Vacuum decay and bubble nucleation in the anti-de Sitter black holes

Black hole phase transition is a very interesting topic, which attracted much attention since Hawking’s discovery of black hole thermodynamics Hawking:1975vcx . One of the well known example of black hole phase transition is the Hawking-Page phase transition Hawking:1982dh . That is there is a first order phase transition between the AdS black hole and the thermal AdS space. Inspired by the studies of the phase transition kinetics in condensed matter physics and polymer physics, the Schwarzschild AdS black hole was treated as a complex system with the horizon radius as the order parameter, based on which the thermodynamics and the kinetics of the Hawking-Page phase transition was investigated Li:2020khm . In ordinary material system admitting first order phase transition, the bubble nucleation is the key mechanism to realize the phase transition process with spatial inhomogeneity Langer:1967 . The thermal activation of the bubble nucleation in Schwarzschild AdS black hole was investigated in Sasaki:2014spa ; Chen:2017suz . Interestingly, the bubble nucleation process that describes the complete evaporation of the Schwarzschild AdS black hole is found to have the same probability with that of the Hawking-Page phase transition from the black hole to the thermal radiation.

In recent years, the thermodynamics of charged AdS black hole in the extended phase space involving the variable cosmological constant, has been widely explored Kubiznak:2014zwa ; Kubiznak:2016qmn . The small/large Reissner-Nordstrom AdS (RNAdS) black hole phase transition, which is another type of first order phase transition and has the similar behavior with the Van der Walls fluid Kubiznak:2012wp , was also extensively investigated by researchers. When the transition is spatial homogeneous, the dynamics of the RNAdS black hole phase transition has been studied based on the underlying free energy landscape Li:2020nsy ; Li:2021vdp . However, the bubble nucleation process by taking the spatial inhomogeneous into account was not investigated until now. In this work, we will address the problems of the vacuum decays and the bubble nucleation in the Schwarzschild AdS (SAdS) black holes and the RNAdS black holes.

The bubble nucleation driven by the vacuum energy difference stems from Coleman’s pioneer work on the false vacuum decay without gravitation Coleman:1977py ; Callan:1977pt , where the $O(4)$ symmetric Euclidean bounce solution is related to the tunneling process and the Euclidean action of the bounce solution gives the tunneling rate. Later, this approach was generalized to take the effect of gravitation into account Coleman:1980aw . Especially, modelling the inhomogeneity by the black hole, the symmetry of the Euclidean bounce solution is reduced to $O(3)$. It is also found that the seed black hole can catalyze the bubble nucleation process Hiscock:1987hn ; Gregory:2013hja ; Burda:2015yfa . In general, if a true vacuum bubble is created inside of the false vacuum, it will collapse into the seed black hole. However, this collapsing bubble can tunnel to the growing bubble that inflates to the spatial infinity or falls through the cosmological horizon. Black hole seeded false vacuum decay has very important applications in early universe Burda:2015isa ; Burda:2016mou ; Cai:2020ndh , as well as in the information paradox of black holes in AdS space Chen:2018aij ; Chen:2021jzx .

In this work, we consider the false vacuum decay and the bubble nucleation in the Schwarzschild AdS black hole and the RNADS black hole. The geometry of our model is described by the spacetime with an uncharged thin wall. The interior and the exterior spacetimes separated by the bubble wall are two different AdS black holes with different cosmological constant or vacuum energy. With this setting, the driving force to generate the bubble nucleation is dominated by the difference of the vacuum energy. Using the Israel junction conditions Israel:1966rt , we derived the equation of motion for the thin bubble wall. It is shown that for the fixed mass of the interior spacetime, there is a mass range for the exterior spacetime when the Euclidean bounce bubble solution exists. It is also shown that for the fixed mass of the exterior (initial) RNAdS black hole, there exists a minimum mass of the interior (final) RNAdS black hole that the bounce solution exists. We derived the analytical expression of the Euclidean action of the bubble wall solution. The tunneling rate can only be calculated numerically. In numerical investigations, we firstly compare our result of the tunneling coefficient from the Minkovski spacetime to AdS space with that of Coleman and De Luccia in Coleman:1980aw . It is shown that the black hole can catalyze the bubble nucleation process. Then, we investigated the bubble nucleation rate in the SAdS black holes and the RNAdS black holes. The similar conclusion is obtained. For the bubble nucleation in the RNAdS black holes, we show that the tunneling rate to the final RNAdS black hole with the minimum critical mass is the highest among all the possible tunneling channels.

