# Can GR be derived by postulating a maximum force?

This paper was published in a peer review journal, and claims the answer is yes. http://arxiv.org/abs/physics/0607090

The derivation in the paper seems more like dimensional analysis hand-waving in the beginning. But then uses some specific results from differential geometry to uniquely get to Einstein's field equations. I can't tell if he is assuming something along the way during the hand-waviness that accidentally garauntee's the solution.

Also, the article states that this can be worded also as postulating a maximum total energy flow/flux through a surface. But in discussion from Is there an energy density limit in GR?, it sounds like components of the stress energy tensor can be arbitrarily large by changing coordinate system.

So can GR be derived by postulating a maximum force, or a maximum total energy flow through a surface?

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@John the inevitable replies you will get will say that Christoph ("Motion Mountain") Schiller is a crackpot etc. etc. It is highly unlikely that more than a handful of people in the physics community are familiar with Schiller's work or Jacobson's for that matter from which Schiller apparently derives his argument. Point being this is a controversial topic that is likely to lead to mudslinging and acrimony. –  user346 Mar 31 '11 at 8:18
Adding to the above, in these papers there is an introduction of $T_{ab}$ - as matter flow - in a vacuum situation. So something non-standard is going on there. Also in the book I have seen that the corresponding minimum length only works if SR is false at that length. –  Roy Simpson Mar 31 '11 at 12:43
@Deepak I think most people know of Jacobson's work, or at least I would consider his math uncontroversial. Although the tentative conclusions and hints some extrapolate from it, like Verlinde and his entropic gravity, are, umm, controversial. Jacobson's work itself is fine. Looking at the linked paper, all of the, umm, 'hand-waving' as John puts it, is in trying to get to the start of Jacobson's work. @John It would be very very difficult to answer this politely, so it almost comes off as a trolling question. I don't think you are going to get many(any?) responses. Maybe Lubos will do it. –  Edward Apr 2 '11 at 1:14
@Edward, yes of course, Jacobson's work is unimpeachable. But there are lots of papers whose calculations are correct but whose conclusions are disregarded by the broader community. Jacobson's 1995 paper and followup work has, unfortunately, not received the attention it deserves. This is because if you really accept the validity of his reasoning then you have no option left but to regard the Einstein equations as an equation of state. Most physicists aren't quite ready to accept the resulting implication that one cannot "quantize" Einstein's equations anymore than you can quantize PV=nRT. –  user346 Apr 2 '11 at 3:19
@John This is a neat idea. I tried to read the paper, but Schiller is simultaneously too vague and contradicts himself, making it practically impossible to figure out what is going on. However there is a mainstream author who notes that GR seems to predict a maximum tension arxiv.org/abs/hep-th/0210109 and he describes what he means by that much more precisely. So there may be a way to reformulate Schiller's idea correctly. As it stands though, I agree with Carl: "it's junk". –  Ginsberg Apr 5 '11 at 22:47

The maximum force / power are obtained in the paper as: $$\textrm{Force} \le \frac{c^4}{4G}\;\;\;\;\;\;\;\;\;\textrm{Power} \le \frac{c^5}{4G}$$ These are just 1/4 the Planck force or Planck mass one would obtain from the Planck length, mass, and time.

First noticeable weirdness is that he keeps talking about momentum and energy in high gravity situations when these concepts are not very well defined in the context of GR. I guess that's okay in that he's deriving GR rather than making a statement about GR. Later he explains that force and power are highly observer dependent.

Page 3 has another weirdness, could be some sort of typo or oversight by the author:

As a result, a maximum force (respectively power) value in nature implies the following statement: one imagines a physical surface and completely covers it with observers; then the integral of all momentum (respectively energy) values flowing through that surface per time, measured by all those observers, never exceeds the maximum value.

The above is weird in that the maximum force should depend on the size of the surface. That is, a surface of twice the size should allow twice the force. Maybe this is incorrect intuition on my side, but it could at least use a better description.

On the same page, he notes that force and power are parts of a 4-vector in relativity. Kind of cool. I never thought about that. Might be nice to see a reference.

Reading further, he makes it clear that the "maximum forces" he's talking about are on the event horizons of black holes. That seems reasonable. In the Schwarzschild coordinates objects freeze on the event horizon but in the coordinates I prefer, Gullstrand-Painleve (GP), objects fall through and one can compute an acceleration at the event horizon. The amusing thing is that I wrote the definitive paper on this subject:

Int.J.Mod.Phys.D18:2289-2294 (2009) Carl Brannen, The force of gravity in Schwarzschild and Gullstrand-Painlevé coordinates
http://arxiv.org/abs/0907.0660

If you look at equation (11) of the above, the acceleration is zero at the event horizon for stationary observers in Schwarzschild or Gullstrand-Painleve coordinates. For the Schwarzschild case this is intuitively obvious. That's part of why particles get stuck on the event horizon. I think his force doesn't make sense.

Ah hah! He has some counterarguments to critiques of his theory. The first one is the "mountatin example". In it, he writes: Nuclei are quantum particles and have an indeterminacy in their position; this indeterminacy is essentially the nucleus–nucleus distance. This is clearly untrue; the electron wave functions define the nucleus-nucleus distance and this is far greater than the size of the nucleus wave function. One's measured in barns, the other's measured in Angstroms.

The counter-argument for "multiple neutron stars" fails to take into account how special relativity deals with simultaneous events. He's claiming that (in a given rest frame) it's impossible to have simultaneous events that are widely separated. This is a confusion of spacelike separation and simultaneity. With his use it eliminates any set of observers to be simultaneous but it seems that was an intrinsic assumption of his argument.

So my verdict is that it's junk, but better junk than most in that I didn't see any violations of units.

He uses arguments that all rely on the Planck units to argue that the Planck units are fundamental. For example, the diameter of a black hole has a close relationship between the mass of the black hole and the gravitational constant. So when you consider the time light takes to cross this distance, of course you're going to get more Planck constant related things. There's plenty of opportunities to get the right constants as there's an infinite number of ways of choosing arbitrary lengths and distances from the geometry of a black hole. No surprises.

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Huh. I never realized till I sat down just now and derived it that the Planck force and Power don't depend on $\hbar$. Weird. –  Jerry Schirmer Apr 7 '11 at 3:50