Extended Thermodynamics of Self-Gravitating Skyrmions
aa r X i v : . [ g r- q c ] S e p Extended Thermodynamics of Self-Gravitating Skyrmions
Daniel Flores-Alfonso , ∗ and Hernando Quevedo , , † Instituto de Ciencias Nucleares,Universidad Nacional Aut´onoma de M´exico,AP 70543, Ciudad de M´exico 04510, Mexico Department of Theoretical and Nuclear Physics,Kazakh National University, Almaty 050040, Kazakhstan
A modification of the hedgehog ansatz has recently led to novel exact black hole solutions withselfgravitating SU(2) Skyrme fields. Considering a negative cosmological constant the black holesare not asymptotically anti-de Sitter (AdS) but rather asymptote to an AdS version of Barriola–Vilenkin spacetime. We examine the thermodynamics of the system interpreting the cosmologicalconstant as a bulk pressure. We use the standard counterterm method to obtain a finite Euclideanaction. For a given coupling of the matter action the system behaves as does a charged AdS blackhole in the fixed charged ensemble. We find that in the limit case when the Skyrme model becomesa nonlinear sigma model the system exhibits a first order phase transition of Hawking–Page type.The universality class of these Einstein–Skyrme black holes is that of van der Waals.
PACS numbers: 04.40.b, 04.70.BwKeywords: Skyrmions, black hole thermodynamics
I. INTRODUCTION
There have been recent developments in black hole thermodynamics by considering the cosmological constant as athermodynamic variable, which can be interpreted as a bulk pressure (for a recent review see [1]). The thermodynamicphase space is extended to include pressure and volume into the fold. In some cases, this thermodynamic volumecan be interpreted as a geometrical volume; however, this is not the case, in general [2]. Taking into account thecosmological constant in this way modifies the first law of thermodynamics. A relation was found by Smarr [3] forasymptotically flat black holes which corresponds to the Gibbs–Duhem equation. This procedure cannot be carriedout in the same manner for AdS black holes, say. Nonetheless, this is redeemed when the cosmological constant gainsthe status of a thermodynamic pressure. In this extended setting, black hole thermodynamics is brought closer to thethermodynamics of ordinary substances as it puts the first law in the same footing as the Smarr (or Gibbs–Duhem)relation. This has been shown to be valid in the case of asymptotically AdS spaces, but it also holds for asymptoticallyLifshitz spacetimes [4]. Moreover, this approach also holds for spacetimes which are only locally asymptotically AdS[5].In theories of semi-classical quantum gravity, where this approach has been applied, the gravitational mass of thesystem corresponds to the thermodynamic enthalpy, rather than the energy. This means that in the limit of vanishingcosmological constant the usual interpretation is recovered. In such a context, the (Euclidean) action serves as thefundamental equation for the thermodynamic system, i.e., all equations of state may be derived from it. In general,gravitational actions diverge; notwithstanding, finite values can be obtained by using surface terms as counterterms inthe calculation [6]. The counterterm approach eliminates the need to carry out background subtractions, which is veryuseful when ambiguities arise regarding the background that should be used to subtract. In some cases, it is entirelyunknown which background should be used. The approach taken in [6] is unique for (locally) asymptotically AdSspaces as the couterterms depend crucially on the AdS curvature scale. The surface terms themselves are universaland depend only on the dimension of spacetime and the cosmological constant.Recently, black hole thermodynamics has been successfully applied to spacetimes which were previously unantici-pated. Some examples include C-metrics and Lorentzian Taub–NUT metrics[7, 8]. Within the Pleba´nski-Demia´nskifamily, which includes all Einstein–Maxwell black holes, the acceleration and NUT parameters have been the mostproblematic. For example, they do not allow for general type D metrics to be written in the usual Kerr–Schild form[9].However, a recent approach has allowed for the incorporation of these spacetimes into the standard thermodynamictreatment. In accelerating black holes, a conical deficit is present, which can be interpreted as the consequence of a ∗ daniel.fl[email protected] † [email protected] cosmic string pulling on the black hole. In Taub–NUT spacetimes, the Misner strings can be thought as being thereason behind conical deficits. The unifying concept for both spacetimes are conical defects. A fresh perspective onthe thermodynamics of spaces with conical defects has