Thermodynamics of the 3-dimensional Einstein-Maxwell system

Recently, I studied the thermodynamical properties of the Einstein-Maxwell system with a box boundary in 4-dimensions [1](JHEP 04 (2024) 083). In this paper, I investigate those in 3-dimensions using the zero-loop saddle-point approximation and focusing only on a simple topology sector as usual. Similar to the 4-dimensional case, the system is thermodynamically well-behaved when $\Lambda<0$ (due to the contribution of the"bag of gold"saddles). However, when $\Lambda=0$, a crucial difference to the 4-dimensional case appears, i.e. the 3-dimensional system turns out to be thermodynamically unstable, while the 4-dimensional one is thermodynamically stable. This may offer two options for how we think about the thermodynamics of 3-dimensional gravity with $\Lambda=0$. One is that the zero-loop approximation or restricting the simple topology sector is not sufficient for 3-dimensions with $\Lambda=0$. The other is that 3-dimensional gravity is really thermodynamically unstable when $\Lambda=0$.


Introduction
Although there are various problems, such as the non-renormalizability problem, the integration contour problem [2], gravitational path integral may be useful for extracting the information or challenging the paradoxes of quantum gravity. 1  The Euclidean path integral representation of the gravitational partition function is the simplest one [10].Formally, its definition is completely parallel to that of quantum field theory except that the external fields are defined on the boundary as the boundary conditions of the bulk fields and we sum over all configurations (and topologies) in the bulk [10,11,12].In practice, we use the saddle-point approximation and consider only the simplest topologies S 1 × R d−1 and R 2 × S d−2 (for d-dimensional spacetime with a spherical boundary), assuming that they give the dominant contributions.It turns out that there exists the Hawking-Page phase structure for the systems with the asymptotically AdS boundary condition [11] and with the box boundary condition [12] for pure gravity theory with Λ ≤ 0. Despite of the ignorance of the other topology sectors, this non-trivial phase structure is widely accepted as true and it is sufficient to consider the simplest topologies to describe the thermal equilibrium of quantum gravity, at least for an appropriate parameter region. 2 (For 2-dimensional gravity theories, however, see an interesting discussion [13] on the importance of the higher topology sectors.)So the task boils down to just finding the solutions g i in the simple topology sectors, evaluating the action of the solutions I E [g i ], and summing over e −I E [g i ] s (or just picking the dominant one).
Even in this simplified task, it has caused problems in the literature.One is the integration contour problem [2].Another, which I would like to treat as a problem, is the problem of the "bag of gold (BG)" saddle [14,1].This problem arises when we consider the box boundary conditions.For both the AdS boundary condition and the box boundary condition, (Euclidean) black hole (BH) geometries become the saddle points in the R 2 ×S d−2 topology sector.Its Lorentzian interpretations may be "a BH in AdS" and "a BH in a box", respectively.However, for some systems with the box boundary condition, there exists another type of saddles in the topology sector, which I called the"bag of gold (BG)" saddle [14].The difference to the BH saddle is that the area of the bolt is larger than that of the boundary sphere.Part of the problem is that, since their Lorentzian picture is that a horizon is located on the "outside" of a box, people have implicitly or explicitly excluded the contributions of BG saddles by "physical" considerations. 3An example where the ignorance of BG saddles causes a problem is the Einstein-Maxwell system [17].In the paper, they showed that, without the contributions of the BG saddles, the free energy behaves somewhat peculiarly, i.e. it becomes a discontinuous function.Recently, I re-examined the thermodynamical properties of the Einstein-Maxwell system in 4-dimensions with the box boundary condition [1] and found that there exist BG saddles and that they contribute to the partition function.As a result, the peculiar behavior of the free energy observed in [17] was resolved and it is shown that the system has a 1 One of the most exciting recent developments is the explanation of the Page curve [3] by using the Euclidean gravitational path integral [4,5,6,7].