Topological interpretation of extremal and Davies-type phase transitions of black holes

Topological arguments are currently being used as a novel scheme to discern the properties of black holes while ignoring their detailed structure and specific field equations. Among various avenues of black hole physics, where this novel approach is being utilized, the phase transition in black hole thermodynamics lies at the forefront. There are several types of phase transition in black holes; such as the van der Waals type phase transition, Davies-type phase transition, extremal phase transition, and Hawking-Page (HP) transition. So far, the topological interpretation, where the critical point has been identified with the non-zero topological charge, has been obtained only for the van der Waals type phase transition and HP transition in different spacetimes. To complete the picture, here we provide the same interpretation for two other phase transitions: Davies-type phase transition and extremal phase transition. The entire analysis is general and is valid for any spacetime where these types of phase transitions are observed. More importantly, our analysis suggests that amid the apparent differences in these phase transitions, they share the same topological characteristics, \textit{i.e.} non-zero topological charge arising from different thermodynamic potentials in different types of phase transition.


Introduction
The initial works of Bekenstein [1] and Hawking [2] first laid the foundation for understanding black holes as thermodynamic objects as it was found that the black holes can be endowed with thermodynamic parameters like entropy and temperature.Furthermore, it was shown that one can obtain the laws of black hole mechanics, which exactly resemble the thermodynamic laws of a conventional thermodynamic system [3].These initial results were stepping stones to identifying black holes as thermodynamic objects.Even though the microstates that are responsible for black hole thermodynamics remain elusive, we are confident about the thermodynamic connection of black holes.
Phase transitions are one of the key aspects of black hole thermodynamics, which has been explored since the seminal work by Davies [4].He identified the critical points as ones with a diverging heat capacity (followed later by [5][6][7][8]).Subsequently, in AdS spacetime, another type of phase transition was observed by Hawking and Page [9], known as the Hawking-Page phase transition.Later it was found that the transition of black holes from being non-extermal to extremal marks another type of phase transition, known as the extremal phase transition [10][11][12][13][14][15][16][17][18].More recently, in AdS spacetime, considering the cosmological constant as a variable (and identifying the cosmological constant as thermodynamic pressure), a van der Waals-type phase transition is observed in black holes where the mass plays the role of enthalpy [19][20][21][22][23][24][25][26][27].
Over the years the phase transition of black holes has been studied quite extensively.Earlier, the thermogeometrical interpretation of phase transitions was provided [16,[28][29][30][31][32] (also see the unified approaches [33][34][35]), where the critical points were identified as ones that are endowed with the diverging Ricci-scalar in the thermodynamic phase space.This (thermogeometrical) interpretation has been found to hold good for all types of phase transitions described above.Recently, the topological interpretation has been provided by researchers, which helps to distinguish the properties of the black holes, without resorting to the dynamical equation of the spacetime or the structural details.This formalism was earlier used in the coordinate space of black holes [36,37], to identify the location of the light rings using null geodesics.It was shown that the location of the light rings correspond to the zero point of a special vector, constructed in the coordinate space of the black hole spacetime.Subsequently, for a black hole in asymptotically flat spacetime with a topologically spherical Killing horizon, it was found that the black holes have at least one light ring outside the horizon for each sense of rotation [37], which was later later generalized to other cases [38][39][40].Later, this topological argument was used in different aspects of black hole thermodynamics  (also see the recent papers [69][70][71][72]).In the black hole phase transition, it was shown that the critical point can have a topological interpretation in terms of non-zero topological charge [42][43][44][45][46][47][48]50].So far, this interpretation has been found to hold for the van der Waals type phase transition [42][43][44][45][46] and the Hawking-Page phase transition [47,48,50].It was first obtained for the van der Waals type phase transition [42], where it was shown that a topological interpretation can be provided using Duan's topological current ϕ-mapping theory [73,74].The same analysis has been extended later to the Hawking-Page transition.However, the topological interpretation is still missing for the other two types of phase transitions in black holes, namely the Davies type phase transition and the extremal phase transition.Therefore, it becomes an obvious question whether the same interpretation exists for these other two types of phase transitions.In this paper, we will explore this issue, i.e. we obtain a topological interpretation for the Davies phase transition and for the extremal phase transition.The entire analysis is general as it is valid for any spacetime which is endowed with these types of phase transitions.
