Thermodynamics of the Einstein-Maxwell system

At first glance, thermodynamic properties of gravity with asymptotically AdS conditions and those with box boundary conditions, where the spatial section of the boundary is a sphere of finite radius, appear similar. Both exhibit a similar phase structure and Hawking-Page phase transition. However, when we introduce a U(1) gauge field to the system, discrepancies in thermodynamic properties between the two cases has been reported in [7] (JHEP 11 (2016) 041). In this paper, by accepting the assumption that all Euclidean saddles contribute to the partition function, I found that these discrepancies are resolved due to the contribution from the"bag of gold (BG),"which is the class of Euclidean geometries whose area of bolt is bigger than that of the boundary. As a result, the Hawking-Page phase structure is restored, with the unexpected properties that the upper bound of thermodynamic entropy is always larger than the boundary area divided by 4G when the chemical potential is non-zero, and that such high entropy states are realized at sufficiently high temperature.


Introduction
It is often said that BH is thermodynamically unstable, but becomes stable when enclosed by a "box".There are two ways of realizing this; one is to impose an asymptotic AdS condition and the other is to impose a box boundary condition, i.e. we actually place a sphere boundary of finite radius.In terms of gravitational thermodynamics, the BH itself is not a thermodynamic system, but is realized as a thermodynamically stable phase in the gravitational system enclosed by the "box".The thermodynamic properties are studied in [1] for the AdS boundary and in [2] for the box boundary (for the case with cosmological constant see [3, 4, 5, 6]), and from their works we know that both cases have similar thermodynamic properties; the empty (or thermal gas) phase is realized at low temperature, the BH phase is realized at high temperature, and there is a phase transition called Hawking-Page phase transition (Fig. 1).Since both cases have a "box" boundary and exhibit similar thermodynamic properties for the case of pure gravity, one might expect this kind of similarity to hold when we add some other fields to the system.However, it has been reported that in the Einstein-Maxwell system there is some discrepancy between them [7].Let's focus on the case of the grand canonical ensemble.For the AdS boundary condition, the system exhibits a phase structure similar to the pure gravity case, i.e. the behavior of the free energy is qualitatively the same as in pure gravity (for sufficiently low chemical potential µ) [8].See Fig. 1.On the other hand, for the box boundary condition, the dominant BH branch does not extend to infinity and terminates at some finite temperature (Fig. 2).The termination point is controlled by the value of the chemical potential µ (and the cosmological constant Λ and the radius of the box r b ).The only branch that exists beyond the temperature is the empty phase, and apparently the free energy has a discontinuity there.In this sense, thermodynamics itself is ill-defined for this system, since we cannot define thermodynamic quantities such as energy or electric charge there.
Figure 1: Qualitative behavior of free energies F for boxed gravity systems.F is defined by the on-shell action times the temperature.This behavior can be seen in the pure gravity system with AdS boundary, that with box boundary (for Λ ≤ 0), and the Einstein-Maxwell system with AdS boundary (for sufficiently low µ).A previous study [7] claimed that this cannot be seen in the Einstein-Maxwell system with box boundary and it behaves like Fig. 2.
Figure 2: Qualitative behavior of the free energy of the empty phase and the free energy of the BH phase in the Einstein-Maxwell system with box boundary.The BH branch terminates at some finite temperature depending on µ (and Λ and r b ).If there are no other dominant saddle points, the partition function would be a discontinuous function and one might say that the thermodynamics of the system is ill-defined.My claim in this paper is very simple: the bag of gold saves the day.(See Fig. 7).
In this paper, I want to resolve the problem of this ill-definedness of the Einstein-Maxwell system by considering "bag of gold (BG)" saddles, the class of Euclidean saddle point geometries whose area of a bolt is larger than that of the boundary [4]. 1 In the previous paper [4] I discussed the role of BG saddles in the thermodynamics of pure gravity with positive Λ.Including them in the path integral leads to the lack of thermodynamic stability and the entropy bound, which are the universal properties of pure gravity with Λ ≤ 0. In a sense, I have shown a bad aspect of the BG saddle.It breaks the seemingly universal properties of gravitational thermodynamics.If one believes that thermodynamic stability or the entropy bound must be universal, one may argue that the BG saddles should be excluded from the path integral, since we do not know which saddle points contribute to the Euclidean path integral of gravity [9]. 2 Here I will show a good aspect of the BG saddle by showing that it leads to the well-defined thermodynamic description of the Einstein-Maxwell system with box boundary condition and similarity to that with AdS boundary condition.The reason is simple.The BG branch appears and is smoothly connected to the BH branch and it extends to infinity (see Fig. 7 and compare with Fig. 2).And in fact, the BG branch is always thermodynamically stable, in contrast to the case of positive Λ.
Throughout this paper I use the standard path integral "definition" of the gravitational partition function and assume that the dominant saddle points are in the following class of metrics and U (1)  gauge fields on a manifold of B × S 1 topology or of S 2 × D topology: These are Euclidean solutions of the Einstein-Maxwell system. 4 The coordinate ranges and parameter values depend on the boundary condition.The fact that M and μ are identified with the total energy E and the chemical potential µ for the AdS (and asymptotically flat) boundary condition but not for the box boundary condition may be familiar.Less familiar is the relation between Q and the total charge Q. Normally Q can be identified with Q ( Q = Q).However, for the BG saddle, Q is not Q, but there is a minus sign (− Q = Q), as I will explain in Section 3.
The organization of this paper is as follows.In Section 2, I review the thermodynamics of the Einstein-Maxwell system with AdS boundary [8].In subsection 3.1, I review the properties of the BH phase branch in the Einstein-Maxwell system with box boundary without Λ, which are systematically studied in [7]. 5 I then study the properties of the BG phase in subsection 3.2 and claim that this leads to the well-defined thermodynamics in the system in subsection 3.3.In Section 4, I generalize the analysis of Section 3 to the case with a cosmological constant Λ.The Λ < 0 case is qualitatively the same as the Λ = 0 case.In the Λ > 0 case, since not only good BGs but also bad BGs seem to appear, one might expect the system to become thermodynamically unstable, as in the pure gravity case.This is certainly true for 0 < √ Gµ < 1.However, for the case of µ > 1, bad BGs no longer exist, but BHs do.As a result, the system is thermodynamically stable in this case.I summarize all the material in this paper in Section 5.
In the main part of this paper I will focus only on grand canonical ensembles, where the corresponding Euclidean action is given by [10, 12].6 However, since canonical and microcanonical ensembles are equally important, I investigate their properties in Appendix A and some of the results are also listed in Section 5.

