Classical Black Hole Thermodynamics

2 Classical Black Hole Thermodynamics

In this section, I will give a brief review of the laws of classical black hole mechanics.

In physical terms, a black hole is a region where gravity is so strong that nothing can escape. In order to make this notion precise, one must have in mind a region of spacetime to which one can contemplate escaping. For an asymptotically flat spacetime $(M, g ) ab$ (representing an isolated system), the asymptotic portion of the spacetime “near infinity” is such a region. The black hole region, $ℬ$ , of an asymptotically flat spacetime, $(M, gab)$ , is defined as

where $ℐ+$ denotes future null infinity and $I−$ denotes the chronological past. Similar definitions of a black hole can be given in other contexts (such as asymptotically anti-deSitter spacetimes) where there is a well defined asymptotic region.

The event horizon, $ℋ$ , of a black hole is defined to be the boundary of $ℬ$ . Thus, $ℋ$ is the boundary of the past of $ℐ+$ . Consequently, $ℋ$ automatically satisfies all of the properties possessed by past boundaries (see, e.g., [55 ] or [99 ] for further discussion). In particular, $ℋ$ is a null hypersurface which is composed of future inextendible null geodesics without caustics, i.e., the expansion, $𝜃$ , of the null geodesics comprising the horizon cannot become negatively infinite. Note that the entire future history of the spacetime must be known before the location of $ℋ$ can be determined, i.e., $ℋ$ possesses no distinguished local significance.

If Einstein’s equation holds with matter satisfying the null energy condition (i.e., if $Tabkakb ≥ 0$ for all null $a k$ ), then it follows immediately from the Raychauduri equation (see, e.g., [99 ]) that if the expansion, $𝜃$ , of any null geodesic congruence ever became negative, then $𝜃$ would become infinite within a finite affine parameter, provided, of course, that the geodesic can be extended that far. If the black hole is strongly asymptotically predictable – i.e., if there is a globally hyperbolic region containing $I− (ℐ+ ) ∪ ℋ$ – it can be shown that this implies that $𝜃 ≥ 0$ everywhere on $ℋ$ (see, e.g., [55 , 99 ]). It then follows that the surface area, $A$ , of the event horizon of a black hole can never decrease with time, as discovered by Hawking [53 ].

It is worth remarking that since $ℋ$ is a past boundary, it automatically must be a $0 C$ embedded submanifold (see, e.g., [99 ]), but it need not be $C1$ . However, essentially all discussions and analyses of black hole event horizons implicitly assume $C1$ or higher order differentiability of $ℋ$ . Recently, this higher order differentiability assumption has been eliminated for the proof of the area theorem [36 ].

The area increase law bears a resemblance to the second law of thermodynamics in that both laws assert that a certain quantity has the property of never decreasing with time. It might seem that this resemblance is a very superficial one, since the area law is a theorem in differential geometry whereas the second law of thermodynamics is understood to have a statistical origin. Nevertheless, this resemblance together with the idea that information is irretrievably lost when a body falls into a black hole led Bekenstein to propose [13 , 14 ] that a suitable multiple of the area of the event horizon of a black hole should be interpreted as its entropy, and that a generalized second law (GSL) should hold: The sum of the ordinary entropy of matter outside of a black hole plus a suitable multiple of the area of a black hole never decreases. We will discuss this law in detail in Section 4.

The remaining laws of thermodynamics deal with equilibrium and quasi-equilibrium processes. At nearly the same time as Bekenstein proposed a relationship between the area theorem and the second law of thermodynamics, Bardeen, Carter, and Hawking [12 ] provided a general proof of certain laws of “black hole mechanics” which are direct mathematical analogs of the zeroth and first laws of thermodynamics. These laws of black hole mechanics apply to stationary black holes (although a formulation of these laws in terms of isolated horizons will be briefly described at the end of this section).

In order to discuss the zeroth and first laws of black hole mechanics, we must introduce the notions of stationary, static, and axisymmetric black holes as well as the notion of a Killing horizon. If an asymptotically flat spacetime $(M, gab)$ contains a black hole, $ℬ$ , then $ℬ$ is said to be stationary if there exists a one-parameter group of isometries on $(M, g ) ab$ generated by a Killing field $ta$ which is unit timelike at infinity. The black hole is said to be static if it is stationary and if, in addition, $a t$ is hypersurface orthogonal. The black hole is said to be axisymmetric if there exists a one parameter group of isometries which correspond to rotations at infinity. A stationary, axisymmetric black hole is said to possess the “ $t– ϕ$ orthogonality property” if the 2-planes spanned by $ta$ and the rotational Killing field $ϕa$ are orthogonal to a family of 2-dimensional surfaces. The $t –ϕ$ orthogonality property holds for all stationary-axisymmetric black hole solutions to the vacuum Einstein or Einstein–Maxwell equations (see, e.g., [56 ]).