been proposed in reference [10].The early universe might have formed cosmic strings, which may be observed due to their gravitational effects onthe cosmic microwave background radiation or gravitational wave experiments. Cosmic strings are a certain typeof topological defect. They have been found to lead to interesting consequences when in the presence of primordialblack holes [11]. In recent years, cosmic strings, domain walls and textures have received considerable attention.These topological defects together with monopoles and skyrmions remain one of the most active fields in modernphysics. Topological defects arise in many elementary particle models and have found applications in cosmology [12].Skyrme was the first to devise a three dimensional topological defect solution arising from a nonlinear field theory[13]. In the gravitational sector, the Einstein–Skyrme system has attracted considerable attention since sphericallysymmetric black hole solutions with a nontrivial Skyrme field were found numerically [14]. This marked the firstcounterexample to the black hole no-hair conjecture. It should be noted that this solution is stable against sphericallinear perturbations [15]. In the present paper, we study the thermodynamic phase structure of a black hole withSkyrme matter found by Canfora and Maeda [16]. This solution possesses solid angle deficits very similar to theconical deficits due to cosmic strings. The thermodynamics of black holes with conical defects has motivated us tostudy the thermodynamics of spacetimes with solid angle deficits.This paper is organized as follows: In section II, we introduce the (AdS-) Barriola–Vilenkin spacetime which is asolution to the Einstein field equations with nonlinear sigma model matter. We continue on to describe the dynamicsof Einstein–Skyrme systems and the Canfora–Maeda solution. In section III, we use the standard counterterm methodto obtain the finite Euclidean action for the system at hand and derive the relevant thermodynamic relations. Wethen provide a detailed description of the phase structure of the Skyrme black hole. In section IV, we close with asummary of our results, highlighting the novel aspects of our work. II. CLASSICAL ASPECTS
Many topological defects are represented by nonlinear sigma models, which are one of the most important nonlinearfield theories. They have a vast application in physics which ranges from statical mechanics to gravitation, especiallystring theory. Some examples are Nambu–Goldstone bosons, the superfluid He and the quantum Hall effect. TheBarriola–Vilenkin spacetime was built to support global monopoles [17]; they are an example of nonlinear sigmamodel matter harbored in Einstein backgrounds. A close configuration is the Gibbons–Ruiz Ruiz black hole [18],which shares its asymptote with that of Barriola–Vilenkin.Scaling arguments stemming from Derrick’s renowned theorem show that nonlinear sigma models do not admitstatic soliton solutions in 3+1 dimensions. Skyrme constructed his model exactly to circumvent this result and did soby adding higher derivative terms to the action. This makes manifest that skyrmions behind horizons share asymptoticbehavior with their corresponding sigma model limit. It should be noted that skyrmions describe an entirely differentclass of topological defects. Skyrmions and other “classical lumps” with horizons have been investigated in theliterature [19, 20]; however, their role in the AdS context is less clear [21].Before continuing any further, we shortly review spacetimes that are asymptotically AdS with a solid angle deficitin comparison to AdS itself. We write the AdS generalization of the Barriola and Vilenkin global monopole spacetimeas d s = − B ( r ′ )d t ′ + A ( r ′ )d r ′ + r ′ (d θ + sin θ d φ ) , (1)with the metric functions given by B ( r ′ ) = A ( r ′ ) − = 1 − α + r ′ l . (2)Hereafter, we refer to this spacetime as AdS Barriola–Vilenkin and abbreviate it as AdS-BV. Here l is the AdScurvature radius and is related to the cosmological constant by Λ = − /l . Now, making a coordinate change r ′ = r (1 − α ) / , (3a) t ′ = t (1 − α ) − / , (3b)the above line element becomesd s = − (cid:18) r l (cid:19) d t + (cid:18) r l (cid:19) − d r + r (1 − α )(d θ + sin θ d φ ) . (4)Note that for vanishing α we recover the AdS geometry. For constant t and r the above metric describes a spherewith solid angle deficit of 4 πα . The Gibbons–Ruiz Ruiz black hole has a line element similar to (1), but with metricfunctions given by B ( r ′ ) = A ( r ′ ) − = 1 − α − m ′ r ′ . (5)The class of asymptotically flat spacetimes with an angle deficit α has been studied by Nucamendi and Sudarsky[22]. Therein, they find an Arnowitt–Deser–Misner (ADM) mass generalization given by m = m ′ (1 − α ) − / . In thispaper, we focus on a recent black hole found by Canfora and Maeda [16], which has Skyrme matter and includes acosmological constant. Their metric has this same type of solid angle deficit and specializes to the Gibbons–Ruiz Ruizspacetime. A. Einstein–Skyrme Systems
