Thermodynamics of Reissner–Nordstr$\ddot{\mathrm{o}}$m black holes in higher dimensions: rainbow gravity background with general uncertainty principle

Abstract

In this paper, we investigate the thermodynamic properties of Reissner–Nordstr$\ddot{\mathrm{o}}$m black holes embedded in higher (d) dimensions in the framework of rainbow gravity incorporating the effects of the generalized uncertainty principle. We also examine all the properties graphically by varying the rainbow gravity parameter $\eta$ and the generalized uncertainty principle parameter $\alpha$. We find the existence of remnant and critical mass of the concerned black hole. We calculate the local temperature, local internal energy and hence we analyse the thermal stability of the black hole by computing the local heat capacity. Further, we study the phase transitions of the aforesaid black hole solution under the effects of generalized uncertainty principle. From the analysis of the specific heat at the horizon, we observe that there are phase transitions for all dimensions but when we analyze the same, measured by the local observer, we find that there exist only two phase transitions.

Appendix

The heat capacity of the black hole is given as:

$$\begin{aligned} C_h= & {} \left[ \pi \sqrt{m^2-4 q^2} \sqrt{1- \frac{\eta }{E_p^n} \left( \frac{1}{2} \sqrt{m^2-4 q^2}+\frac{m}{2}\right) ^{-\frac{n}{d-3}}}\right] \nonumber \\&\left[ 2^{-\frac{d-4}{d-3}} \left( \left( 2^{\frac{d}{d-3}+1} (2 d-5) q^2 \left( \frac{1}{2} \sqrt{m^2-4 q^2}+\frac{m}{2}\bigg )^{\frac{3}{d-3}}\right. \right. \right. \right. \nonumber \\&+\,\left( \sqrt{m^2-4 q^2}+m\bigg )^{\frac{d}{d-3}} \left( (d-3) \sqrt{m^2-4 q^2}+(2-d) m\right) \right) \nonumber \\&\left( \sqrt{m^2-4 q^2}+m\right) ^{-\frac{2(d-1)}{d-3}} \nonumber \\&\quad \left( 1- \frac{\eta }{E_p^n} \left( \frac{1}{2} \sqrt{m^2-4 q^2}+\frac{m}{2}\right) ^{-\frac{n}{d-3}}\right) \nonumber \\&+\,2^{\frac{n}{d-3}-1} \frac{1}{E_p^n} \left( m n \eta \left( \sqrt{m^2-4 q^2}+m\right) ^{-\frac{d+n-2}{d-3}}\right. \nonumber \\&\quad -\,4 n q^2 \eta \left( \sqrt{m^2-4 q^2}+m\right) ^{-\frac{2 d+n-5}{d-3}}\bigg )\bigg )\bigg ]^{-1}. \end{aligned}$$

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The entropy of the black hole is expressed as:

