In 1967 A. D. Sakharov proposed a novel way to model Gravity. Rather than
considering gravity as a fundamental force of nature, the way we think of
electromagnetism, the weak and the strong force, he proposed that gravity
was the result of the collective action of the fundamental Quantum fields.
Since that time, this basic idea of emergent gravity has inspired various
models, generally linked with a thermodynamic description. The recent proposal
of Erik Verlinde, which relates closely to this basic idea, has caused
some excitement recently.
In this spirit I have been exploring a new way to think about gravity. This
idea is not without its problems, some of which I will describe.
Nevertheless, in the hope that a more rigorous treatment might solve these problems,
I will relate this rather speculative approach.
As related in Sakharov's original paper "Vacuum Quantum Fluctuations in
Curved Space and the Theory of Gravitation, " in Einstein's theory of
gravitation the action of space time depends on its curvature.
S= - [ 1/(16*pi*G)] * int dx Sqrt [-g] R
Where R is the trace over the Ricci Tensor.
Sakharov identified this action with the change in the action of Quantum
fluctuations of the vacuum if space time is curved.
In QFT it was assumed, prior to 1998, that the energy momentum tensor of
the Quantum fluctuations of vacuum and the corresponding action, which is
proportional to a divergent integral of the fourth power over the momentum of
the virtual particles of the form
int k^3 dk
was actually zero.
However, the discovery of Dark Energy has called this assumption into
question. Sakharov, based his idea on a proposal by Ya. B. Zeldovich, who had
suggested that the zero energy condition postulated ,when the vacuum
fluctuations are integated over might be due to an equilibrium between two
different modes of vacuum fluctuations. Zeldovich even postulated that we might
see a small cosmological constant if this equilibrium was upset.
Sakharov went on to suggest that shift in the equilibrium might be an
effect of localized mass energy, and that in effect gravity was nothing more
than a shift in the action density of Quantum fluctuations.
Sakharov described the Lagrangian function in a series of powers of the
curvature. ( A and B = approx. 1)
L(R)= L(0) + A*intk dk*R+ B*int dk/k R^2 .....
The first term corresponds to Einstein's cosmological constant (geometric
only), the second term, the action and all remaining terms non linear
corrections.
Based on this G the gravitational coupling is a function of the second term
G= - 1/( 16*pi*A*int k dk)
The density of the vacuum Lagrangian function in this model is based the
ratio of the action density of real particles and what Sakharov called
"Ghost" particles.( Hypothetical particles which give an opposite contribution to
that of the real particles to the R dependent action.)
This "Ghost" sector can be identified with the expansion of the solutions
of the relativistic equations, as proposed by Bob Klauber and myself, which
Klauber called the supplemental states
Abstract: In addition to the two standard solutions of the Quantum field
equations having the form e^{+/-(iwt-ikx)}, there exist two additional
solutions of the form e^{+/-(iwt+ikx). By incorporating these latter solutions,
deemed "supplemental solutions", into the development of quantum field
theory, one finds a simple and natural cancellation of terms that results in an
energy VEV, and a cosmological constant, of zero. This fundamental, and
previously unrecognized, inherent symmetry in quantum field theory shows
promise for providing a resolution of the large vacuum energy problem, simply
and directly, with little modification or extension to the extant
mathematics of the theory. In certain scenarios, slight asymmetries could give rise
to dark energy.
Where k and Chi_a*Chi^1 are parameters I will expand on later.
Here we sum over the "real" and "Ghost" particle vacuum actions. . An
imbalance as expressed by the terms Chi creates a non zero vacuum energy
density.
The question becomes do these integral diverge to infinity or does the
existence of a cutoff provide a finite result for this integration. We shall
come back to this question because it provides a unique prediction for law
mass black holes and acceleration. ( or alternately trouble for this
proposal)
Based on the above it is proposed that we can turn Einstein's model on its
head by modeling gravity strictly in terms of the cosmological constant. (
Vacuum energy density) It is important to understand, that this in no way
proposes a new theory of gravity, insofar as this relates to Einstein's
basic equations. The goal here is to re write the equations in a form more in
line with Sakharov and Zeldovich's proposals. This can be easily
illustrated.
Einstein first proposed that Gravity might be described by
G_mu,nu= 8*Pi*G*T_mu,nu
Where T_mu,nu is the stress energy tensor related to mass energy and
G_mu,nu is the Einstein Tensor.
But Einstein seeing this solution as unstable, requiring either an
expanding or contracting Universe added the now famous cosmological constant term.
G_mu,nu+ Lambda*g_mu,nu = 8*pi*G*T_mu,nu
Where Lambda is the cosmological constant and g_mu,nu is the metric tensor.
This CC terms described a geometric effect of space time which Einstein
hoped would balance exactly the effects of concentrated mass energy. However,
this was a mistake few grad students would make. This Balance sits on a
knife's edge balance, this equation cannot provide any physically realizable
static state. With the discovery of the expansion of the Universe, this
term ceased to have any purpose.
Nevertheless there is no reason to set Lambda to zero, the question now
became, is Lambda equal to zero and if so why.
This question became more acute when Weinberg and Zeldovich, independently
found a new way to look at this constant. They reasoned that one might well
move this term to the right side of Einstein's equation, and equate it
with vacuum energy density. Therefore we get
In Quantum field theory the vacuum energy density is predicted to be due
the various symmetry breaking events, resulting in Fermion condenses and the
energy associated with the zero point fluctuations. In this model the
energy density from the Fermion Condensates is expected to be constant, only
dependent on the energy scale of the symmetry breaking events. The action for
this energy source is given by
These Condensates quickly settle into their lowest energy state with
the kinetic terms at zero.
