One of the first great discoveries of modern physics was Newton writing down the equation for the force of gravity between two massive objects. The discovery was monumental not because it was complicated, but because it was profound. You can state the law in plain English very briefly. "The gravitational force between two objects is equal to their masses multiplied together, divided by the square of the distance between them, and multiplied by a particular universal constant to get the overall scale right."
This is the way things are, classically. But as far as we know there's nothing forcing this to be true. Why not have a universe with the same gravitational law modified to read "divided by the fourth power of the distance between them"? It would be pretty weird and a lot of nice theoretical properties would vanish into the ether. Gauss' law for gravitation would be totally borked, for one. But there's nothing logically prohibiting it.
It might not lead to a very interesting universe. A universe without gravity, for instance, would seem to be one without stars and planets. No stars and planets means no people and very possibly no intelligent life at all, and so sometimes the so-called Anthropic Principle is invoked to explain why the laws of physics are so rich. If they weren't (the argument goes), we wouldn't be here to see them. I'm rather unimpressed by this, but it's an interesting idea. But what about our universe with modified gravity obeying a fourth-power force law?
Are orbits even possible? Check. It's still an attractive central potential, and so it's possible to pick a distance and velocity such that the force is exactly equal to that which is necessary to keep things orbiting in a circle. Which is all well and good, but is the orbit stable? Maybe any arbitrary perturbation will send the planet spiraling out into the distance, or crashing into the sun? This is a trickier question. A bit of a time crunch prevents me from writing out all the math (though we'll do it shortly), but in fact the answer is "no, it isn't". A circular orbit under a 1/r^4 force law will not stay circular. It isn't stable.
Which means the earth couldn't orbit the sun, which means we wouldn't be here to think about it. Which means we don't live in a universe with a 1/r^4 gravitational force law. As in fact we don't.
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More interesting to me is, how far do you have to go from 1/r^2 before orbits become unstable? Is a gravitation with 1/r^2.001 OK, or does it have to be exactly 2? How about if it averages 1/r^2, but varies on timescales of a day or two?
I suspect with variations in exponent you still get stars, but planets are trickier. One planet might be easier to keep orbiting than a stable of them, but maybe extra planets in harmonic orbits could compensate for small instabilities?
It's not clear if galaxies are especially bothered about gravitational details. Regardless, maybe in the absence of planets and galaxies, life forms as patternings in the primordial interstellar plasma soup. In our universe planets have gummed up formation of the most glorious life imaginable.
Or maybe they haven't.
The inner planets of our solar system are only stable because 43 arcseconds per century of Mercuryâs total precession is due to the effect of general relativity.
Without that, Mercury and Jupiter would fall into secular resonance with disastrous results. The Earth would be toast.
http://www.nature.com/nature/journal/v459/n7248/pdf/459781a.pdf
I question whether planets or stars could form under such weak gravity.
Stars would need to be much more massive to support fusion reactions, assuming a gas cloud could collapse against its internal pressure at all.
It would be an interesting exercise to someone with the time/resources to do the physics & math.
General relativity makes it possible to have unstable circular orbits even with 1/r^2.
Two orbiting black holes can't stay in a circular orbit. They must either merge or separate.
Is inverse square law that much unexpected?
It comes naturally from divergence theorem that the total fluxes coming out of any closed surface is constant. It is evident in Coulomb's law, to apply to gravity one just need to admit the existence of graviton.
The surface area of a sphere is proportional to r^2, not r raised to any other power, no matter how close it is to interger 2.
fx: You seem to be assuming Euclidean space.
Thinking of Bertrand's theorem? I remember finding a really pretty derivation in AmJPhys, that you might enjoy.
Nathan: It turns out that we get stable orbits for forces that go as 1/r^n only for n strictly less than 3. Above that the orbits go spiraling away.
hm, that's interesting. i wonder what consequences that has for theories that include extra dimensions on the very small scale? or do we not deal much with 1/r^n forces once we get down to the subatomic?
For a universe with a 1/r^4 gravitational force law, I would also question the stability of celestial bodies themselves.
The gravitational pull toward a proton at a distance of the Bohr's radius would be over 16 orders of magnitude greater than the pull toward an object of Earth's mass concentrated in a tiny ball at a distance of Earth's radius.
Imagining gravity in such a universe is difficult. In our universe, computing gravity toward a perfectly spherical object whose density is a function of distance from the center is straightforward. We apply the concept of Gaussian surfaces and we see that gravity obeys the inverse square law for the whole object. A 1/r^4 gravitational force law would not be duplicated the same way.
An interesting novel that has this fact as a minor plot point is Diaspora by Greg Egan. He even has a web page detailing some of the mathematics, with plots: http://gregegan.customer.netspace.net.au/DIASPORA/15/15.html#WELLS
I wonder whether a 1/r4 gravitational force would allow stable orbits in a universe with five spatial dimensions. (Or whether stars would form at all in five-dimensional space, because gravity would fall off so quickly.)
"This is the way things are, classically. But as far as we know there's nothing forcing this to be true."
I don't remember if it's in Feynman or Goldstein, but there definitely is a result that requires things to come out as inverse square for gravity (and for Coulomb's law as well). Wish I still had copies of the books. Seemed to result from a conservation law analysis, as I recall. Dammit, been out of the field too long.
Heh, please tell me that was intentional.
What specifically do you dislike about the anthropic principle?