Bousso mentions the decay of the vacuum as needed to prevent Boltzmann brains in the distant future. If the universe were to reach a final state as a deSitter vacuum this eternal state would permit Boltzmann brains. The small cosmological constant Λ = 10^{-123} would permit statistical fluctuations as a finite (though very small temperature) to give rise to any configuration, including a...

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Bousso mentions the decay of the vacuum as needed to prevent Boltzmann brains in the distant future. If the universe were to reach a final state as a deSitter vacuum this eternal state would permit Boltzmann brains. The small cosmological constant Λ = 10^{-123} would permit statistical fluctuations as a finite (though very small temperature) to give rise to any configuration, including a brain or whole person --- even a whole planet or solar system with people and so forth. However, the decay of the vacuum state which would otherwise permit this can be seen from rather elementary grounds.

Consider the decay of the de Sitter vacuum by quantum means and the prospect for this as a mechanism for the production of nascent cosmologies or baby universes. The observable universe, under eternal inflation from dark energy, will asymptotically evolve to a de Sitter spacetime. This spacetime is a vacuum configuration with a cosmological constant Λ. The stationary metric for this spacetime is

ds^2 = Adt^2 – A(r)^{-1}dr^2 - r^2dΩ^2, A(r) = (1 - Λr^2/3)

A radial null geodesic with ds^2 = 0 and dΩ^2 = 0 gives the velocity v = dr/dt = A(r), where this pertains to both out and in going geodesics near the cosmological horizon r = sqrt{3/Λ} as measured from r = 0. The total action for the motion of a particle is S = ∫ p_r dr - ∫ Hdt. Consider the bare action of massless particles, across the horizon from r to r',

S = ∫_r^{r'}p_rdr = ∫_r^{r'}∫_0^{p_r}dp_rdr.

The radial velocity of a particle is dr/dt = v = dH/dp_r, which enters into the action as,

S =∫_r^{r'}∫_0^H drdH'/v.

The field defines H’ = ħω'. The integration over frequencies is from E to E - ω, for the ADM energy. The action is properly written as

S = -ħ∫_r^{r'}∫_E^{E-ω}drdω'/v,

where the negative sign indicates the quanta is tunneling across the horizon to escape the Hubble region with radius The radial velocity

v = sqrt{Λ/3}r

defines the action

S = -ħ∫_r^{r’}∫_0^ω dωdr/(1 - sqrt{Λr^2/3}) = sqrt{3/Λ}tanh^{-1}sqrt{Λ/3}r

We now perform a similar analysis above, but instead consider the radiation production according to the transition Λ --- > Λ + δΛ. The transition is considered according to the metric back reaction of the de Sitter vacuum. This tunneling defines the imaginary part of the action

S = ∫_{Λ _0}^Λ pdr.

The velocity term here is the computed from the Hamilton equation {v} = \partial H/\partial p which permits this to be written as

S = ∫_{Λ _0}^Λ ∫_0^RdHdr/v.

The Hamiltonian used is the H = G^{00} computed here, with v = (1 - Λr^2/3) with

S = ∫_{Λ _0}^Λ∫_{r_i}^{r_f}dr(4r/3)^2∫_{Λ _0}^Λ(Λ/(1-Λr^2/3)^2}}(1+Λ r^2/3)/(1 - Λr^2/3)dΛ

This complicated expression is evaluated for Λ = Λ _0 - δ Λ with

S = 8(sqrt{Λ/3}arctanh({Λ /3}r) + r/(1-3/(Λr^2)) + 2/(Λ-3/r^2))|_{r_i}^{r_f}δΛ

Of course this solution exhibits a singularity for the radius extended across sqrt{3/ Λ }. We then consider the integration with respect to the small change in the cosmological constant and the calculus of residues and Cauchy integral formula gives this result in a very simple form for z^2 = Λ r^2 and we then left with the result

δS = (π sqrt{3}/4)δΛ.

The cosmological constant will decrease to zero as r_cc = sqrt{Λ/3} --- > ∞. The decay of this vacuum will be extremely slow, on the order of 10^{10^{100}} years for a half life (cosmological horizon doubles), which should then be still fast enough to prevent the formation of Boltzmann brains. This is something which would have to be verified explicitly.

LC

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Very recently there have been unexpected advances in understanding dark energy. In fact if the claim of the Egyptian Scientist M. S. El Naschie is correct, then there is no more a mystery regarding dark energy. El Naschie’s solution is disarmingly simple and was presented at two conferences which were almost entirely devoted to his work. The first was held in Bibliotheca Alexandrina early October...

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Very recently there have been unexpected advances in understanding dark energy. In fact if the claim of the Egyptian Scientist M. S. El Naschie is correct, then there is no more a mystery regarding dark energy. El Naschie’s solution is disarmingly simple and was presented at two conferences which were almost entirely devoted to his work. The first was held in Bibliotheca Alexandrina early October 2012 and the second was in Shanghai a week or so ago. On both occasions El Naschie presented a revision of Einstein’s theory leading to an equation very similar to that of Einstein’s namely Energy equals mass x the square of the speed of the light. However unlike Einstein’s equation, the result is divided by 22. His explanation of 22 is as follows: As in the old string theory of strong interaction, space time of relativity should have been considered 26 dimensional. Taking 4 only is what Einstein did and that is how he got his famous result. Nevertheless Einstein ignored 22 dimensions. This is a scaling factor following Nottale’s theory as argued by El Naschie. Even in simpler terms, he reasons that Einstein knew only one elementary messenger particle namely the photon. He knew nothing about the other 11 messenger particles of the standard model which were not known in 1905. Adding 11 super partners it turned out that Einstein did not know about an additional 22 elementary particles. These are the particles needed to explain the missing dark energy. In this way El Naschie was able to show that 95.5% of the energy of the Universe is missing. Alternatively this energy was never there to start with because space time is a fractal and although it looks puffed up it boils down to very little similar to cotton candy. In addition the compactified 22 dimensions are the cause for the negative pressure which increases the acceleration of the Universe’s expansion. He claims to have tested his theory using 25 different methods including Witten’s M-Theory and reached the same result. Even more importantly this result agrees completely with observation. In other words mathematics and physics have been substantiated by measurement which led last year to the award of the Nobel Prize to the 3 team who obtained this incredible measurement and data. Click on this link to get more info re the above (under news) http://www.msel-naschie.com/ and also http://mohamed-elnaschie.blogspot.com/.

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