The Black Hole Entropy Distance Conjecture and Black Hole Evaporation
LLMU-ASC 46/20, MPP-2020-210
Prepared for submission to JHEP
The Black Hole Entropy Distance Conjecture andBlack Hole Evaporation
Marvin Lüben, a Dieter Lüst, a,b
Ariadna Ribes Metidieri b a Max-Planck-Institut für Physik (Werner-Heisenberg-Institut),Föhringer Ring 6, 80805 München, Germany b Arnold-Sommerfeld-Center for Theoretical Physics, Ludwig-Maximilians-Universität,Theresienstr. 37, 80333 München, Germany
E-mail: [email protected] , [email protected] , [email protected] Abstract:
We extend the recently proposed Black Hole Entropy Distance Conjecture to the caseof charged black holes in de Sitter space. By systematically studying distances in the space ofblack hole geometries with multiple horizons, we find that the distance is generically related to thelogarithm of the entropy. From the infinite distance conjecture this predicts the appearance of amassless tower of modes in the limit of infinite entropy. Further, we study the evaporation of theseblack holes and relate it to the geometric distance. We find that the corresponding distance to thefinal stage of evaporation is finite. We conclude that evaporation does not lead to the appearanceof a light tower of black hole microstates. a r X i v : . [ h e p - t h ] D ec ontents C.1 Perturbative analysis 30C.2 Thermodynamic considerations 33C.3 Summary 35
The swampland program aims to distinguish those low-energy effective field theories that can becompleted into quantum gravity in the ultraviolet from those that cannot [1]. In order to decidewhether an effective theory is in the swampland or not, a series of criteria have been conjectured.For reviews we refer to [2, 3]. The Distance Conjecture [4] is of particular interest for the presentpaper. It states that at an infinite distance in the moduli space of theories an infinite tower of statesbecomes massless. Therefore, the effective theory is rendered invalid in the infinite distance limit.While the distance conjecture was postulated for moduli, it was generalized to arbitrary fieldsin [5]. Therein, the generalized distance conjecture was applied to background metrics within thecontext of Anti-de Sitter (AdS) space-times. It was found that the flat limit is at infinite distancein field space. This implies that the near-flat limit of pure AdS is accompanied by an infinite towerof light states. The distance between background metrics was further applied to space-times witha single horizon in [6]. The analysis shows that the limit of infinite entropy S is at infinite distance– 1 –n field space. Combined with the distance conjecture [4] it follows that a tower of modes becomesmassless as m ∼ S − c (1.1)with c ∼ O (1) . The limit S → ∞ is identified with Minkowski space-time. This observationprompted the postulation of the Black Hole Entropy Distance Conjecture (BHEDC) [6]. While inthese studies the variation of the background metric was related to the geodesic flow, also othertypes of flow have been studied in the context of the swampland [7, 8]. The conjectures on AdSwere explicitly tested in concrete string theoretic setups, e.g. in [9].Here we generalize [5, 6] to spherically symmetric and static space-times with multiple horizons.In particular, we consider effective theories with positive cosmological constant. Therefore, we studymetrics that belong to the family of Reissner-Nordstrøm-de Sitter (RNdS) solutions. Workingwith multi-horizon space-times that are asymptotically de Sitter poses some challenges such as thedefinition of the entropy. For a space-time with a single horizon, the entropy of the space-time canbe identified with the entropy associated to the single horizon. The entropy is given by the arealaw [10–12] S = A/ , (1.2)where A is the area of the horizon in Planck units. We will briefly discuss how the entropy canbe defined for multi-horizon space-times. On the other hand, within string theory the microscopicorigin of the entropy is best understood for supersymmetric black holes [13], but recently progresswas made for Schwarzschild geometries in the context of soft hairs [14–18].Our strategy for generalizing the BHEDC is as follows. Using the definition of geodesic distancebetween space-times [5, 19, 20], we compute the distance between points in the moduli space ofmetrics. In particular, we identify the infinite distance points. We then relate the geodesic distancesbetween points in moduli space to the entropy of these points. As we will see, the space-timeconfiguration with infinite entropy is at infinite geodesic distance. On the other hand, all space-time configurations with finite entropy are at finite distance from each other. Moreover, our studyshows that the distance ∆ is generically proportional to the logarithm of the entropy of the space-time, ∆ ∼ log S . (1.3)In other words, the BHEDC can be generalized to multi-horizon space-times.This generalization allows to apply the BHEDC to physical phenomena in our universe. In thispaper, we study the evaporation of charged black holes in asymptotically de Sitter space-time dueto Hawking and Schwinger. This is the most general spherically-symmetric and static configurationthat respects the completeness conjecture [21] and the no hair theorems [22, 23]. Further, in [24, 25]RNdS black holes were used to formulate de Sitter versions of the Weak Gravity Conjecture [26].We will take the conjecture of [24] as a basis such that the black hole evaporates quasi-statically.In addition, we consider the cosmological constant to vary in time to circumvent the de Sitterconjectures [27–33]. We implement the variation of the cosmological constant via thermodynamicconsiderations [34]. The Hawking and Schwinger effects accumulate and backreact on the geometry.This can be interpreted as a variation of the background metric, which travels a distance in themoduli space. Hence, we can use our generalized prescription to compute the geometric distancebetween the initial and final point of the evaporation process and relate the inferred distance tothe Generalized Distance Conjecture. As we will see later, evaporation leads to a finite distance inmoduli space. Hence, this toy model of self-similar and quasi-static evaporation satisfies all theseswampland criteria.The organization of this work is as follows: in Sec. 2 we introduce the Reissner-Nordstrøm-deSitter solution of Einstein field equations. The reader familiar with the RNdS solution can safely– 2 –kip this section and only use it as a reference for the notation and unit system used. Sec. 3 isdevoted to the construction of the moduli space of metrics, the solution of the geodesic equationand the computation of moduli space distances between spherically symmetric and static mani-folds. Next, we discuss how to define entropy in the space-times with multiple horizons in Sec. 4.Further, we establish a relation between moduli space distances and entropy. In Sec. 5, we analyzethe evaporation of Reissner-Nordstrøm and Schwarzschild black holes in asymptotically de Sitterbackgrounds and identify the final stage of evaporation. We compute the distances between theinitial and final space-times of the evaporation process. Finally, we conclude in Sec. 6. In this section we review the black hole solution of Einstein gravity with cosmological constantcoupled to Maxwell’s theory. We work in (3+1)-dimensional space-time and use the ( − , + , + , +) -metric signature. The action for the metric tensor g µν and gauge field A µ is given by [35, 36] S = (cid:90) M d x √− g (cid:20) πG ( R − − g F µν F µν (cid:21) where G is Newton’s constant, R is the Ricci scalar, and Λ the cosmological constant. Further, F µν is the electromagnetic field strength tensor and g the U (1) gauge coupling.In Schwarzschild coordinates ( t, r, θ, φ ) the most general, spherically symmetric, and staticsolution for the metric g µν and gauge field A µ is given byd s = − V ( r ) d t + V ( r ) − d r + r d S , A = − g π qr d t (2.1)with d S = d θ + sin θ d φ the metric on the 2-sphere and V ( r ) = 1 − Gmr + g G π q r − r (cid:96) . (2.2)The parameters m and q are related to the mass and charge of the space-time [34, 37], and we willrefer to (cid:96) = (cid:112) / Λ as de Sitter radius with Λ > . This is the Reissner-Nordstrøm-de Sitter (RNdS)solution that describes an electrically charged black hole in an asymptotic de Sitter space-time. Forthe sake of brevity, we define the mass parameters M = Gm , Q = q (cid:114) g G π . (2.3)The metric function then takes the compact form V ( r ) = 1 − Mr + Q r − r (cid:96) . (2.4)The causal structure of RNdS space-time is characterized by three horizons. A causal horizoncan be defined as the null hyper-surface where the metric changes its signature. Since the RNdSspace-time is static, the Killing, apparent and event horizons coincide [38]. Further, for sphericallysymmetric space-times the aforementioned horizons are located at the radial position r = r h wherethe metric function (2.4) vanishes, V ( r ) (cid:12)(cid:12)(cid:12) r = r h = 0 , (2.5)which represents a quartic polynomial. The number of real roots is dictated by the sign of thediscriminant locus D of that quartic polynomial, explicitly given by D
16 = M (cid:96) − Q (cid:96) − M (cid:96) + 36 M Q (cid:96) − Q (cid:96) − Q (cid:96) . (2.6)– 3 –or D ≥ , Eq. (2.5) has a maximum of four real-valued roots, but one of them is always negativeand hence non-physical. The space-time can have a maximum of three causal horizons. These arethe inner black hole horizon r − , the outer black hole horizon r + and the cosmological horizon r c .Only the horizons r + and r c are accessible to an observer outside of the black hole. The explicitexpressions for the horizons r {± ,c } as functions of the parameters (2.3) can be found in AppendixA. Next let us define the phase space of RNdS solutions as the 3-dimensional parameter spacespanned by the mass M , charge Q and de Sitter radius (cid:96) . To respect the Cosmic CensorshipConjecture [39], i.e. to ensure absence of naked singularities , we impose M ≥ and D ≥ . Thelatter condition implies that all three horizons are real and satisfy r c ≥ r + ≥ r − . Further, weimpose (cid:96) ≥ to exclude Anti-de Sitter. We refer to the region that respects these conditions as physical phase space D .The boundary of the physical phase space, that we denote by ∂ D , is characterized by D = 0 .This implies that two or three horizons are degenerate. We distinguish three such cases. The casewhere the black hole and cosmological horizons coincide, r + = r c , is referred to as Nariai solution.If instead the inner and outer black hole horizons coincide, r − = r + , the space-time is calledextremal. Finally, the space-time with all three causal horizons being degenerate, r c = r + = r − , iscalled ultra-cold.In Fig. 1, the physical phase space D and its boundary ∂ D are shown. Since the discriminantlocus (2.6) and the horizons (A.1) depend only on the dimensionless ratios M/(cid:96) and
Q/(cid:96) , the 3-dimensional phase space can be represented in a 2-dimensional plane. Along the Nariai and coldlines, depicted in black, two horizons are degenerate and in thermal equilibrium at zero tempera-ture, while the non-degenerate horizon has finite temperature. The lukewarm line characterized by M = | Q | and indicated by the dotted black line is special because the black hole and the cosmolog-ical horizon are in thermal equilibrium at a finite temperature, even though the horizons are notdegenerate [44]. Finally, the neutral case ( Q = 0 ) corresponds to Schwarzschild-de Sitter. The point M/(cid:96) = Q/(cid:96) = 0 represents pure de Sitter, Schwarzschild and Reissner-Nordstrøm simultanously.For a recent study of these solutions in the context of the Weak Gravity Conjecture, we refer to[25].To finish this section, we introduce the concept of a geodesic observer [45], which is crucialfor this paper. The geodesic observer is an observer that remains at their radial position withoutacceleration. In asymptotically flat space-times, the geodesic observer is at infinity I ± . Forasymptotic de Sitter, future and past infinity are at spatial distance due to the presence of thecosmological horizon. The geodesic observer is located at the position r g , where the gravitationalpull of the black hole and the expansion cancel. Hence, r g is determined by the equation V (cid:48) ( r ) | r = r g = 0 . (2.7)The geodesic observer is crucial to correctly normalize the Killing vector fields. Take the time-likeKilling vector field ξ ( t ) = γ ( t ) ∂ t with constant γ ( t ) . Instead of normalizing at infinity, where ξ ( t ) isspacelike, the Killing vector field is normalized such that ξ t ) | r = r g = − . We thus have to choose γ ( t ) = V ( r g ) − / . This yields an extra redshift factor in the surface gravity, κ h = 12 (cid:112) V ( r g ) | V (cid:48) ( r h ) | , (2.8)where the subscript denotes horizon, h = ± , c . In the limit of vanishing cosmological constant werecover the standard normalization because r g → ∞ and thus V ( r g ) → . See, e.g. [40–43] for some related work on naked singularities. – 4 – .00 0.05 0.10 0.15 0.20 0.250.000.050.100.150.200.25
