Decay of hot Kaluza-Klein space

The nonperturbative instabilities of hot Kaluza-Klein (KK) spacetime are investigated. In addition to the known instability of hot space (the nucleation of 4D black holes) and the known instability of KK space (the nucleation of bubbles of nothing by quantum tunneling), we find two new instabilities: the nucleation of 5D black holes, and the nucleation of bubbles of nothing by thermal fluctuation. These four instabilities are controlled by two Euclidean instantons, with each instanton doing double duty via two inequivalent analytic continuations; thermodynamic instabilities of one are shown to be related to mechanical instabilities of the other. I also construct bubbles of nothing that are formed by a hybrid process involving both thermal fluctuation and quantum tunneling. There is an exact high-temperature/low-temperature duality that relates the nucleation of black holes to the nucleation of bubbles of nothing.

Figure 1

In one-dimensional quantum mechanics there are two pure strategies for traversing obstacles—quantum tunneling through the barrier (rate $\sim \exp [-\frac{1}{\hslash }]$), or thermal fluctuation over the barrier (rate $\sim \exp [-\frac{1}{T}]$). Often the dominant process will be a mixed strategy that fluctuates partway up the barrier and then tunnels through the rest. In this paper we will see that the same is true for the nucleation of bubbles of nothing. The traditional Witten bubble of nothing results from pure quantum tunneling; we will construct bubble-of-nothing instantons that proceed by pure thermal fluctuation, as well as others that are part thermal and part quantum.

Figure 2

To traverse the barrier, a particle may thermally fluctuate to $\overline{x}$, and then quantum mechanically tunnel to $\overline{\overline{x}}$. Since the quantum part of the process conserves energy, $U(\overline{x})=U(\overline{\overline{x}})$. In the semiclassical description, after tunneling the particle appears at $\overline{\overline{x}}$ at rest, before classically rolling out to large $x$.

Figure 3

The Euclidean action of the instantons that traverse the potential of Fig. 2, as a function of the temperature $T$. The tunneling rate is $\exp [-I_E/\hslash ]$, so smaller $I_E$ means faster tunneling. Because of the thermal assist the quantum tunneling instanton has an action that falls with $T$. The intermediate solution chooses the worst possible value of $\overline{x}$—it gives the slowest case—and consequently has an extra negative mode.

Figure 4

Left: The potential energy density $V(\varphi )$ of an example field. Center: A bubble of ${\varphi }_{\mathrm{t}}$ separated by a thin wall from the background $\varphi =0$. Right: The energy $U[R]$ of a static bubble of radius $R$ in flat spacetime.

Figure 5

Thermal (top) and thermally assisted quantum (bottom) flat-spacetime quantum field theory instantons as a function of $\sigma T/\epsilon$; the larger $\sigma T/\epsilon$, the larger the bubble needs to be relative to the thermal circle. (Since we are by assumption in fixed flat spacetime, the periodicity of the thermal circle is fixed at $\beta$; if we turned on gravitational backreaction the thermal circle would shrink near the center of the bubbles due to gravitational blueshifting, à la Tolman-Oppenheimer-Volkoff [20].) The thermal instantons have a U(1) symmetry around the thermal circle; the quantum instantons have only a ${\mathrm{Z}}_2$ symmetry. The field first thermally fluctuates to $\varphi (\overrightarrow{x},\tau =0)$ (the horizontal slice represented by the black lines—this is the analogue of $\overline{x}$), then quantum tunnels to $\varphi (\overrightarrow{x},\tau =\beta /2)$ (the horizontal slice through the center of the bubbles—the analogue of $\overline{\overline{x}}$).

Figure 6

Black strings (top) and “caged” black holes (bottom) as a function of temperature. The shaded region is the interior of the event horizon. At high temperatures the black hole is small and the black string is thin; for $\beta <(1.75\dots )L$ the black string is so thin that it has a mechanical “Gregory-Laflamme” instability to become nonuniform. At lower temperatures the black hole is larger and is elongated in the KK direction. Eventually, for $\beta >(3.39\dots )L$, the would-be black hole is too large to be accommodated and there is no such solution.

Figure 7

Free energy of black strings and black holes as a function of their temperature. (This and all ensuing such diagrams are schematic—the precise curves can be straightforwardly plotted by adapting the numerical solution of Fig. 5 of [8], but the result would be unsightly and unilluminating.) To accentuate the similarity to Fig. 3 we have plotted $F$ in units of $L{T}^2$. This quantity would be independent of $T$ for an uncaged 5D black hole, but for $L<\infty$ the gravitational attraction to the image black holes bends the curve downwards [29]. Between the Gregory-Laflamme point and the merger point there is also a nonuniform black string.

Figure 8

The black-string instanton of Eq. (10). The angular $S^2$ directions have been suppressed.

Figure 9

The black string is given by a constant-$w$ slice through the instanton of Fig. 8.

Figure 10

The free energy of a black string, as a function of its radius $R$.

Figure 11

The thermal bubble of nothing, given by a constant-$z$ slice through the instanton of Fig. 8.

Figure 12

The Euclidean action $I_E$ of the instantons that make a black string and a thermal bubble of nothing, in units of ${\beta }^{3/2}{L}^{3/2}/G_5$ (schematic). The tunneling rate is $\mathrm{\Gamma }\sim \exp [-I_E/\hslash ]$, so smaller $I_E$ means faster tunneling. Above the Gregory-Laflamme temperature the string has an extra negative mode; below $T_{\sout{\text{therm}}}$ the thermal tunneling instanton has an extra negative mode (and so is not even locally the fastest way across the barrier). A $\beta \leftrightarrow L$ duality permutes the decays.

Figure 13

The black-hole instanton. The angular $S^2$ directions have been suppressed.

Figure 14

The five-dimensional caged black hole, given by a constant-$w$ slice through the instanton of Fig. 13.

Figure 15

A snapshot of the (thermally assisted) quantum bubble of nothing at nucleation, given by a constant-$z$ slice through the instanton of Fig. 13. At the moment of nucleation the bubble is at rest; it then expands, accelerating outwards and annihilating spacetime.

Figure 16

The Euclidean action $I_E$ of the instantons that make a caged black hole and a quantum bubble of nothing, in units of ${\beta }^{3/2}{L}^{3/2}/G_5$ (schematic). The tunneling rate is $\mathrm{\Gamma }\sim \exp [-I_E/\hslash ]$, so smaller $I_E$ means faster tunneling. A $\beta \leftrightarrow L$ duality permutes the decays. Black holes only exist for temperatures above the merger point; the quantum bubble-of-nothing instanton only exist for temperatures below $T_{\sout{\text{quant}}}$.

Figure 17

The Euclidean action $I_E$ of the instantons that control each of the four nonperturbative instabilities of hot KK space, in units of ${\beta }^{3/2}{L}^{3/2}/G_5$ (schematic). The tunneling rate is $\mathrm{\Gamma }\sim \exp [-I_E/\hslash ]$, so smaller $I_E$ means faster tunneling. A $\beta \leftrightarrow L$ duality permutes the decays. (Not shown: solutions with extra negative modes, such as the lumpy black string, the black string past the Gregory-Laflamme point, or their corresponding duals.)