The Force of Gravity

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 D, a peer reviewed physics journal (impact factor of 1.87) which specializes in gravitation, astrophysics, and cosmology.

My initial feeling was to submit the paper instead to Foundations of Physics 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 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 black 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 r^2 = x^2 + y^2 + z^2 . 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 Black 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 4\pi r^2 , 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 waves 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.



Filed under physics

28 responses to “The Force of Gravity

  1. Kea

    CONGRATULATIONS! Excellent news. Of course you weren’t going to win a prize. But maybe next year you will …

  2. PhilG

    Well done! That’s an impressive calculation and easily worth the honorable mention at least.

  3. carlbrannen

    Phil, Kea;

    Thanks for the congratulations.

    Maybe I would have done better if I hadn’t called general relativity’s wormhole theory “science fiction”. If so, it was an expensive insult.

  4. Nige

    Congratulations! This sounds very interesting, especially the gravitons flux! Good luck with the publication.

  5. Kea

    Carl, why have they still not announced the results?

  6. Nige

    ‘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.”’

    Ummm. Are you saying that you take a large sphere of space with radius r, containing the usual matter density (due to galaxies, etc.); the surface area of that sphere increases with r^2 but the volume and hence total mass in the sphere increases as r^3. Thus, the total mass per unit surface area of the sphere is directly proportional to the ratio (r^3)/(r^2) = r. If that’s the physics of your calculation, then it’s a nice simple argument, and one which I missed.

    You don’t have to worry about gravitons interacting with themselves in low energy physics, because the coupling constant for gravity is so low, the field is normally weak and doesn’t contain significant energy to produce a lot of gravitons compared to masses. So at low energy (well below Planck scale), the main source for the emission of gravitons is mass, not gravity fields.

  7. Sorry, I’ve just found that the title to the paper is hyperlinked to a PDF file. I’ll read it carefully!

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  9. carlbrannen

    Pandora; maybe we should probably wait until the empire strikes back.

    Kea; the results are up now. As usual, all the winners are professionals.

    Nigel; your point is exactly on target, but as you’ve now discovered, isn’t in the paper. It was limited to 1500 words, which I may have exceeded slightly.

    The calculation suggests that the background of gravitons steadily increases with cosmological time and that consequently, the speed of light should decrease, hence Louise Riofrio’s cosmology on a flat space background.

    For more on the topic of the cosmology, see this blog’s post Louise Riofrio Crushes CMB Anomalies from last year.

    This gravity theory connects up with the other forces discussed around here. The idea is that gravitons are emitted when a left handed particle becomes right handed and vice versa. And that these emitted gravitons interact with other matter by similarly influencing them to switch from left to right handed. It’s like the influence of photons on the emission of more photons in a laser.

  10. Kea

    Carl, I still can’t see the results. Where are they?

  11. carlbrannen

    Kea, they’ve listed my name on their latest “announcements” page.

    Also, lots of people who get honorable mentions publish on arXiv. I will look around for someone who is suitable to endorse. If you happen to know any endorsers in gr-qc (general relativity and quantum cosmology), do tell. I need to modify the LaTeX to include the honorable mention.

  12. Kea

    Carl, I know lots of endorsers. Whether they will endorse YOU, however, is quite another question.

  13. Kea

    Carl, I can still only see 2008, and not your name….

  14. carlbrannen

    You probably need to press “refresh”. If not, I’ll post a screen shot, LOL.

  15. carlbrannen

    Also, they get to endorse the essay, not the person. But they do have to be endorsers for gr-qc.

  16. Kea

    Carl – David Wiltshire, Matt Visser and probably quite a few other Kiwis are endorsers for gr-qc. Email them.

  17. Kea

    Ah! Refresh. Must remember that one!

  18. ervin


    Thanks for your interesting post.

    As you may be aware of, recent tests have confirmed the validity of the gravitational inverse square law all the way down to distance scales as small as 56 microns (see for instance the Case Western Workshop at or one of the current entries of Podolsky’s blog). Unless I am missing something, these results cast doubts on the correctness of your gravitostatic acceleration formula (11) in the limit of sufficiently small distances.

    Another concern I have on your paper is that relies heavily on the concept of graviton. To the best of my knowledge, there is no compelling evidence that gravitons are real, let alone the fact that superluminal quanta are believed to be nonphysical objects.



  19. carlbrannen

    Great comment, Ervin!

    What you are missing is how to use units to make calculations in general relativity. Unit conversions are one of the things that you absolutely have to know backwards and forwards to survive physics grad school. Let’s do an example calculation with an attractive mass M = 100 grams and a distance of 56 microns.

    In general relativity, it is traditional to express all measurements in terms of centimeters. To do this, you put G=c=1. A distance of 56 microns is easy to convert, it’s equal to 5.6×10^-3 cm. This is the “r” in the formulas.

    To convert 100 grams into centimeters, first multiply by the gravitational constant G = 6.674×10^-8 cm^3/gram/sec^2 to get 6.674×10^-6 cm^3/sec^2. Now divide by the speed of light c squared = (2.998×10^+10)^2. We find that 100 grams is equal to 7.425×10^-27 centimeters.

    But rather than using centimeters, my paper is written with M=1. Consequently, we need to convert our centimeters to “M=1” units, in which the distance 7.425×10^-27 cm becomes one “M=1″ unit. That means our distance r= 5.6×10^-3 cm becomes
    r= 5.6×10^-3 cm / 7.425×10^-27 cm/”M=1”
    = 7.5×10^23 “M=1”.

    Therefore for the gravitation problem with mass 100 grams and distance 56 microns, the 2/r^3 term is around 4×10^23 times smaller than the 1/r^2 term. Consequently, deviations from the 1/r^2 law for this sort of experiment are very small.