The paper is organized as follows: In section 2, we describe Duan's topological current theory, adapted to the thermodynamic phase space.In section 3, we discuss the topological interpretation of Davies-type phase transition using the topological current theory discussed in section 2. In section 4 we provide the same interpretation for the extremal phase transition of black holes.We provide the conclusions of our analysis in section 5. Throughout our paper, we have adopted the geometrised unit and have set

Adaptation of Duan's topological current theory
The formalism that we follow is based on Duan's ϕ-mapping topological current theory [73,74], originally defined in the coordinate space, where the coordinates are defined as x µ = {τ, x a }.Thus, in the following discussion, greek indices (µ,ν, ρ etc.) run 0,1,2 and the roman indices run 1,21 .At the onset, one defines a potential Φ(x a ), which is known as Duan's potential.Thereafter, the vector field (ϕ) is defined as ϕ = {ϕ a } = {ϕ 1 , ϕ 2 } where (1) The normalized vector n a can be defined as which has the properties Using the above-normalized vector n a , one can define an anti-symmetric superpotential V µν as Employing the above anti-symmetric superpotential, one can obtain a topological current j µ as Due to the anti-symmetric property of V µν , it is straightforward to find that the current j µ is conserved, i.e.
The zeroth component of j µ can be thought of as the charge density.Therefore, the total total charge Q within a region C can be obtained as The important properties of the charge Q can be unveiled with the following analysis.Substituting (2) in (5), one can obtain One can further obtain Substituting ( 9) on (8), one can obtain the expression of the conserved current as The above expression of topological current can be simplified by introducing the Jacobi tensor, which is given as where is the Jacobian.Thus, the expression of j µ can be obtained in terms of Jacobi tensor as where Using the two-dimensional Laplacian Green function of ϕ space (∆ ϕ ln ||ϕ|| = 2πδ 2 (ϕ)), one can finally obtain the expression of the current j µ as and the total charge inside C is given as Thus, the above expressions suggest that the topological current j µ is non-vanishing only at ϕ = 0 (i.e. on the zeros of ϕ) and the topological charge Q is non-vanishing iff C encloses at lease one zero point of ϕ 2 .If C encloses no zero point, the corresponding charge vanishes.Furthermore, if two boundary curves C 1 and C 2 enclose the same zero points, the corresponding topological charge will be the same.
Furthermore, the above topological charge can be expressed in terms of the winding number as per the following analysis.Suppose there are N isolated zero points of ϕ, and the i-th zero is located at x = z i .Then from the theory of delta function [77], one can obtain Now, we consider that when x goes around the zero point z i once, the number of loops that ϕ a makes in the ϕ vector space is given by the positive integer number β i (which is known as the Hopf index in topology).Then one can further obtain 2 Note that for those zeros of ϕ, which are regular, it can be argued by implicit function theorem [75] that Our analysis is valid for the regular zeros.However, if at the zero point J 0 ϕ x = 0, which might happen if the zeros of ϕ include some branch points, one has to apply the bifurcation theory of topological current.For more details on it, see [76].
On the right hand side β i arises due to accounting the fact that when x goes around the i th zero once, the number of loop made in the ϕ space is given by the Hopf index β i [74].where the Brower degree η i is defined as Finally, the total charge inside C can be obtained as where w i = β i η i is known as the winding number of i-th zero point contained in C .If one further considers that the i-th zero point is enclosed by a smooth and positively oriented closed curve C i and there are no other zero points inside C i , one can obtain the winding number corresponding to the i-th zero point given as where Ω is known as the deflection angle of the vector field along the given contour and it is given by Ω = arctan(ϕ 2 /ϕ 1 ).In terms of the unit vectors n 1 and n 2 , dΩ can be obtained as dΩ = n 1 dn 2 − n 2 dn 1 = ϵ ab n a dn b (the expression which will be used later).
With the above mathematical preliminaries, we can now move on to the main analysis i.e. topological interpretation of critical points of black hole phase transition.Our major goal is to find a suitable thermodynamic potential Φ that yields vectors ϕ a = ∂ a Φ such that ϕ a vanishes on the critical point under consideration.

Davies-type phase transition
As it is now well-known, black holes behave like thermodynamic objects and are endowed with the first law of the form where X i are the generalized displacements like angular momentum (J), electric charge (Q) and Y i corresponds to the generalized forces like angular velocity (Ω), electric potential (Φ E ) etc.
