Copyright © 2006
A 1987 anonymous paper developed a time and space distortion associated with gravity using accepted principles of physics. This was done, in a static context, assuming the existence of an energy potential difference between elevations. Searching for a time and space distortion along similar lines to Special Relativity gave the essential result of:
where represents an energy potential. Using the standard energy potential from Newton
completes the theory1.
I couldn’t find anything wrong with the analysis. As such, I felt compelled to research General Relativity and look more closely at the paper. I continued to find no problems with the paper’s analysis, but I found that the comparisons with GR were very wrong (understatement here). This paper makes a PPN comparison with GR to distinguish the two theories. The new one does not match the recent measurement of . (At the time it was written, there was no such measurement.) Only one place seemed available to make an adjustment. The resulting modified metric eliminates the event horizon and has a perfect limit at . There is also a prediction for the next PPN parameter . This theory has an additional advantage in that it presumes that the universe is fundamentally flat, a recently concluded fact from CBR analysis.
Converting to a metric model:
His analysis represents a distortion of space and time that is embedded in a flat spacetime. This is inconsistent with GR in the general sense. However, it is absolutely comparable to the Schwarzschild metric and also to the PPN approximation. The Schwarzschild metric is an exact solution of GR for the case of a spherically symmetric central mass and the spacetime around that mass. It and the PPN remain the main means of analyzing gravity affects in the solar system.
The following is mostly in spherical coordinates with for the orbital precession analysis.
The proposed scale factors can be put into a standard metric form. Starting with the flat metric:
and using the scale factors as and applied to the above gives:
The Schwarzschild metric, in its original form, has the simplest look:
This would tend to indicate that GR is fundamentally anisotropic where mass causes distortions in the time and radial directions with a plus/minus symmetry.
The PPN approximation of GR and the gravity of this paper:
Underlying the PPN is a flat space and an isotropic gravitational effect. The gravity of this paper takes those as fundamental properties rather than as a part of an approximation technique. This paper only considers the static metric, but we should be able to build the full dynamic metric with only the PPN technique. Simply fill out the off-diagonal elements in the standard PPN way. As such, other features, such as gravity waves, should result from this theory.
The PPN has many terms dealing with dynamics and different forms of energy and their possibly different gravitational effects. Here we will ignore all of them and simply use the terms associated with regular matter.
The PPN reduces to:
where is the same as eq. .
For this paper, we will expand it slightly so that space and time both have the same order terms.
The expansion, I believed, is defined, by convention, so that GR values are 1.
For GR we get the approximation from the Schwarzschild metric:
Changing the radial coordinate makes the metric isotropic2 and ready for a direct PPN approximation.
We can now fill out the values for the coefficients using Taylor series expansions. For GR we get (by default):
For the gravity of this paper (so far):
Radar and laser delay are used to measure the PPN coefficients ,:
NASA now has tens of thousands of ranging data points. These started with planetary radar ranging an 1967 and now include extra high precision laser ranging to planetary probes. At the time of the initial publishing of the base theory in this paper, was measured to be 1.0 and there was no measurement of . In other words, it was as accurate as GR. The latest measurements are from lunar laser ranging. These have nailed at 1.0 to 4 decimal places3. Future gravity measurements are expected to be able to measure .
Considering the basic premise of the theory in this paper, the only place for modification appears to be in the formula for U. It is assumed that gravity is essentially a force with a energy function. Let’s discover the difference between this theory (as it currently stands) and the measured values. Consider this modified form:
Let’s try modifying the potential so that it reads:
What we have is an effective distance for the potential that is affected by the context of the gravity potential itself. Basically, gravity happens in the context of gravity. This is quite reasonable.
Assuming the measurements of the PPN parameters, we can get a long ways toward discovering an that matches the data.
We already know , , and . Also, and .
We find as follows. Starting with eq.,
Also, from eq.,
Here, this author makes a guess at a full function for F. Let
This would be consistent with ,, and . We can now find a modified energy function .
The metric needs to be restated.
is the “effective” radius for calculating the energy function.
Speculation on :
This is a space “dilation” or “expansion” formula. Can something be considered to be moving with velocity ? With special relativity, the distance the traveler travels “constracts”. This is equivalent to the traveler, himself, expanding. It seems difficult to apply such an interpretation.
Without a geometric meaning for the radius, I believe that gravity cannot be a direct effect. To me it seems to be an effect that defines the springiness of space that is required for Quantum Field Theory to work.
