Finely split phase transitions of rotating and accelerating black holes

We investigate the thermodynamic phase behavior of rotating and slowly accelerating anti–de Sitter black holes. While we find some similarities with the nonrotating charged counterparts, such as the peculiar phenomena of “snapping swallowtails” we also find subtle but significant distinctions that can be attributed to a qualitatively modified parameter space of the solution. Consequently the zeroth-order phase transition now occurs over a range of pressures, mini-entropic black holes no longer exist in the regime of slow acceleration, and the “no black hole region” emerges continuously—from a zero temperature extremal black hole. The formerly equal transition pressure now experiences a fine splitting, as the emergence of a no black hole region and the zeroth-order phase transition appear at different pressures, different also from the termination pressure of the first order phase transition. This has the further effect of admitting reentrant phase transitions that can be achieved in two ways, either by varying the temperature at fixed pressure or varying the pressure at fixed temperature.

Figure 1

Admissible parameter space comparison. The admissible parameter space for rotating (left) and charged (right) slowly accelerating black holes is displayed by the shaded region in the $\tilde{A}-\tilde{m}$ plane. The left figure, displayed for fixed $\tilde{a}=0.2$, summarizes the constraints discussed in the main text as follows: the black, red, and blue curves respectively correspond to the boundaries of the Lorentzian metric signature, the existence of a bulk black hole, and the absence of an acceleration horizon; the green line displays $\alpha =0$. The right figure (reproduced from [19]) displays the analogous curves for the charged case with a fixed dimensionless charge parameter $\tilde{e}=eA=0.2$. While in the charged case the point $X$ lies at the intersection of three (red, blue, and green) boundary curves, rotation removes this degeneracy and we now observe two salient intersection points: $X$ and $Y$.

Figure 2

Swallowtail slicing of the parameter space. The arrows display the projections of three important swallowtails (with $J=1$ and $\mu =0.24$) into a single $\tilde{a}=0.4$ plane of the parameter space near the points $X$ and $Y$. The solid black arrow corresponds to $P=P_{nbh}\approx 0.00688$ and terminates at the point $Y$. On the $F-T$ diagram, this corresponds to a curve whose extremal black hole ($T=0$ point) lies just within the admissible parameter space. Note that the arrow does lie within the admissible parameter space for all $\tilde{a}\in [0,0.4]$. This is not obvious in the figure because the admissible region shrinks with increasing $\tilde{a}$. The dashed arrow displays the swallowtail with $P=P_0\approx 0.0062$. Unlike the previous curve, it does not lie fully within the parameter space (and rather terminates at the slow-acceleration boundary). It corresponds to a swallowtail similar to the one shown on Fig. 5, but with the red point lying directly below the free-energy peak of the swallowtail. The dotted arrow is for $P=P_t\approx 0.00601$ and reaches the point $X$. Like the dashed case, it lies partly outside of the admissible region. Strictly speaking, swallowtails with different pressures terminate at different $\tilde{a}$ planes. However, for points $X$ and $Y$ the variation of $\tilde{a}$ is of the order $O({\mu }^5)$ and can be in our figure neglected.

Figure 3

Phase diagram: $\mu =0.1$. For small tensions, the $P-T$ phase diagram is similar to the charged case, with termination of the first order coexistence line (blue), the presence of a zeroth-order transition line (red) and a no black hole (NBH) region delimited by the black lines. The first order transition separates small black hole (SBH) and large black hole (LBH) solutions. The inset shows an enlarged view near $P=P_f$, where the first order phase transition terminates. At this $\mu$, we have $P_0\approx P_f\approx P_{nbh}\approx 0.001194$.

Figure 4

Phase diagram: $\mu =0.245$. For this $\mu$, the $P-T$ diagram clearly illustrates the fine splitting of the transition pressures. Comparing with the $\mu =0.1$ case, the first order phase transition and zeroth-order phase transition curves shrank considerably. However, the inset shows that the zeroth-order transition line has a nontrivial negative gradient, and the gradient of the first order phase transition went from positive to negative. For this tension we have $P_{nbh}\approx 0.007165$, $P_0\approx 0.00637981$, $P_f\approx 0.00637184$, and $P_t^{(J)}\approx 0.006215$, all apparently distinct.

Figure 5

Slow-acceleration bound and its effect on phase transitions. The $F-T$ diagram shows a swallowtail at a pressure $P_f<P<P_0$. The “last” slowly accelerating black hole is highlighted by a red dot—the part of the lower temperature branch of black holes (that develop extra horizons) denoted by a dashed curve is removed. This is the origin of the temperature driven reentrant phase transition as well as of the no black hole region. At any temperature, the solution with the lowest free energy is thermodynamically preferred. Starting from high temperatures, these are solutions lying on the large black hole branch of the swallowtail. As the temperature decreases, we move up the swallowtail, with the black hole radii decreasing continuously in the process. At $T=T_1$, a first order phase transition occurs as the thermodynamically favored state moves from the large black hole branch to the small black hole branch, with a discontinuity in the black hole radius. As the temperature decreases further to $T_0$, a zeroth-order phase transition occurs, when the small black hole transits into an intermediate black hole with a jump in the free energy. At this pressure, the no black hole region starts at $T=T_n$, the temperature of the upper cusp.

Figure 6

Pressure and temperature driven reentrant phase transitions. First order (blue) and zeroth-order (red) coexistence lines are displayed for $\mu =0.19$. For this tension the first order line is double valued, which together with the zeroth-order phase transition, gives rise to a pressure driven SBH/LBH/SBH/LBH reentrant phase transition as we move from the initial point $I$ to the final point $F$ in the diagram. The inset displays the temperature driven LBH/SBH/LBH phase transition, innate to the rotating case, that happens in the tiny range of pressure $P\in (P_f,P_0)$, as we move from the initial point $I$ to the final point $F$ in the inset.