This paper is arranged as follows. In section 2, the geometry of the bubble nucleation model is introduced and the equation of motion of the bubble wall is derived by using the Israel junction conditions. In section 3, we discuss the existence conditions of the Euclidean bounce solution in details. In section 4, the Euclidean action of the bounce solution is calculated analytically. In section 5, the numerical results and the corresponding discussion are presented. The conclusion is summarized in section 6.

In this section, we firstly introduce the geometry of the model Hiscock:1987hn ; Gregory:2013hja ; Burda:2015yfa , and then discuss the Israel junction conditions Israel:1966rt used to determine the equation of motion of the bubble wall. The bubble wall is taken to be thin for simplicity.

The whole spacetime is separated by the thin wall into two parts. On each side of the thin wall $\mathcal{W}$, the spacetime $\mathcal{M}_{\pm}$ is described by the RNAdS metric, which has the form of

$\displaystyle ds_{\pm}^{2}=-f_{\pm}(r_{\pm})dt_{\pm}^{2}+\frac{dr_{\pm}^{2}}{f_{\pm}(r_{\pm})}+r_{\pm}^{2}d\Omega_{\pm}^{2}\;,$

(1)

where

	$\displaystyle f_{\pm}(r_{\pm})$	$\displaystyle=$	$\displaystyle 1-\frac{2M_{\pm}}{r_{\pm}}+\frac{Q_{\pm}^{2}}{r_{\pm}^{2}}+\frac{r_{\pm}^{2}}{L_{\pm}^{2}}\;,$		(2)
	$\displaystyle d\Omega_{\pm}^{2}$	$\displaystyle=$	$\displaystyle d\theta^{2}+\sin^{2}\theta d\phi^{2}\;.$		(3)

Here, $\pm$ denotes the exterior/interior spacetime respectively. Note that on each side of the wall, the angular coordinates $\theta$ and $\phi$ are the same. The interior spacetime is taken to be the bubble nucleation. The SAdS black hole can be obtained by setting $Q=0$, and the equations derived in the following can also be applied to the SAdS black hole.

In the present work, we consider the case that the thin shell is uncharged, and set $G=1$ without loss of generality. Therefore, we take $Q_{+}=Q_{-}=Q$. Considering the symmetry of the metrics, the bubble nucleation with $O(3)$-symmetry is described by the local coordinates on each side of the wall

$\displaystyle X_{\pm}^{a}=\left\{t_{\pm}(\lambda),r_{\pm}(\lambda),\theta,\phi\right\}\;,$

(4)

where $\lambda$ is the proper time of the observer comoving with the wall. Because the wall is made of matter and thus the corresponding trajectory is timelike, the four velocity $u_{\pm}^{a}=\frac{dX_{\pm}^{a}}{d\lambda}$ of the wall must satisfy the normalization condition $g_{ab}u_{\pm}^{a}u_{\pm}^{b}=-1$. Thus, the normalization condition for the four velocity $u_{\pm}^{a}=\left(\dot{t}_{\pm},\dot{r}_{\pm},0,0\right)$ is given by the equation

$\displaystyle f_{\pm}(r_{\pm})\dot{t}_{\pm}^{2}-\frac{\dot{r}_{\pm}^{2}}{f_{\pm}(r_{\pm})}=1\;,$

(5)

where the dot represents the derivative with respect to the proper time $\lambda$. We require that the thin wall or the boundary of the exterior/interior spacetime is parameterized by the trajectory equation