Figure 1 . The phase space of RNdS solutions is shown. The gray shaded region represents the physicalphase space D . The extremal and Nariai solutions ∂ D are indicated by the black solid line and correspondto the boundary of the physical phase space. The lukewarm line M = | Q | is indicated with a dashed line.The phase space outside the gray shaded region is not physical as it gives rise to naked singularities. Each point of the phase space characterizes a metric of the RNdS family. In this paper, we areinterested in measuring the distance between such metrics. The concept of distance between metricswas first introduced by de Witt in [19] and extended by Gil-Medrano and Michor in [20] by con-structing the geometric space of metrics. Further, a prescription for computing the distance alonga path in the moduli space of metrics was given in [6] . Here, we briefly introduce the constructionfor a general moduli space of metrics and quickly specialize to spherically symmetric and staticspace-times.Each point in the moduli space of metrics is a metric itself. We can connect different metricsby a path in moduli space τ (cid:55)→ g ( τ ) where τ is an affine parameter. For every τ , the point g ( τ ) represents a metric. We can assign a distance to such paths using the geometric distance formula [6] ∆ g = c (cid:90) τ f τ i (cid:32) V M (cid:90) M vol ( g ) tr (cid:34)(cid:18) g − ∂g∂τ (cid:19) (cid:35)(cid:33) / d τ (3.1)where c ∼ O (1) is a constant, V M = (cid:82) M vol ( g ) the volume of M and vol ( g ) = (cid:112) | det ( g ) | d n x thevolume element. Eq. (3.1) hence describes the distance between the initial g ( τ i ) and final g ( τ f ) metrics along the path g ( τ ) . The concept of distance allows to define geodesic paths in the modulispace. Specializing to isotropic space-times and minimizing the distance (3.1) leads to the geodesicequation ¨ g − ˙ gg − ˙ g = 0 . (3.2) Also see [7] for a detailed discussion on how to define distances between background fields in terms of entropyfunctionals. – 5 – dot indicates differentiation with respect to the proper time λ , which is related to the affineparameter τ via d λ = d τ (cid:32) V M (cid:90) M √ g tr (cid:34)(cid:18) g − ∂g∂τ (cid:19) (cid:35)(cid:33) / . (3.3)The distance functional (3.1) is not covariant under diffeomorphisms along the metric flow [6].Consequently, the metric path g ( λ ) along which the distance is computed needs to be expressed ina specific coordinate system. It is convenient to work in Eddington-Finkelstein gauge with dimen-sionless coordinates (˜ v, ˜ r, θ, φ ) , where ( v, r ) = α (˜ v, ˜ r ) . Here α is a parameter of mass dimensionone. Updating α → α ( λ ) , the metric flow g ( λ ) is entirely captured by the flow of α ( λ ) , while thedimensionless coordinates (˜ v, ˜ r ) are constant along the flow. In this gauge the line element (2.1)reads ds = − α V ( α ˜ r, α i ) d ˜ v + 2 α d ˜ v d ˜ r + α ˜ r dS . (3.4)The metric potential V ( α ˜ r, α i ) might depend on further parameters α i ( λ ) of mass dimension one,that also flow.Using the above prescription, the geodesic equation (3.2) in dimensionless Eddington-Finkelsteingauge reduces to the following set of equations,dd λ (cid:18) ˙ αα (cid:19) = 0 , d d λ V ( α ˜ r, α i ) = 0 . (3.5)The first equation is solved by α ( λ ) = α i e λ/ , (3.6)where α i is a constant of integration. In section 3.2 we will provide a general solution to the secondequation. However, we can already relate the geometric distance to the proper time λ . Noting that tr( ˙ gg − gg − ) ∝ ( ˙ α/α ) , the geometric distance is given by ∆ = 4 c log (cid:18) α ( λ f ) α ( λ i ) (cid:19) = c | λ f − λ i | (3.7)upon using Eq. (3.6). Hence, we can already conclude that the geometric distance between metricsis infinite, if they are separated by infinite moduli space proper time. This simple relation waspreviously established only for Weyl rescalings [5, 6].The distance conjecture relies on the moduli space of fields having negative curvature [4]. InAppendix B we explicitly show that the moduli space of spherically symmetric and static space-times indeed has negative scalar curvature. In this section we provide a general solution to the geodesic equation (3.2) for isotropic space-timesusing exponential mapping. Our result demonstrates that dimensionless Eddington-Finkelstein(EF) gauge is indeed the most convenient choice for computing geometric distances. From thispoint onward we suppress the explicit dependence of the map g ( λ )(˜ x ) on the dimensionless space-time coordinates for simplicity of notation, since the dimensionless space-time coordinates are fixedalong the flow. See Appendix A of [6] for a detailed discussion on this issue and [7] for a diffeomorphism invariant definition ofthe distance functional based on the Ricci flow. The geodesic flow of a smooth Riemannian manifold is defined as follows [46]: Let t (cid:55)→ γ ( t ) be a geodesicdifferentiable path on M and let T M denote the tangent bundle of M . Hence, a geodesic curve on M determinesa curve t (cid:55)→ ( γ ( t ) , d γ d t ( t )) in T M . The flow of the unique vector filed G on T M whose trajectories are of the form t (cid:55)→ ( γ ( t ) , γ (cid:48) ( t )) is called the geodesic flow on T M . – 6 –et g : [0 , × M → M be a smooth geodesic in moduli space. The evolution of g ( τ ) onlydepends on the initial point g (0) and the tangent vector ˙ g (0) at the initial point for fixed space-time coordinates. Indeed, the moduli space evolution is such that g ( λ ) ∈ M , ˙ g ( λ ) ∈ T g ( λ ) M∀ λ ∈ [0 , [20]. With g i = g ( λ = 0) , the geodesic in M starting at g i in the direction ˙ g (0) is givenby g ( τ ) = g i exp (cid:104) a ( λ ) Id + b ( λ ) H (cid:105) (3.8)where a ( λ ) = c a λ and b ( λ ) = c b λ are smooth functions with some real constants c a and c b . Further,Id is the identity matrix. We have introduced the quantity H = g − ˙ g (0) and its traceless part H = H − n − tr ( H ) Id with n the dimension of the space-time M . We specify to 4-dimensionalLorentzian manifolds, n = 4 . The specific values of c a and c b depend on the initial point g i ∈ M and the initial direction, and hence on the dimensionless space-time coordinates.The general solution (3.8) allows to compute the geometric distance in any gauge. In dimen-sionless Eddington-Finkelstein coordinates (3.4), we have c a = 2 ˙ α/α and c b = ˙ V ( λ ) /V (0) . Here, V ( λ ) denotes the metric potential (2.4) that might depend on λ via the mass parameters and on thedimensionless space-time coordinates. The constant c a is independent of the space-time coordinatessince we consider isotropic space-times. This yields for the geodesic path g ( λ ) = g i exp (cid:34) (cid:18) α ( τ ) α (0) (cid:19) Id + V ( τ ) − V (0)˙ V (0) H (cid:35) . (3.9)This path g ( λ ) smoothly connects g i = g ( λ i ) and g f = g ( λ f ) . In this gauge we find that tr H = 0 .Hence, computing the geodesic distance along this path is remarkably simple. The result is providedby Eq. (3.7).This is not the case in other gauges. As an example, we work out the case of dimensionlessSchwarzschild gauge. We introduce dimensionless coordinates via ( t, r ) = α (˜ t, ˜ r ) . The generalsolution to the geodesic equation is then given by g ( τ ) = ˜ g i exp (cid:34) (cid:18) α ( λ ) α (0) (cid:19) Id + V (0)˙ V (0) log (cid:18) V ( τ ) V (0) (cid:19) H (cid:35) . (3.10)It turns out that the second term does not vanish when evaluating the trace in the distance for-mula (3.1) because tr H (cid:54) = 0 . Since the term containing H encodes how the geodesic path explicitlydepends on space-time coordinates ˜ x , we expect the distance computed in such gauges to not corre-spond to the distance between space-times. However, it would be interesting to explicitly evaluatethe integral in the distance formula to check whether the term proportional to tr H evaluates tozero. So far, we have solved only parts of the geodesic equation. In this section, we solve the secondequation of (3.5). For RNdS metrics, the mass parameters α i that flow with λ are the mass M ,charge Q and de Sitter radius (cid:96) . Hence, we promote them to depend on λ .The second differential equation of (3.5) can be separated into three independent differentialequations by requiring isotropy. In other words, the functions M ( λ ) , Q ( λ ) and (cid:96) ( λ ) must beindependent of ˜ r . Using the general solution for α (3.6), the resulting differential equations are Eq. (3.2) is an ordinary differential equation (ODE). By the existence and uniqueness theorems for ODEs, thesolution of the geodesic equation exists and is unique in an open neighbourhood of a smooth manifold, once a pointand a tangent vector are specified [46]. – 7 –iven by α M − α i ˙ M + ¨ M = 0 , α Q + ˙ Q + Q ( − α i ˙ Q + ¨ Q ) = 0 , α (cid:96) + 3 ˙ (cid:96) − (cid:96) (4 α i ˙ (cid:96) + ¨ (cid:96) ) = 0 . (3.11)We have thus decoupled the system of differential equations and can solve them separately for eachmass parameter. The solutions read M ( λ ) = M i e λ/ (1 + c M λ ) ,Q ( λ ) = Q i e λ/ (cid:112) c Q λ ,(cid:96) ( λ ) = (cid:96) i e λ/ √ c (cid:96) λ , (3.12)where we have chosen the integration constants such that M i = M ( λ = 0) and likewise for the otherparameters w.l.o.g. Further, also c M , c Q and c (cid:96) are integration constants. These general solutionsare the basis for studying the geometric distance between space-times that belong to the RNdS asdone in the next section. In this section, we will systematically compute the distance between different space-times thatbelong to the RNdS family. We are particularly interested in the distance to Minkowski space-time,but we will also study the distances between other points in the phase space D .We found the general solutions for α and the mass paramaters M , Q and (cid:96) as functions of propertime λ . However, α is only a parametrization to study the metric flow, but does not correspond toa mass parameter. Instead, since α must be a smooth function of the mass parameters, they haveto be related in a way that is compatible with Eqs. (3.6) and (3.12). We use a power-law ansatz, α ( λ ) ∝ (cid:88) β,γ,δ M ( λ ) β Q ( λ ) γ (cid:96) ( λ ) δ = (cid:88) β,γ,δ M β i Q γ i (cid:96) δ i (1 + c M λ ) β (1 + c Q λ ) γ/ (1 + c (cid:96) λ ) δ/ e λ ( β + γ + δ ) / , (3.13)where the sum is subject to the condition β + γ + δ = 1 due to dimensional reasons. We immediatelysee that Eq. (3.6) implies the conditions β + γ − δ = 0 and c M = c Q = c (cid:96) if all exponents arenon-zero. Then we can also express the inital value of α in terms of the mass parameters as α i = M β i Q γ i (cid:96) δ i .Specializing to c M = c Q = c (cid:96) = 0 , our general solutions reduce to Weyl rescalings becausethe metric function (2.4) V is independent of λ in that case. Any choice of exponents respectingthe condition β + γ + δ = 1 constitutes a valid parametrization. We hence recover the frameworkof [5, 6]. As an example, we consider the RN black hole. Since the metric function does not dependon (cid:96) we set δ = 0 . The choice γ = 0 leads to α ( λ ) = M ( λ ) so the metric flow is characterized by [6] M ( λ ) = M i e λ/ , Q ( λ ) = Q i (cid:112) c Q λ . (3.14)The other cases discussed in [6] are reproduced by similar choices for β , γ and δ .In the following, we will use different combinations of β , γ and δ and c M , c Q and c (cid:96) to describegeodesics that connect specific space-times to infer the geodesic distance between them.– 8 – .3.1 The distance to Minkowski space-time From the perspective of RNdS space-time, there are two distinct Minkowski limits. The first andtrivial limit is characterized by vanishing mass parameters M = Q = (cid:96) − = 0 . We will denote thislimit as Mink in the following. The second Minkowski limit is arrived at by taking the black holemass to infinity, M → ∞ . To avoid naked singularities, also the other mass parameters have toscale as Q → ∞ and (cid:96) → ∞ . As demonstrated in [6, 16], the near horizon geometry approximatesthat of Minkowski in this limit. Further, the BM S groups at both the black hole and cosmologicalhorizons approach those of future and past null infinity. We will denote this Minkowski limit asMink ∞ in the following. In this section we infer the geometric distance of any space-time in theRNdS family to both Mink , ∞ .First, we study the distance to Mink ∞ . We show that this limit is at infinite distance fromany spherically symmetric and static space-time independently of the parametrization α . FromEq. (3.12) we immediately see that the infinite mass and charge limit is reached only for λ → ∞ .Hence, Eq. (3.7) implies that the distance to Mink ∞ is always infinite, ∆ = c | λ f − λ i | → ∞ . (3.15)Our result generalizes the discussion about the infinite distance limits of [6] to arbitrary sphericallysymmetric and static space-times. Notice that the proof above was independent of the parametriza-tion α , and hence of the specific moduli space geodesic, and of the particular causal structure ofthe initial space-time. Hence, Mink ∞ is a moduli space point that represents Minkowski space-timeand lies at an infinite distance of any other space-time independently of the geodesic path used toreach it.We continue by discussing the distance to Mink . Naively, this limit can be reached in twoways. The first option to arrive at vanishing mass M is λ → −∞ . To avoid naked singularities inthis limit we have to impose c M < , c Q < and c (cid:96) < according Eq. (3.12). However, in thislimit not only M → and Q → , but also (cid:96) → implying that the scalar curvature diverges, R = 12 /(cid:96) → ∞ . Hence, this is not a correct Minkowski limit. The second option to achieve avanishing mass M is given by λ → − /c M . However, Q → and (cid:96) → ∞ need to be reachedsimultaneously. This enforces the integration constants to satisfy c M = c Q = c (cid:96) ≡ k . Then, Mink is reached at proper time λ = − /k . Notice that the conditions M ≥ and Q, (cid:96) ∈ R together withthe absence of naked singularities at the flow start and endpoints restricts the range of the propertime completely to a bounded interval. Setting k = − w.l.o.g. yields the proper time to lie inthe interval λ ∈ [0 , . This means that the geodesics to Mink are unique because all arbitraryconstants in Eq. (3.12) are fixed and the parametrization α ( λ ) is fully specified by Eq. (3.13). Weproceed by explicitly constructing minimal geodesic paths connecting RNdS space-times as well asits various subcases to Mink .Let us consider SdS space-time. This case does not contain Q so we set γ = 0 . Hence, Eq. (3.13)simplifies to α ∝ M / (cid:96) / . (3.16)This coincides with the expression for the position of the geodesic observer in SdS space-time, r g = ( M (cid:96) ) / = ( M i (cid:96) ) / e λ/ , (3.17)as follows from Eq. (2.7). The parametrization is a smooth function of the mass parameters, finite,real-valued and strictly positive for all values of λ ∈ [0 , . Thus, the geodesic path is characterizedby the parametrization α = r g and the mass parameter evolution follows Eq. (3.12) with c M,Q,(cid:96) = − This condition can be imposed simply by requiring that the discriminant locus (2.6) of the metric function ispositive D ≥ . – 9 –nd λ ∈ [0 , . We move on to the RN black hole, for which we set δ = 0 because this case does notcontain (cid:96) . Hence we obtain the parametrization α = Q M = Q M i e λ/ = r g , (3.18)which again coincides with the expression for the geodesic observer for the RN space-time . Finally,the same applies for the RNdS space-time. The parametrization α = r g = r g , i e λ/ defines a smoothpath connecting a