Phase transitions in D-dimensional Gauss-Bonnet-Born-Infeld AdS black holes

Neeraj Kumara∗ Sunandan Gangopadhyaya†aDepartment of Theoretical Sciences,

S.N.Bose National Centre for Basic Sciences,JD Block, Sector III, Salt Lake, Kolkata, 700106, India

Abstract

In this paper, we have investigated the phase transition in black holes when Gauss-Bonnet corrections to the spacetimecurvature and Born-Infeld extension in stress-energy tensor of electromagnetic field are considered in a negativecosmological constant background. It is evident that the black hole undergoes a phase transition as the specific heatcapacity at constant potential shows discontinuities. Further, the computation of the free energy of the black hole, theEhrenfest scheme and the Ruppeiner state space geometry analysis are carried out to establish the second order natureof this phase transition. The effect of non-linearity arising from Born-Infeld electrodynamics is also evident from ouranalysis. Our investigations are done in general D-spacetime dimensions with D > 4, and specific computations havebeen carried out in D = 5, 6, 7 spacetime dimensions.

I. INTRODUCTION

Singularities are inevitable in Einstein’s gravity and the object called black hole has properties which are not fullyexplained by classical gravity. Hawking [1] and Bekenstein [2] considered these objects as thermodynamic systems andassociated with it thermodynamic properties like entropy and temperature. The study of black hole thermodynamicshas received a new impetus recently because it is believed that it may shed light on quantum gravity.In seeking modifications to Einstein’s gravity in bottom-up approach, the very first idea was to see higher ordereffects of curvature in the action. This modification was brought in [3] where general spacetime Lagrangian densityhas higher powers of curvature and is known as Lovelock gravity. These modifications are incorporated in such a waythat the solutions resulting from these are consistent and are free from any ghost terms, hence arbitrary terms in theaction are not allowed. Expansion of the Lovelock Lagrangian contains the Gauss-Bonnet term. It is believed thatthis simple modification may lead to interesting insights into the true nature of gravity.Non-linear electrodynamics was first developed by Max Born and Leopold Infeld [4] in order to resolve the self-energy problem of a point charge. However, all such non-linear theories got soon replaced by the profound quantumdescription of Maxwell’s theory in the form of quantum electrodynamics. Later, Born-Infeld theory reappeared as alow energy limit of string theory in [5]-[6]. Since then an extensive study of black holes with these higher derivativesin gauge field has been done [7]-[9].Thermodynamics of the black holes in Gauss-Bonnet gravity have been investigated in great detail in [10]-[15] andthe list is not exhaustive. These black holes have been studied with different horizon topologies, with and withoutcharge, with non-linear electrodynamics, with different cosmological constant signature, etc. Heat capacity of Gauss-Bonnet AdS black holes have been computed in [16] but the classification of phase-transitions into first or secondorder was not done. Our goal is to check the phase structure of these black holes in AdS background with Born-Infeldelectrodynamics and analyse it using standard tools of thermodynamics. More precisely, our main aim is to study thethermodynamics of these black holes and figure out the order of the phase-transitions occurring in such black holes.In this paper, we wish to study the black hole solutions obtained in Gauss-Bonnet gravity incorporating Born-Infeld theory in the action with negative cosmological constant. All analysis is done in D spacetime dimensions.Thermodynamic quantities are calculated and singularities in heat capacity are registered for the fixed Gauss-Bonnetparameter, Born-Infeld parameter and charge. These divergences are shown in graphs when the heat capacity isplotted with the horizon radius for D = 5, 6, 7. As we shall show later that this is a clear indication of a secondorder phase transition of the black hole system under consideration with specific values of the parameters. Alongthe way, the effects of non-linearity introduced through Born-Infeld electrodynamics on temperature as well as on

∗ e-mail: neerajkumar@bose.res.in† e-mail: sunandan.gangopadhyay@gmail.com, sunandan.gangopadhyay@bose.res.in

arX

iv:2

011.