AdS Boundary Condition
In this section, I briefly review the result of [8].The upshot is that there exists a critical value of chemical potential µ cr ; for µ < µ cr , there exist the AdS phase and the BH phase, and the system exhibits Hawking-Page phase transition, and for µ ≥ µ cr , there is only the BH phase.(Fig. 3) For the AdS boundary case, the saddle points are Euclidean AdS (B×S 1 topology) and Euclidean Reissner-Nordström(RN) AdS BH (S 2 × D topology).The coordinate range of r is [0, ∞) for the former and [r H , ∞) for the latter.The inverse temperature β is equal to the circumference of t.The free energy F (β, µ) is given by where saddle (β,µ) represents a saddle point which satisfies the following conditions; From these conditions, together with f (r H ) = 0, the parameters r H , M , Q, μ are determined.For some β, µ they are not uniquely determined.In this case we get many "free energies" for given β, µ.
For example, for sufficiently high temperatures in Fig. 4 (Left) (or Fig. 1), there are three saddles; one AdS saddle and two BH saddles. 7xplicitly, the free energy for each saddle is given by ) where l − ≡ −3/Λ is the AdS radius.In order for the corresponding real Euclidean geometry to exist, r H > 0. Therefore, from eq. (2.6), we could know the followings; is the minimum temperature.There are two BH saddles when T > T min and one BH saddle when T = T min .Below T min , there are no BH saddles.
• For Gµ 2 ≥ 1, there is only one BH saddle for each temperature.
And with a little algebra, we also know • for Gµ 2 < 1, the BH saddle with the larger horizon radius always thermodynamically stable.Above T tr = 1 πl − 1 − Gµ 2 , it becomes the dominant saddle, and below T tr , the dominant saddle is the AdS saddle.
• for Gµ 2 ≥ 1, the BH saddle is always thermodynamically stable and the dominant one.
Therefore, the behavior of the system is qualitatively different depending on whether µ is less than µ cr ≡ 1/ √ G or not. 8The qualitative behavior of the free energies in each case is shown in Fig. 4, and the phase diagram is shown in Fig. 3.