Therefore
T_mu,nu= - V(theta)*g_mu,nu = - Lambda/(8*pi*G)
and therefore the vacuum energy is given by
rho_v= rho_c +rho_zpf
Where rho_c stands for Fermion Condensates and rho_zpf stands for the
vacuum energy density due the zero point fluctuations. These are the zero
point fluctuations described in the Sakharov-Zeldovich induced Gravity model
and will be our prime concern.
In standard QFT the ZPE is given by
rho_zp(+)f= SUM (all B} (1/4*pi^2)*int { 0 to k_c}
k^2*sqrt[k^2+m^2] dk -
SUM ( all F) ( all B} (1/4*pi^2)*int { 0 to k_c}
k^2*sqrt[k^2+m^2] dk =
int Dw L(+)
Where F and B are the Fermion and Boson degrees of freedom.
If k_c goes to infinity, these integrals diverge. Often the reduced
Planck energy is selected as the cutoff giving
int Dw L(+)= 2E109 J/M^3
due the mass splits between the Fermion and Boson sector
This is an impossibly large energy density, ruled out by the very existence
of the Universe.
However, a different cutoff is suggested by the existence of a different
symmetry, known as Super Symmetry. ( SUSY). Based on the need to stabilize
the Higgs mass, and the convergence of the running coupling constants (The
strong, weak and electromagnetic) this is predicted to break circa 250-300
Gev. With this cutoff we get;
int Dw L(+)= 1E47 J/M^3
This is little help, this value is impossibly large.
So SUSY is no help in solving the cosmological constant problem.
However, we can now include Sakharov's Ghost particle states giving us
rho_zpf(-) = SUM (all F} (1/4*pi^2)*int { 0 to k_c}
k^2*sqrt[k^2+m^2] dk -
SUM ( all F) ( all B} (1/4*pi^2)*int { 0 to k_c} k^2*sqrt[k^2+m^2] dk =
Here we see the value of T_mu,nu is a function of k and Chi_a*Chi^a
What form might this take?
These questions leads us to three different stress energy tensors,
though all are related by the Sakharov-Zeldovich model and to my proposal to
model gravity strictly in terms of a cosmological constant. This will give
us three stress energy tensors.
We can equate a gravity field with a local shift in vacuum energy
density. So we have
G*M/R^2= lambda*c^2*R_H/3
Here R_H is the associated Rinder Horizon. We can re write this
equation as;
G*M/R2= - c^2*sqrt[lambda/3]
Giving us
rho_zpe= -[3/(8*pi*G)*[g^2]
Here g is the gravity field
Given the existence of a ZPE cut off
So we can rewrite the effect of the local gravity field as
This maximum value is because the vacuum energy saturates at the cutoff.
This will predict some strange effects which would seem to make this approach
questionable. I will expand on this problem shortly.
We can continue along with the global cosmological constant which based on
Zeldovich's assertion is proportional to space time curvature.
Rho= [O_v*c^2/(16*pi*G)]*[R]
Where O_v is the vacuum energy density parameter and R is the trace over
the Ricci tensor.
No doubt there are more problems with this model then will identified here.
First of all, this is hardly a very rigorous presentation. A considerable
amount of handwaving was used. The equations for the stress energy tensors
were very much a handwaving operation. They are little more than semi
plausible conjectures based on the logic of this assertion. Of particular
concern are the effect of a low energy SUSY breaking scale. The vacuum energy
density for a black Hole can be shown to be
rho_bh= (3*c^8)/(128*pi*G^3*M^2)
This means that for a SUSY breaking energy scale of circa 250 -300 GEV
Black holes with mass below about 1E24 KG will saturate their gravity fields.
After this point the black hole will lose information via Hawking radiation
by expanding it's local string scale, rather than decreasing the Horizon.
The black will decay away at a constant rate, despite its ever decreasing
mass. It will have a constant temperature. The question is can this make
sense?
Based on a low energy SUSY breaking scale we can expect a maximum possible
acceleration. What can this mean? Perhaps it relates to the shrinking
Rindler Horizon. However, given a low energy SUSY breaking scale we get a
maximum Gibbons Hawking temperature of about 1E-2 degrees Kelvin. Can this make
sense?
Also this model runs into trouble predicting the temperature of the vacuum
when it comes to modeling black holes. The equation for temperature are
T_v = hbar*sqrt 24*pi*G*abs[rho]]/(6*pi*k*c)
This gives the temperature for a black hole as
T_bh= hbar*c^3/ ( 4*pi*G*k*M)
Rather than
T_bh= hbar*c^3/ ( 8*pi*G*k*M)
However, given the crude analysis this factor of 2 problem might not be too
surprising or very fatal. In addition the standard black hole temperature
is the temperature at infinity extrapolated back to the surface of the
Horizon. The actual temperature is given by
Another problem is that given the negative energy density of the gravity
field, the absolute value must be used in the temperature equation. This
lacks justification except for the unphysical nature of imaginary valued
temperature.
Many of these problems, but not all would be alleviated by a high energy
SUSY breaking scale. This possibility has been suggested by several
physicists including Susskind, and they have made the case this would not be fatal
to the running of the coupling constants or the stabilization of the Higgs
mass. This is a question that should be answered at CERN in the next few
years.
The chances that this model is correct are not high. But pushing ideas to
their limit and thinking in unconventional ways can always be justified as
useful in some way or another.