Black holes as bubble nucleation sites

For the non-static bubble, the value of the action agrees precisely with two cases where an equivalent, manifestly regular construction is known: the CDL instanton with zero mass, and a second special case at finite M* where there is a critical, τ-independent bubble wall.

“For the non-static bubble, we find precise agreement for the value of the action with two cases where we are aware of an equivalent, manifestly regular construction: the CDL instanton with zero mass, and a second special case at finite JHEP03(2014)081 M * where there is a critical, τ -independent bubble wall, corresponding to solutions found in [16] .”

• cites [16] J. Garriga and A. Megevand, Decay of de Sitter vacua by thermal activation, Int. J. Theor. Phys. 43 (2004) 883 [ hep-th/0404097 ] [ INSPIRE ]. sourceSupported

There is a critical initial mass above which a black hole remnant survives the process.

“Increasing the initial mass, we find a critical value above which a black hole remnant survives the process.”

The interval in external Euclidean time τ+ runs over a particular range β_CDL, which does not correspond to the regularity condition at the cosmological horizon in general.

“Notice that this interval in the external Euclidean time coordinate, τ + , runs over a particular range, β CDL , which does not correspond to the regularity condition at the cosmological horizon, in general.”

The potential is qualitatively similar for varying κ1, κ2.

“The potential is qualitatively similar for varying κ 1 , κ 2 , and is illustrated in figure 2 for κ 1 = 0.”

For an initial seed black hole with mass M > M_C, the rate is a decreasing function of the mass.

“For an initial seed black hole with mass M > M C , the rate is a decreasing function of the mass”

For any value of κ2, there is a critical value of κ1 for which the maximum of the potential is at zero, delineating bouncing solutions from transmission solutions.

“For any value of κ 2 , there is a critical value of κ 1 for which the maximum of the potential is at zero, delineating bouncing solutions from transmission solutions.”

For sufficiently low initial mass, the original black hole is replaced by flat space during the process, as viewed by a single causal patch observer.

“For a sufficiently low initial mass, the original black hole is replaced by flat space during this process, as viewed by a single causal patch observer.”

For remaining solutions 0 < κ1 < κ*1, there are two solutions corresponding to small-r and large-r bounces.

“Finally, for the remaining solutions,foot_2 0 < κ 1 < κ * 1 , we find that there are two solutions, corresponding to small-r and large-r bounces.”

For a product metric, the limit of regulating a co-dimension two δ-function singularity is well-defined.

“Although in general, one cannot regulate a co-dimension two δ-function singularity in general relativity, [28], for the particular case of a product metric, the limit is well-defined, as the ambiguity occurs due to nontrivial physical content in the transverse components of the energy-momentum tensor, which are not present in the special case of the product metric.”

• cites [28] R.P. Geroch and J.H. Traschen, Strings and Other Distributional Sources in General Relativity, Phys. Rev. D 36 (1987) 1017 [ INSPIRE ]. sourceNot checked yet

The bubble-to-bubble transition rate calculated via the WKB method is related to the spontaneous nucleation rate calculated in the singular instanton approach via a numerical factor depending only on the black hole entropy.

“We present a new action for the bubble wall and show that in general, the bubble-to-bubble transition rate calculated via the WKB method is related to the spontaneous nucleation rate calculated in the singular instanton approach via a numerical factor depending only on the black hole entropy.”

The nucleation rate Γ(M) as a function of initial black hole seed mass M has two regimes delineated by a critical mass GM_C = 32/27 (8πGσl)^3 / (4 + (8πGσl)^2)^2.

“We find that there are two regimes, delineated by a critical mass,”

• Also stated in: § JHEP03(2014)081 · plain

  “GM C = 32 27 (8πGσℓ ) 3 (4 + (8πGσℓ) 2 ) 2 , (1.2)”

The nucleation rate for a bubble of true vacuum which replaces a finite mass SdS black hole is higher than that of the CDL case.

“The nucleation rate for a bubble of true vacuum which replaces a finite mass SdS black hole is actually higher than that of the CDL case.”

The critical value M = M_C marks the point at which the remnant black hole becomes vanishingly small when approached from above, giving way to flat space inside the bubble.

“The critical value, M = M C , marks the point at which the remnant black hole becomes vanishingly small when approached from above, giving way to flat space inside the bubble.”

In cases where ∂τ is a Killing vector, the Euclidean action is always independent of the choice of β once the conical singularities are taken into account.

