To what extent is the entropy-area law universal ?

To what extent is the entropy-area law universal ?

Hiromi Saida111saida@daido-it.ac.jp
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Multi-horizon and multi-temperature spacetime may break the entropy-area law

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It seems to be a common understanding at present that, once event horizons are in thermal equilibrium, the entropy-area law holds inevitably. However no rigorous verification is given to such a very strong universality of the law in multi-horizon spacetimes. Then, based on thermodynamically consistent and rigorous discussion, this paper suggests an evidence of breakdown of entropy-area law for horizons in Schwarzschild-de Sitter spacetime, in which the temperatures of the horizons are different. The outline is as follows: We construct carefully two thermal equilibrium systems individually for black hole event horizon (BEH) and cosmological event horizon (CEH), for which the Euclidean action method is applicable. The integration constant (subtraction term) in Euclidean action is determined with referring to Schwarzschild and de Sitter canonical ensembles. The free energies of the two thermal systems are functions of three independent state variables, and we find a similarity of our two thermal systems with the magnetized gas in laboratory, which gives us a physical understanding of the necessity of three independent state variables. Then, via the thermodynamic consistency with three independent state variables, the breakdown of entropy-area law for CEH is suggested. The validity of the law for BEH can not be judged, but we clarify the key issue for BEH’s entropy. Finally we make comments which may suggest the breakdown of entropy-area law for BEH, and also propose two discussions; one of them is on the quantum statistics of underlying quantum gravity, and another is on the SdS black hole evaporation from the point of view of non-equilibrium thermodynamics.

\PTPindex

451, 454

1 Introduction

Entropy-area law, which claims the equilibrium entropy of event horizon is equal to one-quarter of its spatial area in Planck units [1, 2, 3], is the equation of state of the event horizon in thermal equilibrium. This law has already been verified for spacetimes possessing a single event horizon [4, 5, 7, 6]. Then we may naively expect that the entropy-area law holds also for multi-horizon spacetimes, once every horizon is individually in thermal equilibrium. This expectation is equivalent to consider that the thermal equilibrium of each horizon is the necessary and sufficient condition to ensure the entropy-area law for each horizon. However this expectation has not been rigorously verified in multi-horizon spacetimes. (Comments on existing researches on Schwarzschild-de Sitter spacetime will be given in fourth, fifth and sixth paragraphs in this section. Please wait for those paragraphs if the reader cares about existing researches on multi-horizon spacetimes.) At present, there remains the possibility that the thermal equilibrium may be simply the necessary condition of entropy-area law. If we find an example that some event horizon does not satisfy the entropy-area law even when it is in thermal equilibrium, then we recognize the thermal equilibrium as simply the necessary condition of the entropy-area law.

We can consider a situation in which the entropy-area law may break down in multi-horizon spacetime 222 All discussions in this paper are based on the ordinary general relativity. The other modified theories of gravity are not considered. Even if there is a breakdown of entropy-area law due to exotic fields of modified theory, such a breakdown in modified theory is out of the scope of this paper. . To explain it, it is necessary to distinguish thermodynamic state of each horizon and that of the total system (multi-horizon spacetime) composed of several horizons. Even when every horizon in a multi-horizon spacetime is in an equilibrium state individually, the total system composed of several horizons is never in any equilibrium state if the equilibrium state of one horizon is different from that of the other horizon. For example, if the temperatures of horizons in a multi-horizon spacetime are different from each other, then a net energy flow arises from a high temperature horizon to a low temperature one. Such multi-horizon spacetime can not be understood to be in any equilibrium state, since, exactly speaking, no energy flow arises in thermal equilibrium states. The thermal equilibrium of total system (multi-horizon spacetime) is realized if and only if the temperatures of all constituent horizons are equal. Therefore, when the temperatures of horizons are not equal, the multi-horizon spacetime should be understood as it is in a non-equilibrium state. Here it should be noticed that, generally in non-equilibrium physics, once the system under consideration comes in a non-equilibrium state, the equation of state for non-equilibrium case takes different form in comparison with that for equilibrium case. Especially the non-equilibrium entropy deviates from the equilibrium entropy (when a non-equilibrium entropy is well defined). Indeed, although a quite general formulation of non-equilibrium thermodynamics remains unknown at present, the difference of non-equilibrium entropy from equilibrium one is already revealed for some restricted class of non-equilibrium systems [8, 9]. Hence, for multi-horizon spacetimes composed of horizons of different temperatures, it seems to be reasonable to expect the breakdown of entropy-area law. However, since a “non-equilibrium thermodynamics” applicable to multi-horizon spacetime has not been constructed at present, we need to make use of “equilibrium thermodynamics” to investigate thermodynamic properties of multi-horizon spacetimes.