$\displaystyle r_{\pm}-R(\lambda)=0\;.$

(6)

In terms of the intrinsic coordinates $\zeta^{A}=(\lambda,\theta,\phi)$, the induced metric of the wall is then given by

$\displaystyle ds^{2}=-d\lambda^{2}+R^{2}(\lambda)\left(d\theta^{2}+\sin^{2}\theta d\phi^{2}\right)\;,$

(7)

where the normalization condition of the four velocity of the wall is used. The above induced metric provides the first Israel junction condition Israel:1966rt , which requires the continuity between the interior and the exterior induced metrics on the wall. It is well known that Israel junction conditions should be satisfied by the hypersurface that partitions the spacetime into two regions, provided the Einstein field equations are valid in the two regions Israel:1966rt .

To obtain the equation of motion for the wall, one needs to consider the second Israel junction condition between the interior and the exterior spacetimes Israel:1966rt , which is given by

$\displaystyle[K]_{ab}=K_{ab}^{+}-K_{ab}^{-}=-8\pi G\left(S_{ab}-\frac{1}{2}h_{ab}S\right)\;,$

(8)

where $K_{ab}^{\pm}$ is the extrinsic curvature on each side of the wall, $h_{ab}$ is the induced metric on the wall, and $S_{ab}$ is the energy momentum tensor of the wall. We assume $S_{ab}=-\sigma h_{ab}$, where $\sigma$ is the tension of the wall.

From the trajectory equation of the wall, one can get the normal one-form as

$\displaystyle dn^{\pm}\propto dr_{\pm}-\frac{\dot{R}}{\dot{t}_{\pm}}dt_{\pm}\;.$

(9)

The unit vector normal to each side of the wall is given by

$\displaystyle n_{a}^{\pm}=\left(-\dot{R},\dot{t}_{\pm},0,0\right)\;.$

(10)

It is easy to check that the normalization of the normal vector $n_{a}^{\pm}$ can be guaranteed by Eq.(5). The extrinsic curvature is then defined as

$\displaystyle K_{ab}^{\pm}=\nabla_{a}n_{b}^{\pm}\;.$

(11)

One can calculate the $(\theta,\theta)$ component of the extrinsic curvature

$\displaystyle K_{\theta\theta}^{\pm}=\Gamma_{\theta\theta}^{r}n_{r}^{\pm}=Rf_{\pm}(R)\dot{t}_{\pm}\;,$

(12)

where $\Gamma_{\theta\theta}^{r}$ is the connection coefficient. Therefore, the $(\theta,\theta)$ component of the second Israel junction condition gives

$\displaystyle f_{+}(R)\dot{t}_{+}-f_{-}(R)\dot{t}_{-}=-4\pi G\sigma R\;.$

(13)

The $(\phi,\phi)$ component of the second Israel junction condition gives the same equation as the $(\theta,\theta)$ component. Other non-zero components (for example, $(t,t)$ and $(r,r)$ components) of the second Israel junction condition give no independent equation, but the derivatives of the above equation. Combining with the normalization condition of the unit normal vector $n_{a}^{\pm}$, one can derive the equation of motion of the wall

	$\displaystyle\dot{R}^{2}+U(R)$	$\displaystyle=$	$\displaystyle 0\;,$
	$\displaystyle U(R)$	$\displaystyle=$	$\displaystyle 1-\frac{2M_{+}\left(1+\Delta D\right)}{R}+\frac{Q^{2}}{R^{2}}-\frac{D^{2}M_{+}^{2}L_{+}^{2}}{R^{4}}+\frac{\left(1-\Delta^{2}\right)}{L_{+}^{2}}R^{2}\;,$		(14)

where

	$\displaystyle D$	$\displaystyle=$	$\displaystyle\frac{1}{4\pi\sigma L_{+}}\frac{\left(M_{+}-M_{-}\right)}{M_{+}},$
	$\displaystyle\Delta$	$\displaystyle=$	$\displaystyle\frac{L_{+}}{8\pi\sigma}\left[\frac{1}{L_{-}^{2}}-\frac{1}{L_{+}^{2}}-\left(4\pi\sigma\right)^{2}\right]\;.$		(15)

This equation determines the motion of the thin wall in real time.

