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Thermodynamic stability of black holes surrounded by quintessence

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Abstract

We study the thermodynamic stabilities of uncharged and charged black holes surrounded by quintessence (BHQ) by means of effective thermodynamic quantities. When the state parameter of quintessence \(\omega _q\) is appropriately chosen, the structures of BHQ are something like that of black holes in de Sitter space. Constructing the effective first law of thermodynamics in two different ways, we can derive the effective thermodynamic quantities of BHQ. Especially, these effective thermodynamic quantities also satisfy Smarr-like formulae. It is found that the uncharged BHQ is always thermodynamically unstable due to negative heat capacity, while for the charged BHQ there are phase transitions of the second order. We also show that there are several differences on the thermodynamic properties and critical behaviors of BHQ between the two ways we employed.

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Notes

  1. Here \(\kappa _{eff}\) is negative. Physically acceptable temperature should be positive, so we take the absolute value.

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Acknowledgements

This work is supported in part by the National Natural Science Foundation of China (Grant Nos. 11605107, 11475108) and by the Doctoral Sustentation Fund of Shanxi Datong University (2011-B-03).

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Authors and Affiliations

  1. Department of Physics, Shanxi Datong University, Datong, 037009, China

    Meng-Sen Ma & Ren Zhao

  2. Institute of Theoretical Physics, Shanxi Datong University, Datong, 037009, China

    Meng-Sen Ma & Ren Zhao

  3. Medical College, Shanxi Datong University, Datong, 037009, China

    Ya-Qin Ma

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Correspondence to Meng-Sen Ma.

Appendix A: Complete expressions of \(C_v\) and \(C_p\)

Appendix A: Complete expressions of \(C_v\) and \(C_p\)

The complete forms of Eq. (4.11) is:

$$\begin{aligned} C_v=\frac{2 \pi x^2 (x+1)^2 r_q^2 }{A(x)}\left[ \left( x^6+x^2\right) r_q^2-Q^2 \left( x^6+2 x^5+2 x+1\right) \right] , \end{aligned}$$
(A.1)

where

$$\begin{aligned} A(x)= & {} Q^2 \left( 3 x^{10}+10 x^9+10 x^8-x^6-8 x^5-x^4+10 x^2+10 x+3\right) \nonumber \\&+\,x^2 \left( x^8+4 x^7-6 x^4+4 x+1\right) r_q^2. \end{aligned}$$
(A.2)

The complete forms of Eq. (4.12) is:

$$\begin{aligned} C_p= & {} -\frac{2 \pi x^2 (x+1)^2 r_q^2}{B(x)} \times \left[ Q^2 x^2 \left( -2 x^6+3 x^5+15 x^4+8 x^3+15 x^2+3 x-2\right) r_q^2 \right. \nonumber \\&+ \left. Q^4 \left( x^8+3 x^7-7 x^6-20 x^5-20 x^4-20 x^3-7 x^2+3 x+1\right) \right. \nonumber \\&+ \left. x^4 \left( x^4-4 x^3-4 x+1\right) r_q^4\right] , \end{aligned}$$
(A.3)

where

$$\begin{aligned} B(x)= & {} -\,2 Q^2 x^2 \left( 2 x^{10}+8 x^9+9 x^8-6 x^7-27 x^6-48 x^5-27 x^4-6 x^3+9 x^2+8 x+2\right) r_q^2 \nonumber \\&+\, x^4 \left( x^8+4 x^7-4 x^5-14 x^4-4 x^3+4 x+1\right) r_q^4+Q^4 \left( 3 x^{12}+16 x^{11}+30 x^{10} \right. \nonumber \\&+ \left. 20 x^9-30 x^8-132 x^7-210 x^6-132 x^5-30 x^4+20 x^3+30 x^2+16 x+3\right) \end{aligned}$$
(A.4)

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Ma, MS., Zhao, R. & Ma, YQ. Thermodynamic stability of black holes surrounded by quintessence. Gen Relativ Gravit 49, 79 (2017). https://doi.org/10.1007/s10714-017-2245-4

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