10.3. Entropy and energy of de Sitter spacetime
The best studied spacetimes with horizons are the black hole spacetimes. (For a sample of references, see [297, 298, 299, 300, 301, 302, 303, 304, 305, 306]). In the simplest context of a Schwarzschild black hole of mass M, one can attribute an energy E = M, temperature T = (8 M)-1 and entropy S = (1/4)(AH / LP2) where AH is the area of the horizon and LP = (G / c3)1/2 is the Planck length. (Hereafter, we will use units with G = = c = 1.) These are clearly related by the thermodynamic identity TdS = dE, usually called the first law of black hole dynamics. This result has been obtained in much more general contexts and has been investigated from many different points of view in the literature. The simplicity of the result depends on the following features: (a) The Schwarzschild metric is a vacuum solution with no pressure so that there is no PdV term in the first law of thermodynamics. (b) The metric has only one parameter M so that changes in all physical parameters can be related to dM. (c) Most importantly, there exists a well defined notion of energy E to the spacetime and the changes in the energy dE can be interpreted in terms of the physical process of the black hole evaporation. The idea can be generalized to other black hole spacetimes in a rather simple manner only because of well defined notions of energy, angular momentum etc.
Can one generalize the thermodynamics of horizons to cases other than black holes in a straight forward way ? In spite of years of research in this field, this generalization remains non trivial and challenging when the conditions listed above are not satisfied. To see the importance of the above conditions, we only need to contrast the situation in Schwarzschild spacetime with that of de Sitter spacetime:
All these suggest that to make any progress, one might require a local approach by which one can define the notion of entropy and energy for spacetimes with horizons. This conclusion is strengthened further by the following argument: Consider a class of spherically symmetric spacetimes of the form
If f (r) has a simple zero at r = a with f'(a) B remaining finite, then this spacetime has a horizon at r = a. Spacetimes like Schwarzschild or de Sitter have only one free parameter in the metric (like M or H-1) and hence the scaling of all other thermodynamical parameters is uniquely fixed by purely dimensional considerations. But, for a general metric of the form in (172), with an arbitrary f (r), the area of the horizon (and hence the entropy) is determined by the location of the zero of the function f (r) while the temperature - obtained from the periodicity considerations - is determined by the value of f'(r) at the zero. For a general function, of course, there will be no relation between the location of the zero and the slope of the function at that point. It will, therefore, be incredible if there exists any a priori relationship between the temperature (determined by f' ) and the entropy (determined by the zero of f) even in the context of horizons in spherically symmetric spacetimes. If we take the entropy to be S = a2 (where f (a) = 0 determines the radius of the horizon) and the temperature to be T = | f'(a)| / 4 (determined by the periodicity of Euclidean time), the quantity TdS = (1/2)| f'(a)| a da will depend both on the slope f'(a) as well as the radius of the horizon. This implies that any local interpretation of thermodynamics will be quite non trivial.
Finally, the need for local description of thermodynamics of horizons becomes crucial in the case of spacetimes with multiple horizons. The strongest and the most robust result we have, regarding spacetimes with a horizon, is the notion of temperature associated with them. This, in turn, depends on the study of the periodicity of the Euclidean time coordinate. This approach does not work very well if the spacetime has more than one horizon like, for example, in the Schwarzschild-de Sitter metric which has the form in (172) with
This spacetime has two horizons at r ± with
where cos x = - 33 MH-1. (The parameter x is in the range (, (3/2)] and we assume that 0 27M2H-2 < 1.) Close to either horizon the spacetime can be approximated as Rindler. Since the surface gravities on the two horizons are different, we get two different temperatures T ± = | f'(r ± )| / 4. To maintain invariance under it it + (with some finite ) it is necessary that is an integer multiple of both 4 / | f'(r+)| and 4 / | f'(r-)| so that = (4n ± / | f'(r ± )|) where n ± are integers. Hence the ratio of surface gravities | f'(r+)| / | f'(r-)| = (n+/n-) must be a rational number. Though irrationals can be approximated by rationals, such a condition definitely excludes a class of values for M if H is specified and vice versa. It is not clear why the existence of a cosmological constant should imply something for the masses of black holes (or vice versa). Since there is no physical basis for such a condition, it seems reasonable to conclude that these difficulties arise because of our demanding the existence of a finite periodicity in the Euclidean time coordinate. This demand is related to an expectation of thermal equilibrium which is violated in spacetimes with multiple horizons having different temperatures.