Box Boundary Condition
In this section I consider Einstein-Maxwell thermodynamics in the case of a box boundary without Λ.Considering the box boundary, we can find empty saddles and BH saddles, similar to the previous section.Although one might expect the qualitatively similar behavior to the AdS boundary case, the result is somewhat strange; stable BH saddles cease to exist above a certain temperature (Fig. 2).This is the result obtained by Basu, Krishnan, and Subramanian in [7].9I briefly review their analysis in subsection 3.1.Then, in subsection 3.2, I show that there are other types of saddles, which I call BG saddles [4].And in subsection 3.3, by including their contribution to the path integral, I show that the thermodynamic behavior becomes regular, similar to the AdS boundary case.
The free energy F (β, r b , µ) of each saddle is defined as eq.(2.1) and the parameters of a saddle are determined by boundary conditions (and f (r H ) = 0 for BH saddles); boundary condition Empty saddle : μ and their circumferences of t are given by β and 4π f ′ (r H ) respectively.The corresponding energy E and charge Q are given by 10 thermodynamical quantity Empty saddle : E = 0, Q = 0 (3.5) Euclidean RN BH saddle : The detailed analysis of thermodynamic properties of this system has been done in [7].Probably, an easy way to see the peculiarity found in [7] is to see the relationship between temperature and horizon radius; When the chemical potential is turned off (µ = 0), it reduces to T = 1  becomes the dominant one above the temperature.In the sense that thermodynamic quantities, , energy and charge are defined by , where ξ i is the normalized Killing vector of the U (1) isometry on the boundary and u i is the normal vector of the integration surface.The integrals should be over a surface homologous to the sphere in order to identify them with the conserved charges in Lorentzian theory.If we simply take the sphere as the integration surface, then u = utdt = f (r b )dt.
such as energy and charge, are ill-defined there because of the discontinuity, one might conclude that thermodynamics is ill-defined in this system.However, I would like to argue that this is not true.

good BG saddle
When evaluating a partition function of gravity with a spherical boundary, we usually approximate it with saddles with U (1) × SO(3) isometry.The saddles that dominantly contribute when the phase has finite entropy at zero-loop order can be obtained by Euclideanizing an appropriate part of some static and spherical symmetric spacetime including a horizon.It can be a BH horizon but not necessarily be.The other type of horizon is the inner horizon, which appears inside the BH horizon when a BH has conserved charges other than energy, such as electric charge or angular momentum. 11In this case, we can obtain an Euclidean saddle by Euclideanizing a suitable part between the inner horizon and the time-like singularity.(Fig. 6) This saddle is slightly different from the BH saddle in the sense that the area of the bolt is larger than the area of the boundary sphere.This means that the corresponding entropy exceeds the "boundary Bekenstein-Hawking entropy," which is given by the boundary area divided by 4G.In the previous paper [4], I call this type of saddles "bag of gold(BG)". 12n the Einstein-Maxwell system, BG saddles and the corresponding thermodynamical quantities are simply given by 13 field configuration thermodynamical quantity The form of the metric and gauge field (3.8) is exactly the same as that of the BH (3.2).The only difference is that the radial coordinate r increases from the boundary towards the bolt, as opposed to BH saddles.This fact leads to the following differences from the BH saddle case; • The minus sign appears in front of β in (3.9) because f ′ (r G ) is now negative.
• The minus sign appears in front of Q in (3.10) since n r , which appears in the definition of charge Q ≡ d 2 z √ σ 1 4π n ν F νµ u µ , is now negative, while the other parts are unchanged.