(More precisely, they are the derivation of the island conjecture [8].)For the details, see the nice review [9]./G ≫ 1 for the box boundary condition (where r b is the radius of the boundary sphere). 3The real problem of the BG saddle is that whether all BGs contribute to the partition function or not.As I will explain shortly, it seems that BGs should contribute to the partition function of the Einstein-Maxwell system with Λ ≤ 0 from the thermodynamic point of view [1].However, BGs appearing when Λ > 0 (for both pure gravity and the Einstein-Maxwell) make the system thermodynamically unstable [14,15,16,1].
In this paper, I investigate the thermodynamical properties of the Einstein-Maxwell system in 3-dimensions with the box boundary condition.Previously, the case of Λ < 0 was studied by Huang and Tao [18], but they did not consider the contribution of BG saddles.So I will start with revisiting the case of Λ < 0 and see how their results are modified by including the BG saddle contributions and how they differ from the 4-dimensional case.Although the analysis is completely parallel to that of 4-dimensions, related to the fact that we do not have BH solutions for Λ ≥ 0 in 3-dimensions [19,20], the thermodynamical properties are qualitatively different from the 4-(or higher )dimensional Einstein-Maxwell system when Λ is negative but close to zero or when Λ ≥ 0. A short summary of the result is; • When Λ < 0, the system is well-behaved and thermodynamically stable due to the contribution of the BG saddles.Without them, the free energy shows a strange behavior similar to the 4-dimensional case [17].
• When Λ < 0 and −Λr 2 b is sufficiently large, the phase diagram is similar to the 4-dimensional case [1].(Fig. 4) b is small, the fraction of the BH phase in the phase diagram shrinks and vanishes when Λ = 0.This is in contrast to the 4-dimensional case, where the BH phase still exists when Λ = 0. (Fig. 4) • Therefore, when Λ = 0, only BG saddles4 exist in the R 2 × S 1 sector.(Fig. 5) But they are not thermodynamically stable.So the system is not thermodynamically stable either, at least at zero-loop order.This is a striking difference to the 4-dimensional case.
As I noted above, we usually believe that (i) the (zero-loop) saddle-point approximation, and (ii) the concentration on the simple topology sector, are sufficient to describe the thermodynamics of quantum gravity.This may be true for higher dimensions.However, the last result indicates that, in 3-dimensions with Λ = 0 (at least for the Einstein-Maxwell system), (i) and (ii) are not sufficient.We may have to consider one-loop corrections or the contribution from the complicated topologies. 5he organization of this paper is as follows: In Section 2, I study the thermodynamical properties of the Einstein-Maxwell system with box boundary condition in 3-dimensions when Λ < 0. In subsection 2.1, I review the analysis of Huang and Tao [18], in which they investigated the detailed properties of the empty saddles and the BH saddles of the system.Although the analysis on these saddles was correct, their two additional assumptions lead to the incorrect results on the phase structure of the system.In subsection 2.2, I point out that the Bertotti-Robinson (BR) saddles [21,22,23,24] and the BG saddles were missing in their analysis and investigate their properties.I then show the complete phase structure of the system.In Section 3, I point out a qualitative difference on the behavior of phase structure between the 3-dimensional and the 4-dimensional system when Λr 2 b is close to zero.In Section 4, I investigate the thermodynamical properties of the Einstein-Maxwell system with box boundary condition in 3-dimensions when Λ = 0. Throughout this paper, I concentrate only on the grand canonical ensembles.