$$\begin{aligned} S_h= & {} \frac{1}{4} \pi E_p^{-3 n} \left( r_h^{-7 d-n} \left( -\frac{8 q^8 r_h^{22} \eta E_p^{2 n}}{-7 d-n+22}+\frac{8 q^6 \eta r_h^{2 d+16} E_p^{2 n}}{-5 d-n+16}-\frac{8 q^4 \eta r_h^{4 d+10} E_p^{2 n}}{-3 d-n+10}\right. \right. \nonumber \\&+\,r_h^{2 (d+5)} \left( \frac{8 q^6 r_h^6 \eta E_p^{2 n}}{-5 d-n+16}+\frac{8 q^4 \eta r_h^{2 d} E_p^{2 n}}{3 d+n-10}\bigg )+\frac{8 \eta r_h^{8 d-2} E_p^{2 n}}{d-n-2}\bigg )\right. \nonumber \\&+\,r_h^{-7 d-2 n} \left( -\frac{6 q^8 r_h^{22} \eta ^2 E_p^n}{-7 d-2 n+22}+\frac{6 q^6 \eta ^2 r_h^{2 d+16} E_p^n}{-5 d-2 n+16}-\frac{6 q^4 \eta ^2 r_h^{4 d+10} E_p^n}{-3 d-2 n+10}+r_h^{2 (d+5)}\right. \nonumber \\&\left( \frac{6 q^6 r_h^6 \eta ^2 E_p^n}{-5 d-2 n+16}+\frac{6 q^4 \eta ^2 r_h^{2 d} E_p^n}{3 d+2 n-10}\right) \nonumber \\&+\,\frac{6 \eta ^2 r_h^{8 d-2} E_p^n}{d-2 n-2}\bigg )+r_h^{-7 d-3 n} \left( -\frac{5 q^8 r_h^{22} \eta ^3}{-7 d-3 n+22}+\frac{5 q^6 \eta ^3 r_h^{2 d+16}}{-5 d-3 n+16}\right. \nonumber \\&-\,\frac{5 q^4 \eta ^3 r_h^{4 d+10}}{10-3 (d+n)}+r_h^{2 (d+5)} \left( \frac{5 q^6 r_h^6 \eta ^3}{-5 d-3 n+16}+\frac{5 q^4 \eta ^3 r_h^{2 d}}{3 d+3 n-10}\right) +\frac{5 \eta ^3 r_h^{8 d-2}}{d-3 n-2}\bigg )\nonumber \\&+\,r_h^{-7 d} \left( -\frac{16 q^8 r_h^{20} E_p^{3 n}}{20-7 d}\right. \nonumber \\&+\,\frac{16 q^6 r_h^{2 d+14} E_p^{3 n}}{14-5 d}-\frac{16 q^4 r_h^{4 d+8} E_p^{3 n}}{8-3 d}+r_h^{6 d} \left( \frac{16 q^2 r_h^2 E_p^{3 n}}{2-d}-\frac{16 q^2 r_h^4 E_p^{3 n}}{4-d}\right) \nonumber \\&+\,r_h^{2 (d+5)} \left( \frac{16 q^6 r_h^4 E_p^{3 n}}{4-5 (d-2)}+\frac{16 q^4 r_h^{2 d-2} E_p^{3 n}}{3 d-8}\right) +\frac{16 r_h^{8 d-2} E_p^{3 n}}{d-2}\bigg )\bigg ). \end{aligned}$$

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The quantum corrected temperature of the black hole is given by

$$\begin{aligned} T_G= & {} \frac{d-3}{4\pi }\sqrt{1- \frac{\eta }{E_p^n} \frac{1}{r_h^n}} \times \left\{ \frac{m}{\left( \frac{m}{2} + \frac{\sqrt{m^2-4q^2}}{2} \right) ^{\frac{d-2}{d-3}}} -\frac{2 q^2}{\left( \frac{m}{2} + \frac{\sqrt{m^2-4q^2}}{2} \right) ^{\frac{{2d-5}}{d-3}}}\right\} \nonumber \\&\left\{ 1+\frac{\alpha ^2 l_p^2}{\left( \frac{m}{2} + \frac{\sqrt{m^2-4q^2}}{2} \right) ^{\frac{2}{d-3}}}+\cdots \right\} ^{-1} \end{aligned}$$