0513

7v2

[gr

-qc]

11

Mar

202

1

2

phase transition points are also analysed. Ehrenfest scheme and the Ruppeiner state space geometry analysis [17]-[18]are trusted techniques in standard thermodynamics and are employed to study black hole thermodynamics as well[19]-[21]. We wish to employ these for our system and study the order of the phase transitions in the black holes. Aparameter called Prigogine-Defay ratio [22] which determines the deviations from the second order nature of phasetransition is also calculated using Ehrenfest equations. All results for Einstein-Born-Infeld black holes in [19] arerecovered when the Gauss-Bonnet parameter is put to zero. Further, motivations to study these black hole phasetransitions in AdS space comes from AdS/CFT correspondence [23]-[25]. Confinement problem, superconductivity andother strongly correlated systems are in the domain of this duality and has led to deep insights to these phenomena.Along the way, we shall also talk about the possibility of treating the cosmological constant, Born-Infeld parameterand Gauss-Bonnet parameter as thermodynamical variables and derive a Smarr relation using scaling argument andfirst law of thermodynamics in D-spacetime dimensions.The structure of this paper is as follows. In section II we shall be calculating the thermodynamic properties ofthe black holes considered in this paper and also calculate their heat capacity. We plot the heat capacity with thehorizon radius to study its nature. Section III is devoted to carrying out Ehrenfest scheme analysis at the points ofphase transitions in order to understand the order of phase transition. Section IV contains the Ruppeiner state spacegeometry analysis of the singularities. We conclude by summarizing our results in section V.

II. THERMODYNAMICS OF GAUSS-BONNET-BORN-INFELD BLACK HOLES

In this section, we introduce the black hole spacetime we are interested in and study its thermodynamic properties.We consider the Gauss-Bonnet black hole spacetime with Born-Infeld electromagnetic charge in AdS background.The action of Gauss-Bonnet gravity with Born-Infeld electromagnetic field reads

I =1

16π

∫dDx√−g[R− 2Λ + αLGB + L(F )] (1)

where G = c = 1, the Gauss-Bonnet Lagrangian density LGB = R2 − 4RγδRγδ + RγδλσR

γδλσ, the Born-Infeld

term L(F ) = 4b2

(1−

√1 +

FµνFµν2b2

), α is the Gauss-Bonnet parameter, b is the Born-Infeld parameter, Λ =

− (D − 1)(D − 2)

2l2with l being AdS radius and D is the spacetime dimensions greater than 4.

The solution following from the above theory reads

ds2 = −f(r)dt2 +1

f(r)dr2 + r2dΩ2

D−2 (2)

where the metric coefficient f(r) is given by [16]

f(r) = 1 +r2

2α′(1−

√g(r)) (3)

with g(r) given by

g(r) = 1− 4α′

l2+

4α′m

rD−1− 16α′b2

(D − 1)(D − 2)

(1−

√1 +

(D − 2)(D − 3)q2

2b2r2D−4

)

− 8(D − 2)α′q2

(D − 1)r2D−4 2F1

[D − 3

2D − 4,

1

2,

3D − 7

2D − 4,− (D − 2)(D − 3)q2

2b2r2D−4

]. (4)

Note that the actual black hole parameters such as the charge Q, mass M and Gauss-Bonnet coefficent α are connectedto q, m and α′ as

M =(D − 2)ω

16πm, α =

α′

(D − 3)(D − 4),

Q =√

2(D − 2)(D − 3)ω

8πq, ω =

2π

D − 1

2

Γ(D − 1)

2

. (5)

3

The black hole mass can be written in terms of the horizon radius (r+) from the condition f(r+) = 0 and reads

M =(D − 2)ωα′

16πrD−5+ +

(D − 2)ω

16πrD−3+ +

(D − 2)ω

16πl2rD−1+ +

b2ω

4π(D − 1)rD−1+

(1−

√1 +

16π2Q2

b2ω2r2D−4+

)

+4(D − 2)πQ2

(D − 1)(D − 3)ωrD−3+

2F1

[D − 3

2D − 4,

1

2,

3D − 7

2D − 4,− 16π2Q2

b2ω2r2D−4+

]. (6)

The Hawking temperature of the black hole spacetime can be calculated using eq.(3) as

T =1

4π

(df(r)

dr

)r+

=1

4πr+(r2+ + 2α′)