In this work, we concentrate on the thermodynamic behavior of a black hole with nonlinear scalar matter. Thesystem is a four dimensional Einstein–Skyrme configuration and includes a cosmological constant. The matter contentare scalar fields which fulfill a generalized hedgehog ansatz. This section is dedicated to describing the SU(2) Skyrmemodel; for more details, see [23].We write the basic action as I [ g, U ] = − πG Z d x √− g ( R − − K Z tr (cid:18) A ∧ ⋆A + λ F ∧ ⋆F (cid:19) , (6)where the fundamental field U is an SU(2) − valued scalar field and A is the pullback of the Maurer-Cartan form ofSU(2) by U . It follows that A = U − d U is an su (2) − valued one − form and complies with A ∧ A = − d A . It is alsouseful to define F = A ∧ A . The first term in the matter action of (6) is that of a nonlinear sigma model. Noticethat it is a quadratic term and so the second term can be recognized as a quartic term or as a higher curvatureterm, whichever is more convenient. We emphasize that the second term, proportional to λ , was added by Skyrmeto deform the nonlinear sigma model action. Notice that A is reminiscent of a Yang–Mills pure gauge potential andthat the equation F = − d A might lead to similarities with Maxwell matter. We also comment that in a pion model( λ = 0) K is related to the pion decay constant and since pions are pseudo-Goldstone bosons, then K characterizesthe symmetry breaking present in the system. In a global monopole, K is related to the solid angle deficit α inspacetime — as compared to a Minkowski background.It is standard to parametrize the su (2) directions using Pauli matrices σ i or alternatively by t i = − i σ i . So the su (2) − valued one − form A may be decomposed as A = A i ⊗ t i or as A = A µ ⊗ dx µ . The t i can also be used togetherwith unity as a base for R and so a point y on SU(2) ≈ S can be parametrized by y = Y + Y i t i , (7)as long as Y + Y + Y + Y = 1.Turning to the action once more, the equations of motion areRic( g ) − R g + Λ g = 8 πGT, (8)where the energy-momentum tensor T has components T µν = − K (cid:20)(cid:18) A µ A ν − g µν A α A α (cid:19) + λ (cid:18) F µα F αν − g µν F αβ F αβ (cid:19)(cid:21) , (9)and d ⋆ A − λ A, d ⋆ F ] = 0 . (10)Notice that sigma model solutions, which satisfy d ⋆ A = 0, can be promoted to Skyrme model solutions providedthey comply additionally with [ A, d ⋆ F ] = 0 so that the above equation is satisfied.Following reference [16], we also define a symmetric tensor S = − / A ⊗ A ) with components S µν = δ ij A iµ A jν . (11)So the energy-momentum tensor is now written as T µν = K (cid:20)(cid:18) S µν − g µν S (cid:19) + λ (cid:18) S S µν − S µα S αν − g µν ( S − S αβ S αβ ) (cid:19)(cid:21) , (12)where S is the trace of S . This last depiction explicitly breaks the similarities to Yang–Mills matter and makesthe degree of nonlinearity in Skyrme matter manifest. Although, as pointed out in [16], the Skyrme contribution istraceless very much like Maxwell matter. The matter equations are now portrayed as ∇ µ (cid:2) A µ + λ (cid:0) SA µ − S νµ A ν (cid:1)(cid:3) = 0 . (13)The nonlinearity inherit to these equations makes finding exact solutions difficult. However, the use of simplifyingAns¨atze makes the equations more tractable. B. The Canfora–Maeda Solution
The exact solution under discussion was built by generalizing the hedgehog Ansatz in such a way that the fieldsthemselves need not reflect spherical symmetry, but their energy-momentum tensor does. This ultimately makes themetric resemble that of the Reissner–Nordstr¨om geometry. The fundamental scalar field U is given by the map U : ( t, r, θ, ϕ ) t r = cos θ t + sin θ sin ϕ t + sin θ cos ϕ t . (14)Notice that t r has Frobenius norm -2 as all the other t i ; this justifies our notation. Notice that the t i are used here asin Eq.(7) so that t r represents the unit radial vector in R ; in other words, it is in correspondence with the positionsof S .The geometry generalizes the Schwarzschild-AdS spacetime, which is recovered by setting K = 0, and is given byd s = − f ( r )d t + f ( r ) − d r + (1 − α ) r (d θ + sin θ d φ ) , (15a) f ( r ) = 1 − Gmr + q r + r l . (15b)However, q is not an integration constant as in the Reissner–Nordstr¨om black hole, but is fixed by the couplingconstants of the theory. Here α parametrizes the solid angle deficit 4 πα and is given by α = 8 πGK . Moreover, q isjust shorthand for αλ/ − α ) and m is the Nucamendi–Sudarsky mass. A quasi-local calculation [24–28] yields theAbbott–Deser–Tekin (ADT) mass as M = m (1 − α ) . (16)For asymptotically AdS spacetimes, the ADT mass is calculated as the ADM mass, except that the background metricis not flat and the lapse is not unity. It also yields the standard result for the energy per unit length of a cosmicstring, which is proportional to the angle deficit[20]. The exterior of a cosmic string is Minkowski, but with a solidangle deficit. III. THERMODYNAMICS