A null surface, $𝒦$ , whose null generators coincide with the orbits of a one-parameter group of isometries (so that there is a Killing field $a ξ$ normal to $𝒦$ ) is called a Killing horizon. There are two independent results (usually referred to as “rigidity theorems”) that show that in a wide variety of cases of interest, the event horizon, $ℋ$ , of a stationary black hole must be a Killing horizon. The first, due to Carter [35 ], states that for a static black hole, the static Killing field $ta$ must be normal to the horizon, whereas for a stationary-axisymmetric black hole with the $t– ϕ$ orthogonality property there exists a Killing field $a ξ$ of the form

which is normal to the event horizon. The constant $Ω$ defined by Eq. (2

) is called the angular velocity of the horizon. Carter’s result does not rely on any field equations, but leaves open the possibility that there could exist stationary black holes without the above symmetries whose event horizons are not Killing horizons. The second result, due to Hawking [55 ] (see also [45 ]), directly proves that in vacuum or electrovac general relativity, the event horizon of any stationary black hole must be a Killing horizon. Consequently, if $a t$ fails to be normal to the horizon, then there must exist an additional Killing field, $ξa$ , which is normal to the horizon, i.e., a stationary black hole must be nonrotating (from which staticity follows [84

, 85 , 37 ]) or axisymmetric (though not necessarily with the $t– ϕ$ orthogonality property). Note that Hawking’s theorem makes no assumptions of symmetries beyond stationarity, but it does rely on the properties of the field equations of general relativity.

Now, let $𝒦$ be any Killing horizon (not necessarily required to be the event horizon, $ℋ$ , of a black hole), with normal Killing field $ξa$ . Since $∇a (ξbξb)$ also is normal to $𝒦$ , these vectors must be proportional at every point on $𝒦$ . Hence, there exists a function, $κ$ , on $𝒦$ , known as the surface gravity of $𝒦$ , which is defined by the equation

It follows immediately that $κ$ must be constant along each null geodesic generator of $𝒦$ , but, in general, $κ$ can vary from generator to generator. It is not difficult to show (see, e.g., [99 ]) that

where $a$ is the magnitude of the acceleration of the orbits of $ξa$ in the region off of $𝒦$ where they are timelike, $V ≡ (− ξaξa)1∕2$ is the “redshift factor” of $ξa$ , and the limit as one approaches $𝒦$ is taken. Equation (4

) motivates the terminology “surface gravity”. Note that the surface gravity of a black hole is defined only when it is “in equilibrium”, i.e., stationary, so that its event horizon is a Killing horizon. There is no notion of the surface gravity of a general, non-stationary black hole, although the definition of surface gravity can be extended to isolated horizons (see below).

In parallel with the two independent “rigidity theorems” mentioned above, there are two independent versions of the zeroth law of black hole mechanics. The first, due to Carter [35 ] (see also [78 ]), states that for any black hole which is static or is stationary-axisymmetric with the $t–ϕ$ orthogonality property, the surface gravity $κ$ , must be constant over its event horizon $ℋ$ . This result is purely geometrical, i.e., it involves no use of any field equations. The second, due to Bardeen, Carter, and Hawking [12 ] states that if Einstein’s equation holds with the matter stress-energy tensor satisfying the dominant energy condition, then $κ$ must be constant on any Killing horizon. Thus, in the second version of the zeroth law, the hypothesis that the $t– ϕ$ orthogonality property holds is eliminated, but use is made of the field equations of general relativity.

A bifurcate Killing horizon is a pair of null surfaces, $𝒦A$ and $𝒦B$ , which intersect on a spacelike 2-surface, $𝒞$ (called the “bifurcation surface”), such that $𝒦A$ and $𝒦B$ are each Killing horizons with respect to the same Killing field $ξa$ . It follows that $ξa$ must vanish on $𝒞$ ; conversely, if a Killing field, $ξa$ , vanishes on a two-dimensional spacelike surface, $𝒞$ , then $𝒞$ will be the bifurcation surface of a bifurcate Killing horizon associated with $ξa$ (see [101 ] for further discussion). An important consequence of the zeroth law is that if $κ ⁄= 0$ , then in the “maximally extended” spacetime representing a stationary black hole, the event horizon, $ℋ$ , comprises a branch of a bifurcate Killing horizon [78 ]. This result is purely geometrical – involving no use of any field equations. As a consequence, the study of stationary black holes which satisfy the zeroth law divides into two cases: “extremal” black holes (for which, by definition, $κ = 0$ ), and black holes with bifurcate horizons.