non-Nariai RNdS space-time to Mink . Since the expression is lengthy, we donot display it here explicitly.We conclude that the path λ (cid:55)→ g ( λ ) parametrized by α = r g is positive and smoothly connectsSdS, RN and RNdS space-times to Mink . The corresponding distances are given by ∆ g = 4 c log (cid:18) r g , f r g , i (cid:19) = 4 c , (3.19)where we used that λ i = 0 and λ f = 1 . The distance to Mink is hence always finite.The discussion above applies to space-times with more than one mass parameter, i.e. SdS,RN and RNdS, for which the geodesic observer is located at a finite radial position. We finish byanalyzing the geodesics connecting S and dS to Mink . If we set the constants to c M = 0 or c (cid:96) = 0 ,the geodesics to Mink are parametrized by α ∝ M or α ∝ (cid:96) , respectively, i.e. Weyl rescalings.Hence, there are geodesics connecting both Mink ∞ and Mink to Schwarzschild space-time at infinitedistance, since these limits are obtained as lim λ →∞ M = ∞ , lim λ →−∞ M = 0 . (3.20)For dS, Mink is arrived in the limit lim λ →∞ (cid:96) = ∞ . Hence, the limit Mink is at infinite distancefrom both S and dS. If we instead allow for c M (cid:54) = 0 or c (cid:96) (cid:54) = 0 , respectively, geodesics exist thatconnect both S and dS to Mink at finite distance. Since the horizons are not co-moving with thegeometry, we need to impose extra conditions on the parameters to ensure consistency along theflow. For the case of S, the mass of the black hole has to be positive and any observer outside theblack hole must not cross the black hole horizon. These conditions yield M ≥ ⇒ c M λ ≥ − , (3.21a) r ≥ M = ⇒ c M λ ≤ r i M i − , (3.21b)and imply that the range of proper time must lie in the finite range λ ∈ [ − /c M , ( r i / (2 M i ) − /c M ] .Setting c M = − and specifying to an observer that is initially located on the black hole horizon, r i = 2 M i , the range of λ is λ ∈ [0 , . The mass and radius evolve as M ( λ ) = M i e λ/ (1 − λ ) , r ( λ ) = r i e λ/ , (3.22)so the black hole shrinks to zero while the observer drifts away. This parametrization yields thedistance between S and Mink as ∆ = 4 c log (cid:18) r f r i (cid:19) . (3.23)However, in this case the horizons are not co-moving with the radial coordinates. That means thatthis notion of distance receives an extra contribution, which does not stem from the pure distancebetween space-times. Hence, Eq. (3.23) gives an upper bound on the distance. The discussion for The RN space-time presents two geodesic observers. One is located at infinity because the space-time is asymp-totically flat. The other one is located between the Cauchy and the black hole horizon, given by r g = Q /M . – 10 –e Sitter space-time is completely analogous . We conclude that the distance between the S anddS to Mink is finite and bounded from above by (3.23).To summarize, Mink ∞ is at infinity distance from RNdS and all its subcases. On the otherhand, parametrizing the metric flow in terms of the geodesic observer, α = r g , allows to constructgeodesics connecting RN, dS and RNdS to Mink . This establishes that the distance between thesespace-times is finite. For S and dS, the flow cannot be parametrized in terms of the geodesicobserver, but we find that these space-times are at finite distance from Mink as well. In the specialcase of Weyl-rescalings the distance to Mink turns out to be infinite in all cases. Eq. (3.13) allows to identify different geodesics connecting spherically symmetric and static space-times with non-vanishing mass parameters. These geodesics are not unique, since the choice ofinitial and final space-times only fixes some of the integration constants c M , c Q and c (cid:96) , but notall of them. Hence, different combinations of exponents and constant yield different geodesics. Asimple strategy for identifying geodesic is the following: set one of the relevant constants appearingon Eq. (3.13) to zero and study the allowed range of λ dictated by the remaining mass parameters.Here, we present some relevant examples of geodesics between space-times.We start by analyzing the geodesic paths from RNdS, SdS and RN to S. The geodesics describingthe flow to the Schwarzschild geometry must be such that the mass of the black hole remains finitewhile Q → and (cid:96) → ∞ smoothly. A simple choice is to set c M = 0 implying that α ∝ M as thenthe mass does not vanish for finite values of λ . In the following we identify geodesics connectingthese space-times to S and infer the geometric distance.• RN → S: Let’s start by setting c M = 0 . Then, M = M i e λ/ ∝ α and Q = Q i e λ/ (cid:112) c Q λ .The charge of the black hole needs to be real-valued and the black hole must remain subex-tremal along the flow of mass parameters. Hence, Q ∈ R = ⇒ λ ≥ − /c Q , (3.24a) M ≥ Q = ⇒ λ ≤ c Q (cid:0) M /Q − (cid:1) , (3.24b)so the range of proper time is bounded λ ∈ [ − /c Q , (( M i /Q i ) − /c Q ] . This geodesicdescribes the flow of an extremal RN black hole which loses all its charge until it becomes aSchwarzschild black hole. The distance is finite and given by ∆ = 4 c log (cid:18) r + , f r + , i + r − , i (cid:19) , (3.25)where r + , f = 2 M f and r ± = M ± (cid:112) M − Q . This follows from α = M = ( r + + r − ) / .Alternatively, the geodesic equation can also be solved for the position of the horizons r ± = r ± , i e λ/ , r ∓ = r ∓ , i e λ/ (1 + c ∓ λ ) . (3.26)Choosing r + = r + , i e λ/ enforces the consistency conditions r − ≥ ⇒ λ ≥ − c − , (3.27a) r − ≤ r + = ⇒ λ ≤ c − (cid:18) r + , i r − , i − (cid:19) . (3.27b) For dS impose that the observer must not cross the cosmological horizon and that the de Sitter radius is real-valued. This solution is equivalent to (3.12) by identifying c M = c Q r ( − , i) / (2 M i ) and c − = c Q . – 11 –ence, the solution of the geodesic equation (3.26) also describes a geodesic connecting anextremal or subextremal RN black hole to S. The distance is finite and explicitly given by ∆ = 4 c log (cid:18) r + , f r + , i (cid:19) . (3.28)Notice that the first geodesic is valid for an observer between the interior and exterior blackhole horizons, while the second one considers an observer outside of the exterior black holehorizon.• SdS → S: A simple choice is c M = 0 such that λ ∈ [0 , − /c (cid:96) ] . Then, α ∝ M = r + r c ( r + + r c )2( r + r c + r + r c ) . (3.29)The limit (cid:96) → ∞ yields α ∝ r + . If we instead set c M = 0 . and (cid:96) = − we find that r + = r + , i e λ/ such that the geodesic is parametrized by α ∝ r + as we have checked numerically.• RNdS → S: There exist a geodesic path connecting the RNdS space-time to Schwarzschildcharacterized by the constants c Q = c (cid:96) = k , such that the charge and cosmological constantvanish at λ = − /k . Then, setting c M = 0 implies α ∝ M , α = ( r + + r c )( r + + r − )( r c + r − )2( r + r c + r − + r − r c + r − r + + r c r + ) (3.30)which reduces to α ( λ = − /k ) = r + , f = 2 M f . Alternatively, upon setting c Q = c (cid:96) we canchoose α = (cid:112) Q(cid:96) = (( r + r − r c ( r + + r − + r c )) / , (3.31)which also reduces to α ( λ = − /k ) = r + , f .Next, we consider geodesic paths connecting RNdS to RN. These geodesics describe the diver-gence of the de Sitter radius while the mass and charge of the black hole remain finite. It sufficesto set c (cid:96) = k and consider the proper time interval λ ∈ [0 , − /k ] . There are different possibilitiesfor the constants c M and c Q , so that the geodesic yields the RN space-time. One such possibilityis to set c M = c Q = 0 and consider the parametrization α = ( r + + r c )( r + + r − )( r c + r − )2 (cid:96) + (cid:114) ( r + + r c ) ( r + + r − ) ( r c + r − ) (cid:96) − r + r − r c ( r − + r + + r c ) (cid:96) ∝ e λ/ . (3.32)Then, in the limit λ → − /k , we recover α → r + with r + = M + (cid:112) M − Q the exterior horizonof the RN black hole.Further, the geodesic paths leading to dS are also finite. The dS space-time is reached as thezero mass (and charge) limit of the SdS (or RNdS) geometries. We consider both cases separately:• SdS → dS: The mass of the black hole needs to vanish at the endpoint of the metric flow.Hence, setting c (cid:96) = 0 and c M (cid:54) = 0 yields (cid:96) = (cid:96) i e λ/ and M = M i e λ/ (1 + c M λ ) . Theconsistency conditions imposed by the CCC and the positivity of the black hole mass restrictthe proper time to a finite interval (cid:96) ≥ √ M = ⇒ λ ≤ c M (cid:18) √ (cid:96) i M i − (cid:19) , (3.33a) M ≥ ⇒ λ ≥ − /c M . (3.33b)– 12 –he geodesic characterized by this flow of the mass parameters connects dS to SdS. Then, theparametrization α can be chosen to be α ∝ (cid:96) = (cid:113) r c + r + r + r c , so the distance betweenSdS and dS is given by ∆ = 4 c log (cid:32) r c , f r , i + r c , i + r + , i r c , i (cid:33) . (3.34)The following choice of constants c M = − and c (cid:96) = 0 . . leads to a parametrization propor-tional to the black hole horizon α ∝ r + .• RNdS → dS: Here we provide two different examples of geodesics connecting the RNdS todS. First, the constants c M,Q,(cid:96) can be set such that α ∝ r c . Here we state that the distanceis finite and given by ∆ = 4 c log (cid:18) r c , f r c , i (cid:19) , (3.35)with r c , f = (cid:96) f . Alternatively, choosing c M = c Q = k and c (cid:96) = 0 yields dS as M, Q → and (cid:96) (cid:54) = 0 . This choice of constants implies α = (cid:96) = (cid:113) r − + r + r c + r − r + + r − r c + r + r c . Thedistance is given by ∆ = 4 c log (cid:96) f (cid:113) r − , i + r , i + r c , i + r − , i r + , i + r − , i r c , i + r + , i r , i . (3.36)Finally, it is left to analyze the geodesic connecting the RNdS to SdS. Again, there exist differentcombinations of the mass parameters that yield the desired behaviour for α . The flow from RNdSto SdS is characterized by the discharge of the black hole. Hence, λ ∈ [0 , − /c Q ] with c Q (cid:54) = 0 .A simple choice for the parametrization characterizing this geodesic is c M = 0 , c (cid:96) (cid:54) = c Q suchthat α = M = ( r + + r − )( r c + r − )( r c + r + ) / (2 (cid:96) ) . Alternatively, setting c (cid:96) = 0 and c M (cid:54) = c Q yields α = (cid:96) = (cid:113) r − + r + r c + r − r + + r − r c + r + r c . Furthermore, the choice c M = c (cid:96) yields theparametrization α = ( M (cid:96) ) / . We summarize the results of this section in Fig. 2, where the distances between space-times areschematically represented. First we demonstrated that Mink ∞ is at infinite distance from all thespace-time configurations that fall into the RNdS class. This is represented by the red dashed lines.On the other hand, Mink is at finite distance from all space-time configuration belonging to theRNdS class. This is indicated by the black solid lines. Note the ambiguity in inferring the distancefrom S and dS to Mink discussed previously.Going further, we studied geodesics and the corresponding geometric distances between variousspace-time configurations with different number of mass parameters that belong to the RNdS family.There always exist a family of geodesics connecting different space-time configuration characterizedby the free constants of integration. However, specific choices are more convenient as we havedemonstrated case-by-case. In more detail, to describe the flow between SdS, RN and RNdS theparametrization in terms of the geodesic observer, α = r g is particularly convenient. As a result,we find that all space-time configurations characterized by at least one mass parameter are at finitedistance from each other. This is indicated by the green lines in Fig. 2. In Sec. 3.3 we analyzed the distances between space-times. We have determined that there alwaysexist at least a minimal geodesic that connects the different space-times between them and to the– 13 – symptotically de Sitter Asymptotically flat
RNdSdS SdS RNSMink Mink ∞ r g r g r g ∞ ∞∞ ∞∞∞ α α Figure 2 . Schematic summary of the moduli space distances between space-times. Dashed lines representinfinite distances, while solid lines stand for finite minimal distances. Mink ∞ lies at an infinite distanceof all other space-times. We show in black the geodesic paths leading to Mink and in green the pathsdiscussed in Sec. 3.3.2. zero mass limit Minkowski space-time Mink . Further, we have also identified a point in modulispace Mink ∞ that represents the Minkowski limit when the mass of the black hole grows infinitely.The next step towards the generalized Black Hole Entropy Distance Conjecture lies in determiningwhether the distance in moduli space can always be expressed in terms of the total entropy of thespace-time, as it was originally done in [6].In order to relate the distance in field space to the entropy of the space-time, we first needto define what we mean with "the entropy of the space-time", since only the entropy of a horizonis a priory unambiguously defined. Hence, in asymptotically flat space-times, the entropy of thespace-time is defined just as a quarter of the area of the exterior black hole horizon. This factreflects that an observer outside of the black hole only has access to the micro-state counting of thedegrees of freedom of the black hole, regardless of the presence of a second Cauchy horizon insidethe black hole , undetectable for the exterior observer.Nevertheless, in asymptotically de Sitter space-times, we are forced to consider observers be-tween two horizons. Further, generally the two horizons radiate at different temperatures and henceform a non-equilibrium system, for which a well defined concept of temperature does not exist.Hence, in the following we briefly review some known examples where the temperature andthe entropy of the space-time can be defined unambiguously. We then extend this concept tonon-equilibrium systems to suggest an upper bound for the entropy of the space-time. The entropy of asymptotically flat space-times has been extensively discussed, it is given by aquarter of the area of the exterior black hole horizon [10, 11]. The entropy of asymptotically de As is the case for a Reissner-Nordstrøm black hole or the Reissner-Nordstrøm-de Sitter space-time. – 14 –itter space-times has also been tackled, and exact results have been obtained for space-times inthermal equilibrium [47, 48]. Nevertheless, how to estimate the total entropy of space-times inabsence of thermal equilibrium is still under debate. Usually, the total entropy of the space-time,that we denote