thermodynamics
Here, I will present the thermodynamic properties of the system including the contribution of BG saddles.Before doing so, I want to give a simple check that BGs lead to well-defined thermodynamics.In 3.1, I claimed that the BH branch ceases to exist at some finite temperature because of the relation between T and r H (3.7).From the equations in the last subsection, we know the following relation for BG: This function form is exactly same as (3.7).Combining (3.7) and (3.11), we may be able to regard it as a single (differentiable) function defined on the range 0, r b (1−Gµ 2 ) .Nothing special happens at r b and the temperature goes to infinity when we take the limit r G → r b (1−Gµ 2 ) , similar to the pure gravity case.One can show that the same thing happens for other thermodynamic quantities, including free energy.Therefore, the BH branch and the BG branch can be smoothly connected and we can regard them as a single phase.One might worry that although the thermodynamic quantities are regular at r H = r b , the corresponding geometry may become zero size and singular.But actually this does not happen and the r H (or r G ) → r b limit of BH/BG is (Euclidean) Bertotti-Robinson(BR) geometry, which is a direct product of AdS 2 and S 2 .See Appendix C for more details.
Let me move on to the thermodynamics.Explicitly, the free energy of the BH/BG phase is where R = R(β, r b , µ) is determined by the relation The resulting F -T diagram and phase diagram are shown in Fig. 7.As noted, the BG phase appears and extends to infinity.(Fig. 7 (Left)) Therefore, the system will be empty phase at low temperature and be BH/BG phase at high temperature.This is the main claim of this paper; this kind of "Hawking-Page phase structure" may be common for gravity systems with a sufficiently small magnitude of an external field and I showed that it is true for the Einstein-Maxwell system.
What is non-trivial may not be that it is true, since after all we are only doing a small deformation of pure gravity.What is non-trivial is how it is realized, i.e. that it is realized by the appearance of BG phases (or saddles).Of course, there are a number of differences from the pure gravity system or the Einstein-Maxwell system with AdS boundary conditions.So I list them below.

• New entropy bound and no energy bound
For the case of pure gravity with box boundary, it is known that there is the entropy bound and the energy bound.This is because the area of the bolt of BH saddles is bounded by the area of the boundary 4πr 2 b and, when µ = 0 and taking the limit R → r b of eq.(3.13), E = r b G .In the Einstein-Maxwell system, however, the maximum area of the bolt is 4π

• Transition temperature
In [4], it was shown that the transition temperature depends strongly on the radius of the boundary and weakly depends on other parameters for the case of box boundary: for a given πr b for the gravity-scalar system.As shown in Fig. 7, the transition temperature of the Einstein-Maxwell system can also depend strongly on µ and can be taken to be 0. 14

• Existence of solutions above µ = 1
For the case of the AdS boundary condition, the BH phase exists above µ cr = 1/ √ G (Fig 3), i.e.BH saddles exist for all µ.However, in the case of the box boundary condition, BH saddles and BG saddles exist only below µ cr and we cannot find any non-trivial Euclidean saddles above µ cr (Fig. 7).There may exist complex saddles above it and they may contribute to the partition function.However, I will not pursue this possibility here and leave it for future work.

Box Boundary Condition with Λ
In this section, I introduce a cosmological constant Λ and examine how it changes the properties shown in the previous section.Naively, a negative Λ does not change them qualitatively and a positive Λ does due to the bad BG saddle.(Although there is an unexpected behavior when √ Gµ > 1 in the positive Λ case.)If you are familiar with gravitational thermodynamics and the BG saddle (i.e. one of the references [4, 5, 6]), you can skip this section and go to the final section.Subsection 4.1 is devoted to the case of negative Λ and subsection 4.2 is to the case of positive Λ. 14 If we focus only on the transition temperature between the empty phase and the BH phase, it is still confined around 1 πr b , i.e. the range is 2 .However, in the next section, we will see that even the transition temperature between the empty phase and the BH phase can be taken to be zero.
At the beginning of subsection 4.2, I give a very brief review of the thermodynamical properties of pure gravity with positive Λ.In both cases, the field configuration, boundary condition, and the thermodynamical quantity are given by field configuration thermodynamical quantity Empty saddle : E = 0, Q = 0 (4.7)