As mentioned in the previous section, the Canfora–Maeda black hole generalizes the AdS-Schwarzschild spacetime.This itself motivates investigating its thermodynamics. As Hawking and Page have shown, there are interesting phasetransitions in the AdS-Schwarzschild system [29]. In our thermodynamic treatment, we shall consider the cosmologicalconstant as a thermodynamic variable. The matter content is scalar and so no modifications from the matter sectorare expected for the first law of thermodynamics. However, the Skyrme matter does contribute to the Smarr relation[30] and in an AdS context, considering the cosmological constant as a thermodynamic pressure, gives the generalizedrelation [4].
A. Action and Counterterms
We use the path integral approach to semi-classical quantum gravity and consider the cosmological constant as athermodynamic variable. We study the analytic continuation ( t → i τ ) of the black hole solution (15a) and identifythe period of the imaginary time β with the inverse temperature. The period is fixed by the amount that makes theEuclidean solution regular. Euclidean solutions are required to be regular everywhere so that they serve as a saddlepoint to approximate the path integral. Since the black hole horizon is transmuted into a bolt when the analyticcontinuation is carried out, then the root of the metric function (15b) r + will lead in general to a conical singularity.As mentioned above, this is resolved by fixing the period of the Euclidean time circle by β = 4 πf ′ ( r + ) . (17)Let us recall that upon analytical continuation one must check first for curvature singularities. For the solution athand the Kretschmann scalar is infinite at r = 0. Any Euclidean sheet which is expected to dominate the gravitationalpath integral cannot contain the region r = 0. Moreover, on the Lorentzian sheet null surfaces such as the horizon mayexist without problem. However, when the signature is positive definite a place where the metric function vanishes isa degenerate region. The only way for a positive definite manifold to have a region where, say f ( r + ) = 0, is that itresembles a cone, or cigar. Since the direction which degenerates at the tip of this cone is the Euclidean time, thenit must be this direction which is periodic. This justifies Eq.(17), regularizes the horizon and avoids the curvaturesingularity. Although there is an angular deficit in spacetime which extends to infinity, the only place where its effectis locally observable is at the curvature singularity which (17) avoids.The Skyrme map (14) is unaffected by the analytic continuation and we calculate the Skyrme contribution to theenergy of the system as E = − Kλ β Z M d x √ g tr h F, F i = 2 πKλr + (1 − α ) = q (1 − α )2 Gr + . (18)This expression is like the electric contribution to the energy in a charged black hole with a global monopole [21].In such a black hole, we would have an electric potential difference of Φ = q/ r + and a total electric charge of Q = q (1 − α ) /G . This implies that the Maxwell sector provides and amount of energy given by Φ Q = q (1 − α ) / Gr + .The analogy we are drawing here makes sense because q is the amount that appears multiplying the 1 /r term inthe metric function (15b). Furthermore, the Skyrme term in the action needs no renormalization, as is the case ofMaxwell matter in four dimensions.Before continuing with the action calculation, we recall that in the extended thermodynamics we are consideringthe cosmological constant as a canonical variable and so the effective pressure is given by P = − Λ8 πG = 38 πGl . (19)So the appearance of the cosmological scale in the finite action will point to quantities involving the bulk pressure.When it comes to black hole thermodynamics, the gravitational path integral in the saddle point approximationhas a long history in the quantum gravity literature. Solutions related to AdS were studied in this context in [20, 29]about the same time, but in parallel to the use of the cosmological constant as a pressure [31–33]. The persistentproblem with this approach is that typically the Euclidean action diverges. The counterterm method we focus on isstandard practice[6] and it allows