“In the cases where ∂ τ is a Killing vector -for example in the absence of the bubble wall, or for a τ -independent bubble wall -we show that the Euclidean action is always independent of the choice of β once the conical singularities are taken into account.”

For an initial seed black hole with M < M_C, the nucleation rate is an increasing function of the mass, ∂Γ/∂M > 0.

“For an initial seed black hole with M < M C , the rate is an increasing function of the mass,”

The formula for actions of symmetric singular instantons generalises the 'NUTs and bolts' formula of Gibbons and Hawking.

“Our formula for the actions of symmetric singular instantons generalises the 'NUTs and bolts' formula of Gibbons and Hawking, [15].”

• cites [15] G.W. Gibbons and S.W. Hawking, Classification of Gravitational Instanton Symmetries, Commun. Math. Phys. 66 (1979) 291 [ INSPIRE ]. sourceNot checked — no full text of the source

The periodicity in the time coordinates on each side of the Euclidean wall need not be the same.

“Note that the periodicity in the time coordinates on each side of the Euclidean wall need not be, and indeed in general is not, the same.”

The solution at κ1=κ2=0 is just the CDL solution, but in the causal patch.

“To see that this solution is just the CDL solution, [10], but in the causal patch, we make the coordinate transformation”

• cites [10] S.R. Coleman and F. De Luccia, Gravitational Effects on and of Vacuum Decay, Phys. Rev. D 21 (1980) 3305 [ INSPIRE ]. sourceNot checked yet

The result here is a far larger increase in the bubble nucleation rate than previously found.

“The result here is a far larger increase in the bubble nucleation rate.”

A regular instanton requires positivity of τ+, leading to a lower bound κ1 ≥ κ1,min.

“However, a regular instanton also requires positivity of τ+ , which leads to a lower bound κ 1 ≥ κ 1,min , as derived in (B.4).”

For M > M_C, the dominant process corresponds to bubble nucleation with a vacuum black hole remnant.

“For this range we find that the dominant process corresponds to bubble nucleation with a vacuum black hole, which we shall here term 'the remnant'.”

When κ1=κ2=0 (CDL case), there is only a single Lorentzian bounce solution.

“When κ 1 = κ 2 = 0 corresponding to the CDL case, there is only a single Lorentzian bounce solution.”

For any large negative κ2, one can always find a σ small enough to allow for a bounce.

“On the other hand, for any large negative κ 2 we can always find a σ small enough to allow for a bounce.”

For sufficiently large black holes, the rate eventually becomes subdominant to that of the CDL instanton.

“and for sufficiently large black holes the rate eventually becomes subdominant to that of the CDL instanton.”

For M < M_C, the dominant instanton corresponds to the nucleation of flat space inside the bubble.

“with the dominant instanton corresponding to the nucleation of flat space inside the bubble.”

The paper constructs explicit Euclidean instantons describing the nucleation of a bubble of true vacuum centred on a black hole inhomogeneity.

“Modelling the inhomogeneity by a black hole, we construct explicit Euclidean instantons which describe the nucleation of a bubble of true vacuum centred on the inhomogeneity.”

The contribution from a single conical defect region B_i is given by its area A_i: -∫_{B_i} R + 2∫_{∂B_i} K = -4πA_i.

“Turning to the contribution from the conical singularities, we show in the appendix that the contribution from a single conical defect region B i is given by its area A i”

Using the inverse horizon timescale as a rough estimate (GM_C)^{-1}, Γ* ≈ 2 G^{1/2} e^{-4πG M_C^2}.

“Rather than evaluate the determinant factor, we use the inverse horizon timescale as a rough estimate (GM C ) -1 , then Γ * ≈ 2 G 1/2 e -4πGM 2 C . (6.2)”

• AssumptionThe authors state this as an assumption: The pre-factor A* can be estimated by the inverse horizon timescale (GM_C)^{-1}.

The ratio of the two decay rates is Γ*/Γ_H ≈ 3.9 × 10^3 (G M_C^2)^{3/2} e^{-4πG M_C^2}.

“The ratio of the two decay rates is Γ * Γ H ≈ 3.9 × 10 3 (GM 2 C ) 3/2 e -4πGM 2 C .(6.4)”

Consequently, Γ_b→b = e^{-I - (A_-/4G) - (A_c/4G)}.

“Consequently, Γ b→b = e -I-(A -/4G)-(Ac/4G) , (5.9)”

The seed black hole which would give the highest rate process is given by a critical value M_C where GM_C = 8/27 σγ^2.