If even the simple notion of temperature falls apart in the presence of multiple horizons, it is not likely that the notion of energy or entropy can be defined by global considerations. On the other hand, it will be equally strange if we cannot attribute a temperature to a black hole formed in some region of the universe just because the universe at the largest scales is described by a de Sitter spacetime, say. One is again led to searching for a local description of the thermodynamics of all types of horizons. We shall now see how this can be done.
Given the notion of temperature, there are two very different ways of defining the entropy: (1) In statistical mechanics, the partition function Z() of the canonical ensemble of systems with constant temperature -1 is related to the entropy S and energy E by Z() exp(S - E). (2) In classical thermodynamics, on the other hand, it is the change in the entropy, which can be operationally defined via dS = dE / T(E). Integrating this equation will lead to the function S(E) except for an additive constant which needs to be determined from additional considerations. Proving the equality of these two concepts was nontrivial and - historically - led to the unification of thermodynamics with mechanics.
In the case of time symmetric state, there will be no change of entropy dS and the thermodynamic route is blocked. It is, however, possible to construct a canonical ensemble of a class of spacetimes and evaluate the partition function Z(). For spherically symmetric spacetimes with a horizon at r = l, the partition function has the generic form Z exp[S - E], where S = (1/4)4 l2 and | E| = (l /2). This analysis reproduces the conventional result for the black hole spacetimes and provides a simple and consistent interpretation of entropy and energy for de Sitter spacetime, with the latter being given by E = - (1/2)H-1. In fact, it is possible to write Einstein's equations for a spherically symmetric spacetime as a thermodynamic identity TdS - dE = PdV with T, S and E determined as above and the PdV term arising from the source . We shall now discuss some of these issues.
Consider a class of spacetimes with the metric
where f (r) vanishes at some surface r = l, say, with f'(l) B remaining finite. When dL2 = r2 dS22 with [0 r ], equation (175) covers a variety of spherically symmetric spacetimes with a compact horizon at r = l. Since the metric is static, Euclidean continuation is trivially effected by t = it and an examination of the conical singularity near r = a [where f (r) B(r - a)] shows that should be interpreted as periodic with period = 4 / | B| corresponding to the temperature T = | B| / 4. Let us consider a set of such metrics in (175) with the restriction that [f (a) = 0, f'(a) = B] but f (r) is otherwise arbitrary and has no zeros. The partition function for this set of metrics is given by the path integral sum
where Einstein action has been continued in the Euclidean sector and we have imposed the periodicity in with period = 4 / | B|. The sum is restricted to the set of all metrics of the form in (175) with the behaviour [f (a) = 0, f'(a) = B] and the Euclidean Lagrangian is a functional of f (r). The spatial integration will be restricted to a region bounded by the 2-spheres r = a and r = b, where the choice of b is arbitrary except for the requirement that within the region of integration the Lorentzian metric must have the proper signature with t being a time coordinate. The remarkable feature is the form of the Euclidean action for this class of spacetimes. Using the result R = r2 f - (2/r2)(d / dr)(r(1 - f )) valid for metrics of the form in (175), a straight forward calculation shows that
where Q depends on the behaviour of the metric near r = b and we have used the conditions [f (a) = 0, f'(a) = B]. The sum in (176) now reduces to summing over the values of [f (b), f'(b)] with a suitable (but unknown) measure. This sum, however, will only lead to a factor which we can ignore in deciding about the dependence of Z() on the form of the metric near r = a. Using = 4 / B (and taking B > 0, for the moment) the final result can be written in a very suggestive form:
with the identifications for the entropy and energy being given by:
In the case of the Schwarzschild black hole with a = 2M, the energy turns out to be E = (a/2) = M which is as expected. (More generally, E = (Ahorizon / 16)1/2 corresponds to the so called `irreducible mass' in BH spacetimes .) Of course, the identifications S = (4 M2), E = M, T = (1/8 M) are consistent with the result dE = TdS in this particular case.
