arXiv:gr-qc/0304110 v1 30 Apr 2003

arXiv:gr-qc/0304110 v1 30 Apr 2003

Sitter vacuum with respect to quantum birth of a universe

is well studied [35]. The essential feature is possibility

of multiple birth of causally disconnected universes

from the de Sitter vacuum noted first in 1975 in the Ref

[36]. The global structure of space-time was considered in

1982 by Gott III for the case of creation of an open FRW

universe [37]. In the context of the minisuperspace model

quantum birth of an open or flat universe is possible when

an initial quantum fluctuation contains an admixture of

radiation and strings or some other quintessence with the

equation of state p = −ρ/3 which mimics a curvature

term. In the presence of radiation quantum tunnelling

occurs from a discrete energy level with a nonzero quantized

temperature [38]. An infinite number of regular

cores r = 0 inside a ΛBH enhances essentially the probability

of quantum birth of baby universes inside it as a

result of quantum instability of de Sitter vacuum [34].

their own self-interaction [39]. G-lump holds itself together

by gravity due to balance between gravitational

attraction outside and gravitational repulsion inside of

zero-gravity surface r = r c . For the case of density profile

(4) it is perfectly localized (see Fig.4) [19].















C. Vacuum nonsingular black hole

A ΛBH emits Hawking radiation from both horizons

with the Gibbons-Hawking temperature [3] which for

ΛBH with two horizons is given by [18]

kT = ¯hc [

Rg (r h )

r 2 h


− R′ g(r h )

; r h = r + , r − (27)

r h

The form of the emperature-mass diagram is generic for

de Sitter-Schwarzschild geometry. The temperature on

the BH horizon drops to zero at m = m crit , while the

Schwarzschild asymptotic requires T + → 0 as m → ∞.

The temperature-mass curve has thus a maximum between

m crit and m → ∞. In a maximum the specific

heat is broken and changes its sign testifying to a secondorder

phase transition in the course of Hawking evaporation

and suggesting symmetry restoration to the de Sitter

group in the origin [32].

For particular form of the density profile (4) the temperature

is given by [18]

T h = ¯hc [


− 3r (


1 − r )]



4πkr 0 r h r 0 r g

The mass at the maximum and the temperature of the

phase transition are [18]

m tr ≃ 0.38m Pl

ρPl /ρ 0 ; T tr ≃ 0.2m Pl

ρPl /ρ 0 (29)

D. G-lump

For masses m < m crit de Sitter-Schwarzschild geometry

describes a self-gravitating particle-like vacuum structure,

globally regular and globally neutral. It resembles

Coleman’s lumps - non-singular, non-dissipative solutions

of finite energy, holding themselves together by

FIG. 4. G-lump in the case r g = 0.1r 0 (m ≃ 0.06m crit).

Since de Sitter vacuum is trapped within a G-lump, it

can be modelled by a spherical bubble with monotonically

decreasing density. Its geometry is described by the

metric [19]

ds 2 = dτ 2 −

2GM(r(R, τ))

r(R, τ)

− r 2 (R, τ)dΩ 2 (30)

The equation of motion [34] has the first integral [19]

ṙ 2 − 2GM(r)


= f(R) (31)

which resembles the equation of a particle in the potential

V (r) = − GM(r)


, with the constant of integration f(R)

playing the role of the total energy f = 2E.

A spherical bubble can be described by the minisuperspace

model with a single degree of freedom [40]. Zeropoint

vacuum energy for G-lump, which clearly represents

an elementary spherically symmetric excitation of

a vacuum defined macroscopically by its symmetry (2),

is evaluated as its minimal quantized energy.

By the standard procedure of quantization the equation

(26) transforms into the Wheeler-DeWitt equation

in the minisuperspace [40]

¯h 2 d 2 ψ

− (V (r) − E)ψ = 0 (32)

2m Pl dr2 Near the minimum r = r m the equation (32) reduces to

the equation for a harmonic oscillator with the energy

Ẽ = E − V (r m ), and the energy spectrum is [19]


E n = ¯hω n + 1 )

− GM(r m)

E Pl (33)

2 r m

where ω 2 = Λc 2˜p ⊥ (r m ), and ˜p ⊥ is the dimensionless pressure

normalized to vacuum density ρ 0 at r = 0; for the

density profile (4) ˜p ⊥ (r m ) ≃ 0.2.


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