Dynamics of false-vacuum bubbles

Steven K. Blau; E. I. Guendelman; Alan H. Guth

doi:10.1103/PhysRevD.35.1747

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Dynamics of false-vacuum bubbles

Steven K. Blau and E. I. Guendelman

Alan H. Guth

Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 and Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, Massachusetts 02138

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Phys. Rev. D 35, 1747 – Published 15 March, 1987

DOI: https://doi.org/10.1103/PhysRevD.35.1747

Abstract

The possibility of localized inflation is investigated by calculating the dynamics of a spherically symmetric region of false vacuum which is separated by a domain wall from an infinite region of true vacuum. For a range of initial conditions, the false-vacuum region will undergo inflation. An observer in the exterior true-vacuum region will describe the system as a black hole, while an observer in the interior will describe a closed universe which completely disconnects from the original spacetime. We suggest that this mechanism is likely to lead to an instability of Minkowski space: a region of space might undergo a quantum fluctuation into the false-vacuum state, evolving into an isolated closed universe; the black hole which remains in the original space would disappear by quantum evaporation. The formation of these isolated closed universes may also be relevant to the question of information loss in black-hole formation.

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Comment on ‘‘Dynamics of false-vacuum bubbles’’

Kayll Lake

Phys. Rev. D 38, 3844 (1988)

References (35)