For Davies-type phase transition [4][5][6][7][8], one looks for the divergence of the specific heat, which is defined as Now, using one can obtain Since T ̸ = 0, at the critical point, one obtains In the following, we provide the topological description of Davies-type phase transition adopting Duan's formalism mentioned in the previous section.As we have mentioned earlier, Duan's original formalism is defined in the coordinate space, which we shall use in the thermodynamic space of the black hole.More importantly, here we use the concept of topological charge, which is originally defined in the coordinate space in the thermodynamic phase space.Taking a cue from Eq. ( 24), we define the thermodynamic potential Φ as Here the additional 1/ sin θ factor has been introduced in the thermodynamic potential (adopting the idea from [42]) which simplifies the study of topology.The vector fields can be obtained as Now we can realize the significance of the auxiliary term 1/ sin θ.It serves two major purposes: (i) The vector ϕ is perpendicular to the horizontal lines θ = 0 and θ = π, which can be considered as the natural boundary of the parameter space.(ii) More importantly, this auxiliary term helps to identify the critical point on the S − θ plane.Since ϕ θ vanishes at θ = π/2 ,the zero point of ϕ (where both ϕ S as well as ϕ θ should vanish) would lie on θ = π 2 line.In other words, the zero point of ϕ in the S − θ plane (which can easily be identified on θ = π 2 line due to the presence of the auxiliary term), will correspond to the critical point where ϕ θ vanishes due to θ = π 2 and ϕ S vanishes due to the relation (24) on the critical point.Let us also mention that although we have adopted the auxiliary term as 1/ sin θ following [42], it can also be taken as 1/ cos θ as well.In that case, the natural θ limit would be −π/2 to π/2 and the critical point will be located at θ = 0 line on the S − θ plane.The zero point of ϕ on θ = π/2 line corresponds to the critical point where ϕ S vanishes due to the criticality condition and ϕ θ vanishes as θ = π/2.Thus, the critical point can be attributed to a non-zero topological charge using the formalism mentioned in the previous section.
Thus, from the above general analysis, we can obtain the topological interpretation of the Daviestype phase transition, which will be applicable to any spacetime that has a Davies-type critical point.On the critical point, one can obtain a zero point as per our analysis and one can attribute a non-zero topological charge.
Let us now realize this with a particular example.Let us consider the following action [78] where the first term is the Einstein-Hilbert term, the second term characterizes the dS (Λ > 0), AdS (Λ < 0) or asymptotically flat (Λ = 0) spacetime depending upon the value of Λ and the third term is the coupling of the Maxwell field (for η = 1) or phantom field (for η = −1).The metric solution of the above action is given as [78] where The horizons of the solution can be obtained by solving the condition f (r) = 0, which yields two roots r + and r − and the outer horizon can be identified as the event horizon [78].The mass of the black hole can be expressed in terms of other thermodynamic parameters as where S = πr 2 + is the Bekenstein-Hawking entropy.The Hawking temperature can be obtained as The above expression of the Hawking temperature can also be obtained from the surface gravity (κ) as T = κ/2π, where κ = f ′ (r + )/2 .The specific heat (C Q ) can be obtained as From the above expression, one can obtain the divergence points of specific heat (or the Davies points) when the denominator of (32) vanishes and one obtains the expression of entropy at the Davies points as 3 For simplicity, we fix the value of Q, Λ, η as Q = η = Λ = 1.In that case, the value of entropy on the Davies point can be obtained as S D = 4.0927.
Let us now find the topological interpretation of the Davies-type phase transition.As per our formalism, the thermodynamic potential can be defined as The components of the vector field ϕ can be obtained as The normalized vector can be obtained as n a = ( ϕ S ||ϕ|| , ϕ θ ||ϕ|| ), which is represented in figure 1, which is a pictorial representation of the unit vector n in the S − θ plane.From the figure 1, we find that the zero point (which can be marked as the Davies-type critical point) is located at S = S D and θ = π 2 .Since the winding number or the topological charge is independent of the loops enclosing the zero point, we consider two circular loops C 1 and C 2 where C 1 encloses the critical point and C 2 does not.The two contours are parameterised by ψ and are defined as follows: which represents circle of radius a and centred at (S 0 , π 2 ).For C 1 , we choose a = 0.4 and S 0 = S D , whereas for C 2 , we choose a = 0.4 and S 0 = 5.5.The deflection angle can be measured using the definition provided in the previous section, which can be obtained as and the topological charge can be obtained as For our case, we find that for C 1 , the topological charge can be obtained as Q = −1, and for C 2 , the topological charge can be obtained as Q = 0. Thus, using our formalism, the Davies-type critical points can be given a topological interpretation and the Davies points can be associated with non-zero topological charge.Note that Choosing S 0 = 5.5 is just for illustrative purposes.Any contour that does not include the Davies point will have a zero topological charge.Furthermore, the positivity or the negativity of charges will also depend on the choice of contour as well.In our case, we have considered the contour to be counter-clockwise.If the contour encloses the zero-point clockwise, we would have obtained Q = +1.