“Figure 5 also indicates that the seed black hole which would give the highest rate process is given by a critical value, M C where, GM C = 8 27 σγ 2 .(4.11)”

For M+ < M_C the decay simply nucleates the vacuum bubble with a flat interior.

“For M + < M C the decay simply nucleates the vacuum bubble with a flat interior.”

For this critical case, I_{0,κ*} = -π r_c^2 / G.

“I 0,κ * = - πr 2 c G . (4.7)”

Consequently, Γ_f→bb = e^{-(A_+ - A_-)/4G} Γ_b→b.

“Consequently, we have a relation of the form Γ f →bb = e -(A + -A -)/4G Γ b→b . (5.11)”

If we set Γ_H = Ṁ/M, then Γ_H ≈ 3.6 × 10^{-4} (G^2 M_C^3)^{-1}.

“If we set Γ H = Ṁ /M , then Γ H ≈ 3.6 × 10 -4 (G 2 M 3 C ) -1 (6.3)”

• cites [31] D.N. Page, Particle Emission Rates from a Black Hole: Massless Particles from an Uncharged, Nonrotating Hole, Phys. Rev. D 13 (1976) 198 [ INSPIRE ]. sourceNot checked yet

The rate of false vacuum decay calculated using the instanton method is Γ = e^{-(I - I_SdS)} = e^{-I - (A_+/4G) - (A_c/4G)}.

“However, we can compare this to the rate of false vaccum decay calculated using the instanton method, Γ = e -(I-I SdS ) = e -I-(A + /4G)-(Ac/4G) ,(5.10)”

The largest enhancement of the vacuum decay rate occurs for a black hole with both M-=0 and M+ at the critical value, M_C.

“The largest enhancement of the vacuum decay rate occurs for a black hole with both M -= 0 and M + at the critical value, M C , when the decay rate is”

The solutions that dominate are those which maximise the mass difference, κ2.

“These are the solutions which maximise the mass difference, κ 2 .”

In general, the nucleation processes these instantons describe occur at a rate far higher than that of the M+ = 0 or CDL process.

“In general, the nucleation processes these instantons describe occur at a rate far higher than that of the M + = 0 or CDL process.”

B is a monotonically decreasing function of M at fixed σl.

“In particular, we observe that B is a monotonically decreasing function of M at fixed σℓ”

The bubble nucleation rate calculated by the instanton approach is related to the bubble-to-bubble transition rate obtained in a Lorentzian WKB approach.

“We have also found that the bubble nucleation rate calculated by the instanton approach is related to the bubble-to-bubble transition rate obtained in a Lorentzian WKB approach.”

Tardis-like solutions, where the mass is bigger on the inside, generally have larger action than the CDL case.

“Paradoxically, we find that tardis like solutions, where the mass is bigger on the inside, generally have larger action than the CDL case.”

The exception occurs only for seed black holes which approach the Nariai limit.

“The exception occurs only for seed black holes which approach the Nariai limit.”

B* < B_CDL when compared at the same tension σ.

“Significantly, we find that B * < B CDL when compared at the same tension, σ.”

For M+ > M_C the decay also nucleates a remnant black hole inside the vacuum bubble.

“For M + > M C the decay also nucleates a remnant black hole inside the vacuum bubble.”

Figure 5 shows that for large enough M+ the dominant processes become tardis-like.

“The red shaded region corresponds to solutions where M -< M + , showing that for large enough M + the dominant processes become tardis-like.”

Consistent bubble solutions only exist when σl < √3/2.

“Consistent bubble solutions only exist when σℓ < √ 3/2 (see figure 3).”

For this critical case, B* = π r_h^2 / G.

“B * = πr 2 h G (4.8)”

The decay rate for this largest enhancement is Γ* = A* e^{-B*}.

“Γ * = A * e -B * ,(6.1)”

Figure 4 shows that in all cases, B < B_CDL indicating that the decay rate is higher for a finite mass, at fixed tension.

“In all cases, B < B CDL indicating that the decay rate is higher for a finite mass, at fixed tension.”

The final result for the action of a space-time with a bubble wall separating two regions of possibly different black hole masses and effective cosmological constants is I = -1/(4G)(A_h + A_c) - 1/2 ∫_W σ - 1/(16πG) ∫_W (f+ τ+ - f- τ-).