to solve this problem. In four dimensions, a finite gravitational action is achievedthrough I ren .G = − πG Z M d x √ g (cid:18) R + 6 l (cid:19) − πG Z ∂ M d x √ h K + 18 πG Z ∂ M d x √ h (cid:20) l + l R (cid:21) . (20)Here the geometry of g on M induces a metric h on the boundary ∂ M . In this approach, the boundary at infinityis held fixed to obtain the Einstein field equations. The action is not only the usual Einstein–Hilbert action, but italso contains the Gibbons–Hawking surface term, easily recognized in the above equation by the trace of the extrinsiccurvature K of the boundary as embedded in M . The surface terms at the end of the renormalized action arecounterterms which are unique in the AdS context, as they depend on the AdS scale. These terms are sufficientin four dimensions, but in higher dimensions extra terms are required, which are explicitly known at least up todimensions relevant in the AdS context. Notice that the counterterms are covariant and depend only on, e.g., theRicci scalar R of the boundary. Moreover, it has been shown that this regularization is equivalent to the addition ofa Gauss–Bonnet term[34].For the Canfora–Maeda black hole (15a-15b), we get the following finite gravitational action I ren .G = β G (cid:18) Gm (1 − α ) + r + α − r (1 − α ) l (cid:19) (21)On the other hand, the quadratic part of the matter action diverges. This is the part we have called the sigma modelsegment of the scalar action. So, an additional counterterm must be added to renormalize this divergence. Scalarmatter and global monopole counterterms have been studied before in [35, 36]. However, the counterterm we use forthe matter content is I ct M = K Z ∂ M d x √ h (cid:20) l h A, A i (cid:21) = − K Z ∂ M d x √ h (cid:20) l S (cid:21) . (22)This counterterm is of kinetic type and is reminiscent of the one found for scalars in Lifshitz spacetimes [37]. Since A does not depend on r , there is no ambiguity in the above equation. Notice that written in this way, the counterterm,as written in the left hand side, shares a similar structure as the ones in [38]. Observing the right hand side, we noticethat it has the same structure as the very last term in (20). Finally, we write the action of the Einstein–Skyrmesystem as I ren . = β G (cid:18) Gm (1 − α ) − r (1 − α ) l + q (1 − α ) r + (cid:19) . (23)We now portray the imaginary time period as β = 2 πr (1 − α ) Gm (1 − α ) − q (1 − α ) /r + + r (1 − α ) /l , (24)which means that we can rewrite the action in the following form I = βm (1 − α ) − πr (1 − α ) G . (25)Immediately, we compute the state variables of the system, as the Gibbs free energy is
I/β = H − T SH = (cid:18) ∂I∂β (cid:19) P = m (1 − α ) = M, (26a) S = β (cid:18) ∂I∂β (cid:19) P − I = πr (1 − α ) G = A h G , and (26b) V = 1 β (cid:18) ∂I∂p (cid:19) β = 4 πr (1 − α )3 . (26c)The enthalpy H , entropy S and thermodynamic volume V indeed fulfill the first law of thermodynamicsd H = T d S + V d P. (27)Moreover, the expression (24) takes on the new form H T S − P V + E, (28)which is recognizable as the Smarr relation and generalizes equation (97) in [30] for vanishing angular momentum.Certainly, the path integral approach is consistent with the mass variation approach to thermodynamics and the quasi-local formalism [4, 30, 39]. We recollect that in the perfect fluid interpretation of the cosmological constant, the energydensity is ρ = − P = Λ / πG . Removing a portion of spacetime to form a black hole of volume V thermodynamicallycosts an amount P V . This formation energy is naturally captured in the enthalpy and, from the gravitational pointof view, this is reflected in the black hole mass [2, 39].To sustain global monopoles and skyrmions such as (14), we must cut out a region of spacetime given the angledeficit 4 πα . So, to form a black hole in AdS with an angle deficit, a smaller volume needs to be removed. Thedifference in volumes is, of course, given by the cone over the deficit area 1 / παr ) r + . The solid angle deficit inspacetime affects all extensive thermodynamic quantities. If we recall the holographic principle, this is exactly whatis expected. The entropy is given by the horizon area and so the proportionality (1 − α ) in (26b) and (26c) becomesclear. Since the mass and Skyrme energy can be written in terms of surface integrals [30], they too will have thissame (1 − α ) factor; see (26a) and (18). As we mentioned above, Skyrme energy is comparable to electric energy.The extensive variable for Maxwell configurations is the electric charge which, in general, may be written as a surfaceterm. This is observed in [21], where the electric charge possesses the same proportionality factor as all other globalcharges. B. Black Hole Chemistry