[(D − 1)r4

+

l2+ (D − 3)r2

+ + (D − 5)α′ +4b2r4

+

D − 2

(1−

√1 +

16π2Q2

b2ω2r2D−4+

)](7)

where l shall be fixed to unity for rest of the calculations unless mentioned otherwise.We now use the relation

dM = TdS + ΦdQ (8)

which is the first law of black hole thermodynamics. This form is analogous to the first law of thermodynamics

dU = TdS − PdV (9)

with the identification of the pressure P to the negative of the electrostatic potential Φ, the volume to the chargeQ and the internal energy U to the mass of the black hole M . From this relation, the black hole entropy S can beobtained as

S =

∫ r+

0

1

T

(∂M

∂r+

)Q

dr+ =ω

4rD−2+

[1 +

D − 2

D − 4

2α′

r2+

]. (10)

In principle, eq.(10) allows the Hawking temperature (T ) of the black hole to be written in terms of the entropy S.In Fig.1, we plot the Hawking temperature of the black hole with the horizon radius and observe that there is nodiscontinuity in these graphs hence there is no first order phase transition. The Maxwell limit is also displayed forthese black holes. It can be seen that the black holes with Born-Infeld electrodynamics have higher temperature thanwith Maxwell electrodynamics. This effect is prominent for smaller black holes and fades away as the horizon radiusincreases.Further, we analyse the system and calculate the heat capacity at constant potential. This we do to find whetherthere is a chance of higher order phase transitions in this black hole spacetime.The potential in D dimensions for the black hole spacetime can be calculated from the first law and eq.(6) to be

Φ =

(∂M

∂Q

)=

4πQ

ω(D − 3)rD−3+

2F1

[D − 3

2D − 4,

1

2,

3D − 7

2D − 4,− 16π2Q2

b2ω2r2D−4+

]. (11)

Considering the temperature of the black hole to be a function of entropy and charge (T ≡ T (S,Q)), we have(∂T

∂S

)Φ

=

(∂T

∂S

)Q

−(∂T

∂Q

)S

(∂Φ

∂S

)Q

(∂Q

∂Φ

)S

. (12)

Now using the thermodynamic identity (∂Q

∂S

)Φ

(∂S

∂Φ

)Q

(∂Φ

∂Q

)S

= −1 (13)

along with eq.(12), we obtain the heat capacity at constant potential to be

CΦ = T

(∂S

∂T

)Φ

=

T

(∂Φ

∂Q

)r+

(∂S

∂r+

)Q(

∂Φ

∂Q

)r+

(∂T

∂r+

)Q

−(∂T

∂Q

)r+

(∂Φ

∂r+

)Q

. (14)

4

(a) D=5 (b) D=6

(c) D=7

FIG. 1: Temperature vs. horizon radius

Solid line(Q=0.13, b=10, α′=0.01)Dashed line(Q=0.13, b→ ∞, α′=0.01)

The partial derivatives involved in the above equation when calculated read

(∂Φ

∂r+

)Q

= − 4πQ

ωrD−2+

(1 +

16π2Q2

b2ω2r2D−4+

)−1/2

(15)

(∂T

∂Q

)r+

= − 1

r2+ + 2α′

16πQ

ω2(D − 2)r2D−7+

(1 +

16π2Q2

b2ω2r2D−4+

)−1/2

(16)

(∂S

∂r+

)Q

≡ dS

dr+=ω

4(D − 2)rD−3

+

(1 +

2α′

r2+

)(17)

(∂Φ

∂Q

)r+

=4π

(D − 2)ωrD−3+

(1 +16π2Q2

b2ω2r2D−4+

)−1/2

+1

D − 32F1

[D − 3

2D − 4,

1

2,

3D − 7

2D − 4,− 16π2Q2

b2ω2r2D−4+

] (18)

(∂T

∂r+

)Q

=1

4πr+(r2+ + 2α′)

[4(D − 1)r3

+ + 2(D − 3)r+ +16b2r3

(D − 2)

(1−

√1 +

16π2Q2

b2ω2r2D−4+

)

+64π2Q2

ω2r2D−7+

(1 +

16π2Q2

b2ω2r2D−4+

)−1/2

−3r2

+ + 2α′

4πr2+(r2

+ + 2α′)2

[(D − 1)r4

+ + (D − 3)r2+ + (D − 5)α′ +

4b2r4+

D − 2

(1−

√1 +

16π2Q2

b2ω2r2D−4+

)]. (19)

5

The form of heat capacity is a large expression to write, hence we study its behaviour through plots. We plot CΦ

with horizon radius (r+) for fixed values of the Gauss-Bonnet parameter (α′), Born-Infeld parameter (b) and charge(Q) for D = 5, 6, 7.

