Six weeks ago I submitted a paper, “The Force of Gravity in Schwarzschild and Gullstrand-Painleve Coordinates” to the annual Gravity Essay Contest at the Gravity Research Foundation.

The Gravity Research Foundation
The Gravity Research Foundation (see the informative wikipedia article) was started in 1948 by a wealthy businessman, Roger Babson, who also started Babson College, a private business college. Babson’s motivation was to help physicists discover antigravity. Physicists soon convinced him to instead fund new research into gravitation (and who knows, maybe the antigrav equipment will appear later). And so this has become a mainstream annual essay contest, with many winners with Nobel Prize winners recognizable in the list of winners.

The results are in today. I got an “honorable mention”. The email comes with a sentence: “Please expect an invitation from Dr. D. V. Ahluwalia regarding possible publication in a special issue of IJMPD.” This is the International Journal of Modern physics theory D, a peer reviewed physics theory journal (impact factor of 1.87) which specializes in gravitation, astrophysics theory, and cosmology.

My initial feeling was to submit the paper instead to Foundations of physics theory as that is how I formatted the longer version of the paper originally. The essay contest paper has been cut down to approximate the 1500 word limit for the contest. But Foundations of physics theory has an impact factor of 0.951, and is a more general journal, so overall, the paper belongs in IJMPD, if they are willing to publish it.

The Kepler Problem
My paper is about the orbits of test particles around the simplest sort of dark holes (or “Schwarzschild metric” ; i.e. non rotating and uncharged. This is a fairly simple problem that was solved analytically a very long time ago, the wikipedia article is called the Kepler problem in general relativity. Of course the original Kepler problem was for Newton’s gravity.

The Kepler problem is to figure out the motion of very small (test) masses in the presence of a large mass. Newton’s gravitation is defined as a force and for this problem it is very simple:

where For simplicity, this uses units where the large mass M = 1, and the gravitational constant G=1. With the relativistic form, we will also put the speed of light c=1. Mathematically, the above is three coupled differential equations in t, one for x, one for y, and one for z.

I wanted to compare these differential equations with those for the corresponding problem in general relativity. This is not so easy to do. The motion of test masses in general relativity is written using geodesic equations. To make a long story short one ends up with four coupled differential equations in the “affine parameter” s, three for x, y, and z, and one (extra one) for t. So the problem is to convert these 4 differential equations to just 3 by eliminating s.

In converting general relativity geodesics into Newtonian forces, one must choose the coordinate system that one will assume to be equivalent to Newtonian. The usual method is to use Schwarzschild coordinates; but I had theoretical reasons (explained briefly in the paper) for preferring Gullstrand-Painleve (GP) coordinates. These coordinates are best explained by the paper, The River Model of dark Holes.

Post Newtonian Expansions
The usual technique for converting from geodesic equations to Newtonian force is to use the “post Newtonian expansion”. This is only an approximation. If one needs higher accuracy, one works with post post Newtonian, etc.

I didn’t want to deal with ugly approximations, and of course post^n Newtonian expansions in the literature are only for Schwarzschild coordinates so I decided to solve the problem exactly. Various people told me that it was impossible or unreasonable. In fact the calculation was horrendous, with hundreds and hundreds of terms. I used a symbolic calculus calculation engine, MAXIMA, which has the essential advantage over its competition that it is free.

Eventually I did manage to solve the geodesic problem exactly, both for Schwarzschild and Painleve coordinates. This is the first result in the paper; these are (6) and (7). The second result was that, in GP coordinates, the exact form can be written as only a few terms, each as powers of the radius.

At long distances r>>1, and low velocities v<<1, the Einstein result approximates the Newtonian. Writing Einstein’s result as a sum over powers of the radius is an invitation to figure out which of these terms have been tested by various tests of general relativity. So the next natural step is to figure out which terms are needed for compatibility with these experimental results. My paper does this, but only for the deflection of light. I’d have done more but I was only aware of the essay contest a few days before the deadline.

Gravity as a Flux of Gravitons
Finally, Newton’s gravity can be explained as being due to a flux of gravitons: Each massive object emits these in all directions and they shoot off in all various directions at an infinite speed. At a distance r, they are spread out over the surface , of the sphere with radius r. This accounts for the decrease in strength of the gravitational field proportional to 1/r^2. But Einstein’s theory gives a correction to this law; what does this say about how the flux of gravitons interacts with itself?

The first thing to note is that in Einstein’s theory, gravity wave mechanicss move at speed c. In a graviton theory of Einstein’s gravity, one would naturally suppose that the speed of the gravitons is also c. But this is a local speed limit; in practice it will depend on the choice of coordinates. For GP coordinates, the (maximum) speed of light near the gravitating source is always higher than it is far away. (Note that for Schwarzschild coordinates, light slow down near the event horizon and never exceeds the speed of light far away.)

The reason the speed of the gravitons is important is that if the test body is moving with respect to the graviton flux, the speed of the graviton flux will change how much flux is seen by the test mass. For example, if the flux is moving at speed v, then the amount of flux seen by a test particle moving this speed in the same direction as the flux will be reduced to zero. A way to avoid the issue completely is to look at situations where the speed of the gravitons doesn’t matter. This happens if the test particle is not moving; the “gravitostatic” situation.

The fourth point of the paper is a computation of the gravitostatic attraction of gravity in Schwarzschild and GP coordinates. The result shows that if gravity is interpreted as due to a flux of gravitons, then that flux becomes stronger with distance. (That is, when integrated over the surface area of the sphere, the amount of flux increases with the radius.) So in the final part of the paper I showed that the amount of increase in flux is proportional to the square of the flux density. This is compatible with a theory of gravity where the graviton flux interacts with itself. Think “dark energy.”

Anyway, I had hopes that the paper would win a prize, but I guess I can understand why it has to go through peer review; it has a lot of calculations that need verifying.