negative Λ
When Λ = 0, there always exist the cusp structure as in Fig. 7 and there are no extremal BH/BG states, i.e. the system does not show the behavior as in the right panel of Fig. 4, which can be seen in the case of the AdS boundary.When a negative Λ is turned on, a new type of behavior and the behavior like the right panel of Fig. 4 appear depending on the parameters µ, Λ, r b and, as a result, extremal BH/BG states appear.To see why they do (not) appear in the case of Λ = 0(Λ = 0), let us look again at the relation between T and r H . From (4.5) or (4.6) we get So T can be written as 15 where T is defined by 12) The plot of T for sufficiently small µ is shown in Fig. 8.Because the other factors of temperature Figure 8: The plot of T for √ Gµ < . When , the minimum of T becomes negative and there are two positive roots corresponding to extremal BH or BG states.When 0) also becomes negative and there is only one positive root.Explicitly, the two roots are are always positive, 16 the positivity of T can be used to judge the existence of solution for a given R. Let the radius of the minimum R min , which is given by As shown in the figure, the positivity of T (R min ) and T (0) depends on the parameters µ, r b , Λ; 15 Here, ǫ = 1 for R < r b and ǫ = −1 for R > r b . 16For R < r b 1−Gµ 2 and √ Gµ < 1.
• 0 < √ Gµ < In this range, T (R) is always positive, as shown in Fig. 8.So there are no extremal states and there are always solutions for 0 < R < r b (1−Gµ 2 ) . • The minimum T (R min ) becomes negative but T (0) is still positive.So there are two positive roots, which are given by ) .These two roots represent extremal BHs (or BGs).The one with the larger radius is thermodynamically (locally) stable and the other is unstable.
Since both T (0) and T (R min ) are negative, there is only one positive root.
The qualitative behaviors of the free energies of these cases are shown in Fig. 9.Note that this time, unlike the Λ = 0 case, we may not have the exact expression of the transition temperature between the empty phase and the BH/BG phase.However, we still have that of T end , which is the "transition" temperature between the BH and BG phases: In the second range of µ, the locally stable extremal BH/BG appears but it is not always the Figure 9: Qualitative behaviors of the free energies when Λ < 0. In the case of Λ < 0, there are qualitatively different behaviors depending on the chemical potential.For each type of behavior, the position of the boundary between BH and BG also depends on the chemical potential.For the phase boundary behavior, see Fig. 11.Note that although in the middle figure I show the case where the empty phase has the lowest free energy at low temperature, the global stability of the locally stable BH/BG branch in the figure also depends on the chemical potential.There exists the critical chemical potential, which depends on the combination −Λr 2 b , and empty phase is realized when the chemical potential below it and BH/BG phase is realized when the chemical potential above it at low temperature in the middle case.(See Fig. 10.) dominant saddle.Since the extremal BH/BG in the third range is always the dominant saddle, the exchange of the dominance between the empty phase and the extremal BH/BG phase would occur in the second range.This critical chemical potential, above which extreme BH/BGs become the dominant saddles, is plotted in Fig. 10.This depends only on the combination −Λr 2 b .In the left panel, I also show the upper and lower boundary values of the second range.In the right panel, I have plotted the relative values, relative to the lower boundary value Note that, since this system has a negative Λ, r b /l − → ∞ limit, with a suitable redefinition of the variables, leads to the system of AdS boundary conditions.More precisely, by defining  AdS . 18

