Generalized phase transitions in Lovelock gravity

We investigate a novel mechanism for phase transitions that is a distinctive feature of higher-curvature gravity theories. For definiteness, we bound ourselves to the case of Lovelock gravities. These theories are known to have several branches of asymptotically anti–de Sitter solutions. Here, extending our previous work, we show that phase transitions among some of these branches are driven by a thermalon configuration: a bubble separating two regions of different effective cosmological constants, generically hosting a black hole in the interior. Above some critical temperature, this thermalon configuration is preferred with respect to the finite-temperature anti–de Sitter space, triggering a sophisticated version of the Hawking-Page transition. After being created, the unstable bubble configuration can in general dynamically change the asymptotic cosmological constant. While this phenomenon already occurs in the case of a gravity action with square curvature terms, we point out that in the case of Lovelock theory with cubic (and higher) terms new effects appear. For instance, the theory may admit more than one type of bubble and branches that are in principle free of pathologies may also decay through the thermalon mechanism. We also find ranges of the gravitational couplings for which the theory becomes sick. These add up to previously found restrictions to impose tighter constraints on higher-curvature gravities. The results of this paper point to an intricate phase diagram which might accommodate similarly rich behavior in the dual conformal field theory side.

Figure 1

In (a) we depict a particle in a potential with two local minima. At $x_F$, the false vacuum, while the true vacuum is at $x=x_T$. There is a potential barrier of height $E_0$ between the two vacua. In (b) the Euclidean counterpart of the same system. The potential is now $-V$.

Figure 2

Lagrangian and momentum for the action (3.1). For the same mean velocity $v$, the action is lower for jumps between $v_{\pm }$ (green dot, light gray in b&w) than for constant speed, the minimum action corresponding to the value on the dashed line (effective Lagrangian).

Figure 3

Hamiltonian as a function of the momentum for the action (3.1). Multivalued momenta correspond to the swallowtail part of the curve whereas those corresponding to the effective Lagrangian are the ones without that part of the curve, just the two upper branches. The crossing correspond to the jump depicted in Fig. 2 as a dashed line.

Figure 4

Euclidean section of a thermalon configuration; a bubble pops out in empty AdS space with cosmological constant ${\mathrm{\Lambda }}_{+}$, hosting a black hole belonging to a different branch of the Lovelock theory characterized by ${\mathrm{\Lambda }}_{-}$.

Figure 5

Potential (in blue, thin) of a brane gluing spherically symmetric stable and unstable branches of LGB gravity with parameters $\lambda =0.001$, $L=1$, $M_{-}=1$, and $M_{+}=2,2.4,7,10^2,10^3,10^4$ from left to right. $f_{-}/2$ (in red, thick) is (half the) metric function corresponding to the stable branch. The blue curve slides up the red one as we increase $M_{+}$. For the first two values of $M_{+}$ the blue and red curves do not intersect.

Figure 6

Potential (in blue, thin) of a brane gluing spherically symmetric stable and unstable branches of LGB gravity with parameters $\lambda =-0.01$, $L=1$, $M_{-}=1$, and $M_{+}=1.01,1.4,2,3,5$ from left to right. $f_{-}/2$ (in red, thick) is (half the) metric function corresponding to the stable branch and $f_{+}/2$ (in orange, dashed) is (half) the one corresponding to the unstable branch (for $M_{+}=5$). Singularities are located where the curves end, the potential ending when we find the outermost of them. Before that we encounter a point where $V=f_{+}/2$ (dots) beyond which the potential is unphysical. The physical potential does not even display a maximum for the thermalon to sit.

Figure 7

Schematic Penrose diagram for a nearly static bubble that undergoes collapse from $t=0$ onwards. The thick dashed line represents the trajectory of the bubble with inner black hole and outer naked singularity geometries. The spacelike singularity becomes timelike as the bubble reaches $r=0$. The resulting space-time is a (boosted) naked singularity, dashed lines representing constant $t_{+}$ slices. The lowest one corresponds to the Cauchy horizon introduced by the timelike singularity. The upper and right wedges represent the physical region of the space-time corresponding to a black hole formed by collapse whereas the rest corresponds to the complementary white hole region.

Figure 8

$g_{\pm }^{\star }$ (thick line) and $\mathrm{{\rm Y}}[g_{\pm }^{\star }]\propto M_{\pm }$ (thin line) are plotted as functions of $\lambda$ in the planar case. Dashed/solid lines corresponding to the inner/outer, $g_{-}$/$g_{+}$, branches, respectively.