by S , is just taken to be the sum of the entropies of the horizons surrounding theobserver [34, 49]. Here we argue that this estimate is an upper bound to the total entropy of thespace-time.For the sake of completeness, we start by briefly reviewing the results of [47, 48] regarding thetotal entropy of space-times in thermal equilibrium.In order to define the thermodynamic quantities of a bifurcated horizon we work in the Euclideanpath integral approach. Then, following the prescription of [47], we firstly analytically continue themetric of the space-time to the Euclidean sector by performing the transformation t → iτ . Next, weidentify the periodicity of the Euclidean time such that all conical singularities are removed fromthe metric. Notice that in asymptotically de Sitter space-times, we are forced to consider observersbetween the (exterior) black hole and the cosmological horizons. Hence, in general we have todeal with conical singularities at r = r c and r = r + , which cannot be removed simultaneously [50].Nevertheless, there exist some special cases within the physical phase space of the RNdS and the SdSspace-times such that some of the horizons radiate at the same temperature, and consequently, suchthat the conical singularities of the Euclidean section of the metric can be removed by choosinga certain periodicity for the Euclidean time. We refer to these space-times as being in thermalequilibrium.The space-time in the Euclidean section describes a micro-canonical ensemble, and hence aclosed thermodynamic system at fixed energy [47]. The partition function of such an ensemble isgiven by Z = e − I , where I is the Euclidean action [47] I = − π (cid:90) M d x √ g ( R − − F ) + 18 π (cid:90) Σ d x √ γK , (4.1)and K is the trace of the extrinsic curvature of the hypersurface Σ with induced metric h .The partition function Z shall be interpreted as the density of states, so the entropy is just S = log Z = − I . (4.2)There is a unique Schwarzschild-de Sitter space-time in thermal equilibrium, the Nariai limit .Notice that for the conical singularities at r = r + , c to be simultaneously removed, the surfacegravity of both horizons need to coincide, thus yielding a vanishing temperature of the horizons inthe Nariai limit T c = T + = 0 . Hence, the entropy of the ensemble is given by [51] S = 2 πr g = 2 A g , (4.3)where r g is the position of the geodesic observer (3.17).The RNdS space-time presents four configurations in thermal equilibrium [44]. The explicitcomputation of the entropy of these space-times can be found in [47]. Firstly, the lukewarm solution( M = | Q | line in Fig. 1) is such that the exterior black hole and the cosmological horizons are inthermal equilibrium at finite temperature ( T + = T c ). Then, the conical singularities at r = r + , c can be removed by identifying the Euclidean time periodicity with τ ∼ τ + πκ + , c . This identificationyields S = π(cid:96) ( (cid:96) − M ) = 14 ( A + + A c ) . (4.4) The discriminant locus (2.6) with zero charge reads D = 4 (cid:96) ( (cid:96) − M ) , so the degenerate case occurs when the mass and de Sitter radius are not independent, but satisfy (cid:96) = 3 √ M . – 15 –or the total entropy for the lukewarm space-time. Next, the cold or extremal space-time (upperbranch in Fig. 1) is characterized by a degenerated black hole horizon at zero temperature. Thedegenerate horizon at r = r − = r + is at an infinite proper distance along the space-like directions ofany point r in the space-time. Hence, the cold Euclidean metric only presents one conical singularityclose to r = r c that can be avoided by identifying τ ∼ τ + πκ c , which yields S = πr c = A c . (4.5)Similarly, the ultra-cold space-time (A.4) presents a triple degenerated horizon at null temperature.Hence, expressing the Lorentzian ultra-cold metric as Mink (1 , × S in Rindler coordinates yields S = 2 π Λ , (4.6)while using the usual coordinates leads to the Mink (1 , × S topology for the Lorentzian section [47]and zero entropy.Nevertheless, in absence of thermal equilibrium, there is at least one conical singularity thatcannot be removed from the Euclidean section of the metric. Hence, the Euclidean metric cannot bemade regular and the concept of temperature is ill-defined. This issue can be avoided by isolatingthe black hole and and the cosmological horizons into two distinct ensembles. This procedure wasoriginally described by Hawking [52] and further explored in [53].In order to do so, we place an imaginary perfectly reflecting wall at r c ≥ r B ≥ r + . The observeris located at r B , so it can measure the state variables of both ensembles. After a finite period of time,the isolated regions r ∈ [ r + , r B ] and r ∈ [ r B , r c ] , that we denote by E + and E c , reach the thermalequilibrium with the corresponding horizons. Hence, the periodicity of the Euclidean time is set ineach ensemble as to avoid the conical singularity of the enclosed horizon, i.e., τ ∼ τ + β + ∗ = 2 π/ ˆ κ + for E + and τ ∼ τ + β c ∗ = 2 π/ ˆ κ c for E c . Here we denoted by ˆ κ h = | V (cid:48) ( r h ) | the surface gravitynormalized at infinity. Further, the temperatures of the canonical ensembles E + , c can be computedto be β + = T − = (cid:90) β + ∗ d τ √ g Eττ = 2 π ˆ κ + (cid:112) V ( r B ) , β c = T − c = (cid:90) β c ∗ d τ √ g Eττ = 2 π ˆ κ c (cid:112) V ( r B ) , (4.7)with V ( r B ) the SdS or RNdS metric function evaluated at the position of the heat wall.The Euclidean action and the Helmholtz free energy of the ensembles E + and E c can be computedanalogously to [53]. The thermodynamic state variables follow by using the usual definitions ofentropy and energy [54]. As discussed in [53], the total entropy of the space-time can be evaluatednumerically, thus yielding S ∼ π ( r + r c ) + f ( r + , r c , r B ) , (4.8)where f ( r + , r c , r B ) measures the discrepancy with respect to the usual definition of the total entropyof the space-time when there is thermal equilibrium . This function f ( r + , r c , r B ) ≤ is negativefor arbitrary values of r + , r c and r B and becomes strictly zero when we consider the thermal wallover one of the horizons f ( r + , r c , r B = r i ) = 0 with i = { + , c } . Therefore, this total entropyestimate is bounded by S ≤ π ( r + r c ) . (4.9) The detailed computation yielding this result will be presented elsewhere. This result is compatible with the construction presented in [49], where a thermodynamic ensemble for the SdSspace-time is built by considering the cosmological horizon as a thermal bath. The total entropy of the ensemble isthen defined as the sum of the entropies of the black hole and cosmological horizon. – 16 – ath α Entropy ∆ ∼ log S Weyl rescaling √S S (cid:88) { RNdS, SdS, RN } →
Mink r g ? ( × ) Nariai → Mink r g πr g (cid:88) S → Mink πM × dS → Mink π(cid:96) × RN → S r + πr (cid:88) { SdS, RNdS } → S r + ? ( (cid:88) ) RNdS → RN (cid:113) M + (cid:112) M − Q ? ( (cid:88) ) { RNdS, SdS } → dS r c ? ( (cid:88) ) RNdS → SdS M , (cid:96) ? ? Table 1 . Summary table of the entropy-distance relation for different geodesic paths. The first columndenotes the path under consideration. In the second column, the convenient choice for α is presented, ifapplicable. The third column summarizes the relevant entropy in each case. The last column shows whetherthe entropy-distance relation holds true, as indicated by (cid:88) . The symbol ( (cid:88) ) means that the relation holdstrue in the limit while ( × ) indicates inconclusive results. After having discussed the entropy of multihorizon space-times, we can now proceed and relate theentropy to the distance between them. In the following we will denote the entropy of a space-timeby S . In this section we wish to generalize the entropy-distance relation ∆ ∼ log S (4.10)found for some restricted cases in [6] to arbitrary space-times that belong to the RNdS family. Sincethe entropy of a space-time is not well-defined for systems out of equilibrium, we will test (4.10)case by case. Our results are summarized in table 1. ∞ –limit The limit Mink ∞ is reached for proper time λ → ∞ and hence is at infinite distance for everyparametrization α . For simplicity we first restrict ourselves to those parametrizations such thatthe metric flow corresponds to Weyl rescalings. In this case, all mass parameters depend on λ onlyexponentially as in Eq. (3.12) for c M = c Q = c (cid:96) = 0 . First, we focus on space-times where anobserver has access to only one horizon, i.e. S, dS and RN. Let r h denote the radial position of ahorizon, then r h ∝ e λ/ due to dimensional reasons. Hence, the choice α = r h corresponds to Weylrescalings. In these cases the entropy of the space-time is unambigously given by the Bekenstein-Hawking formula S = 4 πr h . It immediately follows the relation α = (cid:112) S / π and hence Eq. (4.10).Turning to the case where the observer outside the black hole has access to two horizons, i.e. SdSand RNdS, we can draw the same conclusion. The entropy of the space-time must flow with λ as S ∝ e λ/ again by dimensional reasons. Hence, the parametrization α ∝ S / represents Weylrescalings also in these cases. This establishes the entropy-distance relation (4.10) even if the preciseform of S is not known for these multihorizon space-times. We conclude that the entropy-distancerelation (4.10) holds true for every space-time configuration belonging to the RNdS family in thecase of Weyl rescalings. Let us emphasize that Eq. (4.10) holds true in general and not only in thelimit Mink ∞ . – 17 – .2.2 Entropy-distance relation in the Mink –limit We proceed by testing the entropy-distance relation in the limit of Mink . For the cases where anobserver has access to only one horizon that we denote by r h , i.e. S, dS and RN, the parametrization α = r h leads to an infinite distance. As in the case of Mink ∞ , the Bekenstein-Hawking formulaimmediately yields α ∝ S / . Therefore, Eq. (4.10) also holds true in the zero entropy limit.However, other parametrizations are possible, which do not correspond to Weyl rescalings and yieldfinite distance. The entropy-distance relation cannot be established for these types of flows.Moving on to cases where the observer has access to two horizons, i.e. SdS and RNdS, we haveestablished α = r g as appropriate parametrization. To relate r g to the entropy, we first discussthose multihorizon space-times for which a notion of entropy exists, i.e. space-times in thermalequilibrium. As discussed in Sec. 4.1, in the case of Nariai black hole, as well as extremal RN blackhole (for an observer between the interior and exterior black hole horizons), the total entropy canbe written as S = 2 A g / πr g . This indeed implies Eq. (4.10) to be exact for these cases. Forthe lukewarm, cold and ultra-cold space-time however, there is no geodesic to Mink that maintainsthermal equilibrium. Hence, the entropy-distance relation cannot be tested in these cases.Finally, we study the entropy-distance relation for space-times out of thermal equilibrium,where the observer has access to two horizons, i.e. SdS, RN, RNdS. The geodesic observer yieldsthe convenient parametrization here, α = r g , so that the distance of these space-times to Mink isgiven by ∆ = 4 c log (cid:18) r g , f r g , i (cid:19) . (4.11)Since the limit is arrived at after a finite amount of proper time, the distance is finite. The entropy-distance relation (4.10) would hold in these cases, if the entropy of these space-time was related tothe geodesic observer as S = 2 πr g . As an example, consider the path SdS → Mink . The totalentropy of SdS needs to satisfy Eq. (4.9). Now we test whether the estimate S = 2 πr g for the totalentropy of SdS is a good ansatz. Expressing this ansatz in terms of the individual entropies of theblack hole horizon S + and the cosmological horizon S c yields S = 2 (cid:18) S + S c ( S + + S c + 2 (cid:112) S + S c ) (cid:19) / (4.12)This ansatz respects the inequality S ≤ S + + S c as we expect for the total entropy of a multi-horizonspace-time. However, in the limit without black hole, i.e. pure dS, our ansatz does not reduce tothe entropy of the de Sitter horizon, but instead S → as S + → . We conclude that πr g does notcorrespond to the entropy of multi-horizon space-times out of thermal equilibrium. Although thisindicates that the entropy-distance relation cannot be established in these cases, we would need totest the distance (4.11) against the analytical expression of the total entropy of these space-timesto have a fully conclusive result. Here we briefly identify whether the geodesic distance between the space-times with non-zero massparameters described in Sec. 3.3.2 can be related to the total entropy of the space-time.We identified a geodesic connecting the RN and to S parametrized by α = r + . Therefore, itfollows that the distance between these space-times can be expressed as a function of the totalentropy of the space-time ∆ = 2 c log (cid:18) S f S i (cid:19) . (4.13)Here, the final entropy is the one of Schwarzschild space-time, i.e. S f = 4 πM , and the initalentropy is the one of RN for an observer outside the black hole, i.e. S i = π ( M i + (cid:112) M − Q ) .