The Lorentz Transforms can be either an Aether derivation or a Relativity derivation. If the gravity effect is an adjustment of the springiness of a fixed Aether, then the speed-of-light adjustments slow light with respect to the speed of the Aether and this doesn’t work.
If the spacetime is fundamentally a flat Minkowski space with 4 dimensions, we get the natural inclination to use x(t), y(t), z(t) because there is only one of the opposite-type dimensions. Starting with this kind of space and the existence of springiness for Quantum Physics, gravity can be an effect that changes this springiness.
If spacetime is not fundamentally Minkowski, then the background Aether could possibly be a gravity Aether from our universe’s mass and/or external mass. The formula could be a reflection of how gravity effects add together when dynamics are considered.
One nice effect of this “effective” radius is that is now well behaved. As , , not infinity. Furthermore, this is nicely monotonic and is well behaved as well:
With no gradient of at the center of any mass, however large the mass happens to be, there is no force there – no singularity – no traditional black hole.
This nice effect may be the real indicator of what the adjustment means. If the space and time distortions are the result of some mechanism or gravity process, then they must always be finite.
This metric predicts a new PPN value for from the second derivative of the space metric. Here we have to change our labeling. The PPN has by definition. So we will use the letter .
Starting with eq. rewritten here as:
Reading off and, we finally have from eq.
The predicted value for is one third of this (double check via the GR metric):
Although the gravity of this paper is developed from a static model, we can make it fully dynamic. One way is to defer to the PPN. Another is more constructive. We can simply give gravity a speed. The gravity potential becomes simply the time-delayed potential based on that speed. There are two natural speeds that can be considered. One is the speed of light based on the metric itself. The other is to have the speed-of-gravity be the faster speed of light that would occur without gravity. All these variations feel internally consistent.
Breaking the Rules:
Imagine, instead of a metric definition, the following: First consider a glass sphere with an index of refraction. Light bends toward the center of the sphere as it enters. The standard way of calculating the bending of light is to consider light as a wave with a wavefront that is perpendicular to the direction of travel. The process of tracking the bending of light is identical to tracking a geodesic. The tracking works for any entity described as a wave with a wavefront. If there is a side-to-side gradient in the index of refraction then the light - or any particle described as a wave - curves. Replace the glass sphere with an aether that, as you get closer to the central mass, has a stronger index of refraction. With this approach, the classic aether, instead of being dragged around by the masses, is actually created by the masses. How? Perhaps it is zero-mass, zero-spin, zero-energy, infinite speed gravitons emitted from every mass - an infinite number of them forming a graviton density in a fixed 3D space. Perhaps it affects the springiness of space that allows quantum mechanics to work. In any case, an aether formulation needs to be considered again in a rigorous way. Note: a “light clock” embedded in glass will run slower and be the same size. The metric for light inside glass would have g00=1/n, gii=1. If the effective substance of the glass “aether” was made from stuff that is much smaller than an atom, then the change in the speed of light would affect the size of the atoms embedded in it. In other words, the size of the light clock would shrink. We would get a metric at least very similar to the one in this paper.
Cosmic Models and Collapse:
All three theories include the collapse of a neutron star. With GR, the neutron star collapse results in a singularity. This has traditionally been considered a catastrophic problem. In the original heretical theory, when 0.5 < U < 1, objects moving toward the center from an external frame-of-reference appear to be moving away from each other in their local frames-of-reference. He argues that our universe could be a collapsing neutron star. The following highlights that the modified theory also has this inversion. Consider the following:
An object is at a distance R from a central mass. That central mass could be the combined mass of a collapsing neutron star. As an object descends toward R=0, the object gets smaller in each of its x,y,z dimension according to S = S0*(1-U). How many of these objects can be fit in a circle around the center. The “normal” ratio, without any gravitational distortion, gives
As , you can’t fit any objects in a circle around the center. With gravity included, we get
So we get the asymptotic ratio
Comparing with we see a clear inversion. The inversion point appears to be where when is just a little larger than .
1) “Gravity, A Rigorous Derivation of the Gravitational Field” (anonymous), 1987
2) “Gravitation”, Misner Thorne and Wheeler, 1973
3) Lunar Laser Ranging Tests of the Equivalence Principle with the Earth and the Moon, James G. Williams, Slava G. Turyshev, Dale H. Boggs, 2004