The extremal phase transition
In the extremal phase transition, the black hole undergoes a transition from being non-extremal to extremal one and the critical point is located at the extremal limit.In the extremal limit, the surface gravity of the black hole (or the black hole temperature) vanishes.In this case, the desired thermodynamic potential can be identified in terms of the thermodynamic mass of the black hole as Again the auxiliary factor 1/ sin θ has been included for the reasons mentioned earlier.The vector fields can be obtained as Again, we find that the zero point of ϕ on θ = π 2 line is located on the critical point where T = (∂M/∂S) X i vanishes.Therefore, applying the discussion presented in the section 2, this general analysis suggests that the extremal critical point can be attributed to a non-zero topological charge.
The above analysis will be valid for any spacetime which has an extremal critical point, where the Hawking temperature T vanishes.However, we shall again realize it for a given spacetime.Let us consider the BTZ black hole [79,80], which is the solution of Einstein's gravity in (1 + 2) dimensional spacetime.The metric is given as where Here M and J are the mass and angular momentum of the black hole and l is related to the cosmological constant.The horizons can be obtained from the condition that the lapse function (N ) which is given as The extremal condition is given by ∆ −→ 0, in which case the outer horizon (or the event horizon, denoted by r + ) and the inner horizon (denoted by r − ) coincide.The Hawking temperature of the black hole is given as which as mentioned above vanishes in the extremal limit when r + = r − .The expression of entropy is given by [79,80] The expression for the mass (M ) as a function of entropy can be expressed as The thermodynamic potential Φ can be identified as Note that here we have set J = 1 = l for our convenience.From (43), we find that at the extremal limit, the mass is given as M ext = J/l.For J = 1 = l, we find that at the extremal limit, the value of mass and entropy is given as M ext = 1 and S ext = 2 √ 2π = 8.88577.
The components of the vector field ϕ can be obtained as Again, the normal fields can be obtained as n a = ( ϕ S ||ϕ|| , ϕ θ ||ϕ|| ), which is again represented in figure 2, which is a pictorial representation of the unit vector n in the S − θ plane.
From the figure 2, one can find that the critical point is located at S = S ext and θ = π/2 .
To obtain the topological charge, again we choose two loops C 1 and C 2 , which are defined by the parametric relation (36).For C 1 , we choose a and S 0 as a = 0.3 and S 0 = S ext , which implies that C 1 encloses the critical point.For C 2 , we choose a and S 0 as a = 0.3 and S 0 = 7.7, which implies that C 2 does not enclose the critical point.For these loops, we calculate the topological charge (using eq.38).It turns out that for C 2 , the topological charge vanishes but for C 1 , one can obtain Q = 1 .Thus, using our formalism, one can provide the topological interpretation of the extremal phase transition and the extremal critical point can be associated with a non-zero topological charge.Note that for the specific example in Davies type phase transition, we obtained Q = −1, whereas for the present example in extremal phase transition, we have obtained Q = +1.As we discussed earlier, it depends upon the orientation of the ϕ a vector (as it can be seen, the direction of arrows in figure 1 and figure 2 are different whereas the contours are counter-clockwise in every case).Earlier, employing the generalized off-shell free energy, the solutions of black holes were given topological interpretation as the defects in the thermodynamic parameter space [41].In that case, using free energy as the thermodynamic potential in every case, it was discussed that black holes can be classified into three groups based on their topological charges as Q = −1, 0, +1 .We find that the same thermodynamic potential cannot provide the topological interpretation for every type of phase transition in black holes (For Davies-type phase transition, inverse-temperature, and for extremal phase transition black hole mass plays the role of the thermodynamic potential in our case).Furthermore, in different cases, we obtain both Q = −1 and Q = +1.These mark the similarities and differences of our analysis with [41].

Conclusions:
Over the years, the study of phase transitions in black hole thermodynamics has gained a lot of prominence among physicists.It is considered one of the most fascinating parts of the theory of black hole physics.Different types of phase transitions have been observed in black holes: Davies-type phase transitions, Hawking-Page phase transitions, extremal phase transitions, and van der Waalstype phase transitions.In this paper, we have presented the topological interpretation for Davies-type and extremal phase transition, which was missing in the literature thus far.