Motivated by the above consideration, this paper treats Schwarzschild-de Sitter (SdS) spacetime as the representative of multi-horizon spacetimes. We construct two thermal equilibrium systems in SdS spacetime; one of them is for black hole event horizon (BEH) and another is for cosmological event horizon (CEH). Note that, since the temperature of BEH is always higher than that of CEH in SdS spacetime [10], we need a good way to obtain thermal equilibrium systems of BEH and CEH. As will be explained in detail in Sec.2, we will adopt the same way of constructing two thermal equilibrium systems as Gibbons and Hawking have used in calculating the Hawking temperatures of BEH and CEH [10]; it is to introduce a thin wall between BEH and CEH which reflects perfectly the Hawking radiation coming from the horizons. The region enclosed by the wall and BEH (CEH) settles down to a thermal equilibrium state, and we obtain two thermal equilibrium systems separated by the perfectly reflecting wall. Then we will examine the entropy-area law for the two thermal equilibrium systems individually. (Although we are motivated by a non-equilibrium thermodynamic consideration in previous paragraph, the whole analysis in this paper is based on equilibrium thermodynamics and we discuss the two equilibrium thermodynamics for BEH and CEH individually.) As will be explained in Sec.2, our two thermal systems are treated in the canonical ensemble to obtain the free energies of BEH and CEH. Hence we will make use of the Euclidean action method which is regarded as one technique to obtain the partition function of canonical ensemble of quantum gravity [11]. (See Appendix A of this paper or of the previous paper [6].) Then we will find that the free energies are functions of three independent state variables. The existence of three independent state variables for SdS spacetime was not recognized in existing works on multi-horizon spacetimes [10, 12, 13, 14, 15]. But in this paper, thermodynamically rigorous analysis with three independent state variables will suggest a reasonable evidence of breakdown of entropy-area law for CEH. The validity of the law for BEH will not be judged, but we will clarify the key issue for BEH’s entropy. These results imply that the thermal equilibrium of each horizon may not be the necessary and sufficient condition but simply the necessary condition of entropy-area law. The necessary and sufficient condition of the law may be implied via some existing works as noted below:

Let us note that some proposals for thermodynamics of BEH and CEH in SdS spacetime are already given to a case with some special matter fields and for an extreme case with magnetic/electric charge [12]. These examples are artificial to vanish the temperature difference of BEH and CEH, and show that the entropy-area law holds for SdS spacetime if the temperatures of BEH and CEH are equal. However, in this paper, we consider a more general case which is not extremal and does not include artificial matter fields. In all analyses in this paper, the temperatures of BEH and CEH remain different and the discussions in those examples [12] can not be applied. If we find the breakdown of entropy-area law for the case that horizons have different temperatures, then it is suggested that the necessary and sufficient condition of entropy-area law is the thermal equilibrium of the total system composed of several horizons in which the net energy flow among horizons disappears.

Next let us make comments on the case that horizons have different temperatures. The construction of SdS thermodynamics with leaving horizon temperatures different has already been tried in some existing works [10, 13, 14, 15]. Those works assume some geometrical conserved quantities to be state variables of SdS spacetime, and derive the so-called mass formula which is simply a geometrical relation and looks similar to the first law of black hole thermodynamics. However, the thermodynamic consistency has not been confirmed in those works. Here “thermodynamic consistency” means that the state variables satisfy not only the four laws of thermodynamics but also the appropriate differential relations; for example, the differential of free energy with respect to temperature is equivalent to the minus of entropy,  333 There are many other similar differential relations in thermodynamics. Those relations are the ones required in the “thermodynamic consistency”, and necessary to understand thermodynamic properties of the system under consideration; e.g. phase transition, thermal and mechanical stabilities, and equations of states. . If one asserts some theoretical framework to be a “thermodynamics”, the framework must satisfy the thermodynamic consistency. Therefore, exactly speaking, it remains unclear whether those existing works [10, 13, 14, 15] are appropriate as “thermodynamics” 444 Those existing works [10, 13, 14, 15] preserve/assume the entropy-area law without confirming thermodynamic consistency. Hence, if the breakdown of entropy-area law is concluded via the thermodynamic consistency, those existing works can not be regarded as “thermodynamic” theory. . On the other hand, it seems to be preferable that the number of assumptions for thermodynamic formulations of BEH and CEH is as small as possible. In order to introduce the minimal set of assumptions which preserves thermodynamic consistency, we will refer to Schwarzschild thermodynamics formulated by York [4] (and also refer to de Sitter thermodynamics [6] which is also based on York [4]).

Furthermore, motivated by the dS/CFT correspondence conjecture [16], some existing works [14] focus their attention on the future and past null infinities in SdS spacetime (see Fig.1 shown in Sec.2, is the null infinities). Those infinities may be appropriate to discuss some geometrical quantities. However, as implied by the causal structure of SdS spacetime, the future null infinity seems to be inappropriate to discuss thermodynamic properties of BEH and CEH, because any observer near future null infinity (not near the future temporal infinity ) can not “access” BEH 555 The observer going towards the future temporal infinity in Fig.1 can “access” BEH, since the BEH becomes a boundary of the causally connected region of that observer (the region I in Fig.1). .

Hence, contrary to the existing works, the analysis in this paper is based on the following two points:

  • As will be explained precisely in next section, we focus our attention on the region enclosed by BEH and CEH (not on null infinity) in SdS spacetime as the object of thermodynamic interests.

  • We have a high regard to the “thermodynamic consistency” preserved by the minimal set of assumptions without referring to some geometrical conserved quantities and dS/CFT correspondence.

Then, as the result of these two points, a suggestion of breakdown of entropy-area law will be obtained.