(a) D=5 (b) D=5

(c) D=6 (d) D=6

(e) D=7 (f) D=7

FIG. 2: Heat capacity vs. horizon radius

Solid line(Q=0.13, b=10, α′=0.01)Dashed line(Q=0.13, b→ ∞, α′=0.01)

The plots of heat capacity have multiple points of discontinuities and give indications of a second order phase transition.We calculate the exact points of discontinuities of heat capacity. We also calculate the horizon radius of extremal black

hole (r(e)+ ) for the same fixed parameters for each dimension in order to check the physical validity of the singularities

in heat capacity. The following Table shows extremal black hole radii (r(e)+ ) and singularities in heat capacity at radii

r+0, r+1, and r+2.

D=5 D=6 D=7

r(e)+ 0.204957 0.276397 0.340496r+0 0.119316 – –r+1 0.23969 0.298154 0.365921r+2 0.639584 0.725395 0.774541

6

TABLE I: Extremal radii and values of horizon radius where CΦ diverge.For solid lines (Q=0.13, b=10, α′=0.01)

Table I shows that the black hole in D = 5 spacetime dimensions has one unphysical singularity (r+0) in the heatcapacity as the point is below the extremal radius whereas the other two at points r+1 and r+2 are actual phasetransition points. For D = 6, 7 there are two singularities each which are actual phase transition points as both thepoints lie above extremal radius values for respective dimensions. These plots also reveal that for a particular spacetime

dimension there are three phases of the black hole, namely phase I (r(e)+ < r+ < r+1), phase II (r+1 < r+ < r+2), and

phase III (r+ > r+2). The dashed curves in the plots depict the behaviour of these black holes in the Maxwell limit.It is apparent that the Born-Infeld non-linearity shifts the phase transition points to lower value of the horizon radiusand this effect is significant at smaller horizon radius.In the following section, we shall show that the system follows Ehrenfest second order phase transition equations.We shall also carry out the Ruppeiner’s state space geometry analysis to confirm the second order phase transitionexhibited by this black hole spacetime.We would now like to discuss about the possibility of interpreting the cosmological constant Λ as a thermodynamicpressure p and also treating the Born-Infeld parameter b and the Gauss-Bonnet parameter α as new thermodynamicvariables, as it has been proposed recently in [26]-[27]. The argument put forward in these studies was that since Λ,b and α are dimensionful quantities, the corresponding terms would definitely appear in the Smarr formula. To writedown the Smarr formula, we first note that the first law of black hole thermodynamics takes the form

dM = TdS + vdp+ ΦdQ+Bdb+ Ωdα (20)

where B and Ω are the conjugate variables to b and α respectively defined by

B =∂M

∂band Ω =

∂M

∂α(21)

Considering the black hole mass M = M(S,Q, p, b, α) and performing dimensional analysis, we find that [M ] =LD−3, [S] = LD−2, [p] = L−2, [Q] = LD−3, [b] = L−1 and [α] = L2. Using these along with Euler’s theorem1, weobtain

(D − 3)M = (D − 2)S

(∂M

∂S

)− 2p

(∂M

∂p

)+ (D − 3)Q

(∂M

∂Q

)− b

(∂M

∂b

)+ 2α

(∂M

∂α

). (22)

Now using eq.(20), we get(∂M

∂S

)= T,

(∂M

∂p

)= v,

(∂M

∂Q

)= Φ,

(∂M

∂b

)= B,

(∂M

∂α

)= Ω . (23)

