A Different Way to Think About Gravity

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Feb 29, 2016, 5:57:32 AM2/29/16

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In a future post I hope to apply this different way to look at gravity to the localization of gravitational energy for Schwarzschild Black Holes. A more ambitious goal for the more distant future is to generalize this for Kerr Newman Black Holes.

A Different Way to Think About Gravity

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.

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.

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

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 we might postulate a zero vacuum energy density by including “ghost” sector quantum fields when the vacuum fluctuations are integrated over. Zeldovich even postulated that we might see a small cosmological constant if this equilibrium was upset between the normal a ghost sector quantum fields, a shift in equilibrium.

Sakharov went on to suggest that shifts in this 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.

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

The density of the vacuum Lagrangian function in this model is based the ratio of the action density of real particles and "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

Mechanism for Vanishing Zero-Point Energy

Authors: Robert D. Klauber

(Submitted on 24 Sep 2003 (v1), last revised 19 Jul 2007 (this version, v3))

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.

http://arxiv.org/PS_cache/astro-ph/pdf/0309/0309679v3.pdf

Based on this set of ideas, we might write the equation for the vacuum stress energy tensor as;

Where K and and 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 vacuum energy.

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

Where is the stress energy tensor related to mass energy and 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.

Where is the cosmological constant and 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 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

And

So that

We can re write this strictly in terms of vacuum energy.

Where;

Where is vacuum energy density.

In Quantum field theory vacuum energy is predicted to result from various symmetry breaking events based on Fermion condenses and the energy associated with the zero point fluctuations. 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;

Where is the scalar Goldstone field associated with symmetry breaking.

Giving

These Condensates quickly settle into their lowest energy state with the kinetic terms at zero.

Therefore;

The vacuum energy is given by;

Where stands for Fermion Condensates and 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, these will be our prime concern.

In standard QFT the ZPE is given by

The positive energy contribution from the fermions is because fermions in the VBL model are ghost sector quanta.

Where F and B are the Fermion and Boson degrees of freedom.

If goes to infinity, these integrals diverge. Often the reduced Planck energy is selected as the cutoff giving

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, we can now include Sakharov's Ghost particle states giving us

The negative energy contribution from the bosons is because bosons in the VBL model are ghost sector quanta.

So we can write

Here we see the value of is a function of k and and

What form might this take?

These questions lead 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.

We can equate a gravity field with a local shift in vacuum energy density. So we have

Here is the distance associated with the Rindler Horizon. We can re write this equation as;

Giving us

Here is the gravity field

So we can rewrite the effect of the local gravity field as

Where

Where is the maximum possible gravity field.

This maximum value is because the vacuum energy saturates at the cutoff. However, as pointed out by Carroll SUSY must be broken given a positive vacuum energy density. This might allow us to postulate a ZPE cutoff at the Planck scale for gravity.

The global cosmological constant, based on Zeldovich's assertion is proportional to space time curvature.