“Pulling all these pieces together with the previous result for the conical defects we reach our final result, I = - 1 4G (A h + A c ) - 1 2 W σ - 1 16πG W f + τ+ -f -τ-,(3.14)”

From the range of κ2, we conclude that σl ≤ √3/2.

“From the range of κ 2 , we conclude that σℓ ≤ √ 3/2.”

From the results of section 4.2, B* = 4πG M_C^2.

“From the results of section 4.2 we have B * = 4πGM 2 C .”

For the critical static bubble, the decay rate exponent is given by B* = π(r_h^2 - (2GM_-)^2)/G.

“B * = π(r 2 h -(2GM -) 2 ) G .(4.10)”

There is no lower limit to the ratio B/B_CDL.

“Figure 4 also shows that that there is no lower limit to the ratio B/B CDL .”

For the critical static bubble wall with M-=0, κ2=κ*=4/27, there exists a single unstable Lorentzian bubble wall at fixed radius R=2γ/3.

“When M -= 0, this corresponds to the special choice κ 2 = κ * = 4/27 which is when there exists a single unstable Lorentzian bubble wall at fixed radius R = 2γ/3.”

When M+ > M_C the dominant process is the nucleation of a static bubble wall solution.

“Correspondingly, when M + > M C the dominant process is the nucleation of a static bubble wall solution as discussed in section 4.4.”

For all cases with finite M, B < B_CDL when compared at fixed tension σl.

“In all cases with finite M , as with the critical bubble wall case in section 4.2, we find B < B CDL (4.9) when compared at fixed tension, σℓ.”

The static bounce action is B* = 8/27 πl^2/G σ^2 l^2/(3 - 2σ^2 l^2).

“B * = π R 2 N -(2GM -) 2 G = 827 πℓ 2 G σ2 ℓ 2 3 -2σ 2 ℓ 2 (B.5)”

Figure 7 shows that the decay rate reaches its maximum value for the case of critical seed black holes M+ = M_C.

“The decay rate reaches its maximum value for the case of critical seed black holes M + = M C , as given by (4.11).”

For the limit of small seed masses M+ << M_N and small tensions σl << 1, the decay exponent B/B_CDL can be brought arbitrarily close to zero.

“In particular, for the limit of small seed masses M + M N and small tensions σℓ 1, the decay exponent B/B CDL can be brought arbitrarily close to zero.”

Putting all contributions together gives I = -1/(4G) Σ_i A_i.

“Putting all the contributions to the action together gives I = - 1 4G i A i . (3.6)”

The WKB approximation gives the tunnelling rate Γ_b→b = e^{iS[R_b]}.

“If we denote the tunnelling rate by Γ b→b , then the WKB approximation gives, Γ b→b = e iS[R b ] (5.8)”

The dominant decay at a fixed tension σl depends on whether or not M+ exceeds a critical value, M_C.

“The dominant decay at a fixed tension σℓ depends on whether or not M + exceeds a critical value, M C .”

In the range M+ > M_C, the nucleation involves the creation of a black hole of finite mass M-.

“In this range the nucleation involves the creation of a black hole of finite mass M -, indicated in figure 6.”

The enhancement effect is larger than previously believed due to contributions to the action from conical singularities.

“The effect is larger than previously believed due to the contributions to the action from conical singularities.”

For 2σl > 1, the bubble has consumed more than half of the de Sitter hyperboloid.

“Clearly one does not have this restriction in the original CDL approach, however, for 2σℓ > 1, the bubble has now consumed more than half of the de Sitter hyperboloid.”

Hiscock did not take into account contributions from conical singularities.

“there the contributions from conical singularities were not taken into account.”

• cites [5] W.A. Hiscock, Can black holes nucleate vacuum phase transitions?, Phys. Rev. D 35 (1987) 1161 [ INSPIRE ]. sourceNot checked yet

The horizons of the Schwarzschild-de Sitter metric coincide when GM/l = 1/√27, corresponding to the Nariai solution.

“The horizons coincide when GM/ℓ = GM N /ℓ ≡ 1/ √ 27, which corresponds to the Nariai solution, [11,12].”

• cites [11] H. Nariai, On some static solutions of Einstein’s gravitational field equations in a spherically symmetric case, Sci. Rept. Tohoku Univ. 34 (1950) 160. sourceNot checked yet
• cites [12] H. Nariai, On a new cosmological solution of Einstein’s field equations of gravitation, Sci. Rept. Tohoku Univ. 35 (1951) 46.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Γ gives the rate at which bubbles are nucleated in a false vacuum SdS universe, centred on the black hole.