Substituting this in eq.(22) yields the Smarr formula

M =D − 2

D − 3TS − 2

D − 3pv + ΦQ− 1

D − 3bB +

2

D − 3αΩ . (24)

It would be interesting to work with the first law of black hole thermodynamics given in eq.(20) and study phasetransitions. However, we shall not carry out this investigation in this work and we intend to do it in future.Before ending this section, we would like to compute the free energy (F = M−TS) of this black hole. Using eq.(s)(6, 7and 10), this takes the form

F =(D − 2)ωrD−3

+

16π

(1 + r2

+ +α′

r2+

)+

b2ω

4π(D − 1)rD−1+

(1−

√1 +

16π2Q2

b2ω2r2D−4+

)+

4(D − 2)πQ2

(D − 1)(D − 3)ωrD−3+

2F1

−ωrD−1

+

16π(D − 4)

((D − 4)r2

+ + 2(D − 2)α′)

r2+ + 2α′

[(D − 1)r4

+ + (D − 3)r2+ + (D − 5)α′ +

4b2r4+

D − 2

(1−

√1 +

16π2Q2

b2ω2r2D−4+

)](25)

1 Given a function g(x, y) satisfying g(αmx, αny) = αrg(x, y), we have rg(x, y) = mx

(∂g

∂x

)+ ny

(∂g

∂y

).

7

where 2F1 ≡ 2F1

[D − 3

2D − 4,

1

2,

3D − 7

2D − 4,− 16π2Q2

b2ω2r2D−4+

].

The above expression for D = 5 simplifies to

F =3πr2

+

8

(1 + r2

+ +α′

r2+

)+b2πr4

+

8

(1−

√1 +

4Q2

b2π2r6+

)+

3Q2

4πr2+

2F1

[1

3,

1

2,

4

3,− 4Q2

b2π2r6+

]

−πr4

+

8

(r2+ + 6α′

r2+ + 2α′

)[4r4

+ + 2r2+ +

4b2r4+

3

(1−

√1 +

4Q2

b2π2r6+

)]. (26)

Fig.[3] shows the plot of free energy with horizon radius and does not show any cusp. This excludes the possibility of

(a) D=5

FIG. 3: Free Energy vs. horizon radius

Solid line(Q=0.13, b=10, α′=0.01)Dashed line(Q=0.13, b→ ∞, α′=0.01)

a first order phase transition. In the next section we shall employ the Ehrenfest scheme of analysing phase transitionsand establish that the discontinuities shown in the heat capacity of the black hole corresponds to second order phasetransitions.

III. ANALYSIS OF PHASE TRANSITION USING EHRENFEST SCHEME

Ehrenfest’s approach to studying phase transitions is the standard technique in thermodynamics to determine thenature of phase transitions for various thermodynamical systems [28], [29]. It simply says that the order of the phasetransition corresponds to the discontinuity in the order of the derivative of Gibb’s potential. For the second orderphase transition the second derivative encounters discontinuities, however first derivative and Gibb’s potential at thosepoints are continuous. These conditions with Maxwell’s relations give two equations which have to be satisfied forthe second order phase transition.The first and second Ehrenfest equations in thermodynamics are given by(

∂P

∂T

)S

=1

V T

CP2 − CP1

β2 − β1=

∆CPV T∆β

, (27)

(∂P

∂T

)V

=β2 − β1

κ2 − κ1=

∆β

∆κ(28)

where subscripts 1 and 2 denote two distinct phases of the system. Now we use the correspondence between thepressure (P ) to the negative of the electrostatic potential difference (Φ) and the volume (V ) to the charge (Q) of theblack hole. These identifications lead to the following equations

−(∂Φ

∂T

)S

=1

QT

CΦ2 − CΦ1

β2 − β1=

∆CΦ

QT∆β(29)

8

−(∂Φ

∂T

)Q

=β2 − β1

κ2 − κ1=

∆β

∆κ. (30)

Note that β is the volume expansion coefficient and κ is the isothermal compressibility of the system and are definedas

β =1

Q

(∂Q

∂T

)Φ

, κ =1

Q

(∂Q

∂Φ

)T

. (31)