Here let us emphasize that, in next section, we exhibit explicitly the assumptions on which our discussion is based. We think readers can judge the approval or disapproval to every part of our assumptions and analyses. Therefore, even if some part of our discussion and analysis is not acceptable for some reader, we hope this paper can propose one possible issue about the universality of entropy-area law.

This paper is organized as follows: Sec.2 introduces the minimal set of assumptions in which, with referring to Schwarzschild and de Sitter canonical ensembles [4, 6], the two thermal equilibrium systems for BEH and CEH are constructed, and the special role of the cosmological constant is also pointed out. Sec.3 is for the calculation of Euclidean actions of the two thermal equilibrium systems. Sec.4 discusses thermodynamics for BEH and examines the entropy-area law for BEH. In that section, the validity of the law for BEH can not be judged, but the key issue for BEH’s entropy is clarified. Sec.5 proposes a reasonable evidence of the breakdown of entropy-area law for CEH. Finally Sec.6 is for summary and discussions, in which we make some physical comments which may suggest the breakdown of entropy-area law for BEH without rigorous verification, and propose two discussions; one of them is on the quantum statistics of underlying quantum gravity, and another is on the SdS black hole evaporation process from the point of view of non-equilibrium thermodynamics. And four appendices support the main text of this paper: Appendix A and B summarize, respectively, the Euclidean action method and the essence of York’s Schwarzschild thermodynamic, Appendix C exhibits useful differential formulas for calculations of thermodynamic state variables, and Appendix D analyzes the Nariai limit (extremal limit) of our SdS thermodynamics. Throughout this paper we use the Planck units, .

2 Minimal set of assumptions

2.1 Preliminary

Let us dare to start this section with the discussion given already in de Sitter canonical ensemble [6], because this discussion is conceptually essential for thermodynamic consistency.

The aim of this section becomes clear by considering the relation between thermodynamics and statistical mechanics [17, 18]. In statistical mechanics, the partition function can not be expressed as a “function of state variables” unless the appropriate state variables, on which the partition function depends, are specified a priori. To understand this, consider for example an ordinary gas in a spherical container of radius , in which the number of constituent particles is , the mass of one particle is and the mean velocity of particles is . The ordinary statistical mechanics, without the help of thermodynamics, yields the partition function as simply a function of “parameters”, , , and . Statistical mechanics, solely, can not determine what combinations of those parameters behave as state variables. To determine it, the first law of thermodynamics is necessary. (Note that the notion of heat in the first law is established by purely the argument in thermodynamics, not in statistical mechanics.) Comparing the differential of partition function with the first law results in the identification of partition function with the free energy divided by temperature. Then, since the free energy of ordinary gases is a function of the temperature and volume due to the “thermodynamic” argument, the partition function should be rearranged to be a function of temperature and volume , where and for ideal gases due to the law of equipartition of energy 666 When the number of particles changes by, for example, a chemical reaction and an exchange of particles with environment, is also the state variable on which the free energy depends. . (The dependence on is, for example, for ideal gases.)

Here note that the reason why the temperature and volume are regarded as the state variables of the gas is that they are consistent with the four laws of “thermodynamics” and have the appropriate properties as state variable. The appropriate properties are that the state variables are macroscopically measurable, the state variables are classified into two categories, intensive variables and extensive variables, and the extensive variables are additive. Those properties of state variables are specified by purely the argument in thermodynamics, not in statistical mechanics. Therefore, from the above, it is recognized that statistical mechanics can not yield the partition function as a “function of appropriate state variables” without the help of thermodynamics which specifies the appropriate state variables for the partition function.

Turn our discussion to the Euclidean action method for curved spacetimes. Since the Euclidean action method is the technique to obtain the “partition function” of the spacetime under consideration (see Appendix A), it is necessary to specify the state variables before calculating the Euclidean action. In this section, referring to Schwarzschild and de Sitter canonical ensembles [4, 6], we introduce the minimal set of assumptions for SdS thermodynamics, which specify the appropriate state variables for the partition function. Also, the special role of cosmological constant is clarified, which is already found in previous works [6, 15]. The calculation of Euclidean action is carried out not in this section but in next section.

2.2 SdS spacetime

Before introducing the minimal set of assumptions, let us summarize the Lorentzian SdS spacetime in order to prepare some quantities used in the following discussions.

The metric of SdS spacetime in the static chart is

(1)

where is the line element on the unit two-sphere, and

(2)

where is the mass parameter of black hole and is the cosmological constant. The Penrose diagram of SdS spacetime is shown in Fig.1, and the static chart covers the region I.

An algebraic equation has one negative root and two positive roots. The smaller and larger positive roots are, respectively, the radius of BEH and that of CEH . The notion of CEH is observer dependent and the CEH at is associated with the observer going towards the temporal future infinity in region I [10]. The equation is rearranged to , where . Then via a formula, , we get

(3)

where is defined by, . The existence condition of BEH and CEH is . This is equivalent to, , which means

(4)

This denotes that is larger than the Schwarzschild radius and is smaller than the de Sitter’s CEH radius .