positive Λ and bad BG saddle
In the case of pure gravity, when Λ > 0, it has been shown that there exist bad BG saddles and thermodynamics becomes ill-defined due to their contribution [4, 5, 6].Before proceeding to the analysis of the case of the Einstein-Maxwell system, let me briefly review the case of pure gravity.
In the case of pure gravity, the existence of BH saddles and bad BG saddles depends on Λ.Let the radius of the bolt be R.For a given R, we could know the value of f (r b ).If it is not positive, it means that there is no corresponding solution.The value is Therefore the conditions for the existence of the solution are The former corresponds to the BH saddle and the latter to the bad BG saddle.Depending on the value of Λr 2 b , the possible ranges of R for BH and bad BG saddles are different.I classify them into three classes and name them as follows: • {good, bad} I : 0 < Λr 2 b < 1 Both BH and bad BG saddles exist.R ∈ (0, r b ) for BH and R ∈ ( , ∞) for bad BG.
Here good in parentheses stands for BH and bad stands for bad BG.The special case is Λr 2 b = 1, where the maximum bolt radius of BH and the minimum bolt radius of bad BG coincide r b = 18 For large r b /l−, the free energy of the BH can be written as Therefore, in the limit, the horizon radius at the transition is rH = l− 1 − Gµ 2 AdS .Substituting this into T AdS , we get, and R = r b corresponds to Euclidean Nariai space.The reason why I use the term "bad BG" is that their heat capacities are always negative and dominantly contribute to the partition function, making the system thermodynamically unstable [4, 5, 6].Now let's extend the analysis of pure gravity to Einstein-Maxwell, i.e. to the case of µ = 0.As before, we may be able to judge the existence of the solution for a given R and µ by the positivity of f (r b ).This is given by Therefore the conditions for the existence of a solution are The former corresponds to the BH saddle and the good BG saddle, the latter corresponds to the bad BG saddle.This time, depending on the value of Λr 2 b and √ Gµ, the possible ranges of R for BH/good BG and bad BG saddles are different and I classify them into five classes and name them as follows: ) for BH/good BG and R ∈ r b 1−Gµ 2 , ∞ for bad BG.
This time, good in parentheses stands for both BH and good BG.Here, the term "good BG" means that their heat capacities are positive (when they dominantly contribute), as are the BGs that appear when Λ ≤ 0. The above classification is summarized in Fig. 13.Below √ Gµ < 1, it is qualitatively same as the pure gravity case µ = 0, i.e. there always exist bad BG saddles and they b and √ Gµ.Below √ Gµ < 1, it is qualitatively the same as the pure gravity case µ = 0, i.e. there always exist bad BG saddles and they make the system thermodynamically unstable.make the system thermodynamically unstable.On the other hand, something strange happens when √ Gµ ≥ 1, 0 < Λr 2 b < 3. Recall that there do not exist any BH/BG saddles in this range of µ when Λ ≤ 0. This time, however, BH/good BG saddles appear and bad BG saddles that exist when √ Gµ < 1 disappear.Let's focus on the case of 1 ≤ √ Gµ, 0 < Λr 2 b < 3. The behavior of the free energy versus temperature is almost the same as in the case of Λ < 0 (Fig. 9), except for the behavior between the one with cusp (Fig. 9 (Left)) and the one without cusp (Fig. 9 (Right)).As shown in Fig.    • Recently, in [4, 5, 6], it was found that there exists a new type of saddle point geometries and that they cause the thermodynamical instability of pure gravity with a positive Λ.In [4] and here, I call this type of geometry "bag of gold(BG)".BG is similar to BH, but one difference is that the area of the bolt is larger than that of the boundary.So one question is: Are there any BGs that do not lead to thermodynamic instability?
• In [7], it has been reported that the free energy in the Einstein-Maxwell system with box boundary condition shows some peculiar behavior.The BH branch terminates at some finite temperature, and the free energies of the empty saddle and the BH saddle do not show the Hawking-Page phase structure even for sufficiently small chemical potential µ.However, for other cases, such as pure gravity with box boundary condition(Λ ≤ 0), that with AdS boundary condition, and the Einstein-Maxwell system with AdS boundary condition (with sufficiently small µ), they all show the Hawking-Page phase structure.
Can we somehow resolve this discrepancy?
In this paper, I gave a positive answer to these questions at the same time by showing the existence of BGs in the Einstein-Maxwell system and investigating these properties.When Λ ≤ 0, BGs in the Einstein-Maxwell system are good, in the sense that they are thermodynamically stable and dominant in the path integral if they are included.(In contrast to the bad BGs, which cause thermodynamical instability of the system in the case of pure gravity with a positive Λ.)I also showed that the BG branch smoothly connects to the BH branch and, as a result, the system exhibits the Hawking-Page phase structure at small µ.