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The quantum corrected heat capacity of the black hole is given by

$$\begin{aligned} C_G= & {} \left( \pi \sqrt{m^2-4 q^2} \left( \left( \frac{1}{2} \sqrt{m^2-4 q^2}+\frac{m}{2}\right) ^{\frac{2}{d-3}}+l_p^2 \alpha ^2\right) ^2 \right. \nonumber \\&\left. \sqrt{1- \frac{\eta }{E_p^n} \left( \frac{1}{2} \sqrt{m^2-4 q^2}+\frac{m}{2}\right) ^{-\frac{n}{d-3}}}\right) \nonumber \\&\left[ \left( 2^{-\frac{4d+1}{d-3}}\right) \left( \sqrt{m^2-4 q^2}+m\right) ^{-\frac{2(d-2)}{d-3}} \left( n q^2 \frac{\eta }{E_p^n} \left( -2^{\frac{4 d+n}{d-3}}\right) \left( \sqrt{m^2-4 q^2}+m\right) ^{\frac{3-n}{d-3}}\right. \right. \nonumber \\&\left( \left( \frac{1}{2} \sqrt{m^2-4 q^2}+\frac{m}{2}\right) ^{\frac{2}{d-3}}+l_p^2 \alpha ^2\right) \nonumber \\&+\,m n \frac{\eta }{E_p^n} \left( 2^{\frac{2 d+n+6}{d-3}} \right) \left( \sqrt{m^2-4 q^2}+m\right) ^{\frac{d-n}{d-3}} \left( \left( \frac{1}{2} \sqrt{m^2-4 q^2}+\frac{m}{2}\right) ^{\frac{2}{d-3}}+l_p^2 \alpha ^2\right) \nonumber \\&+\,\left( 32^{\frac{d}{d-3}}\right) l_p^2 \alpha ^2 \left( m \left( \frac{1}{2} \sqrt{m^2-4 q^2}+\frac{m}{2}\right) ^{\frac{d}{d-3}}\right. \nonumber \\&\left. -2 q^2 \left( \frac{1}{2} \sqrt{m^2-4 q^2}+\frac{m}{2}\right) ^{\frac{3}{d-3}}\right) \left( 1-\frac{\eta }{E_p^n} \left( \frac{1}{2} \sqrt{m^2-4 q^2}+\frac{m}{2}\right) ^{-\frac{n}{d-3}}\right) \nonumber \\&+\,8^{\frac{d+1}{d-3}} \left( \left( 2^{\frac{2d-3}{d-3}}\right) (2 d-5) q^2 \left( \frac{1}{2} \sqrt{m^2-4 q^2}+\frac{m}{2}\right) ^{\frac{3}{d-3}}\right. \nonumber \\&\left. +\,\left( \sqrt{m^2-4 q^2}+m\right) ^{\frac{d}{d-3}} \left( (d-3) \sqrt{m^2-4 q^2}+(2-d) m\right) \right) \nonumber \\&\left( \left( \frac{1}{2} \sqrt{m^2-4 q^2}+\frac{m}{2}\right) ^{\frac{2}{d-3}}+l_p^2 \alpha ^2\right) \left( 1-\frac{\eta }{E_p^n} \left( \frac{1}{2} \sqrt{m^2-4 q^2}+\frac{m}{2}\right) ^{-\frac{n}{d-3}}\right) \bigg )\bigg ]^{-1}.\nonumber \\ \end{aligned}$$