In order to calculate the Euclidean action of the bounce solution, one needs the equation of motion for the Euclidean bubble wall. By performing the Wick rotation $t_{\pm}\rightarrow-i\tau_{\pm}$ as well as the the Wick rotating $\lambda\rightarrow-i\lambda$ of the proper time of the bubble wall, one can get the equation of motion for the Euclidean bubble wall

	$\displaystyle\dot{R}^{2}+U_{E}(R)$	$\displaystyle=$	$\displaystyle 0\;,$
	$\displaystyle U_{E}(R)$	$\displaystyle=$	$\displaystyle-U(R)$		(16)
		$\displaystyle=$	$\displaystyle-1+\frac{2M_{+}\left(1+\Delta D\right)}{R}-\frac{Q^{2}}{R^{2}}+\frac{D^{2}M_{+}^{2}L_{+}^{2}}{R^{4}}-\frac{\left(1-\Delta^{2}\right)}{L_{+}^{2}}R^{2}\;.$

The bounce solution of this equation determines the semiclassical tunneling rate of the black hole nucleation. The details of this effective potential will be discussed in the following sections.

In our set up, there are six parameters, that is $\left\{M_{+},M_{-},Q,L_{+},L_{-},\sigma\right\}$. Recall that $M_{\pm}$ and $L_{\pm}$ are the mass and the cosmological constant of the exterior/interior black hole. The electric charge of the spacetime is set to $Q$, and $\sigma$ is the tension of the bubble wall.

We take $\left\{L_{+},L_{-},\sigma\right\}$ as the input of the theoretical model. We consider the case that the vacuum energy of the interior spacetime is smaller than that of the exterior spacetime, i.e. $L_{+}>L_{-}$. In the following, we will study the possible parameter range that allows a bounce solution.

Firstly, we consider the conditions of the non-extremal black holes. Without the loss of generality, we take the exterior spacetime as illustration. If the black hole is extremal, the horizon $r_{+}^{*}$ must satisfy the following conditions

	$\displaystyle f_{+}(r_{+}^{*})$	$\displaystyle=$	$\displaystyle 1-\frac{2M_{+}}{r_{+}^{*}}+\frac{Q^{2}}{\left(r_{+}^{*}\right)^{2}}+\frac{\left(r_{+}^{*}\right)^{2}}{L_{+}^{2}}=0\;,$		(17)
	$\displaystyle f_{+}^{\prime}(r_{+}^{*})$	$\displaystyle=$	$\displaystyle\frac{2M_{+}}{\left(r_{+}^{*}\right)^{2}}-\frac{2Q^{2}}{\left(r_{+}^{*}\right)^{3}}+\frac{2r_{+}^{*}}{L_{+}^{2}}=0\;.$		(18)

Fortunately, the equations can be solved analytically, which is given by

$\displaystyle\left(r_{+}^{*}\right)^{2}=\frac{L_{+}^{2}}{6}\left(\sqrt{1+\frac{12Q^{2}}{L_{+}^{2}}}-1\right)\;.$

(19)

Substituting this expression back, one can get the critical mass of the extremal RNAdS black hole as

$\displaystyle M_{+}^{*}=\frac{L_{+}}{3\sqrt{6}}\left(\sqrt{1+\frac{12Q^{2}}{L_{+}^{2}}}+2\right)\left(\sqrt{1+\frac{12Q^{2}}{L_{+}^{2}}}-1\right)^{1/2}\;.$

(20)

When $M_{+}>M_{+}^{*}$, the exterior black hole has two horizons and is non-extremal. The similar condition is also applied to the interior RNAdS black hole, i.e.