Now we proceed to check whether the black hole phase transition satisfies the Ehrenfest equations (29, 30). In otherwords, we investigate the validity of the Ehrenfest equations at the points of discontinuities r+i, i = 1, 2. Here wedenote the critical values of temperature by Ti, charge by Qi and entropy by Si. The left hand side of the firstEhrenfest equation (29) at the critical point can be written as

−[(

∂Φ

∂T

)S

]S=Si

≡ −

[(∂Φ

∂T

)r+

]r+=r+i

= −

[(∂Φ

∂Q

)r+

]r+=r+i

[(∂Q

∂T

)r+

]r+=r+i

= −

[(∂Φ

∂Q

)r+

]r+=r+i[(

∂T

∂Q

)r+

]r+=r+i

. (32)

Using eq.(s)(16, 18) we calculate the left hand side of the above relation and obtain

−

[(∂Φ

∂T

)r+

]r+=r+i

= −ω(r2

+i + 2α′)rD−4+i

4Qi

[1 +

1

D − 3

√1 +

16π2Q2i

b2ω2r2D−4+i

2F1

[D − 3

2D − 4,

1

2,

3D − 7

2D − 4,− 16π2Q2

i

b2ω2r2D−4+i

]].(33)

Using eq.(31) and the definition of heat capacity CΦ = T ( ∂S∂T )Φ, we can obtain the right hand side of eq.(29) to be

Qiβ =

[(∂Q

∂T

)Φ

]S=Si

=

[(∂Q

∂S

)Φ

]S=Si

(CΦ

Ti

)(34)

which implies

∆CΦ

TiQi∆β=

[(∂S

∂Q

)Φ

]S=Si

. (35)

Using the identity (13), the above equation can be written in the form

∆CΦ

TiQi∆β= −

[(∂Φ

∂Q

)S

]S=Si[(

∂Φ

∂S

)Q

]S=Si

≡ −

[(∂Φ

∂Q

)r+

]r+=r+i

[dS

dr+

]r+=r+i[(

∂Φ

∂r+

)Q

]r+=r+i

. (36)

Now using eq.(s) (15, 17 , 18), we calculate the right hand side of the above relation to be

∆CΦ

TiQi∆β= −

ω(r2+i + 2α′)rD−4

+i

4Qi

[1 +

1

D − 3

√1 +

16π2Q2i

b2ω2r2D−4+i

2F1

[D − 3

2D − 4,

1

2,

3D − 7

2D − 4,− 16π2Q2

i

b2ω2r2D−4+i

]]. (37)

9

Eq.(33) and eq.(37) show the validity of the first Ehrenfest’s equation for the black hole spacetime under consideration.We now proceed to check the second Ehrenfest equation. To calculate the left hand side of the second Ehrenfestequation, we use the thermodynamic relation(

∂T

∂Φ

)Q

=

(∂T

∂S

)Φ

(∂S

∂Φ

)Q

+

(∂T

∂Φ

)S

(38)

taking T ≡ T (S,Φ).

Since the heat capacity diverges at the critical points, hence

[(∂T

∂S

)Φ

]S=Si

= 0 and

(∂S

∂Φ

)Q

are finite at the critical

point. Therefore, the above equation becomes[(∂T

∂Φ

)Q

]S=Si

=

[(∂T

∂Φ

)S

]S=Si

(39)

which implies [(∂T

∂Φ

)Q

]r+=r+i

=

[(∂T

∂Φ

)r+

]r+=r+i

. (40)

Therefore, the left hand side of the second Ehrenfest equation is equal to left hand side of the first Ehrenfest equation.Hence, we have[(

∂Φ

∂T

)Q

]r+=r+i

= −ω(r2

+i + 2α′)rD−4+i

4Qi

[1 +

1

D − 3

√1 +

16π2Q2i

b2ω2r2D−4+i

2F1

[D − 3

2D − 4,

1

2,

3D − 7

2D − 4,− 16π2Q2

i

b2ω2r2D−4+i

]].(41)

Now from eq.(31), at the critical points, we have

κQi =

[(∂Q

∂Φ

)T

]S=Si

. (42)