    And yes, gravitons are a speculative part of the paper. However, you should remember that this is an essay limited to 1500 words. Eventually I’ll put out a longer paper that will further this issue. What you’re seeing is a very small portion.

  20. Ervin Goldfain


    Thanks for the explanation, it now makes sense. My suggestion is to include dimensional considerations and the use of natural units in the final version of the paper. It would improve its readability for a larger audience, in my humble opinion.

    Are you aware of this paper:

    “Implications of Graviton-Graviton Interaction to Dark Matter” by A. Deur (arxiv:0901.4005 v2, astro-ph.CO) ?.

    It reaches similar conclusions on dark matter (and possibly dark energy) using the intrinsically nonlinear nature of gravity.

    It would be great if it could be shown that dark matter and dark energy follow directly from GR, without any corrections to Einstein’s equations and without the need to quantize gravity first. But I suspect that a serious attempt along these lines is far from being a trivial exercise.

    Regards and good luck,


  21. carlbrannen


    I like your idea about adding natural units. It’s universal in higher physics to ignore units and then add them back in to compare with experiment as it makes the calculations easier. I’ll do it for later papers, but when I look at the other papers at IJMPD I think the audience is pretty sophisticated.

    Originally, the paper assumed M=1 from beginning to end, it was only when I looked around that I realized that the Schwarzschild metric usually includes M so I added it back in, not so elegantly. I only learned of the contest a few days before the deadline, my paper is a hack job.

    I’m working on a much longer paper targeted to Foundations of Physics that will include a derivation of dark energy. In short, the building graviton background causes light to slow down which we interpret as the big bang. As the background builds up, it interacts with itself (the graviton-graviton interaction) and this causes the graviton background to build up at a steadily increasing rate. Standard cosmology interprets the accelerating expansion as due to dark energy.

    The link you’ve included is interesting. My browser says I looked at it before, but I didn’t take a close enough look at it. It could be interesting for dark matter calcs. They’re ignoring gravitons in turns of curvature, but looking at them in terms of nonlinear effects. I’m attributing the (apparent) curvature to graviton interactions. As far as dark matter goes, these might be compatible.

    About 80% of my physics education is in elementary particles and quantum mechanics. I only took one class in cosmology / GR. So I come at these things from a particle point of view. To get an idea of what is going on behind the graviton assumptions, read Bit from Trit and Lubos’ Booboo, then Lorentz Violation and Feynman’s Checkerboard Model.

    Right now I’m working on an explanation for the gravity anomaly Gravity Anomaly During the Mohe Total Solar Eclipse and New Constraint on Gravitational Shielding Parameter, Astrophysics and Space Science, 282, 245-253. To give a hint on my explanation, think about absorption, stimulated emission, and interference effects, in the context of the links on Bit from Trit and the Checkerboard model. I should get the calculation out this weekend.

  22. Ervin Goldfain


    With regard to graviton physics, I think that a legitimate solution for dark energy and/or dark matter that is derived from standard or expanded GR has to first pass the stability test. Since GR is nonlinear, generic perturbations caused by various flavors (and orders) of matter-gravity interaction may simply collapse that solution entirely or render it chaotic, that is, highly sensitive to initial conditions. The stability argument was used to highlight the inherent weakness of Moffat’s non-symmetric gravity and its interpretation of the dark sector, see

  23. Daniel de França MTd2

    Hi Carl,

    shouldn’t graviton self interaction generate dark matter instead of dark energy?

  24. carlbrannen

    Daniel, the two effects appear in two completely different scales, so it could be both.

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  26. Eirik

    I find this interesting. Are you working on an solver for 3 or more particles now? If you have a different speed of gravity than c, then your solutions for 3 particles would differ from the GR one, right?

    Maybe they also differ for rotating bodies?

  27. carlbrannen


    I used to think that the speed of gravity was likely to be sqrt(3) c, now I think that it is very large (i.e. that the speed of light has been slowed down by the cosmological graviton background). But I’m not at all sure which is the case.

    If the graviton speed is very high, then you get Newtonian gravity with very small adjustments due to graviton-graviton interactions. In that case, the 3-body calculation would be essentially identical to the Newtonian result.

    There are adjustments due to graviton-graviton interactions in 3-body stuff. They should show up when the 3 bodies are aligned. That’s why the next post after this one (on this blog) is about the Allais effect.

    I should probably add that in addition to the calculations given in the gravity paper, I also know that the perihelion of Mercury depends only on the first few terms of the Gullstrand-Painleve coordinates (i.e. the terms that depend on integer powers of the radius as opposed to square root powers). And it seems that calculations of the time delay likewise depend only on the integer powers. So the adjustment of this theory to GP coordinates is mostly to trim off the square root terms. Eventually I’ll write this up.

    Regarding rotating bodies, I’m torn. On the one hand, I’d like to do the exact equations of motion for a tests particle in the equatorial plane of a maximally rotating black hole (Kerr metric). On the other hand, while almost all black holes are thought to be close to maximally rotating, there are no solar system tests that are yet sensitive to the difference between Kerr metric and Schwarzschild.

    On the other side, to make the calculation from (my) theory, I have to understand better how Lorentz invariance appears from graviton exchange. I’ve done some general calculations on this and it is plausible, but I need to do more.

    A related subject is the statistics that are implied by the existence of a preferred reference frame (as would apply to gravitons that move in such a frame). One eventually concludes that there should only be one graviton momentum and energy (i.e. you can’t use Lorentz invariance like you can to conclude that two arbitrarily moving electrons are identical particles). I’m thinking of writing up those calculations, which reuse quantum field theory resummation in a particularly disgusting / beautiful fashion, for Foundations of Physics. Maybe that would give a better clue on how to do the spinning body case.

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