– 18 –or the paths connecting SdS and RNdS to S there is no well-defined notion of entropy. However,in the Schwarzschild-limit of these paths, the parametrization becomes proportional to the mass ofthe black hole, α ∼ M . This implies that α ∝ S / where S = 4 πM asymptotically. Hence, theentropy-distance relation (4.10) holds true in the limit. Finally, we apply our previous findings to the evaporation of black holes. We want to test whethera toy model of black hole evaporation passes the known Swampland criteria [2, 3].As such, we consider a black hole with charge because global symmetries are not allowed [55, 56].We thus study the Reissner-Nordstrøm-de Sitter black hole here. Classically, it is stable [57], butit can decay when quantum effects are taken into account. Specifically, the black hole can loosecharge and mass via the Schwinger effect [58] while both the black hole and cosmological horizongive rise to Hawking radiation [11, 12, 52].The discharge of a black hole through the combined effect of Hawking and Schwinger can bestudied in two regimes. These are controlled by the Schwinger transition rate, Γ ∼ e − m qE , (5.1)where m and q are the mass and charge of the particle produced by the Schwinger effect and E is the electric field due to the charged black hole. In the regime of adiabatic discharge where m (cid:28) qE the Schwinger effect is exponentially enhanced and the black hole can loose all its chargevery rapidly. For initial black holes close to the charged Nariai branch the adiabatic dischargeevolves the space-time towards a superextremal neutral Nariai solution, which is outside the phasespace D . Hence, this regime evolves the space-time from a physical to an unphysical solution inthe sense that the Cosmic Censorship Conjecture gets violated dynamically. The opposite regimeis the quasi-static discharge where m (cid:29) qE . Since the Schwinger production rate is exponentiallysuppressed, the discharge happens very slowly. As a result, any space-time that is initially withinthe phase space D remains within the phase space during the entire evaporation process. In thissense, the quasi-static discharge connects physical solutions of Einstein’s equation with each other.This observation motivated the authors of Ref. [24] to formulate a de Sitter version of the WeakGravity Conjecture (WGC) [26]: every particle in the spectrum must satisfy m > qgM P H in orderto avoid super-extremality. In the following, we take the WGC of [24] as basis such that the blackhole discharges quasi-statically.Finally, we consider the cosmological constant to be nonzero and positive to make contact withour Universe. To respect the de Sitter conjectures [27–33], we allow the cosmological constant tovary in time. On time scales on which the mass and charge of the black hole vary significantly ,the drift of the cosmological constant has to be taken into account [59].It remains to test black hole evaporation against the Distance Conjecture [4] and in particularthe Black Hole Entropy Distance Conjecture [6], which we generalized in the previous section toapply it to evaporation. This is the subject of the present section.We work out the technical details in appendix C. We closely follow the analysis of [24], butextend to a time-varying cosmological constant. We first identify the endpoint of the evaporationprocess and present the corresponding trajectories in the phase space D . We then emulate thesetrajectories by moduli space geodesics. We take the time scale to be of the order of the geometric time scale, i.e. τ ∼ M in Hubble units. As a first orderestimate, a quasi-static process would take δτ ∼ million years for the mass of the black hole to vary δM/M ∼ . for sufficiently massive black holes. – 19 – .1 Phase space trajectories of evaporation We discuss the evaporation trajectories first for a general RNdS black hole and then specify toSdS and Nariai. We leave all technical details to appendix C as we closely follow the perturbativeprocedure of [24]. The accumulated effect of Hawking and Schwinger radiation back-reacts on thegeometry in a quasi-static way such as to preserve staticity and spherical symmetry. In other words,we restrict ourselves to a self-similar evaporation.We extend the analysis of [24] by allowing the cosmological constant to vary in time. We usethe first law of black hole mechanics with multiple horizons as presented in Ref. [34] .In this section we discuss physical evaporation processes. We therefore restore the constantsthat we have set to unity in the beginning. We normalize the length scales to the value of thecosmological horizon today, (cid:96) = 1 . × m. This amounts to define the dimensionless radialcoordinate ˜ r = r/(cid:96) as well as the dimensionless mass parameters ˜ M = M(cid:96) = Gmc (cid:96) , ˜ Q = Q(cid:96) = q (cid:115) g G π(cid:96) , ˜ (cid:96) = (cid:96)(cid:96) . (5.2)In the rest of this section, we will suppress the tilde for the sake of visibility. As summarised in appendix C.3, the system of differential equations can be decoupled. It is con-venient to introduce the variables x = M/(cid:96) and y = Q/(cid:96) . The system of differential equations canthen be written as ˚ x = 4 πr g (cid:96) (cid:34) (cid:32) G (cid:113) V ( r g ) + V ( r g ) r g − x(cid:96) r c − r (cid:33) T g − (cid:32) Qr g + r g − x(cid:96) r c − r (cid:18) Qr + − Qr c (cid:19)(cid:33) J g (cid:35) ˚ y = − πr g (cid:96) (cid:34) y(cid:96) r c − r V ( r g ) T g + (cid:18) − y(cid:96) r c − r (cid:18) Qr + − Qr c (cid:19)(cid:19) J g (cid:35) . (5.3)Here, the operator ˚= d / d t g denotes derivative with respect to the proper time of the geodesicobserver. Further, the Hawking flux is given by T g = σ (4 π ) V ( r g ) ( r c | V (cid:48) ( r c ) | − r | V (cid:48) ( r + ) | ) (5.4)as in Eq. (C.17). The expression for the Schwinger flux J g is too lengthy to display here, but canbe found in Eq. (C.18).This system of first order differential equations can be solved numerically for arbitrary initialspace-time configurations within the phase space region D . In Fig. 3, we depict the evaporationflow of the mass parameters of a RNdS space-time in the M/(cid:96) − Q/(cid:96) phase space as dictated byEq. (5.3). The boundary of the physical phase space ∂ D and the lukewarm line are depicted inblack solid and dotted, resp. Two regimes are appreciable: We derive the dynamics of the cosmological constant from the first law of thermodynamics in asymptotic deSitter space-times. The cosmological constant needs to be considered a state variable, which enters the first law as apressure term conjugated to the volume of the space-time [34, 37]. Although classically the laws of thermodynamicsare only applicable to stationary black holes [60], using the first law of thermodynamics as a dynamical equation isjustified for the following reasons. First, we consider a quasi-static regime of evaporation. Second, the formalismused to derive the first law of thermodynamics in asymptotically de Sitter space-times in [34] is equivalent to treatingone of the horizons as a boundary [61]. Hence, the first law derived in [34] fits into the formalism of IsolatedHorizons [62, 63], which in turn justifies that the thermodynamic equations are enough to consider a consistentHamiltonian evolution [64]. – 20 – For large values of
M/(cid:96) and
Q/(cid:96) a process of anti-evaporation takes place, i.e. the mass of theblack hole increases while its charge decreases. In this regime, the space-time slowly evolves inthe direction of the Nariai branch. The black hole is too large as to compensate the radiationcoming from the cosmological horizon, that increases the mass of the black hole. As the massof the black hole increases, its temperature follows, until the black hole is hot enough as torevert the mass flux coming from the Hawking radiation. However, the thermal equilibriumis never reached for initially non-Nariai space-times. This feature is expected from the thirdlaw of thermodynamics: the zero temperature limit cannot be reached in finite time. Instead,the space-time slowly evolves towards an evaporation regime, where both the mass and chargedecrease.• For smaller charge and mass ratios, the mass parameters evolve toward the lukewarm line.The mass loss dominates the process, while the de Sitter horizon increases. In this regimethe black hole temperature is higher than that of the cosmological horizon, so there is a netmass flux from the black hole to the cosmological horizon. Once the lukewarm line is reached,the exterior black hole horizon and the cosmological horizon reach thermal equilibrium, sothe net Hawking radiation cancels out. Notice that the mass loss of the black hole is notonly due to the emission of neutral pairs, but also due to Schwinger. This is reflected in thefact that Eq. (5.3) presents a term ˚ x ∼ J g . Consequently, the black hole evaporation anddischarge continues along the lukewarm branch until empty de Sitter space-time is reached.This evaporation is driven solely by the Schwinger effect, as it can be seen in Eqs. (5.3).During the whole process of evaporation, the black hole losses its charge until the neutral limitis reached. This is also true along the Nariai and extremal branches, since the Schwinger flux iswell defined along the whole phase space D , also along the Nariai branch. Further, the Schwingercharge flux is negative along the whole physical phase space, so there is a net charge loss that onlystops when the black hole depletes completely.The origin ( M/(cid:96), Q/(cid:96) ) = (0 , not only represents empty de Sitter space, but also the family ofReissner-Nordstrøm black holes in asymptotically flat space-time. Our discussion in section 5.1.2will elucidate that the endpoint of evaporation is indeed empty de Sitter space-time. Hence, wecan summarize this section by stating: initially non-Nariai space-times evolve to empty de Sitterspace-time . Next, let us study the Schwarzschild-de Sitter (SdS) space-time by setting Q = ˙ Q = 0 . We studythe evaporation equations for the mass of the black hole and the cosmological radius according tothe Gibbons-Hawking model presented in Appendix C.In the neutral limit, the evolution equations (C.27) reduce to ˚ M = 4 πr g (cid:32) G (cid:113) V ( r g ) + r g V ( r g ) r c − r (cid:33) T g , ˚ (cid:96) = 4 πr g (cid:96) r c − r V ( r g ) T g . (5.5)The Schwinger effect is absent for neutral black holes. The evolution of mass M and de Sitterradius (cid:96) are depicted in Fig. (4).Let us first discuss the Nariai branch, that is represented by the black solid line in Fig. 4. Boththe black hole and the cosmological horizons have the same temperature such that the system isin thermal equilibrium. Hence, there is no net Hawking radiation between the horizons. FromEqs. (5.5), we see that the mass of the black hole and the cosmological constant remain constantin this limit.However, the Nariai branch is unstable. Space-times infinitely close to the Nariai branch donot evolve towards thermal equilibrium, but loose mass while (cid:96) remains almost constant. As before,– 21 – .00 0.05 0.10 0.15 0.20 0.250.000.050.100.150.200.25 Figure 3 . Phase space evaporation of RNdS black holes in the
M/(cid:96) - Q/(cid:96) -plane. The black solid linerepresents the boundary of the physical phase space ∂ D . The dotted black line indicates the lukewarmline. The blue arrows represent the flow of the mass parameters due to the evaporation by Schwinger andHawking effect. While for large values of these ratios a period of anti-evaporation occurs, the space-timefirst looses mass and then evolves along the lukewarm line in the other regions of D . The endpoint of theevaporation process is de Sitter space-time filled with thermal radiation. Figure 4 . Phase space evaporation of the Schwarzschild-de Sitter black holes in the M − (cid:96) plane. TheSchwarzschild black hole fully evaporates to empty de Sitter space-time. The space-times initially in theNariai limit do not evaporate. – 22 –his behaviour is expected due to the third law of black hole mechanics, non-initially cold blackholes cannot evolve towards cold ones in finite time.Instead, space-times that are initially away from the Nariai branch evolve towards empty deSitter space-time, as can be seen from Fig. 4. The de Sitter radius decreases slightly during theprocess, but it remains finite and positive. Therefore, SdS black holes evaporate completely. Duringthis process the background becomes more accelerated due to the disappearance of the pull of theblack hole. Finally, we discuss the evaporation of charged Nariai space-times. As mentioned before, the inte-grated Schwinger flux (C.18) is not valid along the Nariai branch, for which r + → r c . Nevertheless,the Schwinger flux is well defined along the whole phase space D , including the Nariai branch.The integrated expression for the Schwinger flux along the Nariai branch can be obtained throughinstantonic methods, for instance see [24, 65]. In this section we show that a Nariai space-timedischarges along the Nariai branch until the neutral limit is reached. Firstly, notice that the chargeflux is negative along the whole phase space D . The charged Nariai branch is not stable althoughthe black hole and the cosmological horizon are in thermal equilibrium. Initially Nariai space-timesexperience a mass and charge loss due to the Schwinger effect. The mass loss along the Nariaibranch occurs due to the energy loss of the pairs produced when the charged carriers are repelledfrom the black hole horizon.Along the Nariai branch the evaporation equations (5.3) simplify tod x d y = y ˜ r g ˜ r g − x ˜ r c − ˜ r ˜ r g y (cid:16) y ˜ r + − y ˜ r c (cid:17) − y ˜ r c − ˜ r (cid:16) y ˜ r + − y ˜ r c (cid:17) , (5.6)because the Hawking flux vanishes, T g = 0 . Here, the tildes indicate that the radial quantities arenormalized by the de Sitter radius, i.e., ˜ r a = r a /(cid:96) for a = { + , g , c } . From the definition of thegeodesic observer V (cid:48) ( r g ) = 0 it follows that y = ˜ r g ( x − ˜ r g ) . Plugging this into Eq. (5.6) we findthe following simple relation d x d y = y ˜ r g . (5.7)Along the Nariai branch the black hole and cosmological horizon coincide with the position ofthe geodesic observer, r + = r c = r g . In this limit, the expression for the geodesic observer vastlysimplifies, ˜ r g = 1 √ (cid:113) (cid:112) − y . (5.8)Hence, we can integrate Eq. (5.7) analytically to x − x = √ (cid:90) yy d y (cid:48) y (cid:48) (cid:113) (cid:112) − y (cid:48) = 13 √ (cid:18)(cid:113) y + (1 − y ) / − (cid:113) y + (1 − y ) / (cid:19) , (5.9)which coincides with the expression for the Nariai branch (A.3). This simple computation showsthat initial Nariai space-times characterized by ( x , y ) evolve along the Nariai branch, slowlydischarging until the neutral Nariai limit is reached.Finally, this result together with the evaporation paths in phase space for non-Nariai space-times shows that the physical evolution of RNdS space-time always maps physical space-times tophysical space-times given our assumptions. – 23 – .2 Evaporation paths and moduli space geodesics We are finally in the position to relate the geometric distance between space-time configurationsto the physical evaporation process that connects different space-times. In this section, we willconstruct paths in the moduli space of RNdS space-times that solve the geodesic equation (3.2)and emulate the physical evaporation process as presented in Fig. 3. This allows to determine thegeometric distance between any initial space-time configuration and its final state that is arrivedat by a physical evaporation process.