The inclusion of pressure, volume and enthalpy into black hole thermodynamics has led to a different understandingof gravitational systems. A prevalence of van der Waals universality classes can be found in the literature and blackhole mechanics is comparable to chemical systems.Starting from equation (24) and recalling that f ( r b ) = 0, we may write an expression for the pressure as a functionof its volume P = T r b − πr b + αλ − α ) πr b , (29)given that r b ( V ) = (cid:18) V π (cid:19) / . (30)The Einstein–Skyrme system exhibits the famous van der Waals P − V curve in Figure 1. In charged AdS black holes,a critical charge can be found where the temperature’s turning points appear or disappear. In the Canfora–Maedasolution (14-15b) the equivalent is a critical value of the Skyrme coupling constant. Figure 2 portrays how the systemconducts itself for different couplings. FIG. 1: The bulk pressure of the Skyrme black hole system is displayed as a function of the thermodynamic volume. The plotis in accordance with the critical behavior of a van der Waals fluid. In this figure we have chosen T = 0 .
05 and q = 2 .
1. Hawking–Page transition
In the special limit λ = 0, the matter field reduces to an SU (2) nonlinear sigma model. The equation of motion forthe scalar matter is now d ⋆ A = 0 . (31) FIG. 2: The curves displayed above are seen to be analogous to a P − V diagram of van der Waals fluids. The inverse temperatureis plotted versus the bolt radius for different values of the Skyrme coupling constant. Here we have chosen G = 2 α = 1 and l = 10. The black curve represents the critical value of λ , the dotted curve is supercritical and the red curves are subcritical. The fundamental SU (2) − valued scalar field U is still given by equation (14) and satisfies the above equation of motion.The metric, in this limit, is the AdS version of Gibbons and Ruiz-Ruiz [18] and corresponds to AdS-Schwarzschildwith a solid angle deficit. Its Euclidean counterpart has a bolt radius given by the following quadratic equation3 βr b − πl r b + l β = 0 , (32)which means that given a temperature and a pressure there are two branch solutions r b ± = 2 πl β ± r − β π l ! . (33)We therefore speak of a large ( r b + ) and small ( r b − ) black hole configuration. These branch solutions are plotted inFigure 3. Moreover, there is a minimum available temperature for the black holes to exist, the size of the branch FIG. 3: This figure portrays the bolt radii of the Euclidean AdS black holes versus the inverse temperature; here, we have used l = 1. The union of both branches is shown. They connect at the temperature T = √ / πl . black holes coincide at this temperature, which is given by β max = 2 πl √ . (34)These black holes are precisely the AdS Schwarzschild black holes [29] when we set α = 0. Further, still the AdS-BVspacetime reduces to AdS in this special case. From the AdS perspective, there is a region of spacetime that has beencut away. In other words, this is only appreciated when compared to the AdS space. Even when this point of viewis taken, the black hole behavior remains intact — despite there being a “removed region”. To further demonstratethis, we explore the phase structure of the system.Figure 4 shows the Euclidean action of the black hole branches versus the inverse temperature for various valuesof the pressure. Notice that a zero temperature a solution is only possible when m = 0; this case corresponds toAdS-BV for which the Euclidean action vanishes. Comparing the black hole action values, we see that the large blackhole is always preferred over the small one. This indicates that the small black hole is an unstable configuration ofthe system. There is also a set of temperatures for which the Euclidean action is negative, meaning that it is lowerthan that of AdS-BV. This is to say that for high temperatures the preferred phase of the system is a large black holewhile for low temperatures the AdS-BV phase dominates.We point out that at every fixed pressure, while transitioning between AdS-BV and the large black hole, there is adiscontinuity in the entropy. This labels the phase transition as first order and parallels the Hawking–Page transitionin AdS-Schwarzschild[29] and between Taub–NUT and Taub-Bolt spaces found in [5]. For vanishing cosmologicalconstant our results are consistent with those obtained in the thermodynamics of global monopole systems [40, 41]. FIG. 4: The Euclidean action, which is proportional to the Gibbs free energy, for different values of pressure P = 3 / πGl . Wehave chosen the values l = 1 , , , , G = 2 α = 1. The solid lines from left to right decrease in pressure while the dashedcurve is the zero pressure limit.