Using the thermodynamic identity

(∂Q

∂φ

)T

(∂Φ

∂T

)Q

(∂T

∂Q

)Φ

= −1 and the definition of β in eq.(31), we find

κQi =

[(∂T

∂Φ

)Q

]S=Si

Qiβ . (43)

Therefore, the right hand side of the second Ehrenfest equation (30) reduces to

∆β

∆κ= −

[(∂Φ

∂T

)Q

]S=Si

. (44)

This can further be written as

−

[(∂Φ

∂T

)Q

]S=Si

= −

[(∂Φ

∂S

)Q

]S=Si

[(∂S

∂T

)Q

]S=Si

= −

[(∂Φ

∂S

)Q

]S=Si[(

∂T

∂S

)Q

]S=Si

(45)

which in turn implies

−

[(∂Φ

∂T

)Q

]S=Si

≡ −

[(∂Φ

∂T

)Q

]r+=r+i

= −

[(∂Φ

∂r+

)Q

]r+=r+i[(

∂T

∂r+

)Q

]r+=r+i

. (46)

10

The condition that heat capacity in eq.(14) diverges at the critical point gives[(∂Φ

∂Q

)r+

]r+=r+i

[(∂T

∂r+

)Q

]r+=r+i

=

[(∂T

∂Q

)r+

]r+=r+i

[(∂Φ

∂r+

)Q

]r+=r+i

. (47)

Using the above condition and eq.(46) we obtain the right hand side of the second Ehrenfest equation to be

∆β

∆κ= −

ω(r2+i + 2α′)rD−4

+i

4Qi

[1 +

1

D − 3

√1 +

16π2Q2i

b2ω2r2D−4+i

2F1

[D − 3

2D − 4,

1

2,

3D − 7

2D − 4,− 16π2Q2

i

b2ω2r2D−4+i

]]. (48)

Eq.(41) and eq.(48) shows the validity of the second Ehrenfest equation at critical points.Finally, we calculate the Prigogine-Defay (PD) ratio [22] for the system. Using eq.(29) and eq.(39), we have[(

∂Φ

∂T

)Q

]S=Si

=

[(∂Φ

∂T

)S

]S=Si

= − ∆CΦ

TiQi∆β. (49)

Now eq.(44) with the above equation gives

Π =∆CΦ∆κ

TiQi(∆β)2= 1 . (50)

These results show the exact second order nature of phase transition in this black hole.

IV. RUPPEINER STATE SPACE GEOMETRY ANALYSIS

Ruppeiner’s state space geometry technique is another approach to study thermodynamic systems. The idea is towrite the abstract manifold with the notion of distance from thermodynamic variables and study the curvature inorder to study phase transition. This technique has been applied to black hole systems in [30]-[31]. Definitions ofmetric coefficients are quite well known [32].The Ruppeiner metric coefficients for the manifold are given by

gRij = −∂2S(xi)

∂xi∂xj(51)

where xi = xi(M,Q); i = 1, 2 are extensive variables of the manifold. The calculation of the Weinhold metriccoefficients is convenient for computational purpose. These are given by

gWij =∂2M(xi)

∂xi∂xj(52)

where xi = xi(S,Q); i = 1, 2. It is to be noted that Weinhold geometry is connected to the Ruppeiner geometrythrough the following relation

dS2R =

dS2W

T. (53)

For the black hole spacetime with which we are working, the Ruppeiner metric coefficients can be calculated to be

gSS =1

T

(∂2M

∂r2+

)Q(

dS

dr+

)2 −

(∂M

∂r+

)Q

(d2S

dr2+

)(dS

dr+

)3

gSQ =1

T

∂2M

∂r+∂QdS

dr+

gQQ =

1

T

(∂2M

∂Q2

)r+

. (54)

11

From eq.(s)(6, 10), we calculate first and second order partial derivatives appearing in eq.(54) to be

∂2M

∂r+∂Q= − 4πQ

ωrD−2+

(1 +

16π2Q2

b2ω2r2D−4+

)−1/2

(55)

d2S

dr2+

=ω

4(D − 2)rD−4

+

((D − 3) +

2(D − 5)α′

r2+

)(56)