As a warm-up, we first analyze the simple case of a charged Nariai black hole. As discussed inSec. 5.1.2, an initially Nariai space-time discharges following the Nariai branch until the SdS limitis reached. Hence, we need to find a geodesic path whose projection on the phase space followsthe Nariai branch. The problem reduces to finding an appropriate parametrization α . For Nariaispace-times, the outer black hole and cosmological horizon coincide with the geodesic observer, r + = r c = r g . Here we show that α = r g parametrizes a geodesic along the Nariai branch.Rearranging V ( r g ) = 0 we find that the mass and the charge are related to the location of thedegenerate horizons or the geodesic observer as M = r g (cid:32) − r g (cid:96) (cid:33) , Q = r g (cid:32) − r g (cid:96) (cid:33) . (5.10)Hence, it is enough to show that the mass, charge and de Sitter radius evolution (3.12) is compatiblewith Eq. (5.10) when α = r g . Using the ansatz r g = r g , i e λ/ and rearranging Eq. (5.10) yields M = M i e λ/ (cid:32) − r g , i c (cid:96) (cid:96) − r g , i λ (cid:33) , Q = Q i e λ/ (cid:115) − r h, i c (cid:96) (cid:96) − r g , i λ (5.11)where M i = r g , i (1 − r g , i /(cid:96) ) and Q = r g , i (1 − r g , i /(cid:96) ) . Further, identifying the constants c M = − r h, i c (cid:96) (cid:96) − r h, i and c Q = − r h, i c (cid:96) (cid:96) − r h, i , we recover Eq. (3.12).The initial mass, charge and de Sitter radius need to be related through Eq. (A.3). The largestpossible values for the ratios x = M/(cid:96) and y = Q/(cid:96) correspond to the ultra-cold point,
M(cid:96) = 13 (cid:114) , Q(cid:96) = 1 √ . (5.12)Hence, in order to define a geodesic which connects the ultra-cold point to the neutral Nariai limitwe set ( x i , y i ) to (5.12). This geodesic describes the complete discharge of the black hole along theNariai branch. Hence, the proper time ranges over λ ∈ [0 , − /c Q ] such that Q ( λ = − /c Q ) = 0 .This choice of constants fully specifies the geodesic connecting the ultra-cold and the neutral Nariaispace-times.Summarizing, the parametrization α = r g defines a geodesic that evolves Nariai solutions toNariai solutions. The neutral Nariai black hole is at finite distance from the ultra-cold Nariaiblack hole (as well as any other charged Nariai black hole) when following the physical evaporationtrajectory. During this process of evaporation the horizon radius r g increases monotonically. Withthe parametrization α = r g , Eq. (3.7) implies for the geometric distance ∆ = 2 c log (cid:18) r g , f r g , i (cid:19) , (5.13)with r g , f and r g , i the final and initial value of the degenerate horizons and the geodesic observer.Therefore, any charged Nariai space-time is at finite distance of its neutral limit.– 24 – .00 0.05 0.10 0.15 0.20 0.250.000.050.100.150.200.25 Figure 5 . Comparison of the moduli space geodesics (red lines) given by Eqs. (5.14) for arbitrary initialconditions and the phase space evaporation in phase space for the family of RNdS space-times (blue arrows).
Next, we generalize to non-Nariai space-time configurations. For the evaporation we do not havethe analytical expression describing the trajectories in phase space. However, since the geometricdistance (3.7) does only depend on the initial and final points, we use that empty de Sitter is the finalstage of evaporation. The general solution of the geodesic equation imposes the conditions (3.5) forthe parametrization α and the mass parameters. Conditions (3.5) were already solved in section 3.2,leading to Eq. (3.12) that determine the dependence of the mass parameters with the moduli spaceproper time. In order to fully specify the moduli space geodesic, we still need to choose α such thatthe observer at r = ˜ rα does not hit a curvature singularity along the evolution.The evaporation process can be described in terms of the ratios x = M/(cid:96) and y = Q/(cid:96) . Hence,in order to ease the comparison between the evaporation paths and the projection of the modulispace geodesics, we work in terms of the ratios x and y . These evolve along the flow as x ( λ ) = x i (1 + c y λ ) (cid:112) c (cid:96) λ , y ( λ ) = y i (cid:112) c y λ (cid:112) c (cid:96) λ . (5.14)The endpoint of evaporation is empty de Sitter space, which is arrived at λ = − /c y . The constant c y can be absorbed in the definition of the proper time λ , so the geodesics depend only on one freeparameter c (cid:96) .The cosmological horizon is defined along the whole evaporation process, so it smoothly in-terpolates between the initial cosmological horizon r c , i (cid:54) = (cid:96) i and the de Sitter radius in the purede Sitter limit lim λ → r c = (cid:96) f . Hence, we choose the cosmological horizon as our parametrization α = r c16 . The projection of the resulting geodesic paths onto the phase space are represented inFig. 5 by solid red lines for different initial conditions. As can be seen, these geodesics stay withinthe physical region D for the entire interval between the initial and final point of evaporation.Hence, we can utilize this class of geodesics to infer that the geometric distance between any initial We fix the free parameter c (cid:96) through the requirement that lim λ → r c = (cid:96) f is satisfied. – 25 – .00 0.05 0.10 0.15 0.20 0.25 0.30 0.350.00.51.01.52.0 Figure 6 . Comparison of the moduli space geodesics (red lines) given by Eqs. (3.12) for arbitrary initialconditions and the phase space evaporation in phase space for the family of SdS space-times (blue arrows). and final point of black hole evaporation is given by ∆ = 4 c log (cid:18) (cid:96) f r c , i (cid:19) = c . (5.15)Here r c , i is the cosmological horizon of the initial space-time and (cid:96) f the cosmological horizon of thefinal space-time, which is simply given by the de Sitter horizon as the black hole has evaporatedcompletely. Note that the distance is independent of c (cid:96) .Different choices of the integration constants of (3.12) yield different geodesics parametrizedby distinct functions α of the mass parameters. For these other possible families of geodesics weexpect that the distance to empty de Sitter space is again given by Eq. (5.15).Next, we discuss the Schwarzschild-de Sitter case. Again, the cosmological horizon providesa convenient parametrization, α = r c . It yields smooth, real-valued paths in moduli space alongthe evaporation trajectories. In Fig. 6, we show the projection of the moduli space geodesics overthe phase space (red lines) together with the evaporation paths (blue arrows) leading to emptyde Sitter space. Notice that in this case the projected geodesics parametrized by r c stay withinthe physical region. Similarly to the RNdS case, the evaporation paths do not coincide with themoduli space geodesics. Nevertheless, they can be used to compute the distance from SdS to dS.The distance is again finite and explicitly given by Eq. (5.15), with r c , i the cosmological horizon ofthe SdS geometry. The path with parametrization α = r c is smooth and real-valued for λ ∈ [0 , .Summarizing, our toy model describes the evaporation of a general RNdS black hole. The finalpoint of evaporation is empty de Sitter space-time. For the special case of charged Nariai black holes,our model predicts discharging along the Nariai branch until the neutral Nariai limit is reached. Inthe neutral limit, the model predicts that the black hole losses mass and evaporated towards emptyde Sitter space. By emulating the evaporation trajectories by geodesics in the moduli space, weassigned a geometric distance to the evaporation process. Our analysis shows that this distance isfinite. For the Nariai case, the geodesics coincide exactly with the evaporation trajectory in phasespace. – 26 – Summary and conclusions
In the first part of the paper, we reviewed to concept of the moduli space of metrics. Specializingto spherically-symmetric and static space-times, we further refine the prescription of [6] to computedistances along geodesic paths in the moduli space. Let λ be the proper time of the geodesic path,then we demonstrated that the geometric distance ∆ is always given by ∆ = c | λ f − λ i | , (6.1)where c ∼ O (1) . Here, λ f , i refer to the proper time at the final and initial point that are connectedby a geodesic path, respectively. This result establishes that space-times are at infinite distancefrom each other if and only if they are separated by infinite proper time.Building over this, we moved on to study distances between various space-time configurationsthat belong to the family of Reissner-Nordstrøm-de Sitter, which is the most general, sphericallysymmetric and static space-time. We find that all space-times with at least one mass parameter areat finite distance from each other. We further find that the Minkowski limit of infinite mass (andcharge) is at infinite distance from every spherically-symmetric and static space-time. On the otherhand, the Minkowski limit, where all mass parameters vanish identically, can be reached at finitedistance. However, due to the freedom to parametrize the geodesics, this same Minkowski limit canalso be reached at infinite distance. In particular, geodesics that represent Weyl rescalings yieldinfinite distance to both aforementioned Minkowski limits.As next step, we related the distances to the entropy of the space-time, which we denote by S .We first discuss how to define the entropy of multihorizon space-times. An unambiguous definitionis possible only for those space-times that are in thermal equilibrium. In a thorough case by casestudy we find that the distance can be related to the entropy as ∆ ∼ c log S . (6.2)However, for the cases where a multi-horizon space-time flows towards a single-horizon space-time,this relation is valid only asymptotically. For the cases where a single- or multi-horizon space-timeout of thermal equilibrium flows towards the zero-mass Minkowski limit, this relation could not beestablished. For all other cases and in particular for Weyl rescalings, the relation holds true.Finally, we applied our results to black hole evaporation via the combined Hawking andSchwinger effect. By employing a model based on perturbative and thermodynamic considera-tions, we describe self-similar evaporation of RNdS black holes. We find that a general RNdS blackhole evaporates towards de Sitter space-time filled with thermal radiation. The special case of acharged Nariai black hole discharges until it becomes a neutral Nariai black hole. Combined withour previous results, we find that an evaporating RNdS black hole travels a finite distance to reachits final stage. We hence expect the entire evaporation process to lie on the landscape.As a next step, it would be interesting to establish a concept of entropy for the case of multi-horizon space-times that are out of thermal equilibrium. A precise notion of entropy in these caseswill allow to further test the entropy-distance relation. Acknowledgments
