Stability and fluctuations in black hole thermodynamics

I examine thermodynamic fluctuations for a Kerr-Newman black hole in an extensive, infinite environment. This problem is not strictly solvable because full equilibrium with such an environment cannot be achieved by any black hole with mass $M$, angular momentum $J$, and charge $Q$. However, if we consider one (or two) of $M$, $J$, or $Q$ to vary so slowly compared with the others that we can regard it as fixed, instances of stability occur, and thermodynamic fluctuation theory could plausibly apply. I examine seven cases with one, two, or three independent fluctuating variables. No knowledge about the thermodynamic behavior of the environment is needed. The thermodynamics of the black hole is sufficient. Let the fluctuation moment for a thermodynamic quantity $X$ be $\sqrt{⟨(\Delta X{)}^2⟩}$. Fluctuations at fixed $M$ are stable for all thermodynamic states, including that of a nonrotating and uncharged environment, corresponding to average values $J=Q=0$. Here, the fluctuation moments for $J$ and $Q$ take on maximum values. That for $J$ is proportional to $M$. For the Planck mass it is $0.3990\hslash$. That for $Q$ is $3.301e$, independent of $M$. In all cases, fluctuation moments for $M$, $J$, and $Q$ go to zero at the limit of the physical regime, where the temperature goes to zero. With $M$ fluctuating there are no stable cases for average $J=Q=0$. But, there are transitions to stability marked by infinite fluctuations. For purely $M$ fluctuations, this coincides with a curve which Davies identified as a phase transition.

Figure 1

$M^2{p}_3$ as a function of $J/M^2$ for several values of $Q/M$. $p_3$ is negative for all values shown. The curves all diverge at the end of the physical regime $\alpha +\beta =1$.

Figure 2

Fluctuation regimes for $(M,J,Q)$ all free to fluctuate. The curve marks the end of the physical regime $\alpha +\beta =1$. The minus signs (signs of $p_3$) indicate that $S_{\mathrm{tot}}$ has no local maximum anywhere in the physical regime.

Figure 3

Stable fluctuation regime for $(J,Q)$ fluctuating at fixed $M$. The plus signs indicate the signs of both $p_1^{\prime }$ and $p_2^{\prime }$. Clearly, $S_{\mathrm{tot}}$ may have a local maximum at any point in the physical regime.

Figure 4

$p_1$ as a function of $J/M^2$ for several values of $Q/M$. $p_1$ may be positive or negative. The curves all diverge at the end of the physical regime.

Figure 5

$p_2^{\prime \prime }$ as a function of $J/M^2$ for several values of $Q/M$. $p_2^{\prime \prime }$ may be positive or negative. The curves all diverge at the end of the physical regime.

Figure 6

Stable fluctuation regime for $(M,Q)$ at fixed $J$. The physical regime where $p_1$ and $p_2^{\prime \prime }$ are both positive is indicated by $+$ signs.

Figure 7

$M^2{p}_2$ as a function of $J/M^2$ for several values of $Q/M$. $p_2$ may be positive or negative. The curves all diverge at the end of the physical regime.

Figure 8

Stable fluctuation regime for $(M,J)$ fluctuating at fixed $Q$. The physical regime where $p_1$ and $p_2$ are both positive is indicated by $+$ signs.

Figure 9

Stable fluctuation regime for $M$ fluctuating at fixed $J$ and $Q$. The physical regime where $p_1$ is positive is indicated by $+$ signs.

Figure 10

Stable fluctuation regime for $J$ fluctuating at fixed $M$ and $Q$. The physical regime where $p_1^{\prime }$ is positive is indicated by $+$ signs.

Figure 11

Stable fluctuation regime for $Q$ fluctuating at fixed $M$ and $J$. The physical regime where $p_1^{\prime \prime }$ is positive is indicated by $+$ signs.

Figure 12

$Y_1$ (for $dM$), $Y_2$ (for $dJ$), and $Y_3$ (for $dQ$) as functions of $J/M^2$ and $Q/M$ for a Planck mass black hole $M=M_p$. Values outside the physical regime are set to zero. It is seen that $dM$ has a negligible effect on $-2\Delta S_{\mathrm{tot}}$, indicating that $(J,Q)$ fluctuations are the most significant here.

Figure 13

$Y_1$ (for $dM$), $Y_2$ (for $dJ$), and $Y_3$ (for $dQ$) as functions of $J/M^2$ and $Q/M$ for a solar mass black hole $M=M_s$. It is seen that $dJ$ and $dM$ have negligible effect on $-2\Delta S_{\mathrm{tot}}$, indicating that $Q$ fluctuations are the most significant here.