Figure 1: Penrose diagram of SdS spacetime. are the future/past null infinity, is the future temporal infinity. BEH is at , and CEH at . Spacetime singularity is at . Static chart covers the region I. Semi-global black hole chart covers the regions I, II, III and IV. Semi-global cosmological chart covers the regions I, II, III and IV. The maximal extension is obtained by connecting the two semi-global charts alternately.

SdS spacetime has a timelike Killing vector , where is a normalization constant [19]. This becomes null at BEH and CEH. This means those horizons are the Killing horizons of . The surface gravity of BEH and that of CEH are defined by the equations, , . The surface gravity depends on . Throughout this paper we take the normalization for BEH, and for CEH to make positive 777Even if for CEH, we can keep consistency of our analysis by changing the signature of appropriately.. Then the surface gravities become equal to the absolute value at each Killing horizon,

(5)

where Eq.(3) is used in the second equality in each equation. From the inequality in Eq.(4), we get

(6)

This implies the Hawking temperature of BEH is higher than that of CEH, which will be verified by Eqs.(43) and (62).

For later use, let us show some differentials,

(7a)
(7b)
and
(7c)
(7d)

where we used a formula, , and the differentials, , and , obtained from the definition of , .

The metric in semi-global black hole chart is given by the coordinate transformation from to :

(8)

where which means

(9)

We get by this transformation,

(10)

where

(11)

The transformation (8) implies the range of coordinates, and , which covers the region I in Fig.1. By extending to the range, and , the semi-global black hole chart covers the regions I, II, III and IV in Fig.1. In these regions we find since .

The metric in semi-global cosmological chart is given by the coordinate transformation from to :

(12)

where is given in Eq.(9). By this transformation we get

(13)

where

(14)

The transformation (12) implies the range of coordinates, and , which covers the region I in Fig.1. By extending to the range, and , the semi-global cosmological chart covers the regions I, II, III and IV in Fig.1. In these regions we find since .

The maximally extended SdS spacetime is obtained by connecting the two semi-global charts alternately as shown in Fig.1.

2.3 Minimal set of assumptions and working hypothesis

As mentioned in Sec.2.1, we introduce the minimal set of assumptions with referring to Schwarzschild thermodynamics formulated by York [4]. Those assumptions should construct thermal systems for SdS thermodynamics and give us enough state variables which appear as independent variables in the free energy of our thermal system. There are three key points in Schwarzschild thermodynamics from which we can learn about the way to ensure the “thermodynamic consistency” in SdS thermodynamics. Those key points are the same from which the basic assumptions of de Sitter thermodynamics are introduced in previous paper [6], and we summarize those three key points in Appendix B of this paper.

Here we must comment on Anti-de Sitter (AdS) black holes [7]. AdS black hole thermodynamics has a conceptual difference from the other black hole thermodynamics. The difference appears, for example, in the definition of temperature. While the temperatures in asymptotic flat black hole and de Sitter thermodynamics include the Tolman factor [4, 5, 6, 20], the temperature assigned to AdS black hole [7] does not include the Tolman factor, where the Tolman factor [20] expresses the gravitational redshift affecting the Hawking radiation propagating from horizon to observer (see for example Eq.(43) in Sec.4.1 and the key point 3 of Schwarzschild thermodynamics in Appendix B). The temperature in AdS black hole thermodynamics can not be measured by a thermometer of the physical observer who are outside the black hole. This implies that the state variables in AdS black hole thermodynamics are defined not by the observer outside the black hole, but defined just on the black hole event horizon on which no physical observer can rest. In this paper we do not refer to AdS black hole thermodynamics, since it seems to be preferable to expect that state variables are defined according to a physical observer. Hence we refer to Schwarzschild thermodynamics, which is based on a physical observer (See Appendix B).

2.3.1 Zeroth law and independent variables

We will construct two thermal equilibrium systems for BEH and CEH, but place only one observer who can measure the state variables of BEH and CEH. Such observer is in the region I, (see Fig.1). However, as mentioned at Eq.(6), Hawking temperature of BEH is higher than that of CEH. This temperature difference implies that, when the region I constitutes one thermodynamic system, the thermodynamic state of region I is in a non-equilibrium state. Therefore, by dividing the region I into two regions, we construct two thermal equilibrium systems for BEH and CEH individually which are measured by the same observer. To do so, we adopt the following assumption as the zeroth law:

Assumption 1 (Zeroth law)

Two thermal “equilibrium” systems for BEH and CEH in SdS spacetime are constructed by the following three steps:

  1. Place a spherically symmetric thin wall at in the region I, , as shown in Fig.2. This wall has negligible mass, and reflects perfectly Hawking radiation coming from each horizon. We call this wall the “heat wall” hereafter. The BEH side of heat wall is regarded as a “heat bath” of Hawking temperature of BEH due to the reflected Hawking radiation. Also the CEH side of heat wall is regarded as a heat bath of Hawking temperature of CEH.

  2. The region enclosed by BEH and heat wall, , is filled with Hawking radiation emitted by BEH and reflected by heat wall, and forms a thermal equilibrium system for BEH. Similarly the region enclosed by CEH and heat wall, , forms a thermal equilibrium system for CEH. Hence we have “two” thermal equilibrium systems separated by the heat wall. In the statistical mechanical sense, these two thermal equilibrium systems are described by the canonical ensemble, since those systems have a contact with the heat wall.