We thank Gia Dvali and Gerben Venken for very useful discussions. M.L. acknowledges supportfrom a PhD grant from the Max Planck Society. The work of D.L. is supported by the OriginsExcellence Cluster. – 27 –
Details on the structure of RNdS space-time
In this appendix, we provide some explicit expressions for the RNdS space-time.The location of the causal horizons are given by the roots of the quartic polynomial V ( r ) | r = r h =0 with the metric function V ( r ) given by Eq. (2.4). It is straight forward to find that three of theroots are explicitly given by r − = (cid:96) √ (cid:32) − (cid:114) ρ Θ + Θ + (cid:115) − ρ Θ − Θ + 12 √ M/(cid:96) (cid:112) ρ Θ + Θ (cid:33) ,r + = (cid:96) √ (cid:32)(cid:114) − ρ Σ − Σ − (cid:115) ρ Σ + Σ − √ M/(cid:96) (cid:112) − ρ Σ − Σ (cid:33) ,r c = (cid:96) √ (cid:32)(cid:114) − ρ Σ − Σ + (cid:115) ρ Σ + Σ − √ M/(cid:96) (cid:112) − ρ Σ − Σ (cid:33) (A.1)where we introduced the short-hand notation ρ = 1 − (cid:18) Q(cid:96) (cid:19) , Σ = − (cid:16) λ + (cid:112) λ − ρ (cid:17) / , Θ = (cid:16) λ − (cid:112) λ − ρ (cid:17) / , λ = − (cid:18) M(cid:96) (cid:19) − (cid:18) Q(cid:96) (cid:19) . The fourth root is given by r o = − ( r − + r + + r c ) . If the mass parameters are such that thediscriminant locus is positive, D > , all four roots are real-valued. The roots in Eq. (A.1) arepositive and satisfy r c > r + > r − > while the forth root is negative, r o < , and hence unphysical.If the discriminant locus is negative, D < , some of the roots are complex-valued, the space-timeadmits only one causal horizon and exhibits a naked singularity. Since this case violates the CosmicCensorship Conjecture, we consider this configuration as unphysical.Special cases arise when the discriminant locus vanishes, D = 0 . Two out of the three causalhorizons are degenerate. We can distinguish three cases, which specify further here:• Extremal space-time:
In this case, the outer and inner black hole horizon are degenerate, r + = r − . The degenerate horizons are in thermal equilibrium at zero temperature. Theirmass, charge and de Sitter radius are not independent, but related by the analytic relation M(cid:96) = 13 √ (cid:115) (cid:18) Q(cid:96) (cid:19) − ρ / . (A.2)For fixed values of the cosmological radius (cid:96) the near horizon geometry is AdS × S .• Nariai space-time:
The cosmological and the outer black hole horizon are coincident, r c = r + .Both horizons are in thermal equilibrium at zero temperature. Again, the mass parametersare not independent, but follow the relation M(cid:96) = 13 √ (cid:115) (cid:18) Q(cid:96) (cid:19) + ρ / . (A.3)The near horizon geometry for a fixed value of the cosmological radius is dS × S .• Ultra-cold space-time:
The intersection of the aforementioned Nariai and extremal space-timesis characterized by a triple root of the polynomial equation D = 0 . The external and Cauchy– 28 –lack hole horizons coincide with the cosmological one, r − = r + = r c . All three horizons arein thermal equilibrium at zero temperature. The ultra-cold space-time is located at M(cid:96) = 13 (cid:114) , Q(cid:96) = 12 √ . (A.4)This can also be seen in Fig. 1. B Scalar curvature of the moduli space of spherically symmetric andstatic space-times
The distance conjecture [4] states that the scalar curvature of the moduli space of fields should bestrictly negative when the dimension of the moduli space is larger than one. As detailed in Sec. 3,we construct the moduli space of spherically symmetric and static space-times, so our moduli spaceis constructed out of tensor instead of scalar fields. Here we show that the scalar curvature ofthis moduli space is negative and hence that it is compatible with the statements of the DistanceConjecture [4].From the geodesic equation (3.2), we can extract the Christoffel symbol
Γ : M× T g M× T g M → T g M from the symmetrization of ¨ g = Γ( ˙ g, ˙ g ) , (B.1)thus leading to Γ g ( h, k ) = 12 hg − k + 12 kg − h + 14 tr ( g − hg − k ) g − tr ( g − h ) k − tr ( g − k ) h − (cid:104) tr ( g − hg − k ) (cid:105) g + 14 (cid:104) tr ( g − h ) (cid:105) k + 14 (cid:104) tr ( g − k ) (cid:105) h , where g ∈ M and h, k ∈ T ∗ M in the vector bundle of all symmetric (0 , -tensors on M .The curvature of a connection R ( X, Y ) s = ([ ∇ X , ∇ Y ] − ∇ [ X,Y ] ) s (B.2)is called the Riemann curvature tensor for the Levi-Civita connection for the vector fields X, Y ∈ X ( N ) and a vector field s : N → T M along f : N → M . Here X ( N ) denotes the space of smoothvector fields over the smooth manifold N and f is a smooth mapping. In local coordinates, theRiemann curvature tensor reads R ( h, k ) l = d Γ( k )( h, l ) − d Γ( h )( k, l ) + Γ( h, Γ( k, l )) − Γ( l, Γ( h, l )) . (B.3)The Riemannian curvature for the canonical Riemannian metric on the manifold M can be com-puted to be g − R g ( h, k ) l = −
14 [ L, [ H, K ]]+ n (cid:18) H ( tr ( KL ) − (cid:104) tr ( KL ) (cid:105) − (cid:104) tr ( KL ) − (cid:104) tr ( KL ) (cid:105)(cid:105) ) − K ( tr ( HL ) − (cid:104) tr ( HL ) (cid:105) − (cid:104) tr ( HL ) − (cid:104) tr ( HL ) (cid:105)(cid:105) ) (cid:19) − (cid:18) K ( tr ( L ) tr ( H ) − (cid:104) tr ( L ) (cid:105)(cid:104) tr ( H ) (cid:105) − (cid:104) tr ( L ) tr ( H ) − tr ( L ) (cid:104) tr ( H ) (cid:105) − tr ( H ) (cid:104) tr ( L ) (cid:105) ) − H ( tr ( L ) tr ( K ) − (cid:104) tr ( L ) (cid:105)(cid:104) tr ( K ) (cid:105) − (cid:104) tr ( L ) tr ( K ) − tr ( L ) (cid:104) tr ( K ) (cid:105) − tr ( K ) (cid:104) tr ( L ) (cid:105) ) (cid:19) − (cid:18) ( tr ( HL ) − (cid:104) tr ( HL ) (cid:105) ) ( tr ( K ) − (cid:104) tr ( K ) (cid:105) ) − ( tr ( KL ) − (cid:104) tr ( KL ) (cid:105) ) ( tr ( H ) − (cid:104) tr ( H ) (cid:105) ) (cid:19) Id , – 29 –here we have introduced the short-hand notation H = g − h , K = g − k , L = g − l . Further, n = dim M is the dimension of M . For isotropic space-times ( M, g ) the Riemannian curvaturetensor simplifies to g − R g ( h, k ) l = −
14 [ L, [ H, K ]] . (B.4)It is not possible to define the Ricci curvature for the manifold ( M , G ) , but a point-wiseRicci-like curvature can be defined [20]. Taking the point-wise trace of the Riemann curvaturetensor (B.4), that we denote with the subscript x , yieldsRic g ( h, l )( x ) = tr ( k x (cid:55)→ R g ( h x , k x ) l x ) = −
14 ( tr ( H ) tr ( L ) − n tr ( HL )) . (B.5)This amounts to compute the point-wise Ricci scalar as R = − n (cid:18) n ( n + 1)2 − (cid:19) . (B.6)Hence, the manifold ( M , G ) has a negative scalar curvature if the dimension of M is n ≥ . Notethat the scalar curvature is constant and does not depend on moduli space coordinates. C Quasi-static evaporation of RNdS black holes
In this appendix, we present technical details of the evaporation model. Following [24], we perturbthe Einstein equations to linear order, relate the energy-momentum tensor to the Hawking andSchwinger fluxes and extract evolution equations for M , Q and (cid:96) . The resulting system of differentialequations is under-determined. By considering the first law of black hole mechanics we supplementthe system by an additional differential equation. C.1 Perturbative analysis
In order to relate the Hawking and Schwinger particle fluxes to the parameters M , Q and (cid:96) , wefollow the analysis of [24] and solve Einstein’s equation perturbatively. We introduce an order (cid:15) perturbation in the time and radial components of the RNdS metric (2.1) as δ d s = (cid:15) ( δB ( (cid:15)t, r )) d t + δA ( (cid:15)t, r )) d r ) , (C.1)where δA and δB are arbitrary functions that depend on the radial coordinate and on the “slowtime scale” (cid:15)t . This is the time scale on which the geometry evolves, i.e. M = M ( (cid:15)t ) , Q = Q ( (cid:15)t ) and (cid:96) = (cid:96) ( (cid:15)t ) .The change in the geometry is due to the accumulated effect of the Hawking and the Schwingerradiation. The quantum fluxes source the slowly varying geometry as ( δG ab − πGδT cl ab + Λ δg ab )( (cid:15)t ) = 8 πGδT q ab ( t ) (C.2)where δT cl ab ( (cid:15)t ) refers to the classical and hence slow variation of the energy momentum tensor and δT q ab = (cid:104) δT ab (cid:105) to quantum perturbations of energy-momentum . Since both the black hole chargeand the background metric vary, the electric field induced by the charged black hole also changes,which we model through an order (cid:15) perturbation of the field strength. F → F + (cid:15) (cid:114) g πl G δF ( (cid:15)t, r ) d t ∧ d r . (C.3) Notice that we have suppressed the term (cid:15)δ Λ( (cid:15)t ) g ab from Eq. (C.2), since we consider the perturbation of thisequation to first order i (cid:15) and the dependence of the slow time scale of the variation of the cosmological constantgives rise to second order terms in (cid:15) . – 30 –xplicitly, the first order perturbation of the metric (2.1) results in a first order variation of theEinstein tensor δG ab , computed by expanding δG ab = ˜ G ab − G ab to first order in (cid:15) , where ˜ G ab is theperturbed Einstein tensor. The perturbation of the classical energy momentum tensor is computedas δT cl ab = ˜ T cl ab − T cl ab . The stress-energy tensor of classical electromagnetism is given by T cl ab = 14 π(cid:15) (cid:18) F aµ F µb − g ab F µν F µν (cid:19) . (C.4)Let us introduce the time-like vector ξ ( t ) = γ ( t ) ∂ t and space-like vector v = 1 /r ∂ r , where γ ( t ) = V ( r g ) / is normalized at the position of the geodesic observer (see Sec.2). It is convenient toproject the linearized Einstein eq. (C.2) onto these vectors yielding δ i = 8 πGη i , i = 1 , . . . . (C.5)Here, δ i correspond to the projection of the classical part as δG ab − πGδT cl ab + Λ δg ab = δ ξ a ξ b + δ v a v b + δ ξ ( a v b ) + δ g ab (C.6)to first order in (cid:15) . The functions δ i are obtained throught the expansion of Eq. (C.2) to first orderin (cid:15) , thus yielding δ = 12 γ t (cid:32) δ A (cid:0) (cid:96) (cid:0) r (cid:0) M + M r − r (cid:1) − M Q r + 2 Q (cid:1) + r (cid:0) r (2 r − M ) + 2 Q (cid:1)(cid:1) r − r (cid:96) ( r ( r − M ) + Q ) − rδ B (cid:48) (cid:96) (cid:0) r ( r − M ) + 2 Q (cid:1) ( r − (cid:96) ( r ( r − M ) + Q )) − δ F Q(cid:96) r − (cid:96) ( r ( r − M ) + Q ) − δ B (cid:96) (cid:0) r (cid:96) (cid:0) r (7 M − r ) − Q (cid:1) + (cid:96) (cid:0) M Q r + M r ( M − r ) − Q (cid:1) + r (cid:1) ( r − (cid:96) ( r ( r − M ) + Q )) + δ A (cid:48) (cid:0) r ( r − M ) + 2 Q (cid:1) r + r δ B (cid:48)(cid:48) (cid:96) r − (cid:96) ( r ( r − M ) + Q ) (cid:33) (C.7a) δ = 12 r (cid:96) (cid:0) − δ A r (cid:0) r − (cid:96) (cid:0) r ( r − M ) + Q (cid:1)(cid:1) × (cid:0) − (cid:96) (cid:0) M r − M (cid:0) Q + 3 r (cid:1) + 2 Q r + r (cid:1) + r (cid:96) (cid:0) Q − M r (cid:1) + 2 r (cid:1) +8 δ F Qr (cid:96) (cid:0) r − (cid:96) (cid:0) r ( r − M ) + Q (cid:1)(cid:1) + r δ B (cid:48)(cid:48) (cid:0) − (cid:96) (cid:1) (cid:0) r − (cid:96) (cid:0) r ( r − M ) + Q (cid:1)(cid:1) + 2 δ B r (cid:96) (cid:0) r (cid:96) (cid:0) M r − Q (cid:1) + (cid:96) (cid:0) Q r (2 r − M ) + M r (5 M − r ) + Q (cid:1) − r (cid:1) r − r dS ( r ( r − M ) + Q )+ δ B (cid:48) (cid:0) r (cid:96) ( M − r ) + 2 r (cid:96) (cid:1) − δ A (cid:48) (cid:0) (cid:96) ( M − r ) + 2 r (cid:1) (cid:0) r − r(cid:96) (cid:0) r ( r − M ) + Q (cid:1)(cid:1) (cid:1) (C.7b) δ = 2 rV ( r ) γ ( t ) ˙ V ( r ) = − γ ( t ) V ( r ) (cid:18) ˙ M − Qr ˙ Q − r (cid:96) ˙ (cid:96) (cid:19) (C.7c) δ = 12 (cid:32) δ B (cid:48)(cid:48) + 4 δ F Qr + δ B (cid:48) (cid:96) (cid:0) r ( r − M ) + 2 Q (cid:1) r − r(cid:96) ( r ( r − M ) + Q ) − δ A (cid:48) (cid:0) (cid:96) ( M − r ) + 2 r (cid:1) (cid:0) r − (cid:96) (cid:0) r ( r − M ) + Q (cid:1)(cid:1) r (cid:96) + 2 δ B (cid:0) r (cid:96) (cid:0) r (7 M − r ) − Q (cid:1) + (cid:96) ( r − M ) (cid:0) Q − M r (cid:1) + r (cid:1) r ( r − (cid:96) ( r ( r − M ) + Q )) + 2 δ A (cid:18) r ( r (4 r − M )+ Q ) (cid:96) − r (cid:96) + ( r − M ) (cid:0) Q − M r (cid:1)(cid:19) r . (C.7d)– 31 –he dot denotes the derivative with respect to the slow time scale while the prime stands for theradial derivative of the relevant quantity. Notice that only δ ∼ ˙ V ( r ) contains derivative terms withrespect to the slow time (cid:15)t . Consequently, only the liniarized Einstein equation for δ gives rise toa dynamical equation. The functions δ j for j = 1 , , constrain the functions δA, δB and δF .The terms η i correspond to the projection of the quantum perturbation of the energy-momentumtensor δT q ab onto the same vectors, δT q ab = η ξ a ξ b + η v a v b + η ξ ( a v b ) + δ g ab . (C.8)It is convenient to define the projection of the perturbed energy momentum tensor on the time-likeand space-like vector fields like T ≡ δT q ab ξ a v b , E ≡ δT q ab ξ a ξ b , S ≡ δT q ab v a v b , T ≡ g ab δT q ab (C.9)Thus, the different η i are the components of the quantum variation of the energy momentum tensorand can be expressed in terms of T , E , S and T by contracting δT q ab with ξ a and v a . Solving thesystem of linear equations, we obtain η = rV γ t ) (cid:0) T − r S (cid:1) + 3 E r V γ t , η = 12 r (cid:18) r S V − rT V − E γ t (cid:19) ,η = − r γ t ) T , η = 12 (cid:18) r ( −S ) + E rV γ t + T (cid:19) . (C.10)We can now extract the first dynamical equation describing the evolution of the mass parame-ters. The i = 3 component of Eq. (C.5) reads ˙ M − Qr ˙ Q − r (cid:96) ˙ (cid:96) = 4 πGr V ( r ) γ ( t ) T . (C.11)The background parameters M , Q and (cid:96) are evolving with the slow time scale (cid:15)t so their timederivatives are defined as ˙ M = d M d t ( (cid:15)t ) | t =0 , ˙ Q = d Q d t ( (cid:15)t ) | t =0 , ˙ (cid:96) = d (cid:96) d t ( (cid:15)t ) | t =0 . Further, the perturbation to the electromagnetic field strength (C.3) results in an electriccurrent, because the perturbed electrical current density must satisfyd ∗ ˜ F = ∗ ˜ j (C.12)to first order in the perturbation. Expanding Eq. (C.12) to first order in (cid:15) yields the evolution ofthe charge of the black hole sourced by the Schwinger current J ˙ Q = − π J . (C.13)For further details on the derivation of (C.13) we refer to [24].Summarising, the dynamical evolution of the system is dictated by Eqs. (C.2) and (C.13)explicitly given by ˙ M − Qr ˙ Q − r (cid:96) ˙ (cid:96) = 4 πGr V ( r ) γ ( t ) T , ˙ Q = − π J . (C.14)Setting ˙ (cid:96) = 0 and γ ( t ) = 1 , we recover the framework of Ref. [24].It remains to specify the quantities T and J . In order to correctly define the quantum fluxes,let us move to a local inertial frame of reference, i.e. to the position of the geodesic observer.Introducing space-like unit vector n = (cid:112) V ( r g ) ∂ r , we define T g = δT q ab ξ a ( t ) n b , J g = δj a g ab n b . (C.15)– 32 –hese are related to the previously introduced components as T = 1 (cid:112) V ( r g ) r T g , J = 2 π (cid:112) GV ( r ) r J g . (C.16)Working on the inertial frame defined by the geodesic observer simplifies the computations, whilephysics remains independent of the coordinate frame.The mass flux T g is sourced by the outgoing Hawking radiation from the black hole horizonand the incoming Hawking radiation from the cosmological horizon. The lack of thermodynamicequilibrium generates a net flux of thermal radiation through the horizon of the black hole that isrelated to the quantum perturbation of the energy momentum tensor in the direction perpendicularto the generator of the horizon, i.e. the rt − component of the quantum metric perturbation. Also,notice that there is no net flow of charge through the horizon of the black hole, since the samenumber of outgoing and incoming charge carriers are generated. Therefore, the term T only receivesa contribution from the Hawking flux. At the position of the geodesic observer, it is simply givenby the Stefan-Boltzmann law T g = σ ( A c T c − A + T ) = σ (4 π ) V ( r g ) (cid:0) r c | V (cid:48) ( r c ) | − r | V (cid:48) ( r + ) | (cid:1) , (C.17)with σ = π k B c (cid:126) = π Boltzmann’s constant in Hubble units. Here A c and A + as well as T c and T + are the areas and temperatures associated to the cosmological and black hole horizons evaluatedat the position of the geodesic observer. The red-shift factor ∝ V ( r g ) arises from normalizing theKilling vector field at the position of the geodesic observer Eq. (2.8).Finally, we quantify the flux J , which can be obtained though the integration per unit volumeof the Schwinger pair production rate Γ( r ) [24], thus yielding J g = G(cid:96) (cid:112) V ( r g ) r g r c r r c + r (cid:90) S sin( θ ) d θ d φ (cid:90) r c r + d r (cid:48) r (cid:48) Γ( r (cid:48) )= G(cid:96) (cid:112) V ( r g ) r g r c r r c + r q Q π (cid:34) (cid:18) r + e − r QQ − r c e − r c QQ (cid:19) + √ π √ QQ (cid:18) Erf (cid:18) r + √ QQ (cid:19) − Erf (cid:18) r c √ QQ (cid:19)(cid:19) (cid:35) , (C.18)where Erf is the error function. Here Γ( r ) denotes the Schwinger particle production rate in flatspace-time Γ( r ) = ( qE ) π (cid:126) exp (cid:18) − r QQ (cid:19) , (C.19)with E = Q/r the electric field, Q = (cid:126) q/πm and m and q are the mass and charge of theproduced pair of particles. This holds true as long as the charged pair is created in approximatelyflat space-time, i.e. for E ( r ) (cid:29) R ( r ) . This is the case over the whole phase space D (due tothe assumption of a quasiestatic discharge), except at the neutral line where E = 0 . Taking intoaccount the exponential suppression of Eq. (C.19), the leading contribution to the current is dueto the electron, the lightest charged particle of the Standard Model, so we can approximate q ≈ e and m ≈ m e yielding the value of Q used in the calculation of the Schwinger flux. C.2 Thermodynamic considerations
We have found two differential equations for the three parameters M , Q and (cid:96) . We thus need onemore equation to specify the system. The first laws for multihorizon space-times as studied in [34]will form the basis for our purpose. – 33 –e start with a quick review of [34]. The cosmological constant contributes a pressure-liketerm to the first law as P = − Λ8 π = − π(cid:96) < . (C.20)The conjugate variable is the thermodynamic volume V . The first law can be written in differentversions. Evaluating the first law at the black hole horizon leads to δM = T + δS + + (Φ + − Φ ∞ ) δQ + V + δP for r + < r < ∞ , (C.21a)while the first law evaluated at the cosmological horizon reads δM = − T c δS c + (Φ c − Φ ∞ ) δQ + V c δP , r c < r < ∞ . (C.21b)The first law valid in the bulk between the outer black hole and cosmological horizon then reads T + δS + + T c δS c + (Φ + − Φ c ) δQ − V δP , r + < r < r c . (C.21c)The subscripts indicate where the thermodynamic potentials are to be evaluated. Contrary to ourdiscussion in section 2, the Killing vector field is normalized to unity at null infinity and hence γ ( t ) = 1 . Therefore, the thermodynamic potentials appearing in Eq. (C.21) are given by Φ + = Qr + , Φ c = Qr c , Φ ∞ = 0 , S + = πr , S c = πr c ,T + = 14 πr + (cid:18) − Q r − r (cid:96) (cid:19) , T c = − πr c (cid:18) − Q r c − r c (cid:96) (cid:19) , (C.22)with Φ the electric potential, S = A/ the entropy, and T = | κ | / π the temperature. The ther-modynamic temperature lacks the red-shift factor with respect to the one appearing in (C.17).Finally, the thermodynamic volumes V + and V c denote the volume comprised between the blackhole horizon and infinity and between the cosmological horizon and infinity. The volume V denotesthe volume of the bulk, V = V c − V + .In the limit of pure de Sitter, Eq. (C.21b) reduces to − T c δS c + V c δ (cid:18) − Λ8 π (cid:19) . (C.23)The cosmological constant has to decrease to respect the second law of black hole mechanics. Then,the de Sitter radius evolves according to Stefan-Boltzmann’s law ,d (cid:96) d t = T c d S c d t = σA c T c = σ π (cid:96) . (C.24)This implies that a pure de Sitter space-time evaporates to Minkowski.In the Schwarzschild limit we want to recover the evaporation of the black hole horizon dueto the emission of Hawking radiation. In this limit the first law of thermodynamics (C.21b) reads δM = T + δS + . The variation of entropy of the black hole horizon is due to the variation of its mass,so we obtain that d M d t = T + d S + d t = − σA + T . (C.25)We demand the evolution equation for the de Sitter radius to satisfy the first law (C.21c) andto implement the limits discussed above: Gibbons-Hawking radiation of the cosmological horizonin the dS limit and Hawking evaporation of the Schwarzschild black hole. Hence, for the general Notice that in the pure de Sitter limit V c = 4 π(cid:96) / , so the volume factor V c cancels out with the pressure termwhen we express it as a function of δ(cid:96) . – 34 –NdS case we plug Eqs. (C.24) and (C.25) into the first law (C.21c) valid in the bulk, which yieldsthe following dynamical equation:d (cid:96) d t = 4 πr g V dS V c − V + (cid:0) V ( r g ) T g − (Φ + − Φ c ) J g (cid:1) , (C.26)where V dS = 4 π(cid:96) / is simply the volume enclosed by the de Sitter radius. The factor of πr accounts for the normalization at the position of a given observer and the term V ( r g ) arises fromconsidering the red-shift factor that appears in Eq. (C.17) with respect to the temperature definitionin Eq. (C.22). Further, we have used that the evolution of the charge of the black hole is exclusivelysourced by the Schwinger radiation (C.13). C.3 Summary
To summarize, Eqs. (C.14) and (C.26) lead to the following system of differential equations, ˚ M = 4 πr g (cid:34) (cid:32) G (cid:113) V ( r g ) + r g V ( r g ) r c − r (cid:33) T g − (cid:32) Qr g + r g r c − r (cid:18) Qr + − Qr c (cid:19)(cid:33) J g (cid:35) , ˚ Q = − πr g J g , ˚ (cid:96) = 4 πr g (cid:96) r c − r (cid:18) V ( r g ) T g − (cid:18) Qr + − Qr c (cid:19) J g (cid:19) . (C.27)The differential operator ˚= d / d t g denotes the derivative with respect to the proper time of thegeodesic observer, which is related to the Schwarzschild time through d t g / d t = (cid:112) V ( r g ) . Theexpression for the Hawking and Schwinder fluxes are given by Eq. (C.17) and (C.18), resp. References [1] C. Vafa,
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