Λ < 0 case
In this section, I study the thermodynamical properties of the 3-dimensional Einstein-Maxwell system with box boundary and Λ < 0. In 3-dimensions, it is often said that there is no BH when Λ ≥ 0, and only when Λ < 0, BTZ BHs can exist.Therefore it can be a good starting point for studying thermodynamics in 3-dimensions.And indeed, in [18], they considered this case and investigated thermodynamic properties.In subsection 2.1, the basic setups and the properties of empty saddles and BH saddles are reviewed, which are systematically studied in [18].For empty saddles and BH saddles, their results are correct.But for other states, they made wrong statements.As a result, some part of their results on the phase diagram are incorrect.In subsection 2.2, I point out that BR saddles and BG saddles were missing in their investigation and examine their properties.Finally, I show the complete phase diagram.

empty saddles and BH saddles
The Euclidean action for grand canonical ensembles [25,26] I focus only on the metric and the gauge field of the form 2) and assume that they are the dominant contributions to the Euclidean path integral representation of the grand canonical partition function Z(β, µ, r b ) of gravity.And I also assume that they are purely "Euclidean", meaning that the metric is the real Euclidean metric and the gauge field is purely imaginary. 7Here, r is the areal radius and the boundary at r = r b .Within this class of metrics and gauge fields, there are at least two types of solutions.One is the empty saddle; Empty saddle : The last term becomes the standard counterterm for asymptotically AdS spacetime [27,28] when Λ < 0, i.e.
AdS where l AdS is the AdS radius.I used |Λ| instead of 1/l AdS because it can be used for any value of Λ.Note that all quantities in this paper do not have divergences because the boundary sphere is of finite volume.So the last term is not necessary.It is just a constant shift of free energy and energy and does not affect other thermodynamical quantities.However, only if there exists the counterterm, we can take a limit r b /l AdS → ∞ with suitable rescaling of thermodynamical quantities and recover the thermodynamics of asymptotically AdS boundary condition.(See subsection 4.1 of [1].) 7 In order to keep the electric charge real in the Euclidean picture, the boundary value of At must be imaginary on the boundary.Therefore the simplest Euclidean solutions may be of real Euclidean metric and purely imaginary gauge field, since these correspond to the well-known Lorentzian solutions after the Wick rotation.These may be the analog of purely Euclidean solutions for the pure gravity case.For a little more detail, see the footnote 11 of [29], where the similar problem of complex configurations for the rotating grand canonical ensemble was discussed.
The other is the black hole (BH) saddle. 8  BH saddle : where μ, Q and r H ∈ (0, r b ) are parameters that depend on β, µ, r b through the boundary condition of the path integral for the grand canonical partition function Z(β, µ, r b ) [25]; boundary condition Empty saddle : μ In the following, the function f will be used only for the BH (and BG) saddles.Substituting these field configurations to the action (2.1), we get free energy Empty saddle : Again, I want to emphasize that the parameters r H and Q are functions of β, µ, r b .For the empty saddle, since it has no T dependence and µ dependence, the thermodynamical quantities are given 8 Although in the standard parametrization of solutions I should say that there is another type of saddles At(r) = −iμ which corresponds to the Poincare patch of AdS, we can obtain this from rH → 0 limit of BH saddles in the parametrization in the grand canonical ensembles.As I will explain shortly, the parameter Q in (2.6) and (2.7) can be written as For the BH saddle, we can obtain them by using the relations Alternatively, since we also have finer-grained information about the saddle point geometries, we can obtain them by where τ ij and j i are the boundary currents of the metric and the gauge field, defined by The former current is known as the Brown-York tensor [30].)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.Using these fine-grained definitions and the relation S = β(E − µQ − F ), we get thermodynamical quantity BH saddle : Although they did not insist on this point in [18], they showed that there exists a maximum temperature for BHs that can be reached by the limit of the BH horizon approaching the boundary, and that there are no BHs above this temperature.Since they did not show the analytical expression of the temperature, I will derive it here.From the boundary conditions, the temperature can be rewritten as a function of r H (and µ), and its square is In order to apply L'Hôpital's theorem, I define functions N T 2 (r H ) and D T 2 (r H ) by As we can easily check, the fourth derivatives of these functions satisfy lim Using L'Hôpital's theorem, lim . Therefore, for fixed µ, r H → r b limit of temperature is The existence of a maximum temperature of charged BHs in the Einstein-Maxwell system in a box was pointed out in [17,1] for 4-dimensions, and in [31] for higher dimensions.Similarly, we can derive the maximum charge of BH saddles; Note that T (r b ) does not depend on r b .On the other hand, Q(r b ) does not depend on µ.
In addition, they have made two claims in [18] that I would like to claim to be false in the next subsection.
Their claim [18] (which I will claim to be false) • When taking the r H → r b limit of the BH saddles, it corresponds to the "M state".Its entropy is

And its temperature T and its chemical potential µ can be arbitrary values.
• There is a critical temperature, above which there are no BH saddles.Above this temperature, "M states" give the dominant contribution.
According to their paper, the free energy of the "M state" is They expressed this state as a state of "the black hole merging with the boundary".An example of the behavior of the free energies of the empty, BH, and M states, and the phase diagram are shown in Fig. 1.In the next subsection, I will show that the horizon does not merge with the boundary in the r H → r b limit and can even be larger than the boundary.

BR saddles, BG saddles, and thermodynamics
Instead of their two claims, I would like to claim that the following two are correct;

