by quickfur » Fri May 03, 2024 11:54 pm
It's well known that if gravity obeys an inverse cube law, as it would in 4D if we assume gravity is the result of the exchange of (possibly virtual) gravitons, then orbits are inherently unstable. The consequences have been worked out here on this forum before; there are 5 cases:
1) The planet's path diverges, i.e., there is no orbit, it flies off into space.
2) The planet's path spirals outwards, each iteration being a constant distance farther than the previous. Eventually it will also fly off into space, but in a more controlled way.
3) The planet's path is a perfect circle. This is the only case where there is a stable orbit, but it's a local maximum, meaning that the slightest perturbation will destroy its stability and send it into the other 4 unstable cases. I.e., a speck of dust landing on the planet from space will knock it off its perfect circular orbit into one of the other cases. Not to mention the extreme unlikelihood that a perfectly circular orbit would arise spontaneously in a hypothetical star formation scenario.
4) The planet's path spirals inwards, each iteration being a constant distance closer than the previous. Eventually, it will collide with the star it orbits.
5) The planet's path converges to the star, i.e., there is no orbit, it just crashes into the star.
The cause of the lack of stable orbits (besides the impractical perfect circular case) is that the 1/r^3 gravity well can never be perfectly balanced by the mv^2 component of the orbiting body's momentum. There is always a leftover term that will cause the orbit to be unstable. As long as gravity obeys a 1/r^3 law, stable orbits are inherently impossible.
The only way around this is to somehow force gravity to obey a 1/r^2 law instead. It would likely require an unnatural by-fiat imposition of some arbitrary law that violates the flux law or otherwise postulates a completely different mechanism for gravity. Assuming this is done (and this is a huge, huge, huge assumption), then we could have stable orbits in 4D in the analogous way to 3D orbital systems (e.g., orbital paths are conic sections, the stable ones among which would be the circular and elliptical cases).
One possibility that I've come up with in seeking a solution to this conundrum is Einstein's idea behind general relativity: in our 3D universe, it seems awfully convenient that acceleration follows a square law, and gravity also follows a square law. (And furthermore, inertial mass equals gravitational mass, even though there's no a priori reason for such an equivalence.) Einstein thus made the leap of considering what if acceleration is gravity, and gravity is acceleration. Thus, he arrived at general relativity, where gravity isn't a conventional force per se, but is caused by a curvature of space, and the path of an object under gravitational acceleration is actually its inertial path in curved spacetime. If we take this idea in its most radical form, we could postulate that if in our hypothetical 4D universe a similar thing holds, where 4D gravity is the consequence of a quadratically-varying curvature of space, then we could imagine a scenario where gravity actually obeys an inverse square law rather than the inverse cube law implied by the flux theory of force. I.e., our 4D gravity would be exactly the same as motion in an inertial frame, and it's the curvature of the space itself that causes the 1/r^2 shape of the gravity well. This would give us stable 4D orbits, and possibly also give rise to a bunch of unusual consequences that could be rather interesting to explore.