A. H. Guth, Phys. Rev. D 23, 347 (1981).
A. D. Linde, Phys. Lett. 108B, 389 (1982).
A. D. Linde, Phys. Lett. 114B, 431 (1982).
A. Albrecht and P. J. Steinhardt, Phys. Rev. Lett. 48, 1220 (1982).
A. A. Starobinsky, Phys. Lett. 117B, 175 (1982).
A. H. Guth and S.-Y. Pi, Phys. Rev. Lett. 49, 1110 (1982).
S. W. Hawking, Phys. Lett. 115B, 295 (1982).
J. M. Bardeen, P. J. Steinhardt and M. S. Turner, Phys. Rev. D 28, 679 (1983).
R. Brandenberger, R. Kahn and W. H. Press, Phys. Rev. D 28, 1809 (1983).
R. Brandenberger and R. Kahn, Phys. Rev. D 29, 2172 (1984).
K. Sato, H. Kodama, M. Sasaki and K. Maeda, Phys. Lett. 108B, 103 (1982).
K. Sato, M. Sasaki, H. Kodama and K. Maeda, Prog. Theor. Phys. 65, 1443 (1981); H. Kodama, M. Sasaki, K. Sato and K. Maeda, ibid. 66, 2052 (1981); K. Sato, ibid. 66, 2287 (1981); H. Kodama, M. Sasaki and K. Sato, ibid. 68, 1979 (1982).
A. H. Guth and E. J. Weinberg, Nucl. Phys. B212, 321 (1983) have shown that such trapped regions of false vacuum are highly improbable in the context of the original inflationary model. The enormous diversity in the sizes of true-vacuum bubbles makes very unlikely to have any configuration which includes two or more true-vacuum bubbles of comparable sizes. This argument, however, is not relevant in the present context of the new inflationary universe model.
K. Maeda, K. Sato, M. Sasaki and H. Kodama, Phys. Lett. 108B, 98 (1982).
W. Israel, Nuovo Cimento 44B, 1 (1966); ibid. 48B, 463(E) (1967).
J. Ipser and P. Sikivie, Phys. Rev. D 30, 712 (1984).
V. A. Berezin, V. A. Kuzmin and I. I. Tkachev, Phys. Lett. 120B, 91 (1983).
V. A. Berezin, V. A. Kuzmin, and I. I. Tkachev, Quantum Gravity, proceedings of the Third Seminar, Moscow, 1984, edited by M. A. Markov, V. A. Berezin, and V. P. Frolov (World Scientific, Singapore, 1985), p. 605.
A. Aurilia, G. Denardo, F. Legovini and E. Spallucci, Phys. Lett. 147B, 258 (1984); Nucl. Phys. B252, 523 (1985).
We follow the sign conventions of Ref. 21, in which the key definitions are given by Γ ${}_{λ σ}$ ${}^{μ}$ = 1/2 $g^{μ ν}$ ( ${partial}_{λ}$ $g_{ν σ}$ + ${partial}_{σ}$ $g_{ν λ}$ - ${partial}_{ν}$ $g_{λ σ}$ ), $V^{μ}$ ${}_{{;}ν$ } = ${partial}_{ν}$ $V^{μ}$ + Γ ${}_{ν λ}$ ${}^{μ}V^{lambda}$ , $V_{μ; ν}$ = ${partial}_{ν}$ $V_{μ}$ - Γ ${}_{μ ν}$ ${}^{λ}V_{lambda}$ , $R^{μ}$ ${}_{{ν}α$ β} = ${partial}_{α}$ Γ ${}_{ν β}$ ${}^{mu}$ - ${partial}_{β}$ Γ ${}_{ν α}$ ${}^{μ}$ + Γ ${}_{σ α}$ ${}^{μ}Γ$ ${}_{ν β}$ ${}^{σ}$ - Γ ${}_{σ β}$ ${}^{μ}Γ$ ${}_{ν α}$ ${}^{sigma}$ , and $R_{μ ν}$ = $R^{α}$ ${}_{{μ}α$ ν}.
C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (Freeman, San Francisco, 1973).
M. D. Kruskal, Phys. Rev. 119, 1743 (1960); G. Szekeres, Publ. Mat. Debrecen 7, 285 (1960).
G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15, 2738 (1977).
We thank Sidney Coleman for suggesting this method of solution.
The coordinate transformation can be represented most simply by introducing the lightlike coordinates $ζ_{+ -}$ == V +- U. The line element then has the form ${ds}^{2}$ = f( $ζ_{+}$ , $ζ_{-}$ )d $ζ_{+}$ d $ζ_{-}$ + $r^{2}$ ( $ζ_{+}$ , $ζ_{-}$ )d $Ω^{2}$ , a form which is preserved by any transformation of the form $ζ_{+}^{sprime}$ = g( $ζ_{+}$ ), $ζ_{-}^{sprime}$ =h( $ζ_{-}$ ). This transformation can be chosen so that each line in the de Sitter region corresponding to $ζ_{+}$ =const is assigned the value of $ζ_{+}^{sprime}$ which is equal to the value of $ζ_{+}$ at the point where this line intersects the Schwarzschild region, and similarly for each line corresponding to $ζ_{-}$ =const. The values of $ζ_{+ -}$ for lines which do not intersect the Schwarzschild region can be chosen arbitrarily.
S. Coleman and F. De Luccia, Phys. Rev. D 21, 3305 (1980). See also Sec. 6 of Ref. 13, which includes a discussion of a spacetime region which was omitted in the original reference.
A. Vilenkin, Phys. Lett. 117B, 25 (1982); Phys. Rev. D 27, 2848 (1983); ibid. 30, 509 (1984).
J. B. Hartle and S. W. Hawking, Phys. Rev. D 28, 2960 (1983); see also I. G. Moss and W. A. Wright, ibid. 29, 1067 (1984); S. W. Hawking and J. C. Luttrell, Phys. Lett. 143B, 83 (1984).
A. D. Linde, Pis'ma Zh. Eksp. Teor. Fiz. 38, 149 (1983) [JETP Lett. 38, 176 (1983)]; Phys. Lett. 129B, 177 (1983); A. B. Goncharov and A. D. Linde, ibid. 139B, 27 (1984).
E. Farhi and A. H. Guth, Phys. Lett. 183B, 149 (1987).
This question is academic, since the energies that are required are totally inaccessible. We found in Sec. V that the minimum mass necessary for what we called a monotonic trajectory is about $10^{28}$ GeV, and we presume that an energy of this order would be necessary to produce an inflationary universe by any laboratory mechanism.
We thank Bill Press for helping us clarify the issues involved in this question.
The quantum instability of Minkowski space has also been discussed by E. Gunzig and P. Nardone, Gen. Relativ. Gravit. 16, 305 (1984), but they are writing about a different kind of instability.
L. F. Abbott, Phys. Lett. 150B, 427 (1985).
G. 't Hooft, Nucl. Phys. B256, 727 (1985); in Proceedings of the Symposium on Anomalies, Geometry, and Topology, Argonne, Illinois, 1985, edited by W. A. Bardeen and A. R. White (World Scientific, Singapore, 1985), pp. 33–49.