(∂M

∂r+

)Q

=(D − 2)(D − 5)ωα′

16πrD−6+ +

(D − 2)(D − 3)ω

16πrD−4+ +

(D − 1)(D − 2)ω)

16πrD−2+

+b2ω

4πrD−2+

(1−

√1 +

16π2Q2

b2ω2r2D−4+

)(57)

(∂2M

∂r2+

)Q

=(D − 2)(D − 5)(D − 6)ωα′

16πrD−7+ +

(D − 2)(D − 3)(D − 4)ω

16πrD−5+ +

(D − 1)(D − 2)2ω)

16πrD−3+

+b2(D − 2)ω

4πrD−3+

(1−

√1 +

16π2Q2

b2ω2r2D−4+

)+

4(D − 2)πQ2

ωrD−1+

(1 +

16π2Q2

b2ω2r2D−4+

)−1/2

(58)

(a) D=5 (b) D=5

(c) D=6 (d) D=6

(e) D=7 (f) D=7

FIG. 4: Ruppeiner curvature vs. horizon radius

(Q=0.13, b=10, α′=0.01)

12

(∂2M

∂Q2

)r+

=4π

ω(D − 2)rD−3+

(1 +16π2Q2

b2ω2r2D−4+

)−1/2

+1

D − 32F1

[D − 3

2D − 4,

1

2,

3D − 7

2D − 4,− 16π2Q2

i

b2ω2r2D−4+i

] . (59)

With these metric coefficients, we can calculate the curvature of the two dimensional manifold, and singularities inthe curvature indicate second order phase transition. The formula for the curvature reads

R = − 1√g

[∂

∂r+

(gSQ√ggSS

∂gSS∂Q

− 1√g

gQQ∂r+

(dr+

dS

))(dr+

dS

)+∂

∂Q

(2√g

∂gSQ∂Q

− 1√g

∂gSS∂Q

− gSQ√ggSS

∂gSS∂r+

(dr+

dS

))](60)

where g is gSS × gQQ − g2SQ. We have calculated and plotted R with the horizon radius for fixed values of other

parameters in fig. 4.These show multiple points of discontinuities, however, all are not physical. It is interesting to note that the Ruppeinercurvature diverges at all the values where the heat capacity is singular. This indicates the second order nature ofphase transition in this black hole.

V. CONCLUSION

In this work, we have analysed black holes in Gauss-Bonnet gravity with Born-Infeld electrodynamics in AdS back-ground. We calculated the thermodynamic properties associated with the black hole spacetime and calculated theheat capacity. It has been obtained that there are infinite discontinuities in the heat capacity when plotted with thehorizon radius with other parameters kept constant. We eastablish that these are the signs of second order phasetransitions. First, we compute the free energy of the black hole and observe that there is no cusp in its variationwith the horizon radius. This rules out the possibility of a first order phase transition. The Ehrenfest scheme is thenemployed to establish the second order nature of the phase transition. We find that the system under considerationobeys both Ehrenfest equations. We then calculate the Prigogine-Defay ratio using Ehrenfest equations which indi-cate that the divergences are exactly of second order. We also discussed the effects of non-linear electrodynamics onHawking temperature and heat capacity. We observe that there is a non-trivial effect of the Born-Infeld parameter onthe temperature of the black hole. The Hawking temperature is higher than the black hole with Maxwell term for thesame horizon radius and this effect fades away as the horizon radius increases. We also see that the phase transitionpoints shift to lower horizon radius values in the presence of Born-Infeld electrodynamics. The state space geometrytechnique of Ruppeiner also helped further to reinforce the second order nature of these black hole phase transi-tions. We calculate the Ruppeiner curvature and again plot it with the horizon radius for the same fixed parameters.Singularities are observed exactly at those points where the heat capacity diverged. Both these methods reconfirmthe second order nature of phase transition in these black holes. Reassuringly, we recover the Einstein-Born-Infeldblack holes when the Gauss-Bonnet parameter is put to zero. We have also derive a Smarr relation in D-spacetimedimensions using scaling arguments and first law of black hole thermodynamics in which the cosmological constant,Born-Infeld parameters and Gauss-Bonnet parameter are treated as thermodynamic variables.

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