  3. Set our observer at the heat wall. When the observer is at the BEH side of heat wall, the observer can measure the state variables of thermal equilibrium system for BEH. The same is true of CEH. Then the state variables of two thermal equilibrium systems are defined by the quantities measured by the observer at heat wall.

Note that the two thermal equilibrium systems constructed in this assumption have already been used by Gibbons and Hawking [10] to calculate Hawking temperatures of BEH and CEH. Also the above step 3, which gives a criterion of defining state variables, has already been adopted in the consistent thermodynamics of single-horizon spacetimes [4, 5, 6]. We can regard this assumption as a simple extension of the key point 1 of Schwarzschild thermodynamics shown in Appendix B.

Figure 2: Our thermal equilibrium systems for BEH () and CEH (). State variables of them are defined at the heat wall.

It is expected that state variables of the thermal equilibrium system for BEH depend on BEH radius and/or BEH surface gravity . Similarly, state variables of CEH depend on and/or . These imply that the state variables of BEH and CEH depend on and , since the horizon radii and surface gravities depend on and via Eqs.(3) and (5). Furthermore, by the step 3 in assumption 1, there should be -dependence in state variables of BEH and CEH, since the observer is at . Therefore the state variables of BEH and CEH depend on three parameters , and .

The existence of three parameters may imply that the CEH is regarded as a source of external gravitational field which affects the thermodynamic state of BEH. Also, BEH is a source of external gravitational field affecting the thermodynamic state of CEH. Here it is instructive to compare qualitatively our two thermal equilibrium systems of horizon with a magnetized gas. The gas consists of molecules possessing a magnetic moment, and its thermodynamic state is affected by an external magnetic field. The qualitative correspondence between the magnetized gas and our thermal equilibrium systems of horizon is described as follows; the gas corresponds to the system  (), and the external magnetic field corresponds to the external gravitational field produced by CEH (BEH). Then, what we should emphasize is the following fact of the magnetized gas: When the gas is enclosed in a container of volume and an external magnetic field is acting on the gas, the free energy of the gas is expressed as a function of three independent state variables (see for example §52, 59 and 60 in Landau and Lifshitz [18]),

(15)

where is the temperature of the gas. According to this fact of the magnetized gas, it is reasonable for our two thermal equilibrium systems of horizons to require that the free energies are also functions of three independent state variables. For the BEH, the free energy is

(16)

where is the temperature of BEH, is the state variable of system size, and is the state variable which represents the effect of CEH’s gravity on the BEH. And for the CEH, the free energy is

(17)

where is the temperature of CEH, is the state variable of system size and is the state variable which represents the effect of BEH’s gravity on the CEH. Indeed, it will be proven in Secs. 4.1 and 5.1 that the thermodynamic consistency never hold unless the free energies are functions of three independent variables as shown in Eqs.(16) and (17).

Now we recognize that, because free energies are functions of three independent variables (as will be verified in Secs. 4.1 and 5.1), the following working hypothesis is needed:

Working Hypothesis 1 (Three independent variables)

To ensure the thermodynamic consistency of our thermal equilibrium systems constructed in assumption 1, we have to regard the cosmological constant as an independent working variable. Then the three quantities are regarded as independent variables, and consequently the free energies and of our thermal equilibrium systems are functions of three independent state variables as shown in Eqs.(16) and (17). On the other hand, when we regard the non-variable as the physical situation, it is obtained by the “constant process” in the consistent thermodynamics for BEH and CEH which are constructed with regarding as a working variable.

This working hypothesis will be verified in Secs. 4.1 and 5.1, and we can not preserve thermodynamic consistency without regarding as an independent working variable. This implies that, as already commented in Sec.V of previous paper [6], the thermal equilibrium states of event horizon with positive may construct the “generalized” thermodynamics in which behaves as a working variable and the physical process is described by the constant process.

2.3.2 Scaling law and system size

In thermodynamics of ordinary laboratory systems, all state variables are classified into two categories, extensive state variables and intensive ones. The criterion of this classification is the scaling behavior of state variables under the scaling of system size. However, as explained by the key point 2 of Schwarzschild thermodynamics shown in Appendix B, the state variables in thermodynamics of single-horizon spacetimes [4, 5, 6] have its own peculiar scaling behavior classified into three categories, and the state variable of system size is not a volume but the surface area of heat bath. Although the scaling behavior differs from that in thermodynamics of ordinary laboratory systems, the peculiar scaling behavior in thermodynamics of single-horizon spacetimes retains the thermodynamic consistency as explained in the key point 3 in Appendix B. Then, we assume that the key point 2 of Schwarzschild thermodynamics is simply extended to our two thermal equilibrium systems constructed in the assumption 1:

Assumption 2 (Scaling law and system size)

All state variables of our thermal equilibrium systems are classified into three categories; extensive variables, intensive variables and thermodynamic functions, and satisfy the following scaling law: When the length size (e.g. horizon radius) is scaled as with an arbitrary scaling rate , then the extensive variables (e.g. system size) and intensive variables (e.g. temperature) are scaled respectively as and , while the thermodynamic functions (e.g. free energy) are scaled as . This implies that the thermodynamic system size of our thermal equilibrium systems should have the areal dimension, since the system size is extensive in thermodynamic argument. Then we assume that the surface area of heat wall, , behaves as the consistent extensive variable of system size for our thermal equilibrium systems for BEH and CEH. This denotes to set in Eqs.(16) and (17).