[c1] A. H. Guth, Phys. Rev. D 23, 347 (1981).

[c2] A. D. Linde, Phys. Lett. 108B, 389 (1982).

[c3] A. D. Linde, Phys. Lett. 114B, 431 (1982).

[c4] A. Albrecht and P. J. Steinhardt, Phys. Rev. Lett. 48, 1220 (1982).

[c5] A. A. Starobinsky, Phys. Lett. 117B, 175 (1982).

[c6] A. H. Guth and S.-Y. Pi, Phys. Rev. Lett. 49, 1110 (1982).

[c7] S. W. Hawking, Phys. Lett. 115B, 295 (1982).

[c8] J. M. Bardeen, P. J. Steinhardt and M. S. Turner, Phys. Rev. D 28, 679 (1983).

[c9] R. Brandenberger, R. Kahn and W. H. Press, Phys. Rev. D 28, 1809 (1983).

[c10] R. Brandenberger and R. Kahn, Phys. Rev. D 29, 2172 (1984).

[c11] K. Sato, H. Kodama, M. Sasaki and K. Maeda, Phys. Lett. 108B, 103 (1982).

[c12] K. Sato, M. Sasaki, H. Kodama and K. Maeda, Prog. Theor. Phys. 65, 1443 (1981); H. Kodama, M. Sasaki, K. Sato and K. Maeda, ibid. 66, 2052 (1981); K. Sato, ibid. 66, 2287 (1981); H. Kodama, M. Sasaki and K. Sato, ibid. 68, 1979 (1982).

[c13] A. H. Guth and E. J. Weinberg, Nucl. Phys. B212, 321 (1983) have shown that such trapped regions of false vacuum are highly improbable in the context of the original inflationary model. The enormous diversity in the sizes of true-vacuum bubbles makes very unlikely to have any configuration which includes two or more true-vacuum bubbles of comparable sizes. This argument, however, is not relevant in the present context of the new inflationary universe model.

[c14] K. Maeda, K. Sato, M. Sasaki and H. Kodama, Phys. Lett. 108B, 98 (1982).

[c15] W. Israel, Nuovo Cimento 44B, 1 (1966); ibid. 48B, 463(E) (1967).

[c16] J. Ipser and P. Sikivie, Phys. Rev. D 30, 712 (1984).

[c17] V. A. Berezin, V. A. Kuzmin and I. I. Tkachev, Phys. Lett. 120B, 91 (1983).

[c18] V. A. Berezin, V. A. Kuzmin, and I. I. Tkachev, Quantum Gravity, proceedings of the Third Seminar, Moscow, 1984, edited by M. A. Markov, V. A. Berezin, and V. P. Frolov (World Scientific, Singapore, 1985), p. 605.

[c19] A. Aurilia, G. Denardo, F. Legovini and E. Spallucci, Phys. Lett. 147B, 258 (1984); Nucl. Phys. B252, 523 (1985).

[c20] We follow the sign conventions of Ref. 21, in which the key definitions are given by Γ ${}_{λ σ}$ ${}^{μ}$ = 1/2 $g^{μ ν}$ ( ${partial}_{λ}$ $g_{ν σ}$ + ${partial}_{σ}$ $g_{ν λ}$ - ${partial}_{ν}$ $g_{λ σ}$ ), $V^{μ}$ ${}_{{;}ν$ } = ${partial}_{ν}$ $V^{μ}$ + Γ ${}_{ν λ}$ ${}^{μ}V^{lambda}$ , $V_{μ; ν}$ = ${partial}_{ν}$ $V_{μ}$ - Γ ${}_{μ ν}$ ${}^{λ}V_{lambda}$ , $R^{μ}$ ${}_{{ν}α$ β} = ${partial}_{α}$ Γ ${}_{ν β}$ ${}^{mu}$ - ${partial}_{β}$ Γ ${}_{ν α}$ ${}^{μ}$ + Γ ${}_{σ α}$ ${}^{μ}Γ$ ${}_{ν β}$ ${}^{σ}$ - Γ ${}_{σ β}$ ${}^{μ}Γ$ ${}_{ν α}$ ${}^{sigma}$ , and $R_{μ ν}$ = $R^{α}$ ${}_{{μ}α$ ν}.

