus a suitable volume integral . We stress that the entropy is no longer proportional to the horizon area but instead is given by the expression
where denotes the determinant of , the induced metric on the black hole horizon , and the Lovelock terms are evaluated on that surface. The case is Einstein gravity and yields the usual value of one-quarter the horizon area. Note also that the Lovelock coupling constants are regarded as thermodynamic variables in (37). Their conjugate potentials (whose physical meaning has yet to be explored) were denoted by . For spherical Lovelock black holes they can be explicitly computed .
The parameter represents the maximal electromagnetic field strength. This quantity can be related to the string tension in the context of string theory , with . Promoting to a thermodynamic variable adds an extra term in the first law (37) 
where is the thermodynamic conjugate to the coupling . Noting that has units of electric field and the enthalpy has units of energy, the quantity thus has units of electric polarization per unit volume. Consequently has been referred to as ‘Born–Infeld vacuum polarization’ . These results straightforwardly extend to higher dimensions, and there have been further investigations into Born–Infeld electrodynamics in extended phase space [87, 84, 88, 89, 90, 91, 92]. This yields the following generalized Smarr formula:
The extended Smarr formula and the first law in the more general setting of ‘variable background fields’ was recently studied using the covariant formalism ; this approach offers a new perspective on variable from this more general viewpoint.
3 What is a volume of a black hole?
How do we describe and characterize the geometry of horizons, and what are their general geometrical properties? A standard answer is offered by studying a relationship between horizon area (an intrinsic horizon property) and dynamical quantities such as the total energy or angular momentum, and results in the so called Penrose (isoperimetric) inequalities that are closely related to cosmic censorship and Thorne’s hoop conjecture . As we have seen in Sec. 2, extended phase space thermodynamics enables one to define a new “intrinsic” quantity — thermodynamic volume — associated with the (black hole) horizon. It is the purpose of this section to study its physical meaning and characteristic properties, and in particular the associated isoperimetric inequality.
3.1 Thermodynamic volume
The black hole thermodynamic volume is a quantity with dimensions of (length) (in other words, a spatial volume) that characterizes a spacetime and is entirely derived from thermodynamic considerations. For an asymptotically AdS black hole spacetime it is the quantity thermodynamically conjugate to
as defined in (16).
Originally this conjugate variable was interpreted geometrically as a “… finite, effective volume for the region outside the AdS black hole horizon” .
Later, it was pointed out that (16) is
independent of any geometric volume  for most black holes , and should be
regarded as a thermodynamic volume. Furthermore, although the definition (43) was originally coined
for asymptotically AdS black hole spacetimes,
where is the black hole horizon radius. This result does not explicitly depend on the value of (or the charge ) and so can be taken to be valid for . Amusingly, the result is the same as if the black hole were a ball of radius in Euclidean space.
A recent contrasting viewpoint  is that thermodynamic volume should be replaced with a more general notion of gravitational tension
that describes the extra energy associated with the presence of gravitational fields surrounding a black hole. Gravitational tension vanishes in the flat-space limit and is proportional for a Schwarzschild-AdS black hole.
The relationship of this approach to the concept of thermodynamic volume we describe here remains to be explored.
In the presence of rotation, additional charges, and other thermodynamic parameters, the formula for the thermodynamic volume gets more complicated. By now a wide variety of explicit expressions for the thermodynamic volume have been found for black holes for which the exact solution with cosmological constant is known and their thermodynamics is well defined. These include higher-dimensional rotating black holes , charged black holes of various supergravities , superentropic black holes [60, 96], accelerated black holes [97, 98], or ‘ultraspinning black rings’ obtained in the blackfold approximation [27, 95]. For example, for the -dimensional Kerr-AdS black hole (given in App. B), the formula (43) yields
where are various (up to ) rotation parameters, are the associated angular momenta, and
is the horizon area. The total number of spacetime dimensions with in even and in odd dimensions; in even dimensions .
Equation (45) demonstrates that in general a simple expression for with intuitive geometrical meaning does not hold. It is therefore natural to ask if the quantity , defined by (43), obeys properties that one would like to associate with the volume of black hole. A characteristic property for the volume of a simply connected domain in Euclidean space is that it obeys an isoperimetric inequality. We investigate this property for the thermodynamic volume in the next section.