Accepting this assumption, the length size scaling in our thermal equilibrium systems for BEH and CEH should be specified. Here recall that, due to the working hypothesis 1, the fundamental independent variables in our thermal equilibrium systems are , and . Therefore the fundamental length size scaling for our thermal equilibrium systems is composed of the following three scalings;

(18)

where is defined by in Eq.(2), and is an arbitrary scaling rate. The extensivity and intensivity of each state variable of our thermal equilibrium systems should be defined under these fundamental length size scalings as explained in the assumption 2. 888 As explained in Appendix B in previous paper [6], when we regard as a state variable of system size, the scaling of system size should be restricted to the homothetic one, which is the spherical scaling due to the spherical symmetry of SdS spacetime. The fundamental length size scaling (18) is consistent with this restriction. See Appendix B in previous paper [6] for details of such restriction.

2.3.3 Euclidean action method and how to obtain state variables

We need to specify how to get the state variables. As noted in the step 2 in assumption 1, thermodynamics of our two thermal equilibrium systems should be constructed in the canonical ensemble. Therefore we use the Euclidean action method which is the technique to calculate the partition function of quantum gravity [11]. Indeed, the Euclidean action method has already made successes to obtain the partition function of canonical ensemble for the thermodynamics of single-horizon spacetimes [4, 5, 7, 6]. The key point 3 in Appendix B explains how to use the Euclidean action and to define state variables for Schwarzschild thermodynamics. Then, referring to the key point 3, we adopt the following assumption:

Assumption 3 (State variables and Euclidean action method)

Euclidean actions and of our two thermal equilibrium systems yield the partition functions of canonical ensembles by Eq.(80) of Appendix A. And the free energies and are defined by Eq.(82), where the temperatures are defined by Eq.(81). Then, once and are determined, all state variables of BEH and CEH are defined from and as for thermodynamics of ordinary laboratory systems. For example, BEH entropy is defined by , where is the temperature of BEH.

As seen in Eq.(79), the Euclidean action is produced from the Lorentzian action. The Lorentzian Einstein-Hilbert action is

(19)

where is the spacetime region under consideration, is the Ricci scalar, is the determinant of metric, and in the second term are respectively the determinant of first fundamental form (induced metric) and the trace of second fundamental form (extrinsic curvature) of the boundary , and is the integration constant of and called the subtraction term. The second term in Eq.(19) is required to eliminate the second derivatives of metric from the action [21]. The is independent of the metric in , and does not contribute to the Einstein equation obtained by . The Einstein equation for SdS spacetime gives the relation .

When we use the Euclidean action method, it is necessary to specify the integration constant . It is natural to require that our thermal equilibrium systems for BEH and CEH should reproduce, respectively, the Schwarzschild thermodynamics in the limit and the de Sitter thermodynamics in the limit . Then the following working hypothesis is naturally required:

Working Hypothesis 2 (Integration constants in Euclidean action)

For the thermal equilibrium system for BEH, the integration constant in Euclidean action is determined with referring to Schwarzschild canonical ensemble formulated by York [4]. For the thermal equilibrium system for CEH, the integration constant in Euclidean action is determined with referring to de Sitter canonical ensemble formulated in previous paper [6].

We introduce this working hypothesis as if this is a statement separated from the assumption 3. However the determination of integration constant accompanies necessarily the Euclidean action method. The working hypothesis 2 is regarded as a part of the assumption 3.

2.3.4 Effects of external gravitational fields

By the assumption 3 together with the working hypothesis 2, the concrete functional form of free energies and can be determined as functions of three independent working variables . However, since the form of the state variables and have not been specified yet, we can not rearrange and to functions of independent state variables, and .

As explained at Eqs.(16) and (17), the CEH (BEH) is regarded as the source of external gravitational field which affects the thermodynamic state of BEH (CEH). This means that the state variables and represent, respectively, the thermodynamic effect of CEH’s gravity on BEH and that of BEH’s gravity on CEH. Then it is reasonable to expect that depends on the quantity characterizing the CEH’s gravity, and depends on the quantity characterizing the BEH’s gravity. Moreover, due to the step 3 in assumption 1, and should be measurable for the observer at . Then, we can offer two candidates for the pair of dimensionless characteristic quantities of BEH’s and CEH’s gravities;

  • First candidate pair consists of and , where is for BEH’s gravity and is for CEH’s gravity. This pair means that both of BEH’s and CEH’s gravities are characterized by three quantities , since and depend on and .

  • Second candidate pair consists of and , where is for BEH’s gravity and is for CEH’s gravity. This pair means that the BEH’s gravity is characterized by , and the CEH’s gravity is characterized by .

