Thermodynamics of a Higher Dimensional Noncommutative Inspired Anti-de Sitter-Einstein-Born-Infeld Black Hole

Abstract

We analyze noncommutative deformations of a higher dimensional anti-de Sitter-Einstein-Born-Infeld black hole. Two models based on noncommutative inspired distributions of mass and charge are discussed and their thermodynamical properties such as the equation of state are explicitly calculated. In the (3 + 1)-dimensional case the Gibbs energy function of each model is used to discuss the presence of phase transitions.

Appendices

Appendix A: Calculation of the Temperature

Here we derive a expression for the temperature of the modified black hole in (3+1)-dimensions used in the main text; its generalisation to higher dimensions is straightforward. The metric is assumed to be

$$ ds^2 = - N^2(r) dt^2 + f^{-2}(r) + r^2 (d\theta^2 + \sin^2 \theta d\phi^2), $$

(81)

with N(r) = f(r). Then the temperature at r = R is [35]

$$\begin{array}{@{}rcl@{}} T(R) &=& \left( \frac {\partial E}{\partial S} \right) \\ &=& \frac {\partial}{\partial(\pi {r_{H}^{2}})} {\int}_{S^{2}} d^{2} x \sqrt{\sigma} \left[ \frac 1{8\pi} \left( - \frac {2f(R)}R \right) - \epsilon_{0} (R) \right] \\ &=& -\frac {4\pi R^{2}}{8\pi R} \frac 1{f(R)} \frac {\partial f^{2}(R)}{\partial(\pi {r_{H}^{2}})} = -\frac R{4\pi} \frac 1{N(R)} \frac {\partial N^{2}(R)}{\partial r_{H}}. \end{array} $$

(82)

Let us assume that

$$ N^2(r) = 1 - M W(r) + V(r, Q, {\Lambda}), $$

(83)

where W and V are arbitrary functions of their arguments. Then the condition N²(r _H ) = 0 implies

$$ M = \frac 1{W(r_H)} [1 + V(r_H, Q, {\Lambda})]. $$

(84)

In consequence

$$ T(R) = \frac 1{N(R)} \frac {R \,W(R)}{4\pi r_H} \frac {\partial}{\partial r_H} \frac 1{W(r_H)} [1 + V(r_H, Q, {\Lambda})]. $$

(85)

On the other hand, we have

$$\begin{array}{@{}rcl@{}} \frac {\partial N^{2}(r)}{\partial r} &=& -M W_{,r} + V_{,r} \\ &=& - \frac 1{W(r_{H})} [1 + V(r_{H}, Q, {\Lambda})] W_{,r} + V_{,r}, \end{array} $$

(86)

and evaluation at r = r _H gives

$$ \frac {\partial N^2(r)}{\partial r} |_{r_H} = W(r_H) \frac {\partial}{\partial r_H} \frac 1{W(r_H)} [1 + V(r_H, Q, {\Lambda})]. $$

(87)

It follows that

$$\begin{array}{@{}rcl@{}} T(R) &=& \frac 1{N(R)} \frac {R \,W(R)}{r_{H} W(r_{H})} \frac 1{4\pi} \frac {\partial N^{2}(r)}{\partial r} |_{r_{H}} \\ &=& \frac 1{N(R)} \frac {R \,W(R)}{r_{H} W(r_{H})} \frac {\kappa_{H}}{2\pi}. \end{array} $$

(88)

This expression was used to obtain (55) in the main text.

Appendix B: Integrals for Small Noncommutative Corrections

The quantitiy I₁ in the main text is defined by the integral

$$ I_1 := {\int}_r^{\infty} ds \frac {H(s) s^3 \rho_Q(s)}{\sqrt{s^4 + H^2(s)/b^2}}. $$

(89)

For I₁ it suffices to set H = Q due to the presence of the exponentially decaying ρ _Q ; then

$$ I_1 = \frac {Q^2}{(4\pi\theta)^{3/2}} {\int}_r^{\infty} ds \frac {s^3 e^{-s^2/4\theta}}{\sqrt{s^4 + Q^2/b^2}}. $$

(90)