	$\displaystyle M_{-}$	$\displaystyle>$	$\displaystyle M_{-}^{*}\;,$
	$\displaystyle M_{-}^{*}$	$\displaystyle=$	$\displaystyle\frac{L_{-}}{3\sqrt{6}}\left(\sqrt{1+\frac{12Q^{2}}{L_{-}^{2}}}+2\right)\left(\sqrt{1+\frac{12Q^{2}}{L_{-}^{2}}}-1\right)^{1/2}\;.$		(21)

The critical mass of the extremal black hole can be shown to be the increasing function of the AdS radius. Therefore, it can be concluded that $M_{+}^{*}<M_{-}^{*}$ when $L_{+}>L_{-}$.

Now, we consider the general behaviors of the potential $U_{E}(R)$ in the limits of $R\rightarrow 0$ and $R\rightarrow+\infty$. The shapes of the effective potential are plotted in Sec.5. When $D\neq 0$, i.e. $M_{+}\neq M_{-}$, for $R\rightarrow 0$ limit, $U_{E}(R)$ is approximated by

$\displaystyle U_{E}(R)\simeq\frac{D^{2}M_{+}^{2}L_{+}^{2}}{R^{4}}\;.$

(22)

Therefore, $\left.U_{E}(R)\right|_{R\rightarrow 0}>0$ is always guaranteed.

For $R\rightarrow+\infty$ limit, $U_{E}(R)$ can be expanded as

$\displaystyle U_{E}(R)\simeq-1-\frac{\left(1-\Delta^{2}\right)}{L_{+}^{2}}R^{2}\;.$

(23)

Therefore, the necessary condition for the existence of the bounce solution is $\left.U_{E}(R)\right|_{R\rightarrow+\infty}>0$, which gives us the condition $\Delta>1$. This condition can be further expressed as

$\displaystyle L_{-}<\frac{1}{4\pi\sigma}\;,\;\;\;L_{+}>\frac{L_{-}}{1-4\pi\sigma L_{-}}\;.$

(24)

Under these conditions, it can be shown that for the fixed interior black hole mass $M_{-}$, there exists a parameter range of $M_{+}$, in which the potential allows the bounce solution. This will be further discussed in Sec.5.

When $D=0$, i.e. $M_{+}=M_{-}$, the effective potential becomes

$\displaystyle U_{E}(R=-1+\frac{2M_{+}}{R}-\frac{Q^{2}}{R^{2}}-\frac{\left(1-\Delta^{2}\right)}{L_{+}^{2}}R^{2}\;.$

(25)

The condition that $\Delta>1$ should also be imposed. Due to the different behavior near $R\rightarrow 0$, it can be shown that when

$\displaystyle Q<\frac{L_{+}}{\sqrt{12(\Delta^{2}-1)}}\;,$

(26)

there exists a critical mass $M_{+}^{**}$, which is given by

$\displaystyle M_{+}^{*}=\frac{L_{+}}{3\sqrt{6}\left(\Delta^{2}-1\right)^{1/2}}\left(\sqrt{1+\frac{12\left(\Delta^{2}-1\right)Q^{2}}{L_{+}^{2}}}+2\right)\left(1-\sqrt{1+\frac{12\left(\Delta^{2}-1\right)Q^{2}}{L_{+}^{2}}}\right)^{1/2}\;.$

(27)

When $M_{+}>M_{+}^{**}$, the effective potential has three intersection points, i.e. there exists bounce solution. Comparing this critical mass with the critical mass of the extreme black hole, one can conclude that $M_{+}^{**}$ is always smaller than $M_{+}^{*}$. This implies that when $D=0$, i.e. $M_{+}=M_{-}$, there always exists a bounce solution provided Eq.(26) is satisfied.

In summary, the necessary conditions for the existence of Euclidean bounce solution are (1) the interior/exterior RNAdS black hole is non-extremal; (2) $\Delta>1$. For the special case of $D=0$, The existence of the bounce solution restricts the parameter range of the electric charge $Q$.