Another objective is to investigate the thermodynamical properties of the Einstein-Maxwell system under the assumption that all Euclidean saddles contribute to the partition function.I have studied the grand canonical ensembles in the main text and the canonical and microcanonical ensembles in Appendix A. Some of the properties related to the existence of BGs are summarized in Table 1 for grand canonical ensembles and Table 2 for (micro)canonical ensembles.In general, a positive Λ causes the existence of bad BGs and the appearance of bad BGs causes the entropy stability BG entropy bound : Some thermodynamical properties of the Einstein-Maxwell system in canonical ensembles (and microcanonical ensembles).In the column of BG, good means that both good BGs and BHs exist, BH means that only BHs exist, as in Table 1.Stability means thermodynamical stability in canonical ensembles.Q cr (Λ, r b ) is given by (A.5) in Appendix A.
bound to be ∞, except for the two cases in the tables.One is the case of 0 < Λr 2 b < 3, √ Gµ > 1 in grand canonical ensembles.In this case, although Λ is positive, there are no BGs and the system is thermodynamically stable.The other is the case of Λ ≤ 0, √ GQ < r b 1 − Λr 2 b in (micro)canonical ensembles.In this case, although Λ is not positive, there exist bad BGs.However, the entropy bound is still finite and, remarkably, does not depend on the boundary size r b .At the moment, I have no idea about the implications of these cases.
Until recently, the importance (or sometime existence) of BG saddles in gravitational thermodynamics has been overlooked in the literature, perhaps because of their peculiarity. 20Also, in the recent study [4, 5, 6], the role of BGs is examined for pure gravity case but it turned out that they have a bad property, i.e. they cause thermodynamical instability.So the importance of BG saddles was unclear and one might argue that they should always be excluded from the path integral since they all seem to have bad properties and we do not know the true integration contour of the Euclidean gravitational path integral [9].In this paper, I show the existence of good BGs and their necessity for well-defined thermodynamics in the Einstein-Maxwell system.Presumably, from the results in this paper, there is little doubt that some BGs play important roles in gravitational thermodynamics.However, it is still unclear whether all BGs must be included in the path integral or not.
Therefore, as in the grand canonical case, the necessary conditions for the existence of solutions are and examine its parameter dependence on positivity.We can easily check that h And the equation h(R max ) = 0 gives the critical value of Q; , (the larger positive root of h(R) = 0) for BH and R ∈ (r b , ∞) for bad BG.
• {bad} : The other region Only bad BG saddles exist.R ∈ (r b , ∞) for bad BG.
The notation is similar to that of the grand canonical case.However, since there are no good BGs, I used BH instead of good. 22n the (micro)canonical case with a positive Λ, there always exist bad BGs.So the system may be thermodynamically unstable.The behaviors of {bad, bad} I and {bad, bad} II , which the system exhibits when √ GQ is relatively small compared to the dS length, are not seen in the pure gravity system.Probably the {bad, bad} I case is the more peculiar, since the lowest entropy can be arbitrarily large if we tune Λ and Q.For the pure gravity case, the lowest entropy is Standard near horizon and near extremal limit of Reissner-Nordström BH The BR geometry often appears in the context of the near horizon geometry of the RN BH.Firstly, let us review how BR geometry appears using the standard near horizon and near extremal limit.The metric and gauge field of the RN (A)dS BH is written as 24 If we consider a geometry of r b , TH but whose coordinate range of r is r ∈ (0, a] for some positive number a, we can easily check, by a suitable coordinate change, that it is equivalent to that of r b , T H a . 25If we take the coordinate range (0, ∞) for r, it still solves the Einstein-Maxwell equation.However, since (only) the tt component of the metric diverges, it cannot be a solution under either Dirichlet type boundary conditions or conformal Dirichlet type (or asymptotically AdS) boundary conditions.Under appropriate types of boundary conditions, it will be a solution and TH may be interpreted as the temperature measured at infinity.Then, by taking ε → 0 limit, the metric becomes a BR geometry.Let me emphasize that taking this limit means not only a near horizon limit (i.e.zooming up some region of some geometry) but also a near extremal limit (i.e.changing the geometry itself).Only in the case of T H = 0, the limit becomes a near horizon limit of an extremal RN BH.