Recently, it was found that the thermodynamic properties of black holes can be deduced when one uses a phenomenological theory of topology in the thermodynamic phase space.At first [41], it was shown that all the black hole solutions can be identified in three groups based on the topological charges defined in the thermodynamic parameter space i.e.Q = −1, 0, and +1.In addition, it was shown that the black holes with the same topological nature exhibit similar thermodynamic properties.Later, this topological argument has been extended to explore the phase transitions in black holes.Firstly, it was shown in [42] that the critical point in the Van der Waals type phase transition of the black holes can have a topological interpretation using Duan's ϕ-field formalism in the thermodynamic parameter space.In that case, the critical point can be attributed to a non-zero topological charge.Later, this argument has been extended to the Hawking-Page phase transition [47].However, the topological interpretation is still missing in literature for the other two types of phase transitions i.e. the Davies-type phase transition and the extremal phase transition.Thus, inspired by [42], we have adopted Duan's ϕ-mapping topological current theory to provide the topological interpretation for Davies-type and extremal phase transitions.Thereby, we show that the topological interpretation can indeed be provided for these two types of phase transitions by making judicious choices of the thermodynamic potential Φ in every case.For the Davies-type phase transition, which is characterized by the divergence of heat capacity, the inverse Hawking temperature (T −1 ) plays the role of the thermodynamic potential to obtain the topological interpretation.On the other hand, for the extremal phase transition, we found that the mass of the black hole plays the role of the thermodynamic potential.In both cases, we have chosen the thermodynamic potential in such a way that the critical point lies on the zero-point of the vector field ϕ a , which is obtained from the thermodynamic potential Φ.Despite the entire description being provided from a general standpoint, we also have provided specific examples to support our claim and have shown that the topological argument holds good for the black hole phase transition.In all cases, we find that the critical point can be endowed with non-zero topological charges using the ϕ-field formalism.More importantly, we show that the recent development of topological interpretation of black hole phase transitions, which has been so far explored for van der Waals and Hawking-Page phase transition, can be extended for other criticalities as well.Thus, the topological interpretation can act as a standard formalism to describe the phase transition in black holes.
Finally, as we have mentioned earlier, the topological argument is provided in order to describe the properties of the black holes without involving the field equation and detail structure.Hence, the formalism mentioned here can be applicable beyond Einstein's GR and to any black holes that show these types of phase transitions.Let us explain it in a bit more detail.The thermodynamic properties of black holes were initially established in general relativity [1][2][3].However, later it was found that the thermodynamic properties are valid well beyond general relativity [81][82][83][84][85][86][87][88][89][90] (for a recent review on this matter, see [91]).Phase transition, which is a key aspect of black hole thermodynamics, has been found beyond GR as well.For instance, Van der Waals-type phase transition has been observed in higher curvature gravity and scalar-tensor gravity [92][93][94][95].Similarly, Davies-type phase transition has been observed beyond general relativity such as in Horava-Lifshitz gravity [96], Gauss-Bonnet gravity [97], Lovelock gravity [98], f (R) gravity [99], and so on.Furthermore, one can also find extremal black hole solutions beyond GR [100] and one can expect to find extremal phase transition beyond general relativity as well.We emphasize that our analysis will continue to be valid for all those cases as our topological argument has been formulated in the thermodynamic parameter space and does not involve any gravitational equation.
Since our topological argument is formulated in the thermodynamic parameter space of a black hole (BH), we admit that the observational connection with our analysis is not direct and we need to investigate much deeper to comment on the potential connection with black hole shadow, accretion disc etc.However, our analysis distinguishes certain black holes from others, which can have significance from the observational viewpoint.For example, it distinguishes extremal black holes (non-zero topological charge) from the non-extremal ones.Therefore, the extremal black holes can be identified from our analysis.Furthermore, we discussed that for Davies-type phase transition, heat capacity diverges for certain black holes at a specific horizon radius.A closer investigation (see [101] for example) shows that the larger BHs show positive heat capacity and the smaller BHs show negative heat capacity and at the critical limit, heat capacity diverges.Therefore, smaller BHs are relatively unstable.From our topological argument, one can obtain the stable limit (non-zero topological charge) of different black holes.To obtain further connection with other observable features, we need to investigate further, which we keep as a potential future work.

Figure 1 :
Figure 1: The red arrow represents the vector n a in the S − θ plane.The Davies critical point has been denoted by "CP" (black dot), which is located at S = S D , θ = π 2 .The two contours (blue circles) C 1 and C 2 are the two closed loops where C 1 encircles the critical point CP and C 2 does not.

Figure 2 :
Figure 2: The red arrows represent the vector n a in the S − θ plane.The extremal critical point has been denoted by "CP" (black dot), which is located at S = S ext , θ = π 2 .The two contours (blue circles) C 1 and C 2 are the two closed loops where C 1 encircles the critical point CP and C 2 does not.