My claim
• When taking the r H → r b limit of the BH saddles, it corresponds to the Bertotti-Robinson(BR) saddles.• There is a critical temperature, above which there are no BH saddles.Above this temperature, "bag of gold(BG)" saddles give the dominant contribution.
Firstly, let's show the limit r H → r b does not correspond to the situation where "the black hole merges with the cavity" but to the Bertotti-Robinson (BR) geometry [21,22]. 9Define a small parameter ε by r H = r b − ε.The first and second derivatives of f (r By defining a new coordinate r = r H + εr, t = t ε , the metric becomes Then, ε → 0 limit (i.e.near horizon and near extremal limit) leads to which is the Euclidean version of the 3-dimensional Bertotti-Robinson (BR) geometry.The gauge field can be obtained in a similar way, i.e, As with the standard argument for BHs, the absence of a conical singularity at r = 0 leads to the unique temperature; which of course is the same as (2.24).We can also check that the parameter µ is actually the chemical potential of this saddle A t(r = 1) √ g tt = µ.Free energy and other thermodynamical quantities are given by free energy BR saddle : So far, we have seen that the r H → r b limit of BH saddles is not a singular geometry of zero size but a BR saddle, whose circumference of the transverse circle remains constant in the radial direction.Since the limit is not singular, one might wonder whether a further deformation of saddles (i.e. the extension of the parameter range of r H beyond r b of BH saddles) is possible.In fact, it is possible.
[1] (Fig. 2) For the metric and gauge field (2.6), (2.7), we still have regular geometries when r H > r b .These are "bag of gold(BG)" saddles.The role of these saddles in pure gravity is discussed in [14,15,16] and their importance for the 4-dimensional Einstein-Maxwell system was recently shown in [1].To distinguish them from BHs, let's use r G for the horizon radius of BG saddles.The expression of boundary conditions, thermodynamical quantities, and free energy are all the same except for the following two points [1]; (i) Since r is increasing toward the bolt, f ′ (r G ) is negative.So there is a minus sign in the relation between β and f ′ (r G ) (2.38).(ii) Since the r component of the normal vector of the boundary is negative for the same reason as before, 3 Difference between 3-dim.and 4-dim.(Λ < 0) There are many qualitative and quantitative differences between the Einstein-Maxwell system in 3-dimensions and that in 4-dimensions. 10What I want to emphasize here the most is the behavior 10 For example, • There is no upper bound on µ for the BH and BG saddles in 3-dim., but √ Gµ = 1 is the upper bound in 4-dim.
• In both cases, there exists the maximum horizon radius for fixed µ. rG,max = r b e 2Gµ 2 for 3-dim.and rG,max = r b 1 1−Gµ 2 for 4-dim.• In 4-dim., there can exist another small BH phase in the low temperature region, separated from the main BH of the BH phase (and the BG phase) in the phase diagram versus the value of Λ.In 4-dimensions, when we take Λ → 0 limit or Λ = 0, the phase diagram is still similar to the left figure in Fig. 4, i.e. there is a BH phase [1].However, in 3-dimensions, if we decrease the value of −Λr 2 b , the BH phase shrinks.This can be seen from the fact that the boundary curve between the BH phase and the BG phase is given by (2.33) and the curve becomes close to the T axis.This indicates that the BH phase disappears when Λ = 0.The disappearance of the BH phase is not surprising, since, under some suitable conditions, including the dominant energy condition, it is proved that there are no BHs in 3-dimensions when Λ = 0 [19].Even for the system with a box boundary, this statement still be true [32].What may be surprising is the existence of the BG phase even for the Λ → 0 limit.This seems to imply that even for Λ = 0, the BG phase exists, possibly as a thermodynamically stable state.However, I will show in the next section that BG saddles do exist, but they cause thermodynamic instability.