[c21] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (Freeman, San Francisco, 1973).

[c22] M. D. Kruskal, Phys. Rev. 119, 1743 (1960); G. Szekeres, Publ. Mat. Debrecen 7, 285 (1960).

[c23] G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15, 2738 (1977).

[c24] We thank Sidney Coleman for suggesting this method of solution.

[c25] The coordinate transformation can be represented most simply by introducing the lightlike coordinates $ζ_{+ -}$ == V +- U. The line element then has the form ${ds}^{2}$ = f( $ζ_{+}$ , $ζ_{-}$ )d $ζ_{+}$ d $ζ_{-}$ + $r^{2}$ ( $ζ_{+}$ , $ζ_{-}$ )d $Ω^{2}$ , a form which is preserved by any transformation of the form $ζ_{+}^{sprime}$ = g( $ζ_{+}$ ), $ζ_{-}^{sprime}$ =h( $ζ_{-}$ ). This transformation can be chosen so that each line in the de Sitter region corresponding to $ζ_{+}$ =const is assigned the value of $ζ_{+}^{sprime}$ which is equal to the value of $ζ_{+}$ at the point where this line intersects the Schwarzschild region, and similarly for each line corresponding to $ζ_{-}$ =const. The values of $ζ_{+ -}$ for lines which do not intersect the Schwarzschild region can be chosen arbitrarily.

[c26] S. Coleman and F. De Luccia, Phys. Rev. D 21, 3305 (1980). See also Sec. 6 of Ref. 13, which includes a discussion of a spacetime region which was omitted in the original reference.

[c27] A. Vilenkin, Phys. Lett. 117B, 25 (1982); Phys. Rev. D 27, 2848 (1983); ibid. 30, 509 (1984).

[c28] J. B. Hartle and S. W. Hawking, Phys. Rev. D 28, 2960 (1983); see also I. G. Moss and W. A. Wright, ibid. 29, 1067 (1984); S. W. Hawking and J. C. Luttrell, Phys. Lett. 143B, 83 (1984).

[c29] A. D. Linde, Pis'ma Zh. Eksp. Teor. Fiz. 38, 149 (1983) [JETP Lett. 38, 176 (1983)]; Phys. Lett. 129B, 177 (1983); A. B. Goncharov and A. D. Linde, ibid. 139B, 27 (1984).

[c30] E. Farhi and A. H. Guth, Phys. Lett. 183B, 149 (1987).

[c31] This question is academic, since the energies that are required are totally inaccessible. We found in Sec. V that the minimum mass necessary for what we called a monotonic trajectory is about $10^{28}$ GeV, and we presume that an energy of this order would be necessary to produce an inflationary universe by any laboratory mechanism.

[c32] We thank Bill Press for helping us clarify the issues involved in this question.

[c33] The quantum instability of Minkowski space has also been discussed by E. Gunzig and P. Nardone, Gen. Relativ. Gravit. 16, 305 (1984), but they are writing about a different kind of instability.

[c34] L. F. Abbott, Phys. Lett. 150B, 427 (1985).

[c35] G. 't Hooft, Nucl. Phys. B256, 727 (1985); in Proceedings of the Symposium on Anomalies, Geometry, and Topology, Argonne, Illinois, 1985, edited by W. A. Bardeen and A. R. White (World Scientific, Singapore, 1985), pp. 33–49.

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Dynamics of false-vacuum bubbles

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