The first law of black hole mechanics is simply an identity relating the changes in mass, $M$ , angular momentum, $J$ , and horizon area, $A$ , of a stationary black hole when it is perturbed. To first order, the variations of these quantities in the vacuum case always satisfy

In the original derivation of this law [12

], it was required that the perturbation be stationary. Furthermore, the original derivation made use of the detailed form of Einstein’s equation. Subsequently, the derivation has been generalized to hold for non-stationary perturbations [84 , 60

], provided that the change in area is evaluated at the bifurcation surface, $𝒞$ , of the unperturbed black hole (see, however, [80 ] for a derivation of the first law for non-stationary perturbations that does not require evaluation at the bifurcation surface). More significantly, it has been shown [60 ] that the validity of this law depends only on very general properties of the field equations. Specifically, a version of this law holds for any field equations derived from a diffeomorphism covariant Lagrangian, $L$ . Such a Lagrangian can always be written in the form

where $∇a$ denotes the derivative operator associated with $gab$ , $Rabcd$ denotes the Riemann curvature tensor of $gab$ , and $ψ$ denotes the collection of all matter fields of the theory (with indices suppressed). An arbitrary (but finite) number of derivatives of $R abcd$ and $ψ$ are permitted to appear in $L$ . In this more general context, the first law of black hole mechanics is seen to be a direct consequence of an identity holding for the variation of the Noether current. The general form of the first law takes the form

where the “...” denote possible additional contributions from long range matter fields, and where

Here $nab$ is the binormal to the bifurcation surface $𝒞$ (normalized so that $ab nabn = − 2$ ), and the functional derivative is taken by formally viewing the Riemann tensor as a field which is independent of the metric in Eq. (6

). For the case of vacuum general relativity, where $√ --- L = R − g$ , a simple calculation yields

and Eq. (7

) reduces to Eq. (5

The close mathematical analogy of the zeroth, first, and second laws of thermodynamics to corresponding laws of classical black hole mechanics is broken by the Planck–Nernst form of the third law of thermodynamics, which states that $S → 0$ (or a “universal constant”) as $T → 0$ . The analog of this law fails in black hole mechanics – although analogs of alternative formulations of the third law do appear to hold for black holes [59 ] – since there exist extremal black holes (i.e., black holes with $κ = 0$ ) with finite $A$ . However, there is good reason to believe that the “Planck–Nernst theorem” should not be viewed as a fundamental law of thermodynamics [1 ] but rather as a property of the density of states near the ground state in the thermodynamic limit, which happens to be valid for commonly studied materials. Indeed, examples can be given of ordinary quantum systems that violate the Planck–Nernst form of the third law in a manner very similar to the violations of the analog of this law that occur for black holes [102 ].

As discussed above, the zeroth and first laws of black hole mechanics have been formulated in the mathematical setting of stationary black holes whose event horizons are Killing horizons. The requirement of stationarity applies to the entire spacetime and, indeed, for the first law, stationarity of the entire spacetime is essential in order to relate variations of quantities defined at the horizon (like $A$ ) to variations of quantities defined at infinity (like $M$ and $J$ ). However, it would seem reasonable to expect that the equilibrium thermodynamic behavior of a black hole would require only a form of local stationarity at the event horizon. For the formulation of the first law of black hole mechanics, one would also then need local definitions of quantities like $M$ and $J$ at the horizon. Such an approach toward the formulation of the laws of black hole mechanics has recently been taken via the notion of an isolated horizon, defined as a null hypersurface with vanishing shear and expansion satisfying the additional properties stated in [4 ]. (This definition supersedes the more restrictive definitions given, e.g., in [5 , 6 , 7 ].) The presence of an isolated horizon does not require the entire spacetime to be stationary [65 ]. A direct analog of the zeroth law for stationary event horizons can be shown to hold for isolated horizons [9 ]. In the Einstein–Maxwell case, one can demand (via a choice of scaling of the normal to the isolated horizon as well as a choice of gauge for the Maxwell field) that the surface gravity and electrostatic potential of the isolated horizon be functions of only its area and charge. The requirement that time evolution be symplectic then leads to a version of the first law of black hole mechanics as well as a (in general, non-unique) local notion of the energy of the isolated horizon [9 ]. These results also have been generalized to allow dilaton couplings [7 ] and Yang–Mills fields [38 , 9 ].

In comparing the laws of black hole mechanics in classical general relativity with the laws of thermodynamics, it should first be noted that the black hole uniqueness theorems (see, e.g., [56 ]) establish that stationary black holes – i.e., black holes “in equilibrium” – are characterized by a small number of parameters, analogous to the “state parameters” of ordinary thermodynamics. In the corresponding laws, the role of energy, $E$ , is played by the mass, $M$ , of the black hole; the role of temperature, $T$ , is played by a constant times the surface gravity, $κ$ , of the black hole; and the role of entropy, $S$ , is played by a constant times the area, $A$ , of the black hole. The fact that $E$ and $M$ represent the same physical quantity provides a strong hint that the mathematical analogy between the laws of black hole mechanics and the laws of thermodynamics might be of physical significance. However, as argued in [12 ], this cannot be the case in classical general relativity. The physical temperature of a black hole is absolute zero (see Section 4.1 below), so there can be no physical relationship between $T$ and $κ$ . Consequently, it also would be inconsistent to assume a physical relationship between $S$ and $A$ . As we shall now see, this situation changes dramatically when quantum effects are taken into account.