In this section, we derive the semiclassical tunneling rate of bubble that nucleated in the RNAdS black hole. In general, if a Lorentzian bubble is created inside of the false vacuum, it will collapse into the original black hole. However, considering the quantum effects, this collapsing bubble can tunnel to the growing bubble that inflates to the AdS spatial infinity. In the last section, we have discussed the necessary conditions that the Euclidean bounce solution exists. It is well known that the Euclidean bounce solution oscillates between the tunneling points of the corresponding Lorentzian bubble solution and the semiclassical tunneling rate of Lorentzian bubble is related to the Euclidean action of the bounce solution Coleman:1977py ; Callan:1977pt ; Coleman:1980aw .

Now, we calculate the Euclidean action of the bounce solution. We start with Einstein-Hilbert action Gibbons:1976ue

	$\displaystyle I_{E}=$	$\displaystyle-$	$\displaystyle\frac{1}{16\pi}\int_{\mathcal{M_{+}}}\left(\mathcal{R}_{+}-2\Lambda_{+}\right)\sqrt{g}d^{4}x-\frac{1}{16\pi}\int_{\mathcal{M_{-}}}\left(\mathcal{R}_{-}-2\Lambda_{-}\right)\sqrt{g}d^{4}x$		(28)
		$\displaystyle-$	$\displaystyle\frac{1}{16\pi}\int_{\mathcal{M_{+}\cup M_{-}}}F_{ab}F^{ab}\sqrt{g}d^{4}x+\int_{\mathcal{W}}\sigma\sqrt{h}d^{3}x$
		$\displaystyle+$	$\displaystyle\frac{1}{8\pi}\int_{\partial M_{+}}K^{+}\sqrt{h}d^{3}x-\frac{1}{8\pi}\int_{\partial M_{-}}K^{-}\sqrt{h}d^{3}x\;,$

It is obvious that the trace of the energy-momentum tensor for the electromagnetic field is zero. For the RNAdS solution, we have $\mathcal{R}_{\pm}=4\Lambda_{\pm}$ in the exterior and interior spacetimes. The Israel junction condition at the thin wall gives $K^{+}-K^{-}=-12\pi G\sigma$. Note that for the AdS spacetime, we have $\Lambda_{\pm}=-\frac{3}{L_{\pm}^{2}}$. The Euclidean time $\tau_{\pm}$ in the action integral has the period $\beta$, which is selected to be the period of the bounce solution rather than the period $\beta_{-}$ that can remove the canonical singularity at the horizon of the interior spacetime.

For the interior spacetime $\mathcal{M}_{-}$, the gravitational bulk term contains the extra contribution from the canonical singularity Gregory:2013hja ; Burda:2015yfa . It can be shown that this contribution is proportional to the horizon area times the deficit angle of the canonical singularity Fursaev:1995ef ; Solodukhin:2011gn ; Nishioka:2018khk . Therefore, we have

	$\displaystyle I_{\mathcal{M}_{-}}$	$\displaystyle=$	$\displaystyle-\frac{1}{16\pi}\int_{\mathcal{M_{-}}}\left(\mathcal{R}_{-}-2\Lambda_{-}\right)\sqrt{g}d^{4}x$		(29)
		$\displaystyle=$	$\displaystyle-\frac{1}{4}\left(1-\frac{\beta}{\beta_{-}}\right)\mathcal{A}_{-}-\frac{1}{4}\int\frac{2}{3}\Lambda_{-}\left(R^{3}-r_{h}^{3}\right)d\tau_{-}\;\;,$

where $\beta_{-}$ and $\mathcal{A}_{-}$ are the inverse temperature and the horizon area of the interior black hole, respectively. By taking the differential of the metric function $f_{-}(R)$ and using the relation among the mass, the area and the temperature of the RNAdS black hole, one can get