Λ = 0 case
In this case, there exist two types of saddles, the one is the empty saddle, And the other type of saddles can be obtained by setting Λ = 0 in (2.6), (and replacing the parameter r H with r G ,) • When µ = 0, the transition temperature is always given by T = 1 2πr b in 3-dim.and does not depend on Λ, as shown in [18].In 4-dim.it also depends on Λ.
Since for the coordinate range r ∈ [r b , r G ), f (r) > 0 and f (r G ) = 0, this is the BG saddle, which is not singular.And here I also set Q = −Q since these are BGs.The boundary conditions are similar to those in subsection 2.2.From the boundary conditions, we get the following expression; At this point, if we admit a general fact that the Bekenstein-Hawking formula S = πr G 2G is true, 11 we can easily see that the entropy does not change no matter how we increase or decrease the temperature for fixed µ (i.e., from (4.4) and the Bekenstein-Hawking formula, the entropy S(T, µ) can be written as S = πr G 2G = πr b 2G e 2Gµ 2 which does not depend on T ).This implies that the heat capacity is zero.Therefore, if these saddles give dominant contribution, it leads to thermodynamical instability of the system. 12Explicitly, free energy and thermodynamical quantities are given by free energy empty saddle : Since, in this case, the BG free energy (4.7) is a linear function of T , the "transition temperature" can be easily found as The "phase diagram" is shown in Fig. 5. Finally, let's discuss what happens when we turn off µ.
11 Strictly speaking, we do not have to use the Bekenstein-Hawking formula here and we can confirm this relation S = πr G 2G by calculating the free energy and using thermodynamic relations, as I will derive in (4.12). 12Precisely, it is neither stable nor unstable, it is marginal.However, if it is not stable, the system may not reach thermal equilibrium.It is in this sense that I refer to this situation as unstable.
Figure 5: "Phase diagram" of the case where Λ = 0. Since the BG "phase" is not thermodynamically stable and the corresponding saddles give the dominant contribution, the system itself is not thermodynamically stable either.
From (4.4), we see that µ → 0 limit corresponds to r G → r b limit.Again, this limit does not lead to the merging of horizon and boundary but leads to a regular geometry, which is the direct product of the 2-dimensional Rindler space and S 1 .By defining a small parameter ε by ε = Gµ 2 , the first derivative of .Then, the ranges of these new coordinates will be r ∈ [0, 1], t ∈ 0, 1 4πT 2 in the ε → 0 limit.Then, in the ε → 0 limit, the metric and gauge field become  13 An alternative way is to take a double scaling limit of the BR geometry (2.30): 4πT = −Λ Gµ 2 : fixed, Λ → 0, Gµ 2 → 0 The free energy of the Rindler × S 1 saddles is given by simply setting µ = 0 in the free energy of the BG saddles (4.7), and we can easily check that it will be dominant above the temperature T = 1 2πr b .(Fig. 5) At first glance, the phase structures of the Einstein-Maxwell system in 3-dimensions and 4dimensions are similar when −Λr 2 b is sufficiently large.There are three phases, the empty phase, the BH phase, and the BG phase (Fig. 4).However, when it is close to zero, differences appear; For 4-dimensions, the BH phase exists when Λ → 0 or Λ = 0, and the system is thermodynamically stable.For 3-dimensions, however, it shrinks in the Λ → 0 limit and ceases to exist at Λ = 0.So in 3-dimensions, when a negative Λ is switched off, the boxed systems no longer have the saddles with the smaller horizon (i.e.BH saddles), as previously proved [19,32], but still have the ones with the same size or larger horizon (i.e.BR saddles and BG saddles).But these saddles make the system thermodynamically unstable. 14If we also turn off µ, the system does not allow the saddle with the larger horizon (BG saddle), but still allows the ones with the same size horizon (BR saddle, or Rindler × S 1 saddle), which again makes the system thermodynamically unstable. 15his thermodynamical instability offers two options for how we think about the thermodynamics of 3-dimensional gravity with Λ = 0.One is that the zero-loop approximation or restricting the simple topology sector is not sufficient for 3-dimensions with Λ = 0.The other is that 3-dimensional gravity is really thermodynamically unstable when Λ = 0.

2 G
≫ 1 for the asymptotically AdS boundary condition and r d−2 b r H With this expression, we can show that (the last term in (2.6)) → 0 At → −i √ −Λr b µ = −iμ In addition, unlike the BH saddles, this saddle can have any temperature.However, the free energy of this saddle is always larger than that of the empty saddle (i.e.FP oincare = 0, Fempty = − √ 1−Λr 2 b 4G + √ |Λ| 4G r b < 0.) So I will ignore this saddle in this paper.

Figure 1 :
Figure 1: Incorrect F − T diagram and phase diagram.(For the empty and BH phases, they are correct.)In [18], they claimed that there are two phase transitions; one is the Hawking-Page transition between the empty and BH phases, the other is a second order phase transition between the BH phase and the M state.The transition point is the same as the termination point of the BH branch and it is given by (2.24) and marked with the red dot in the left figure.(2.24) also represents the boundary between the BH phase and the M states in the right figure.The boundary between the empty phase and the M state is given by the relation T = √ 1−Λr 2 b and its energy is E = √ −Λ 4G r b .And its temperature T and its chemical potential µ are NOT arbitrary values.

Figure 2 :
Figure 2: Spatial section of the BH saddle, BR saddle, and BG saddle.They are characterized by the radius of the horizon (or bolt); whether it is smaller or larger than the radius of the boundary.

Figure 3 :
Figure 3: Behavior of the free energies.The terminal point of the BH branch corresponds to the BR saddle.Above the terminal temperature, there are BG phases which are thermodynamically stable.For low µ, there is a Hawking-Page phase transition between the BH/BG phases as shown in the left figure.For high µ, the Hawking-Page phase transition does not occur and the system is in the BH/BG phase for all T as shown in the right figure.The left figure is the correct version of Fig. 1.

Figure 4 :
Figure 4: Examples of phase diagrams.The behavior of the phase diagram depends on the value of −Λr 2 b and is classified into three cases; the case of 0 < −Λr 2 b < 1 (Left), −Λr 2 b = −1 (Middle), and 1 < −Λr 2 b (Right).The left figure is the correct version of Fig. 1.