In this post I will be considering the relationship between the Bekenstein
Bound and the geometry of Universes. We will look at the three possible
cases, K= -1, 0, and +1.
But first a look at each case above in terms of the energy condition in
each Hubble volume will be illustrative. I will also briefly relate these
energy conditions to the full set of solutions of the relativistic equations.
These are the physical solutions, Positive energy flowing forward in time
(matter) Negative energy flowing backward in time (anti matter) and the two
unphysical solutions Positive energy flowing backward in time (negative
anti matter) and negative energy flowing forward in time (negative matter).
This will related in passing to the BIVERSE model, the creations of Universe
in CPT inverse pairs as postulated by many cosmological models such as
Sakharov's Multisheet model, the Carroll Chen Model, the Hawking Hartle no
boundary model and Linde's twin Universe hypothesis.
We take a look at a simple calculation we can use to define the energy
condition of the Universe as a function of geometry. Based on the Friedmann
equations we have;
Rho_crit= 3*H^2/(8*pi*G)
Where, H is the Hubble parameter, G is the gravity constant and
Rho_crit is the critical density in a given Hubble Volume to achieve K=0, a
Euclidean metric.
Therefore here we have;
Rho_Hubble=Rho_crit
And
K=0 and Omega=1
Where K is the curvature parameter and Omega is the energy density
parameter.
When we have K=1 we have a non Euclidean spherical metric where
Rho_Hubble > Rho Crit
And when we have K=-1 we have a non Euclidean hyperbolic metric where;
Rho_Hubble < Rho_Crit
Therefore
Rho_crit= 3*H^2/(8*pi*G)
M_crit= Rho_crit*V_Hubble
V_hubble = 4*pi*R_Hubble^3/3
Since R_hubble = c/H
V_Hubble= 4*pi*c^3/(3*H^3)
M_crit= c^3/(2*H*G)
Therefore
E_mass= SUM {all i} gamma_i*m_i*c^2
Here we are summing over all relativistic masses in the Hubble
Volume.
However the negative gravitational energy must be included
therefore
E_grav= - G*gamma_i*m_i*M_unv/ R_i
Given a homogenous and isotropic Universe we can set
Now we can relate each of these geometries to the Bekenstein
Bound.
S_Bek= < 2*pi*R*E/(hbar*c) = A_Hubble/(4*L_plk^2)
Where S_Bek is the maximum entropy allowed, A_Hubble is the
surface area of the Hubble volume and L_plk is the Planck length.
Therefore
S_Bek= pi*R^2*c^2/(hbar*G)
E_Bek=R*c^4/(2*G)
Rho_Bek( energy) = 3*H*c^2/(8*pi*G)
Rho_Bek(mass)= 3*H^2/(8*pi*G)
From this we can see that the critical mass energy density condition
for any Hubble Volume saturates the entropy of that volume based on the
Bekenstein Bound.
Looking at the Friedmann equations we can see that
K=(a^2/c^2)*(8*pi*G*rho(mass)/3-H^2)
Where a is the scale factor
For K= -1 E_unv>0 we have
Rho(mass) = 3*( H^2-c^2/a^2)
Therefore
S_Hubble< S_Bek
For K=0 E_unv=0 we have
Rho(mass)= 3*H^2/(8*pi*G)
S_Hubble= S_Bek
For K=+1 E_unv < 0 we have
Rho(mass)= 3*(H^2+c^2/a^2)
S_Hubble> S_Bek
Therefore we find that for a closed Universe the Bekenstein bound is
violated for a given Hubble Volume. We can relate this to the BIVERSE model
by noting that K=1 is the unphysical relativistic solution, negative energy
flowing forward in time. ( and its CPT partner a positive energy Universe
flowing backward in time).
This fact was discovered in a more rigorous treatment by W.
Fischler and L. Susskind.
Holography and Cosmology
Authors: W. Fischler, L. Susskind
(Submitted on 4 Jun 1998 (v1), last revised 11 Jun 1998 (this version, v2))
Abstract: A cosmological version of the holographic principle is proposed.
Various consequences are discussed including bounds on equation of state
and the requirement that the universe be infinite.
Fischler and Susskind write with regard to closed Universe K=1
"Depending on the equation of state, the bound will reached
either while the Universe is still growing, for example when the energy density
is dominated by the non relativistic matter or during recollaspse like in a
radiation dominated Universe. This seems to indicate that positively
curved closed Universes are inconsistent with the Holographic principle. We do
not know what new behavior sets in to accommodate the Holographic principle
or if this violation of the Holographic principle just excludes these
Universes as inconsistent?"
End quote.
Based on this I would offer the following conjecture.
Under the assumption that the creation of Universes is properly
described by the general inflationary paradigm the only Universes ever created are
zero energy Universes in which each Hubble volume saturates the Bekenstein
entropy bound. Strictly speaking open Universe without dark energy are not
precluded from existing based on this analysis. However, based on some
string theory formulations a non zero CC is needed to break SUSY. Therefore
assuming that the general Super String SUSY paradigm is correct we might
exclude such Universes from existing too. It might further be noted that this
postulated zero sum Energy conditions for the creations of Universes does
not violate energy conservation even when the "seed" energy needed to launch
inflation is included in the various BIVERSE models.