3.2 Reverse isoperimetric inequality
In Euclidean space , the isoperimetric inequality for the volume of a connected domain whose area is states that the ratio
obeys , where
is the volume of the unit -sphere. Equality holds if and only if the domain is a standard round ball.
It was conjectured in  that a reverse isoperimetric inequality,
holds for any asymptotically AdS black hole, upon identifying with the horizon area and with the associated thermodynamic volume,
the bound being saturated for Schwarzschild-AdS black holes. In other words, for a fixed thermodynamic volume the
entropy of the black hole is maximized for the Schwarzschild-AdS spacetime.
employing the arithmetic/geometric (AG) inequality . Since and , the reverse isoperimetric inequality (49) follows.
For a broad variety of (charged and/or rotating) spherical black holes , as well as for example (thin) ultraspinning black rings with toroidal horizon topology , the conjecture (49) has been shown to be valid. For more complicated black holes (49) has been confirmed numerically. Recently a class of exotic black hole spacetimes was found to violate (49). These are studied in the next subsection.
3.3 Super-entropic black holes
Super-entropic black holes describe an exotic class of rotating AdS black hole solutions with noncompact event horizons and finite horizon area, whose entropy exceeds the maximum implied from the conjectured reverse isoperimetric inequality (49).
First obtained by taking a particular limit of the Carter metric , there is now an entire class of rotating and/or charged super-entropic black holes in four and higher dimensions [100, 60, 96, 101] for which
does not hold
The simplest example of such a black hole can be obtained by applying a new type of (singular) ultraspinning limit to the Kerr-AdS metric (see App. B) in which the rotation parameter approaches the AdS radius [60, 96]. The resulting metric reads
The thermodynamic charges are
with the parameter denoting the periodicity of the coordinate . The isoperimetric ratio is straightforwardly computed
and obviously violates the conjecture (49).
The metric (3.3) exhibits many exotic properties [60, 96]: it describes a black hole whose horizon has the topology of a sphere with two punctures. Fixed sections are non-compact and near the axis of symmetry approach Lobachevsky space. The axis itself is removed from the spacetime, and the coordinate becomes null as . The geometry of the horizon can be visualized by embedding it in Euclidean 3-space as illustrated in Fig. 1.
Super-entropic black holes indicate that the reverse isoperimetric inequality as stated in Sec. 3.2 cannot be entirely correct, thereby motivating the following more stringent version :
Conjecture (Revised reverse isoperimetric inequality). For an AdS black hole with thermodynamic volume and with compact horizon of area , the ratio (47) satisfies .
The proof of this conjecture remains an interesting open question for further study.
3.4 Negative volume: Taub-NUT solution
So far we have limited ourselves to applications of the extended phase space thermodynamics to black hole spacetimes. However if taken seriously it should apply to all geometries and spacetimes, even those without horizons . In this subsection we look into its applications to the Taub-NUT-AdS class of solutions.
The Euclidean Taub-NUT-AdS metric
and represents a ‘gravitational analogue’ of magnetic monopole, with the NUT charge playing the role of the dyonic charge to gravitational mass . In order to ensure the invisibility of Misner strings, the Euclidean time has to be identified with periodicity . Asymptotically, the solution approaches a squashed 3-sphere, written as an fibration over with first Chern class .
Concentrating on thermodynamics, the most peculiar feature of the solution is that the entropy
where the latter equality follows from regularity requirements and imposes a restriction
Of particular interest is the Taub-NUT case , which yields
or in other words a negative thermodynamic volume! First noted in , this peculiar feature has been interpreted as the fact that in the Taub-NUT case, it is the environment that has to do work on the system to create the solution while the universe has to increase its volume. This is in contrast to the black hole case where part of the universe had to be removed to ‘make a place’ for the black hole.
Picking up the threads of , the thermodynamic properties and possible phase transitions of Taub-NUT-AdS solutions and their generalizations have been further studied [113, 112, 101] and extended to include rotation  and deformations to dyonic black holes . It is somewhat remarkable that extended phase space thermodynamics provides a sensible framework for the study of these unusual solutions and that a plausible explanation may exist for objects characterized by negative volume.