	$\displaystyle\frac{2}{3}\Lambda_{-}R^{3}=2M_{-}-\frac{2Q^{2}}{R}-R^{2}f_{-}^{\prime}\;,$
	$\displaystyle\frac{\mathcal{A}_{-}}{\beta_{-}}+\frac{2}{3}\Lambda_{-}r_{h}^{3}-2M_{-}+\frac{2Q^{2}}{r_{h}}=0,$

Using the the two equations, we can get

	$\displaystyle I_{\mathcal{M}_{-}}$	$\displaystyle=$	$\displaystyle-\frac{\mathcal{A}_{-}}{4}+\frac{\beta}{4}\left[\frac{\mathcal{A}_{-}}{\beta_{-}}+\frac{2}{3}\Lambda_{-}r_{h}^{3}-2M_{-}\right]+\frac{1}{4}\int\left[R^{2}f_{-}^{\prime}+\frac{2Q^{2}}{R}\right]\dot{\tau}_{-}d\lambda\;$		(30)
		$\displaystyle=$	$\displaystyle-\frac{\mathcal{A}_{-}}{4}-\frac{\beta Q^{2}}{2r_{h}}+\frac{1}{4}\int\left[R^{2}f_{-}^{\prime}+\frac{2Q^{2}}{R}\right]\dot{\tau}_{-}d\lambda\;.$

The analytical derivation can only reach this result and the numerical method should be invoked to deal with the rest.

For the exterior spacetime $\mathcal{M}_{+}$, we have to add a cutoff boundary $r_{0}$, and subtract the divergent volume contribution from the pure AdS space. However, one has to require that the metrics of the two geometries at the cutoff surface are the same. The trick is to require that the time-periodicity in the counter term agrees with that at the cutoff boundary Witten:1998qj ; deHaro:2000vlm

$\displaystyle\beta_{0}=\beta\frac{f_{+}^{1/2}}{\left(1-\Lambda_{+}r_{0}^{2}/3\right)^{1/2}}\simeq\left(1+\frac{3M_{+}}{\Lambda_{+}r_{0}^{3}}\right)\beta\;.$

(31)

The action integral of the exterior spacetime with the counterterm is given by

	$\displaystyle I_{\mathcal{M}_{+}}$	$\displaystyle=$	$\displaystyle-\frac{1}{4}\int\frac{2}{3}\Lambda_{+}\left(r_{0}^{3}-R^{3}\right)d\tau_{+}+\frac{1}{4}\int\frac{2}{3}\Lambda_{+}r_{0}^{3}d\tau_{0}\;$		(32)
		$\displaystyle=$	$\displaystyle\frac{1}{2}\beta M_{+}+\frac{1}{4}\int\frac{2}{3}\Lambda_{+}R^{3}d\tau\;$
		$\displaystyle=$	$\displaystyle\beta M_{+}-\frac{1}{4}\int\left[R^{2}f_{+}^{\prime}+\frac{2Q^{2}}{R}\right]\dot{\tau}_{+}d\lambda\;,$

where in the last step, we use the fact

$\displaystyle\frac{2}{3}\Lambda_{+}R^{3}=2M_{+}-\frac{2Q^{2}}{R}-R^{2}f_{+}^{\prime}\;.$

For the contribution of the electromagnetic field, using the fact that the gauge potential is given by $A=\frac{Q}{r}dt_{\pm}$, one can directly calculate the action integral as Gibbons:1976ue

$\displaystyle I_{EM}=\frac{\beta Q^{2}}{2r_{h}}\;.$

(33)

For the thin wall $\mathcal{W}$, using the Israel junction conditions, we have

	$\displaystyle I_{\mathcal{W}}$	$\displaystyle=$	$\displaystyle\int_{\mathcal{W}}\sigma\sqrt{h}d^{3}x+\frac{1}{8\pi}\int_{\mathcal{W}}\left(K^{+}-K^{-}\right)\sqrt{h}d^{3}x$		(34)
		$\displaystyle=$	$\displaystyle-\frac{1}{2}\int_{\mathcal{W}}\sigma\sqrt{h}d^{3}x$
		$\displaystyle=$	$\displaystyle\frac{1}{2}\int\left(Rf_{+}\dot{\tau}_{+}-Rf_{-}\dot{\tau}_{-}\right)d\lambda\;.$