T → T end limit in grandcanonical ensemble
In the main text, I claimed that, when the temperature is equal to T end (4.13), the BR geometry appears in the grand canonical ensemble.Essentially, the T → T end limit is a near horizon and near extremal limit similar to the one above.However, in the grand canonical ensemble, we do not fix the horizon radius r H and Q is not a control parameter.Therefore, for the sake of clarity, I will briefly describe how the BR geometry appears in the T → T end limit.
In grand canonical ensemble, the control parameters are µ, r b , and T .Let's fix µ and r b .And for simplicity, I use the fact that, near T end , r H is a monotonically increasing function of T and approaches r b .Therefore I use r H instead of T as a control parameter.By introducing a parameter ε as a control parameter by

Figure 3 : 1 −
Figure 3: Phase diagram of the Einstein-Maxwell system with AdS boundary condition.For four dimensions, the critical value of the chemical potential is given by µ cr = 1/ √ G as shown in the figure.The phase boundary is given by the function T = 1 πl − 1 − Gµ 2 .(l − is the AdS radius and
4πr H r b r b −r H and will diverge as the horizon radius r H approaches the box radius r b .Therefore, the BH branch extends to infinity in the F -T diagram and the behavior of the free energy is similar to the AdS boundary case as shown in Fig. 1 or Fig. 4 (Left) [2].However, once the chemical potential is turned on (µ > 0), the denominator in the square root of eq.(3.7) never reaches zero for r H ≤ r b , i.e. the temperature remains finite when we take the limit r H → r b .This leads to the peculiar F -T diagram and phase diagram as in Fig. 5.At T = 1−Gµ 2 4π √ Gr b µ , the BH branch ceases to exist and the empty saddle

Figure 5 :
Figure 5: Incorrect F − T diagram and phase diagram.

Figure 6 :
Figure 6: (Left) Penrose diagram of the Reissner-Nordström BH.By Euclideanizing the colored region inside the inner horizon, we obtain a BG geometry.(Right) Schematic picture of the BG geometry.The thick closed curve on the right represents the boundary Euclidean time circle.The area of the bolt is larger than that of the boundary sphere.

r 2 b( 1 − 2 b G 1 ( 1 − 2 bG
Gµ 2 ) 2 for a given µ < 1.From eq. (3.13), we know that the energy can be arbitrarily large.Therefore, for a fixed µ < 1, there is a new entropy bound πr Gµ 2 ) 2 , which is always larger than that of pure gravity πr , and no energy bound.

Figure 7 :
Figure 7: (Left) Qualitative behavior of the free energy versus temperature when Λ = 0.The BH branch can be extended beyond T end = 1 4πr b 1−Gµ 2 √ Gµ and this branch comes from the contribution of BG saddles.(Right) Phase diagram.Although I do not know whether the BH phase and the BG phase should be considered as different phases, I will tentatively treat them as different phases in this paper.

Figure 10 :.
Figure 10: (Left) The plot of the critical chemical potential above which the extremal BH/BGs become the dominant saddles.Green and red dots represent the values.It depends on the combination −Λr 2 b .I also show the upper and lower boundary values of the second range, that is, the dashed curve (upper) represents √ Gµ = 3 3−Λr 2 b

Figure 12 :
Figure 12: The enlarged figure near the disconnected BH phase of the Middle panel of Fig 11.

Figure 13 :
Figure 13: Classification of the existence and the possible range of the bolt radius R. It depends on Λr 2b and √ Gµ.Below √ Gµ < 1, it is qualitatively the same as the pure gravity case µ = 0, i.e. there always exist bad BG saddles and they make the system thermodynamically unstable.

Figure 14 :
Figure 14: Qualitative behaviors of the free energy when 0 < Λr 2 b < 3. The position of the boundary between BH and BG depends on µ and Λ.Because of this dependence, the phase diagram may be classified as Fig. 15.

2 b 3 3−Λr 2 b.
, the BH/good BG branch is smoothly connected to the empty branch at T = Λr b 2π(3−Λr 2 b ) As far as I know, this kind of second order phase transition between empty stability BG entropy bound

Figure 17 : 1 π
Figure 17: The parameter dependence of Q c and the corresponding temperature T c .The combinations √ GQc l − and T c r b depend only on −Λr 2 b .(Left) √ GQc l − .The dashed line represents that of AdS boundary condition √ GQ c,AdS l − = 1 6 ≃ 0.1667 [8].(Right) T c r b .The dashed line represents T c,AdS l − = 2 3

πr 2 bG 2 bG
when the BH branch is absent.Actually, the lowest entropy of the lower branch for the {bad, bad} I case is πr .However, in this case, there always exists another branch that gives the dominant contribution.

8 ) 9 ) 11 )
Besides the theory parameter Λ, this class of geometries are controlled by Q and r H . Let's fix r H by r H = r b .(Here, r b is some positive constant and is identified with r b in the equation (C.1).)The first and second derivatives of f (r) at r = r b are f ′ (r b )Instead of Q, let's parameterize this geometry by two parameters T H and ε (although it seems that one of them is fictitious).They are defined by the relation 4πT

Table 1 :
Some thermodynamical properties of the Einstein-Maxwell system in grand canonical ensembles.In the column of BG, good means that both good BGs and BHs exist, BH means that only BHs exist.N/A means that there are no Euclidean saddles in this parameter region.