Here, purely logically, we can consider the “inverse” pair of second one, where is for BEH’s gravity and is for CEH’s gravity. This means that the BEH’s gravity is characterized by , and the CEH’s gravity is characterized by . However this is physically unacceptable, since we do not expect that the BEH does not depend on and the CEH does not depend on . Therefore, the reasonable candidates for the pair of characteristic quantities of BEH’s and CEH’s gravities are the two candidates listed above. Then, should be a function of or , and should be a function of or .

On the other hand, as will be mathematically verified in Secs.4.3 and 5.3, the state variables and are the extensive variables and proportional to . The proportionality to is consistent with the scaling law of extensive variables denoted in assumption 2. And, according to the previous paragraph, the factor of proportionality should be a function of the characteristic quantity of BEH’s or CEH’s gravity. Although the verification of the extensivity of and are shown later, we accept it in the following assumption 4 for the simplicity of our discussion.

From the above, it is reasonable to adopt the following assumption:

Assumption 4 (Extensive variable of “external field”)

The state variables and in Eqs.(16) and (17) are the extensive variables. (This will be verified in Secs.4.3 and 5.3). Then, there are two candidates for the functional forms of and . One of them is based on the quantities :

(20)

where and are arbitrary functions of single argument, whose functional forms are not specified at present. Another candidate of and is based on the quantities :

(21)

At least for the present author, no criterion to choose one of these candidates is found, and the way for determining the functional forms of and are also unknown. An obvious constraint on and is that they never be constant to make and independent of the state variable of system size .

In Sec.6.1, we will make some comments on the issue which of Eqs.(20) and (21) is valid. Those comments will suggest that Eq.(20) may be appropriate, but we do not have mathematical verification to choose Eq.(20) as the general form of and . Therefore, to retain the logical strictness of this paper, we list the two possibilities (20) and (21) in the assumption 4.

From the above, we recognize that the minimal set of assumptions for “consistent” thermodynamics of our two thermal equilibrium systems should be composed of four assumptions. However, the determination of functions and remains as a future task and we can not find concrete functional forms of them. Although the state variables and are not specified in this paper, the existence of them enables us to examine the validity of entropy-area law in SdS spacetime as shown in Secs. 4 and 5.

3 Euclidean actions

Referring to the assumption 3 and working hypothesis 2, we calculate Euclidean actions for the two thermal equilibrium systems for BEH and CEH constructed in the assumption 1.

3.1 Euclidean action for BEH

Euclidean space of thermal equilibrium system for BEH is obtained by the Wick rotations in the static chart and in the semi-global black hole chart. These Wick rotations are equivalent, because the coordinate transformation (8), , implies that the imaginary time in the semi-global chart is defined by , where is the imaginary time in the static chart. Euclidean metric in the static chart is

(22)

Euclidean metric in the semi-global black hole chart is

(23)

where is defined in Eq.(11). About the semi-global chart, we get from the coordinate transformation (8),

(24)

Then, because the thermal equilibrium system for BEH is the region , , in Lorentzian SdS spacetime, we find the topology of Euclidean space of thermal equilibrium system for BEH is . Because at , the center of -part is at the BEH , and the boundary of -part is at the heat wall . The topology of heat wall boundary is , where is along the -direction.

Because the Lorentzian SdS spacetime is regular at , the Euclidean space is also regular at . To examine the regularity of Euclidean space at , we make use of the static chart (22). Let us define a coordinate and a function by

(25)

We get , from which we find

(26)

This means , and near the BEH, . Therefore the Euclidean metric near BEH is

(27)

It is obvious that the Euclidean space is regular at BEH if the imaginary time has the period defined by

(28)

Throughout our discussion, has the period in the Euclidean space of thermal equilibrium system for BEH.

Now let us proceed to the calculation of the Euclidean action of thermal equilibrium system for BEH. Following the working hypothesis 2, we use the same integration constant as Schwarzschild canonical ensemble [11, 4], which gives us

(29)

where is the Lorentzian Einstein-Hilbert action for flat spacetime which reduces to only the surface term (the second term in Eq.(19) ) due to and for flat spacetime, and is the trace of second fundamental form of in flat spacetime [11, 4]. Here note that, the integral element in should be given by that of SdS spacetime when is used as the integration constant of action integral of SdS spacetime, because the background spacetime on which the integral in is calculated is SdS spacetime.

For our thermal equilibrium system for BEH in SdS spacetime, the region in is , , and its boundary is at . There is another boundary at in the Lorentzian region . However we do not need to consider it, because the points at in the Euclidean space do not form a boundary but are the regular points when has the period (28). Then the first fundamental form () of in the static chart is

(30)

where

(31)

Here, since is the region enclosed by BEH and heat wall, the direction of unit normal vector to is pointing towards CEH, . Then the second fundamental form of in the static chart is

(32)

where diag. means the diagonal matrix form.

On the other hand, the second fundamental form of a spherically symmetric timelike hypersurface of radius in flat spacetime is given by setting and in Eq.(32),

(33)

This gives .

From the above, applying the Wick rotation to the Lorentzian action in Eq.(19), we obtain the Euclidean action of the thermal equilibrium system for BEH via Eq.(79),

(34)

where the relation for SdS spacetime is used in the first equality, the relation due to is used in the second equality, is the quantity evaluated on Euclidean space, and is the Euclidean region denoted by , , and . This corresponds to in Eq.(82) of Appendix A, which yields the partition function of our thermal equilibrium system for BEH.