At last, we have to calculate Euclidean action of the background RNAdS black hole, which gives Chamblin:1999hg ; Caldarelli:1999xj

$\displaystyle I_{B}=-\frac{\mathcal{A}_{+}}{4}+\beta M_{+}\;,$

(35)

where $\mathcal{A}_{+}$ is the horizon area of the exterior sapcetime. Note that the integral is also performed on the period of the bounce solution and the contribution from the canonical singularity has been taken into account.

By combining the previous results, we can get the tunneling coefficient $B=I_{E}-I_{B}$ as ¹¹1In principle, one does not have to subtract the action of the background spacetime. The background part is just a constant and not relevant to the bubble solution.

	$\displaystyle B$	$\displaystyle=$	$\displaystyle I_{\mathcal{M}_{+}}+I_{\mathcal{M}_{-}}+I_{\mathcal{W}}+I_{EM}-I_{B}$		(36)
		$\displaystyle=$	$\displaystyle\frac{1}{4}\left(\mathcal{A}_{+}-\mathcal{A}_{-}\right)+\frac{1}{2}\int\left[\left(R-3M_{+}\right)\dot{\tau}_{+}-\left(R-3M_{-}\right)\dot{\tau}_{-}\right]d\lambda\;.$

This result indicates that the tunneling coefficient $B$ depends on the bounce solution, which is only solved numerically. Although the integral is performed on one period of the bounce solution, this result does not depend on the period $\beta$ explicitly.

The semiclassical tunneling rate is then given by Coleman:1977py ; Callan:1977pt

$\displaystyle\Gamma\propto e^{-B}\;.$

(37)

This is the semi-classical tunneling rate that the bubble is nucleated in a false vacuum of the RNAdS black hole spacetime. It can also be applied to calculate the tunneling rate of the bubble nucleation in the SAdS black holes.

In this work, we have studied the false vacuum decay and the bubble nucleation in the Schwarzschild AdS black holes and the RNADS black holes. Starting from the geometric setting of the bubble wall spacetime and using the Israel junction conditions, we derived the equation of motion for the bubble wall. For all the cases that we have considered, if the mass of the black hole inside the bubble wall is fixed, there is a mass range for the exterior black hole spacetime that the Euclidean bounce bubble solution exists. It is also shown that for the fixed mass of the exterior (initial) RNAdS black hole, there exists a minimum mass of the interior (final) RNAdS black hole that the bounce solution exists.

We then derived an analytical expression of the Euclidean action for the bounce solution. However, the tunneling coefficient can only be calculated numerically. For the numerical results, we firstly compare our result of the tunneling coefficient from the Minkovski spacetime to AdS space with that of Coleman and De Luccia. It is found that the numerical result perfectly reproduce the CDL’s result when the mass of the initial Schwarzschild black hole approaches zero. Then, we considered the bubble nucleation in the SAdS black hole and the RNAdS black hole. It is shown that the black hole can catalyze the bubble nucleation process. In particular, for the bubble nucleation in the RNAdS black holes, we show that the tunneling rate to the final RNAdS black hole with the minimum critical mass is the highest among all the possible tunneling channels.

At last, we should point our the thin wall model considered in the present work is uncharged. In general, the bubble wall produced in the charged spacetime background should be charged also Kuchar:1968 ; Lemos:2021jtm . For the future direction, it is interesting to consider the charged thin wall that separates the interior and the exterior spacetimes. Another interesting aspect is to study the effect of inhomogeneity by the black hole on vacuum decay and bubble nucleation in higher derivative gravity and modified gravity theories Cai:2008ht ; Charmousis:2008ce ; Salehian:2018yoq .