Let us perform the change of variable u = s²/4𝜃, then

$$\begin{array}{@{}rcl@{}} I_{1} &=& \frac12 \frac {Q^{2}}{(4\pi\theta)^{3/2}} {\int}_{r^{2}/4\theta}^{\infty} du \frac {u e^{-u}}{\sqrt{u^{2} + Q^{2}/(4\theta b)^{2}}} \\ &=&\frac12 \frac {Q^{2}}{(4\pi\theta)^{3/2}} {\int}_{r^{2}/4\theta}^{\infty} du e^{-u} \left[ 1 + \left( \frac Q{4\theta b u} \right)^{2} \right]^{-1/2} \\ &=&\frac12 \frac {Q^{2}}{(4\pi\theta)^{3/2}} {\int}_{r^{2}/4\theta}^{\infty} du e^{-u} \sum\limits_{k = 0}^{\infty} \binom{-1/2}{k} \left( \frac Q{4\theta b u} \right)^{2k} \\ &=&\frac12 \frac {Q^{2}}{(4\pi\theta)^{3/2}} \sum\limits_{k = 0}^{\infty} \binom{-1/2}{k} \left( \frac Q{4\theta b} \right)^{2k} {\int}_{r^{2}/4\theta}^{\infty} du u^{-2k} e^{-u} \\ &=&\frac12 \frac {Q^{2}}{(4\pi\theta)^{3/2}} \sum\limits_{k = 0}^{\infty} \binom{-1/2}{k} \left( \frac Q{4\theta b} \right)^{2k} {\Gamma}\left[ 1-2k, \frac {r^{2}}{4\theta} \right]. \end{array} $$

(91)

Using now the asymptotic behaviour for the upper incomplete gamma function [40]

$$ {\Gamma}(n, x) \sim x^{n-1} e^{-x}, \qquad x \to \infty, $$

(92)

we have

$$\begin{array}{@{}rcl@{}} I_{1} &=&\frac12 \frac {Q^{2}}{(4\pi\theta)^{3/2}} \sum\limits_{k = 0}^{\infty} \binom{-1/2}{k} \left( \frac Q{4\theta b} \right)^{2k} \left( \frac {r^{2}}{4\theta} \right)^{-2k} e^{-r^{2}/4\theta} \\ &=& \frac12 \frac {Q^{2} e^{-r^{2}/4\theta}}{(4\pi\theta)^{3/2}} \sum\limits_{k = 0}^{\infty} \binom{-1/2}{k} \left( \frac Q{b r^{2}} \right)^{2k} \\ &=& \frac12 \frac {Q^{2} e^{-r^{2}/4\theta}}{(4\pi\theta)^{3/2}} \left[ 1 + \left( \frac Q{b r^{2}} \right)^{2} \right]^{-1/2} \\ &=& \frac {Q^{2}}2 \frac {e^{-r^{2}/4\theta}}{(4\pi\theta)^{3/2}} \frac {r^{2}}{\sqrt{r^{4} + Q^{2}/b^{2}}}. \end{array} $$

(93)

We now evaluate

$$ I_2 := {\int}_r^{\infty} ds \frac {H^2(s)}{\sqrt{s^4 + H^2(s)/b^2}}, $$

(94)

for small values of 𝜃. In this case we use

$$ H(r) = Q \left[ 1 - \frac r{\sqrt{\pi\theta}} e^{-r^2/4\theta} \right], $$

(95)

valid for small values of 𝜃. We have then

$$\begin{array}{@{}rcl@{}} I_{2} &=& Q^{2} \left\{ {\int}_{r}^{\infty} ds \frac 1{\sqrt{s^{4} + Q^{2}/b^{2}}} - \frac 4{(4\pi\theta)^{1/2}} {\int}_{r}^{\infty} ds \frac {s e^{-s^{2}/4\theta}}{\sqrt{s^{4} + Q^{2}/b^{2}}} \right. \\ &&\quad\quad\left. + \frac {Q^{2}}{b^{2}} \frac 2{(4\pi\theta)^{1/2}} {\int}_{r}^{\infty} ds \frac {s e^{-s^{2}/4\theta}}{(s^{4} + Q^{2}/b^{2})^{3/2}} \right\}. \end{array} $$

(96)