Figure 9

Planar bubble potential in LGB gravity for $\lambda =0.1$, $a_{\star }=1$, and $d=5$, 6, 7, 10, 50 (from top to bottom). The nonplanar profile is qualitatively the same.

Figure 10

$\mathrm{{\rm Y}}[-x^2]/x$ (thick blue line) and ${\mathrm{{\rm Y}}}^{\prime }[-x^2]$ (thin gray line) for cubic Lovelock theory with parameters $L=1$, $\lambda =0.17$ and $\mu =0.02$. The paired points with the same color—green (left), blue (middle), and red (right) on each branch—correspond to each of the three solutions of the static junction conditions (6.2). The area below the gray curve between each couple of points vanishes according to the second equation.

Figure 11

Bubble potentials for cubic Lovelock theory with $\lambda =0.17$ and $\mu =0.02$. We just show the relevant branch of the potential connected to the physical equilibrium points, those with positive mass. The upper potential corresponds to the stable configuration (in green in Fig. 10)—notice that the relevant extremum is the minimum where the potential vanishes—while the other is the unstable one (in blue also in Fig. 10). The origin of this plot corresponds to the horizon of the inner EH geometry.

Figure 12

Free energy versus temperature for $\lambda =0.035$, 0.05, 0.12, 0.249. The dashed black curve corresponds to the unstable branch of black holes while the thick black one represents its extremal free energy. The thick red curve (in gray in the printed version) corresponds to the bubble with the same asymptotics and its extremal limit is indicated by the black dot. The thin red curve corresponds to the free energy of such extremal state when vanishing entropy is assumed. The dots indicate the locus of different kinds of possible phase transitions even though just some of them may take place in each case. The black dot when relevant represents a phase transition from an extremal to a nonextremal bubble, and the red one a transition from the extremal black hole to the nonextremal bubble. The green dot (in lighter gray) indicates the minimal temperature for the existence of nonextremal bubbles when these are divided in two branches, one thermodynamically stable and one unstable.

Figure 13

In red (gray in the printed version) the bubble states, light for the extremal, and in gray the extremal (BD-unstable) black hole. The red dot indicates the value of $\lambda$ for which the transition between the extremal unstable black hole and the nonextremal bubble (thick red line) begins to exist. The black line corresponds to the bubble extremal limit and the green one (lighter gray) to the minimal temperature of these configurations when this does not correspond to the extremal one. The colors of the different lines are the same used in Fig. 12 to indicate the points in the plot those lines represent.

Figure 14

Free energy versus temperature for $\lambda =0.05$, 0.11, 0.2 (and zoom). In red (gray in the b&w version), the bubble states and in black the black holes in the stable branch. The thin black line corresponds to the extremal geometry once assumed the vanishing of the entropy. The red (gray) dots indicate the locus of extremal bubble to black hole transitions and viceversa. The dashed (red) line corresponds to the extension for $a<\frac{1}{\sqrt{2}}$ of ${\mathcal{F}}_{+}$. These are not viable bubble solutions in the same way as its extremal extension represented by the thin red (gray) line. Acceptable bubble geometries appear for $\lambda \ge {\lambda }^{\star }$ (see the two lower figures) represented by the thick red line and they also have a well-defined extremal extension. The blue dot indicates nonextremal to extremal bubble transitions.

Figure 15

The phase diagram is divided in two regions in this case both corresponding to black holes, either extremal (white) or nonextremal (gray). The thick black line corresponds to an extrema to nonextremal phase transition, the rest of the lines representing would be transitions involving bubbles. The dots where the dashed line ends indicate the value of lambda for which such bubble configurations actually exist. The colors of the different lines are the same used in Fig. 14 to indicate the points in the plot those lines represent.

Figure 16

Oscillation period for a bubble in 5d LGB gravity for $\lambda =0.1$, $M_{-}=M_{-}^{th}$ and $M_{+}=M_{+}^{th}+\mathrm{\Delta }{M}_{+}$, $M_{\pm }^{th}$ being the mass parameters corresponding to $a_{\star }=1$. The horizontal lines, from top to bottom, correspond to ${\beta }_{-}/n$ for $n=1$, 2, 3, 4, 5, 10, 20, 50. In this specific example we have an infinite tower of smooth manifolds with increasing $n$ and $M_{+}$.