2. Skyrme AdS black holes
After examining the sigma model limit of the system, we turn back to the investigation of the Skyrme couplingconstant λ . For fixed values of α , the q constant which enters the metric (15a) is tuned by the Skyrme parameter λ . The geometry is of Reissner–Nordstr¨om type with the exception that q is not an integration constant, but isrelated to the coupling constant. The Canfora–Maeda solution is like the charged AdS black holes investigated in[42, 43]. However, a chief difference is the manipulation of the q parameter. For charged black holes, a fixed potentialensemble has been studied as well as a fixed charge ensemble. The behavior of the present system is comparable toAdS black holes with a fixed charge.The bolt radii of the black hole must comply with the following quartic equation3 βr b − πl r + l βr − l q β = 0 . (35)Notice that when q = 0, we recover equation (32). In this limit, two of the four solutions of the quartic equationbecome repeated and are null. It turns out that one of the four solutions is in general negative and so unphysical; thisis one of the solutions which is nullified together with q . The other solution is physical and in that limit vanishes andrepresents AdS-BV. Algebraically, all this can be deduced from the discriminant of the quartic equation. Moreover,0only when the discriminant is positive, we will have four real roots which represent three physical solutions: a small,large and intermediate black hole.For β → ∞ one of the roots approaches r e = 16 q − l + 6 l p l + 12 q . (36)Hawking radiation will be absent for a black hole of this size. This represents a zero temperature black hole witha single degenerate horizon. This is the lowest entropy configuration of the system and corresponds to an extremalblack hole such that lim β →∞ (cid:18) Iβ (cid:19) = M e , (37)meaning that the free energy is given by the extremal black hole’s mass. Notice that taking q = 0 yields r e = 0 andso the extremal black hole turns into AdS-BV in the nonlinear sigma model limit. Finite temperature black holes arean excitation of this extremal configurations. Thus, we consider an action ˜ I = I − I e = β ( M − M e ) − S which yieldsthermodynamic equations of state H = ∂ ˜ I∂β ! P = M − M e , (38a) S = β ∂ ˜ I∂β ! P − ˜ I = A h G , and (38b) V − V e = 1 β ∂ ˜ I∂p ! β = 4 π r b − r e ) . (38c)The thermodynamic variables fulfill the first law in the form d H = T d S + ( V − V e )d P . Using the extremal blackhole as a starting point and cutting out spherical regions, the horizon forms a finite temperature black hole where theenergy of formation is given by P ( V − V e ). In figure 5, the action difference I − I e is plotted an shows a distinctiveswallowtail behavior indicating a region where phase transitions occur. Paralleling the λ = 0 behavior, we can readoff from figure 5 that the intermediate black hole branch always has higher free energy than any of the other branches.This phase is never statistically preferred by the system, it is unstable. For a fixed pressure, the large and small blackhole phases have coinciding free energies at a single temperature, the coexistence temperature.Figure 5, which has q = 1, is generic in the sense that higher values of q pull the graph toward more negativevalues when the swallowtail is present. Figure 2 encapsulates the information that there is a critical range where theswallowtails appear. As q gets smaller but remains positive, the swallowtail doe rise toward positive values remainingon the lower half of the plane. As expected, the more one approaches zero the more the swallowtail will morph into theHawking–Page characteristic curve. Up until now we have maintained q positive since it couples a (quartic) kineticterm to the rest of the action. To avoid instabilities and ghosts it must remain so. However, thinking of an analogueof the Mexican hat potential, we explore negative values of q . This sector shows a completely different dynamics thanthe one described up until now. A small and a large black hole are present, but no intermediate black hole and henceno van der Waals transition. Since AdS-BV is also not an available phase, there is also no Hawking–Page transition.One of the phases is always favored thermodynamically and so the sector is devoid of phase transitions.Consider a nearly extremal (small) black hole and raise its temperature until the coexistence temperature is reached.Raising the temperature even further will cause the black hole to transit into a large black hole. The size of the blackhole blows up discontinuously and as a consequence so does its entropy. The thermodynamic system exhibits a firstorder phase transition at the coexistence temperature. The entropy’s behavior is presented in figure 6. IV. CONCLUSIONS