In the above expression, the first integral is nothing but the standard BI electromagnetic contribution to the metric. The second and third integral correspond to exponentially decaying noncommutative corrections and can be evaluated in the same fashion as I₁ with the final result

$$\begin{array}{@{}rcl@{}} I_{2} \!&=&\! Q^{2} \left\{ \frac 1r \,{~}_{2} F_{1} \left( \frac 12, \frac 14; \frac 54; - \frac {Q^{2}}{b^{2} r^{4}} \right) \,-\, \frac 4{(4\pi\theta)^{1/2}} \!\times\! 2\theta \frac {e^{-r^{2}/4\theta}}{\sqrt{r^{4} + Q^{2}/b^{2}}} \right. \\ &&\quad\quad \left. \!+ \frac {Q^{2}}{b^{2}} \frac 2{(4\pi\theta)^{1/2}} \!\times\! 2\theta \frac {e^{-r^{2}/4\theta}}{(r^{4} + Q^{2}/b^{2})^{3/2}} \right\} \\ \!&=&\! Q^{2} \left\{ \frac 1r \,{~}_{2} F_{1}\left( \frac 12, \frac 14; \frac 54; - \frac {Q^{2}}{b^{2} r^{4}} \right) \,-\, \sqrt{\frac \theta\pi} \frac {4 e^{-r^{2}/4\theta}}{\sqrt{r^{4} + Q^{2}/b^{2}}} \,+\, \frac {Q^{2}}{b^{2}} \sqrt{\frac \theta\pi} \frac {2 e^{-r^{2}/4\theta}}{(r^{4} + Q^{2}/b^{2})^{3/2}} \right\}. \\\end{array} $$

(97)

Appendix C: Limit r → 0

We consider the case of (d + 1)-dimensions. Using the fact that [40]

$$ \gamma(s, z) = \sum\limits_{k = 0}^{\infty} \frac {z^s}{k!(s + k)}, $$

(98)

the function H(r) is approximated for small values of r as

$$ H(r) = \frac {2Q}{\pi^{d/2-1}} \left[ \frac 2d \left( \frac {r^2}{4\theta} \right)^{d/2} + {\dots} \right] \sim \frac {Q}{2^{d-2}d} \frac {r^d}{\pi^{d/2-1}\theta^{d/2}}, $$

(99)

where … denotes terms with higher powers of r. Direct substitution of this expression into the equation of motion given by (36) leads to

$$\begin{array}{@{}rcl@{}} (r^{d-2} e^{-2\nu})_{,r} \!&=&\! (d - 2) r^{d-3} - {\Lambda} r^{d-1} + 2b^{2} \left( r^{d-1} - \sqrt{\frac {Q^{2} r^{2d}}{2^{2d-4} d^{2} \pi^{d-2} \theta^{d} b^{2}} + r^{2(d-1)}} \right)\\ \end{array} $$

(100)

$$\begin{array}{@{}rcl@{}} \!&=&\! (d - 2) r^{d-3} - {\Lambda} r^{d-1} + 2b^{2} r^{d-1} \left( 1 - \sqrt{1 + \frac {Q^{2} r^{2}}{2^{2d-4} d^{2} \pi^{d-2} \theta^{d} b^{2}}} \right) \end{array} $$

(101)

$$\begin{array}{@{}rcl@{}} \!&=&\! (d - 2) r^{d-3} - {\Lambda} r^{d-1} - \frac {Q^{2} r^{d + 1}}{2^{2d-4} d^{2} \pi^{d-2} \theta^{d}} + \dots, \end{array} $$

(102)

where … denotes terms with higher powers of r. Integration of this expression gives

$$ r^{d-2} e^{-2\nu} = b_0 + r^{d-2} - \frac {\Lambda}{d} r^d - \frac {Q^2 r^{d + 2}}{2^{2d-4} (d + 2) d^2 \pi^{d-2} \theta^d} + \dots, $$

(103)

where b₀ is a constant that can be set equal to − 2M. Therefore

$$ e^{-2\nu} = 1- \frac{2M}r - \frac{\Lambda}{d} r^2 - \frac {Q^2}{2^{2d-4} (d + 2) d^2 \pi^{d-2} \theta^d} r^4 + \dots $$

(104)

We see that the singularity due to the electromagnetic field has been smoothed out.