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The quantum corrected entropy of the black hole is obtained as:

$$\begin{aligned} S_G= & {} \frac{1}{8} E_p^{-3 n} \left( \left( \left( \frac{32 q^6 r_h^{14} E_p^{3 n}}{14-5 d}+\frac{16 l_p^2 q^6 r_h^{12} \alpha ^2 E_p^{3 n}}{12-5 d}\right) r_h^{2 d}\right. \right. \nonumber \\&+\,\left( -\frac{32 q^4 r_h^8 E_p^{3 n}}{8-3 d}-\frac{16 l_p^2 q^4 r_h^6 \alpha ^2 E_p^{3 n}}{6-3 d}\right) r_h^{4 d}\nonumber \\&+\,\left( -\frac{32 q^2 r_h^4 E_p^{3 n}}{4-d}-\frac{16 l_p^2 q^2 \alpha ^2 E_p^{3 n}}{d}-\frac{16 q^2 r_h^2 \left( l_p^2 \alpha ^2-2\right) E_p^{3 n}}{2-d}\right) r_h^{6 d}\nonumber \\&+\,\left( \frac{16 l_p^2 \alpha ^2 E_p^{3 n}}{(d-4) r_h^4}+\frac{32 E_p^{3 n}}{(d-2) r_h^2}\right) r_h^{8 d} \nonumber \\&+\,\left( \left( \frac{16 l_p^2 q^4 \alpha ^2 E_p^{3 n}}{3 (d-2) r_h^4}+\frac{32 q^4 E_p^{3 n}}{(3 d-8) r_h^2}\right) r_h^{2 d}+\frac{32 q^6 S^{3 n} r_h^4}{4-5 (d-2)}+\frac{16 l_p^2 q^6 E_p^{3 n} \alpha ^2 r_h^2}{2-5 (d-2)}\right) r_h^{2 (d+5)}\nonumber \\&-\,\frac{32 q^8 E_p^{3 n} r_h^{20}}{20-7 d} -\frac{16 l_p^2 q^8 E_p^{3 n} \alpha ^2 r_h^{18}}{18-7 d}\bigg ) r_h^{-7 d}\nonumber \\&+\,\left( \left( \frac{10 q^6 \eta ^3 r_h^{16}}{-5 d-3 n+16}+\frac{5 l_p^2 q^6 \alpha ^2 \eta ^3 r_h^{14}}{-5 d-3 n+14}\right) r_h^{2 d}+\left( -\frac{10 q^4 \eta ^3 r_h^{10}}{10-3 (d+n)}-\frac{5 l_p^2 q^4 \alpha ^2 \eta ^3 r_h^8}{8-3 (d+n)}\right) r_h^{4 d}\right. \nonumber \\&+\,\left( \frac{5 l^2 \alpha ^2 \eta ^3}{(d-3 n-4) r_h^4}+\frac{10 \eta ^3}{(d-3 n-2) r_h^2}\right) r_h^{8 d}+\left( \left( \frac{5 l^2 \alpha ^2 \eta ^3 q^4}{(3 d+3 n-8) r_h^4}+\frac{10 \eta ^3 q^4}{(3 d+3 n-10) r_h^2}\right) r_h^{2 d}\right. \nonumber \\&+\,\frac{10 q^6 \eta ^3 r_h^4}{-5 d-3 n+16}+\frac{5 l_p^2 q^6 \alpha ^2 \eta ^3 r_h^2}{-5 d-3 n+14}\bigg ) r_h^{2 (d+6)}-\frac{10 q^8 \eta ^3 r_h^{22}}{-7 d-3 n+22}-\frac{5 l_p^2 q^8 \alpha ^2 \eta ^3 r_h^{20}}{-7 d-3 n+20}\bigg ) r_h^{-7 d-3 n} \nonumber \\&+\,\left( \left( \frac{12 q^6 r_h^{16} \eta ^2 E_p^n}{-5 d-2 n+16}+\frac{6 l_p^2 q^6 r_h^{14} \alpha ^2 \eta ^2 E_p^n}{-5 d-2 n+14}\right) r_h^{2 d}+\left( -\frac{12 q^4 r_h^{10} \eta ^2 E_p^n}{-3 d-2 n+10}-\frac{6 l_p^2 q^4 r_h^8 \alpha ^2 \eta ^2 E_p^n}{-3 d-2 n+8}\right) r_h^{4 d}\right. \nonumber \\&+\,\left( \frac{6 l_p^2 \alpha ^2 \eta ^2 E_p^n}{(d-2 n-4) r_h^4}+\frac{12 \eta ^2 E_p^n}{(d-2 n-2) r^2}\right) r^{8 d}+\left( \left( \frac{6 l_p^2 q^4 \alpha ^2 \eta ^2 E_p^n}{(3 d+2 n-8) r^4}+\frac{12 q^4 \eta ^2 E_p^n}{(3 d+2 n-10) r^2}\right) r^{2 d}\right. \nonumber \\&+\,\frac{12 q^6 E_p^n \eta ^2 r^4}{-5 d-2 n+16}+\frac{6 l_p^2 q^6 E_p^n \alpha ^2 \eta ^2 r^2}{-5 d-2 n+14}\bigg ) r^{2 (d+6)}-\frac{12 q^8 E_p^n \eta ^2 r^{22}}{-7 d-2 n+22}-\frac{6 l_p^2 q^8 E_p^n \alpha ^2 \eta ^2 r^{20}}{-7 d-2 n+20}\bigg ) r^{-7 d-2 n} \nonumber \\&+\,\left( \left( \frac{16 q^6 r^{16} \eta E_p^{2 n}}{-5 d-n+16}+\frac{8 l_p^2 q^6 r^{14} \alpha ^2 \eta E_p^{2 n}}{-5 d-n+14}\right) r^{2 d}+\left( -\frac{16 q^4 r^{10} \eta E_p^{2 n}}{-3 d-n+10}-\frac{8 l_p^2 q^4 r^8 \alpha ^2 \eta E_p^{2 n}}{-3 d-n+8}\right) r^{4 d}\right. \nonumber \\&+\,\left( \frac{8 l_p^2 \alpha ^2 \eta E_p^{2 n}}{(d-n-4) r^4}+\frac{16 \eta E_p^{2 n}}{(d-n-2) r^2}\right) r^{8 d}+\left( \left( \frac{8 l_p^2 q^4 \alpha ^2 \eta E_p^{2 n}}{(3 d+n-8) r^4}+\frac{16 q^4 \eta E_p^{2 n}}{(3 d+n-10) r^2}\right) r^{2 d}\right. \nonumber \\&+\,\frac{16 q^6 E_p^{2 n} \eta r^4}{-5 d-n+16}+\frac{8 l_p^2 q^6 E_p^{2 n} \alpha ^2 \eta r^2}{-5 d-n+14}\bigg ) r^{2 (d+6)}-\frac{16 q^8 E_p^{2 n} \eta r^{22}}{-7 d-n+22}-\frac{8 l_p^2 q^8 E_p^{2 n} \alpha ^2 \eta r^{20}}{-7 d-n+20}\bigg ) r^{-7 d-n}\bigg )\pi .\nonumber \\ \end{aligned}$$

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