Note that should reproduce the Euclidean action of Schwarzschild canonical ensemble as required in the working hypothesis 2. To check if this is satisfied, take the limit ,

(35)

This coincides with the Euclidean action of Schwarzschild canonical ensemble formulated by York [4].

3.2 Euclidean action for CEH

Euclidean space of thermal equilibrium system for CEH is obtained by the Wick rotations in the static chart and in the semi-global cosmological chart. These Wick rotations are equivalent, because the coordinate transformation (12), , implies that the imaginary time in the semi-global chart is defined by , where is the imaginary time in the static chart. Euclidean metric in the static chart is given by Eq.(22). Euclidean metric in the semi-global cosmological chart is , where is defined in Eq.(14). About the semi-global chart, we get from the coordinate transformation (12),

(36)

Then, because the thermal equilibrium system for CEH is the region , , in Lorentzian SdS spacetime, we find the topology of Euclidean space of thermal equilibrium system for CEH is . Because at , the center of -part is at the CEH , and the boundary of -part is at the heat wall . The topology of heat wall boundary is , where is along the -direction.

Because the Lorentzian SdS spacetime is regular at , the Euclidean space is also regular at . Using the static chart (22) and defining a coordinate and a function by and , we obtain the Euclidean metric near CEH,

(37)

It is obvious that the Euclidean space is regular at CEH if the imaginary time has the period defined by

(38)

Throughout our discussion, has the period in the Euclidean space of thermal equilibrium system for CEH.

Now we proceed to the calculation of Euclidean action of the thermal equilibrium system for CEH. Lorentzian action is defined in Eq.(19) for the spacetime region , . The calculation of Euclidean action is parallel to that of except for the direction of unit normal vector to and the integration constant in . Concerning the vector , since is the region enclosed by CEH and heat wall, the direction of is pointing towards BEH, .

Concerning the integration constant, following the working hypothesis 2, we determine the integration constant for CEH with referring to the de Sitter canonical ensemble formulated in previous paper [6]. At the limit , the term should reduce to the integration constant of the de Sitter canonical ensemble,

(39)

where is the action of flat spacetime used in Eq.(29). Note that the CEH radius in de Sitter spacetime is , and the factor is the ratio of heat wall radius to CEH radius. Hence we set for the CEH in SdS spacetime,

(40)

where it should be emphasized that the signature of in the integrand of (shown in Eq.(29)) should be reversed, because the direction of normal vector is reversed as mentioned in previous paragraph.

Then we obtain the Euclidean action of our thermal equilibrium system for CEH via Eqs.(19) and (79),

(41)

where the relation due to is used, and the overall minus signature comes from the direction of normal vector to . This corresponds to in Eq.(82), which yields the partition function of our thermal equilibrium system for CEH.

Note that should reproduce the Euclidean action of de Sitter canonical ensemble as required in the working hypothesis 2. To check if this is satisfied, take the limit ,

(42)

This coincides with the Euclidean action of de Sitter canonical ensemble formulated in previous paper [6].

4 Black hole event horizon

We examine whether the entropy-area law holds for “consistent” thermodynamics of our thermal equilibrium system for BEH.

4.1 Temperature and free energy of BEH

By the assumption 3, the temperature of BEH is defined by Eq.(81) of Appendix A, which relates to the proper length in the imaginary time direction at the boundary (the direction along part of boundary topology in Euclidean space),

(43)

where is the imaginary time period (28) and is in Eq.(31). Under the length size scaling (18), this temperature is scaled as . Therefore, by the assumption 2, is an intensive state variable of BEH.

Note that this coincides with the Hawking temperature of BEH derived originally by Gibbons and Hawking [10], and the factor is the so-called Tolman factor [20] which expresses the gravitational redshift affecting the Hawking radiation propagating from BEH to observer at . Therefore this is the temperature measured by the observer at heat wall.

By the assumption 2, the extensive state variable of system size for our thermal equilibrium system is the surface area of heat wall,

(44)

By the assumption 3, the free energy of BEH in Eq.(16) is defined by Eq.(82),

(45)

Under the length size scaling (18), this free energy satisfies the scaling law of thermodynamic functions, . As discussed at Eq.(16), is regarded as a function of three independent state variables , and the state variable of CEH’s gravitational effect on BEH. However, since is not specified as mentioned in the assumption 4, the form of as a function of remains unknown. Instead, Eq.(45) shows as a function of independent parameters .

Let us verify that the free energy is a function of three independent state variables. We use the reductive absurdity: Assume that only two (not three) state variables are independent. This assumption means that, as for the ordinary non-magnetized gases, is a function of and , . Here it is obvious from Eq.(43) that depends on three parameters , while depends only on . Then, via Eq.(90) of Appendix C, a mathematical relation must hold if is a function of . However we find this relation does not hold, , via Eq.(47) shown below. Hence the assumption of two independent state variables is denied by the reductive absurdity. Now the working hypothesis 1, which assumes to be a function of three independent state variables, is verified.

To support the discussion in previous paragraph and for later use, we show some differentials:

(46a)