“We claim that Γ gives the rate at which bubbles are nucleated in a false vacuum SdS universe, centred on the black hole.”

The paper proposes that this relation represents a type of crossing relation for the amplitude describing the bubble-to-bubble transition.

“We propose that this represents a type of crossing relation for the amplitude describing the bubble-to-bubble transition.”

This independence removes a possible ambiguity in tunnelling rates when using singular instantons.

“This removes a possible ambiguity in the tunnelling rates when using singular instantons.”

We interpret this as a type of crossing relation between spontaneous production of bubbles and bubble transitions, with the relative factor depending only on the black hole entropies.

“We interpret this as a type of crossing relation between spontaeous production of bubbles and bubble transitions, with the relative factor depending only on the black hole entropies.”

For M+ > M_C the rate is given by B* in (4.10).

“For M + > M C the rate is therefore given by B * in (4.10).”

Since this ratio is small for black holes larger than the Planck mass GM_C^2 > 1, these black holes decay before they can nucleate the false vacuum decay.

“Since this ratio is small for black holes larger than the Planck mass GM 2 C > 1, these black holes decay before they can nucleate the false vacuum decay.”

In our case, there will be a single zero mode representing the time translation symmetry.

“In our case, there will be a single zero mode representing the time translation symmetry.”

According to Callan and Coleman, the pre-factor is made up from a factor of (B*/2π)^{1/2} for each translational zero mode of the instanton and a determinant factor.

“According to Callan and Coleman, [2], this pre-factor is made up from a factor of (B * /2π) 1/2 for each translational zero mode of the instanton and a determinant factor.”

• cites [2] C.G. Callan Jr. and S.R. Coleman, The Fate of the False Vacuum. 2. First Quantum Corrections, Phys. Rev. D 16 (1977) 1762 [ INSPIRE ]. sourceNot checked yet

One interpretation of this result is that the Euclidean instanton calculation actually describes the production of two true-vacuum bubbles in the fully extended Schwarzschild-de Sitter spacetime, only one of which is present in a single static patch.

“One interpretation of this result is that the Euclidean instanton calculation actually describes the production of two true-vacuum bubbles in the fully extended Schwarzshild-de Sitter spacetime, only one of which is present in a single static patch.”

The difference between our result and Hiscock's is due to Hiscock not including the contribution to the action from the conical singularities.

“The difference is due to the fact that Hiscock did not include the contribution to the action from the conical singularities.”

The presence of the inhomogeneity generically acts to enhance the rate of true-vacuum bubble nucleation.

“Our main result is that the presence of the inhomogeneity generically acts to enhance the rate of true-vacuum bubble nucleation.”

The nucleation rate is monotonically increasing with the mass of the black hole, M.

“hence the nucleation rate is monotonically increasing with the mass of the black hole, M .”

The consistent treatment of conical singularities in the Euclidean instanton calculation faithfully reproduces existing results which have been obtained using manifestly regular constructions.

“The salient technical point of our analysis is the consistent treatment of conical singularities in the Euclidean instanton calculation, 4 which faithfully reproduce existing results which have been obtained using manifestly regular constructions.”

The enhanced nucleation rates in the presence of a black hole could rescue some particle models which might otherwise get stuck in an early metastable state and never be able to decay into a radiation dominated universe.

“The enhanced nucleation rates in the presence of a black hole could rescue some particle models which might otherwise get stuck in an early metastable state and never be able to decay into a radiation dominated universe.”

The space-time outside of the matter is made up of a finite number of the regions from the fully extended SdS space-time.

“The space-time outside of the matter is made up of a finite number of the regions from the fully extended SdS space-time.”

• AssumptionThe authors state this as an assumption: The black hole can be used as an approximation to a spherical lump of matter.

The connection with the bubble-to-bubble transition rate can be explained as a type of crossing relation between bubble-to-bubble and the production of two bubbles from the false vacuum.

“The connection with the bubble-to-bubble transition rate can then be explained as a type of crossing relation between bubble-to-bubble and the production of two bubbles from the false vacuum.”

As a consequence, a possible source of ambiguity in using singular instantons has been annulled.

“As a consequence, a possible source of ambiguity in using singular instantons has been annulled.”

The underlying reason for the existence of the critical value is that the static walls only exist when M+ ≥ M_C.

“The underlying reason for the existence of this critical value is that the static walls only exist when M + ≥ M C .”