3.5 Black hole compressibility
Having defined the concept of black hole volume, one can start studying its associated physical quantities. One of them, the black hole adiabatic compressibility  has attracted attention in connection with black hole stability [26, 117, 118] (see also [119, 120]).
Adiabatic compressibility is defined as
and for Kerr-AdS black holes is manifestly positive and regular; non-rotating black holes are adiabatically incompressible. In four dimensions reaches its maximum in the extremal black hole case, while in higher dimensions the ‘softness’ of the equation of state can be used as an indicator of the ultraspinning instability .
Associated with the adiabatic compressibility is the “speed of sound” . Defining an average density, , this reads
where the last equality applies to the 4-dimensional Kerr-AdS black hole. One can think of as a velocity of a “breathing mode” due to changing volume at constant . It has been speculated that a collection of primordial black holes might affect the speed of sound through the medium in the early universe in a manner similar to how a suspension of compressible spheres affects the speed of sound in a fluid .
3.6 Killing co-potential volume
The concept of thermodynamic volume was given a geometric footing when the first law of black hole mechanics was extended to include a cosmological constant . It stems from the following simple idea.
Consider a Killing horizon generated by the corresponding Killing vector . Due to the Killing equation such a vector is divergence-less, , and hence (at least locally) there must exist a Killing co-potential (defined up to a co-closed 2-form), such that . The arguments in App. A then yield
as a candidate definition for the volume of a black hole. There are likewise expressions for the variations of the conserved mass (183) and angular momenta (184) respectively. One would like to integrate these relations to obtain expressions for the total energy and angular momenta of the black hole.
Unfortunately the definition (62) is not unique, since (where ) also satisfies and so is an equally valid co-potential. This renders ambiguous the definition of energy based on (183). The best that can be done is to make a gauge choice for such that 
is the total mass (with the timelike Killing vector), which itself must be determined by other means. This in turn implies from (62) that
is the geometric definition of volume.
The conformal approach to calculating conserved charges [122, 123, 124] provides the most straightforward means for computing mass and angular momenta. It has a great advantage over other methods (such as that of Abbott and Deser ) insofar as it involves an integration at infinity of a finite quantity computed from the Weyl tensor. No infinite subtraction of a pure AdS background is required.
3.7 Other definitions of black hole volume
We conclude this section with an overview of existing alternate approaches towards defining black hole volume:
Geometric volume, due to Parikh , is probably the first ever notion of black hole volume. To find geometric volume, one essentially integrates the full -dimensional volume element over a slice, yielding
which is independent of the choice of ‘stationary time-slicing’ . The lower bound of integration is identified with the ‘position of the singularity’ and becomes problematic to define in the case of rotating black holes. Geometric volume was further studied in [127, 128] and has been compared to thermodynamic volume in , and to vector volume  and Hayward’s volume  in . The volume (65) seems to satisfy the standard isoperimetric inequality, , . It was also implicitly used for the study of horizon thermodynamics [45, 131, 132], the subject of Sec. 6.
Vector volume, due to Ballik and Lake , is a more mathematically rigorous formulation of geometric volume.
Dynamical volume is due to Christodoulou and Rovelli  and is based on the following simple observation in Euclidean space: the volume inside a two-sphere is the volume of the largest spacelike spherically symmetric 3d surface bounded by . Generalizing to curved space, the horizon of a spherically symmetric black hole is foliated by (spacelike) spheres , labeled by the null coordinate (setting at collapse time). At a given ‘time’ , the spacelike slice bounded by of maximal volume is taken to correspond to the volume of the black hole. Interestingly, the dynamical volume grows with and quickly approaches ‘large’ asymptotic values
providing ‘plenty of room’ to store information . Obviously the definition leads to a complicated maximization problem, the resulting volume being very different from the thermodynamic volume and the other three volumes mentioned in this subsection. For this reason we will not discuss this notion any further in this paper and refer interested readers to the original paper  and to recent studies [134, 135, 136]. We just mention that dynamical volume is closely related (with the main difference corresponding to a different choice of boundary conditions) to the time-dependent volume of an Einstein–Rosen bridge that is conjectured to describe the computational complexity of the dual quantum state [137, 138].
