Consistency between dynamical and thermodynamical stabilities for charged self-gravitating perfect fluid

Abstract

The entropy principle shows that, for self-gravitating perfect fluid, the Einstein field equations can be derived from the extrema of the total entropy, and the thermodynamical stability criterion are equivalent to the dynamical stability criterion. In this paper, we recast the dynamical criterion for the charged self-gravitating perfect fluid in Einstein–Maxwell theory, and further give the criterion of the star with barotropic condition. In order to obtain the thermodynamical stability criterion, first we get the general formula of the second variation of the total entropy for charged perfect fluid case, and then obtain the thermodynamical criterion for radial perturbation. We show that these two stability criterions are the same, which suggest that the inherent connection between gravity and thermodynamic even when the electric field is taken into account.

Appendix A: Explicit expression for each term of \(\delta ^2S\)

In this appendix, we would give the explicit expression of $\delta ^2S$ in spherical symmetric case. Note that in the following calculation, we use the integration by parts and drop the boundary terms, and also consider that the Tolman’s law is also valid, $T^{-1}=\chi =e^{\Psi }$. We denote that the n-th term of the right hand of Eq. (35) as $P_n$, and simply write the integral $\int _0^R dr$ as $\int _r$. Together with Eqs. (36) and (38), we obtain

$$\begin{aligned} \begin{aligned} P_1&= \int _C\frac{2}{T}\delta \rho \delta \sqrt{h} \\&=4\pi \int _r 4\pi \frac{1}{r}e^{\Psi +3\Lambda }\frac{\partial }{\partial {r}}\left[ r^2\xi \left( p+\rho \right) \right] ^2 -8\pi {re^{\Psi +4\Lambda }}nqQ\xi ^2(p+\rho ) \\&=4\pi \int _r -48\pi ^2{r^4}e^{\Psi +5\Lambda }(p+\rho )^3\xi ^2+4\pi {r^4}e^{\Psi +3\Lambda } (p+\rho )^2\xi ^2\left( \frac{2\Psi '}{r}+\frac{1}{r^2}\right) \\&\quad -8\pi {re^{\Psi +4\Lambda }}nqQ\xi ^2(p+\rho ). \end{aligned}\nonumber \\ \end{aligned}$$

(A.1)

Considering Eq. (12) and the relation $\Phi '=-e^{\Psi +\Lambda }\frac{Q}{r^2}$, we get

$$\begin{aligned} \begin{aligned} P_2&=\int _C2q\Phi \delta {n}\delta \sqrt{h}\\&=4\pi \int _r4\pi \frac{q\Phi }{nr}e^{2\Psi +3\Lambda }\left( p+\rho \right) \frac{\partial }{\partial {r}}\left( r^2e^{-\Psi }n\xi \right) ^2\\&=4\pi \int _r4\pi {r}nqQe^{\Psi +4\Lambda }\xi ^2\left( p+\rho \right) +4\pi {r^3}q\Phi \xi ^2n'e^{3\Lambda }\left( p+\rho \right) \\&\quad +4\pi {r^2}nq\Phi \xi ^2e^{3\Lambda }\left( p+\rho \right) \\&\quad -4\pi {r^3}nq\Phi \xi ^2e^{3\Lambda }\left( 2\Psi '+3\Lambda '\right) \left( p+\rho \right) -4\pi {r^3}nq\Phi \xi ^2e^{3\Lambda }\left( p+\rho \right) ', \end{aligned} \end{aligned}$$

(A.2)

and

$$\begin{aligned} \begin{aligned} P_3&= \int _C\sqrt{h}q\delta \Phi \delta {n} \\&=4\pi \int _r-qe^{\Psi +\Lambda }\delta \Phi \frac{\partial }{\partial {r}}\left( r^2e^{-\Psi }n\xi \right) \\&=4\pi \int _r-2rnqe^{\Lambda }\delta {\Phi }\xi -{r^2}n'qe^{\Lambda } \delta \Phi \xi +{r^2}nqe^{\Lambda }\delta \Phi \xi \Psi '-{r^2}nqe^{\Lambda }\delta \Phi \xi '. \end{aligned} \end{aligned}$$

(A.3)