In this paper we investigate the extended thermodynamic behavior of a self-gravitating skyrmion in Einstein–Skryme theory. The spacetime is interpreted as a black hole which has swallowed a skyrmion, in a special limit thesystem becomes a black hole with a global monopole in its interior. The approach we use is standard. We use thecounterterm method to obtain the Euclidean action — which in turn yields the free energy of the system. However,there is a new aspect to the renormalization performed here. The Skyrme sector is only regularized with the term inequation (22), which we did not find in the existing literature. The term is of kinetic type since the Skyrme action1
FIG. 5: The Euclidean action of the Einstein–Skyrme system as a function of the inverse temperature for G = 2 α = q = 1and different values of the pressure P = 3 / πGl . The black lines correspond to l = 6 (dotted) and l = 6 . l = 7 . , is also of this type. Common counterterms for scalar fields cannot be used here because of the model’s non-scalingproperties.The thermodynamics of the Canfora–Maeda solution we investigate has not been fully explored previously. TheEuclidean approach yields an enthalpy for the Einstein–Skyrme system. We use the covariant quasi-local ADT methodto obtain a value for the systems mass and find it to coincide with the enthalpy of the system. This further supportsthe idea that the cosmological constant enters thermodynamics through the formation energy of the system. This iscontrary to the approach taken in references [7, 8, 10], where related thermodynamics are carried out. There, theconformal completion method is used to obtain the mass. For the sake of comparison we have calculated the massthrough this method and find it to coincide with the ADT energy. In these works the thermodynamics of variousspaces with conical defects are carried out. It is in this way they relate to spaces with solid angle defects such asthe ones present here. The present Skyrme system has a background metric which is of Reissner–Nordstr¨om typewith the exception that the factor in the r − term in the metric function is not an integration constant. It does notrepresent any electromagnetic charge and is fixed by the coupling constants of the theory. This contributes to thesimilarity between the dynamics of the Skyrme system and charged black holes. The areal deficit in the spacetime2is comparable to spacetimes which possess cosmic strings. There the string tension is directly related to the conicaldeficit. In the references mentioned just above a first law of thermodynamics is posed where the string tension variesand gives rise to a new thermodynamic potential. Here the solid angle deficit is fixed by the coupling constants andso this procedure cannot be carried out.The thermodynamics of nonlinear matter have been studied before for example in Einstein–Skyrme systems butalso recently in other types of nonlinear scalar models. The latter in an extended setting considering thermodynamicvolume and the cosmological constant as a bulk pressure. To our knowledge this is the first time an extended Smarrrelation has been derived for spacetimes containing Skyrme matter. We generalize previous results and find that thethermodynamic volume allows for a geometric interpretation. In other words, the asymptotically locally AdS solutionwe consider can be thought of as formed from a ground state spacetime by removing a portion of its volume. Thisencapsulates forming a black hole where there was none, making it larger and also removing a wedge from it entirely— which corresponds to the solid angle deficit. We have extended the original sense of the formation energy concept[39] to apply for “missing” wedges coming from topological defects.We also mention that charged monopole black holes have been studied recently in the literature and their behaviorhas been found to be of van der Waals type. This further supports the idea that the black hole behavior within thesetting of extended thermodynamics is universal. Although very similar to charged black holes lacking scalar mattersome central differences may found such as the location of critical points. It has been found that these points dependon the exact value of the symmetry breaking parameter, K in our notation. Nonetheless, aspects such as the law ofcorresponding states are blind to this parameter. Comparing this to our current work, we find the result to hold aswell. At this point we stress that the Skyrme system at hand is electromagnetically neutral yet the nonlinearity ofthe scalar matter yields behavior present in charged AdS black holes with and without global monopoles. Acknowledgements
DFA would like to thank Eloy Ay´on–Beato for many enlightening conversations and acknowledges support fromCONACyT through Grant No. 404449. This work was partially supported by UNAM-DGAPA-PAPIIT, Grant No.111617, and by the Ministry of Education and Science of RK, Grant No. BR05236322 and AP05133630. We thankthe anonymous referees for their critical and helpful comments.
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