MOTIVATION: 60 years of work to unify general relativity (GR) and quantum electrodynamics (QED) have produced many fanciful ideas, none of which have advanced either theory. (Which is not to say that neither theory has advanced.) While GR has been tested to 4-5 orders of magnitude, QED has had success to over 10 orders of magnitude. This suggests that GR is most likely to be in error.
If gravity could be modeled as a field in a Minkowskian space-time, the unification with the standard model would be greatly simplified. GR has been tested to the parametrized post-Newtonian formalism, which any competing theory should satisfy. The Noether theorem, and the familiar laws concerning conservation of energy and momentum, satisfy all but two of these parameters. This leaves the Eddington–Robertson–Schiff parameters, and .
The first is of these is satisfied if the alternative theory predicts the same gravitational time dilation as GR. This, combined with the precession of nearly circular orbits, shows agreement with the second.
This theory aims to satisfy these requirements, and become an alternative to GR as the continuous limit for a future quantum theory of gravity.
THEORY: The following can be taken as the founding principles of the theory:
1) Space-time is a Minkowskian 3+1 metric.
2) Energy (not just rest mass) IS gravitational mass, including the (negative) energy of the gravitational field.
3) A gravitational field has an index of refraction impacting all dynamics, proportional to the ratio of the energy of an object at that point to its energy at infinity.
The fact that space-time is really an expanding DeSitter space will not impact the solar system tests considered here.
SOUNDNESS: So, how does this theory satisfy the PPN limit? Glad you asked!
Newtonian Limit: We begin by modifying Gauss's law for gravity, where is the energy density and is the acceleration per :
From here we can derive the energy density of the field itself.
Taking a spherically symmetric energy distribution, the source of which is contained within a distance R of the origin, we can use the flat space-time to define the limit of the total energy of the source and its field outside R as we go to infinity. This in tern can be used to express the total energy within some r>R.
For a test object free falling in this field, with energy e(r), we have the following central force law:
Clearly this reverts to Newton's law when the energy is due almost entirely to the rest mass of the objects, using . More interesting is that this can be used to calculate the work needed to take the rest mass to infinity, and find e(r) in terms of this value.
This produces both the energy gain for non-relativistic massive objects, and for photons observed in the Rebka-Pound experiment, to the PPN limit.
Time Dilation: If a gravitational field is composed of gravitons, the interactions with these gravitons may slow processes down as they pass through the field. This theory assumes this is proportional to the ratio of the potential energy to the energy at infinity.
Clearly this is for the spherically symmetric case.
But this last expression is the time dilation from GR.
Before moving on, it should be noted that this alters the special relativistic time dilation as follows:
Also, in cylindrical coordinates (as a reminder):
Precession of Mercury: In classical mechanics the Lagrangian is defined to be the kinetic energy minus the potential energy.
The Euler-Lagrange equation relative to the angle of rotation gives the following constant of motion:
Using the substitution , and evaluating the Euler-Lagrange equation relative to r gives us the following:
For a nearly elliptical orbit, we can find L at the semi-latus recum, , and find the estimated ellipse.
For nearly circular orbits we have:
This is also the value of the precession for GR.
Differences: So, what does this theory predict that is not mainstream? Let me tell you!
The index of refraction is already shown above to differ from GR in the second order term. This could be measurable by the proposed LATOR mission.
The singular surface a gravitationally collapsed object is expected to be 1/4 the size of an event horizon from GR. This could be observed in planned radio imaging of the black hole at the center of the Milky Way. This is confounded by the matter surrounding the singularity, variables in the models of its geometry and the expected enlarging due to the refraction of light.
Real singularities exist. There is no cosmic censorship. All singularities are all naked.
Academic Pedigree: It seems en vogue to stake a claim of my publications and degrees, to support the idea that I be taken seriously.
I graduated Top Waffle from Waffle House University (Norcross, Georgia) in 2005, and used to manage a sports blog dedicated to coverage of the BCS.