The above analysis also provides an interpretation of entropy and energy in the case of de Sitter universe. In this case, f (r) = (1 - H2 r2), a = H-1, B = - 2H. Since the region where t is time like is "inside" the horizon, the integral for AE in (177) should be taken from some arbitrary value r = b to r = a with a > b. So the horizon contributes in the upper limit of the integral introducing a change of sign in (177). Further, since B < 0, there is another negative sign in the area term from B B / | B|. Taking all these into account we get, in this case,
giving S = (1/4)(4 a2) = (1/4)Ahorizon and E = - (1/2)H-1. These definitions do satisfy the relation TdS - PdV = dE when it is noted that the de Sitter universe has a non zero pressure P = - = - E/V associated with the cosmological constant. In fact, if we use the "reasonable" assumptions S = (1/4)(4 H-2), V H-3 and E = - PV in the equation TdS - PdV = dE and treat E as an unknown function of H, we get the equation H2(dE / dH) = - (3EH + 1) which integrates to give precisely E = - (1/2)H-1. (Note that we only needed the proportionality, V H-3 in this argument since PdV (dV/V). The ambiguity between the coordinate and proper volume is irrelevant.)
A peculiar feature of the metrics in (175) is worth stressing. This metric will satisfy Einstein's equations provided the the source stress tensor has the form Ttt = Trr ((r) / 8); T = T (µ(r) / 8). The Einstein's equations now reduce to:
The remarkable feature about the metric in (175) is that the Einstein's equations become linear in f (r) so that solutions for different (r) can be superposed. Given any (r) the solution becomes
with a being an integration constant and µ(r) is fixed by (r) through: µ(r) = + (1/2)r '(r). The integration constant a in (182) is chosen such that f (r) = 0 at r = a so that this surface is a horizon. Let us now assume that the solution (182) is such that f (r) = 0 at r = a with f'(a) = B finite leading to leading to a notion of temperature with = (4 / | B|). From the first of the equations (181) evaluated at r = a, we get
It is possible to provide an interesting interpretation of this equation which throws light on the notion of entropy and energy. Multiplying the above equation by da and using = 8 Trr, it is trivial to rewrite equation (183) in the form
Let us first consider the case in which a particular horizon has f'(a) = B > 0 so that the temperature is T = B / 4. Since f (a) = 0, f'(a) > 0, it follows that f > 0 for r > a and f < 0 for r < a; that is, the "normal region" in which t is time like is outside the horizon as in the case of, for example, the Schwarzschild metric. The first term in the left hand side of (184) clearly has the form of TdS since we have an independent identification of temperature from the periodicity argument in the local Rindler coordinates. Since the pressure is P = - Trr, the right hand side has the structure of PdV or - more relevantly - is the product of the radial pressure times the transverse area times the radial displacement. This is important because, for the metrics in the form (175), the proper transverse area is just that of a 2-sphere though the proper volumes and coordinate volumes differ. In the case of horizons with B = f'(a) > 0 which we are considering (with da > 0), the volume of the region where f < 0 will increase and the volume of the region where f > 0 will decrease. Since the entropy is due to the existence of an inaccessible region, dV must refer to the change in the volume of the inaccessible region where f < 0. We can now identify T in TdS and P in PdV without any difficulty and interpret the remaining term (second term in the left hand side) as dE = da/2. We thus get the expressions for the entropy S and energy E (when B > 0) to be the same as in (179).
Using (184), we can again provide an interpretation of entropy and energy in the case of de Sitter universe. In this case, f (r) = (1 - H2 r2), a = H-1, B = - 2H < 0 so that the temperature - which should be positive - is T = | f'(a)| / (4) = (- B) / 4. For horizons with B = f'(a) < 0 (like the de Sitter horizon) which we are now considering, f (a) = 0, f'(a) < 0, and it follows that f > 0 for r < a and f < 0 for r > a; that is, the "normal region" in which t is time like is inside the horizon as in the case of, for example, the de Sitter metric. Multiplying equation (184) by (- 1), we get
The first term on the left hand side is again of the form TdS (with positive temperature and entropy). The term on the right hand side has the correct sign since the inaccessible region (where f < 0) is now outside the horizon and the volume of this region changes by (- dV). Once again, we can use (185) to identify  the entropy and the energy: S = (1/4)(4 a2) = (1/4)Ahorizon; E = - (1/2)H-1. These results agree with the previous analysis.