The fourth term $P_4$ and the fifth term $P_5$ can easily be showed as

$$\begin{aligned} P_4= & {} \int _C-\frac{\sqrt{h}\delta {p}\delta \rho }{T(p+\rho )} =4\pi \int _r-r^2e^{\Psi +\Lambda }\frac{\delta {p}\delta \rho }{p+\rho } .\end{aligned}$$

(A.4)

$$\begin{aligned} P_5= & {} \int _C-\frac{\sqrt{h} n q \delta \Phi \delta \rho }{p+\rho } \nonumber \\= & {} 4\pi \int _r2rnqe^{\Lambda }\delta {\Phi }\xi +r^2nqe^{\Lambda } \delta {\Phi }\xi '\nonumber \\&+\frac{r^2nqe^{\Lambda }\delta {\Phi }}{p+\rho }\xi \left( p+\rho \right) '- \frac{e^{2\Lambda }n^2q^2Q\xi \delta {\Phi }}{p+\rho } . \end{aligned}$$

(A.5)

Using Eqs. (39), we have

$$\begin{aligned} \begin{aligned} P_6&=\int _C\frac{1}{T}\left( p+\rho +q\Phi {Tn}\right) \delta ^2\sqrt{h}\\&=4\pi \int _r 48\pi ^2r^4e^{\Psi +5\Lambda }\xi ^2\left( p+\rho \right) ^3-4\pi {r^3} e^{\Psi +3\Lambda }\xi \left( p+\rho \right) \left( \delta {p}+\delta \rho \right) \\&\quad -4\pi {r^3}e^{\Psi +3\Lambda }\left( p+\rho \right) ^2\delta \xi \\&\quad +48\pi ^2r^4e^{5\Lambda }nq\Phi \xi ^2\left( p+\rho \right) ^2 -4\pi {r^3}e^{3\Lambda }nq\Phi \xi \left( \delta {p}+\delta \rho \right) \\&\quad -4\pi {r^3}e^{3\Lambda }\left( p+\rho \right) nq\Phi \delta \xi . \end{aligned} \end{aligned}$$

(A.6)

Together with Eqs. (14) and (40), we get

$$\begin{aligned} \begin{aligned} P_7&= \int _C\sqrt{h}q\Phi \delta ^2{n} \\&= 4\pi \int _r4\pi {r}n^2q^2\Phi {Q}\xi ^2{e^{4\Lambda }} +4\pi {r^3}nq{\Phi }e^{3\Lambda }\delta p \xi \\&\quad -4\pi {r^4}e^{3\Lambda }\xi ^2\left( p+\rho \right) nq\Phi \left( \frac{2\Psi '}{r} +\frac{1}{r^2}\right) \\&\quad +q\Phi \xi 'e^{\Psi +\Lambda }\frac{\partial }{\partial {r}}\left( r^2e^{-\Psi }n\xi \right) +q\Phi {\xi }e^{\Psi +\Lambda }\frac{\partial ^2}{\partial {r^2}}\left( r^2e^{-\Psi }n\xi \right) \\&\quad -{q\Phi }e^{\Psi +\Lambda }\frac{\partial }{\partial {r}}\left( r^2e^{-\Psi }n\delta \xi \right) . \end{aligned} \end{aligned}$$

(A.7)

And Eq. (37) yields

$$\begin{aligned} \begin{aligned} P_8&=\int _C\frac{\delta ^2\rho }{T}\sqrt{h}\\&=4\pi \int _r4{\pi }e^{\Psi +3\Lambda }r^3\xi \left( \delta {p}+\delta \rho \right) (p+\rho )\\&\quad +4{\pi }e^{\Psi +3\Lambda }r^3\delta \xi (p+\rho )^2 +e^{\Psi +2\Lambda }nqQ\delta \xi \\&\quad -4{\pi }e^{\Psi +3\Lambda }r^2n^2q^2\xi ^2-4 \pi {r}nqQ\xi ^2e^{\Psi +4\Lambda }(p+\rho )\\&\quad -\frac{1}{r^2}e^{2\Phi +2\Lambda }qQ\xi \frac{\partial }{\partial {r}}(r^2e^{-\Psi }{n\xi }). \end{aligned} \end{aligned}$$

(A.8)