With a variety of definitions for the black hole volume one may ask if there is any consensus and connection among them. Interestingly, it turns out that all the volumes (apart from the dynamical one) coincide for the simple spherically symmetric case, but produce different results for more complicated spacetimes, for example in the presence of rotation. Concretely, for the Kerr-AdS black hole the geometric volume, or the vector volume , both read
This is to be compared with the expression (45) for the thermodynamic volume. We observe that the two differ by the presence of a “rotational” part and coincide in the limit .
4 Black hole chemistry
With thermodynamic pressure and volume defined, we can extend the thermodynamic phase space and study the thermodynamics of black holes in a new framework, sometimes referred to as Black Hole Chemistry . This change of perspective has been shown to have a number of remarkable consequences—black holes now seem to behave in ways that are analogous to a variety of “everyday” chemical phenomena, such as Van der Waals behavior, solid/liquid phase transitions, triple points, reentrant phase transitions, and heat engines. These will be described in this section, employing the following thermodynamic machinery:
We shall study the thermodynamics of charged and/or rotating AdS black holes in a canonical (fixed or ) ensemble. This will then be related to fluid thermodynamics, by comparing the “same physical quantities”: cosmological pressure is identified with the pressure of the fluid, thermodynamic volume of a black hole with the volume of the fluid, temperature of the black hole with the temperature of the fluid, and so on. Although quite natural, note that this “dictionary” is quite different from the extended AdS/CFT dictionary discussed in the next section.
The thermodynamic potential of interest is the Gibbs free energy
The equilibrium state corresponds to the global minimum of .
Local thermodynamic stability corresponds to positivity of the specific heat
The aim of this program is to construct phase diagrams, find critical points, study their critical exponents, and determine whatever other interesting transitional behavior might arise.
We shall start from the simple example of a Schwarzschild-AdS black hole and its associated Hawking–Page transition, and then proceed to more complicated spacetimes that demonstrate more elaborate phase phenomena.
4.1 A new look at the Hawking–Page transition
The spherically symmetric ansatz (11) in has the metric function
valid for spherical (), planar ), or hyperbolic ) horizon geometries. The thermodynamic quantities are similar to the versions of those in (25) and read
where is the area of the constant-curvature space.
Concentrating on the spherical case, we display the Gibbs free energy in the left part of Fig. 2. We observe two branches of black holes that meet at a cusp. The upper branch displays ‘small’ () thermodynamically unstable black holes with negative specific heat, the lower branch corresponds to ‘large’ black holes with positive specific heat. Large black holes with have negative Gibbs free energy (which is lower than that of an AdS space filled with hot radiation) and represent the globally preferred state. This means that at there is a first order Hawking–Page  phase transition between thermal radiation and large black holes. As noted in Sec. 2, this can be interpreted as a confinement/deconfinement phase transition in the dual quark gluon plasma .
Considering the fluid interpretation in an extended phase space, the coexistence line of thermal radiation/large black hole phases, determined from , reads
One can easily verify (taking and ) that its slope satisfies the Clausius–Clapeyron equation
a result not previously noted in the literature. From the right side of Fig. 2 we see that the corresponding phase diagram for this black hole has no terminal point, indicating that this phase transition is present for all pressures. It is reminiscent of a solid/liquid phase transition, with the radiation phase playing role of a solid .
where we have, in this section only, explicitly restored the Planck length . The quantity plays the role of a ’specific volume’ [73, 27], given by the thermodynamic volume divided by the ‘number of states’ associated with the horizon, . Note that for planar black holes we obtain the ideal gas law, .
4.2 Charged AdS black holes and Van der Waals fluids
Can we go beyond the ideal gas law and obtain a more realistic equation of state? Consider adding charge to the black hole, which implies the metric function becomes
for . This is an exact solution to the Einstein–Maxwell-AdS equations, corresponding to a charged-AdS black hole (2.2).
Charged AdS black holes allow for a first order small-black-hole/large-black-hole (SBH/LBH) phase transition, in a canonical (fixed charge) ensemble [69, 70, 71] (see also [139, 140]), which is in many ways reminiscent of the liquid/gas transition of a non-ideal fluid described by the Van der Waals equation (78). In extended phase space this analogy becomes even more complete since it allows proper identification between intensive and extensive variables [72, 73]. From (25) the thermodynamic quantities then read
giving rise to the following equation of state: