Surface gravity in N dimensions and projectile trajectory

[quickfur]

Today I just realized something fascinating: regardless of dimension, if we consider a free-falling object such as a projectile close to the surface of an n-dimensional planet, its trajectory ought to be approximately parabolic. This is because near the surface of the planet, the height differences in the trajectory are negligible compared to the radius of the planet, so the force of gravity will be approximately constant. Consequently, the projectile will experience (approximately) constant acceleration downwards by gravity, and the resulting trajectory will be more-or-less a parabola.

Only when the height difference of the trajectory becomes significant compared to the radius of the planet, will the inverse power law become significant, i.e., 1/r^2 in 3D, 1/r^3 in 4D, and so on (under the flux law assumption). Very interestingly, 3D is the only dimension where the inverse power law of gravity has the same polynomial degree (quadratic) as the curve traced out by acceleration under a constant force. In 4D, gravity, assuming the flux law of gravity, the force of gravity will be 1/r^3, and so the larger the difference in height in a trajectory, the more the trajectory will deviate from a parabola and take on the properties of a cubic curve. But when the height difference is negligible, the trajectory will be for all practical purposes the familiar old parabola.

This leads to the very interesting theory that if, as Einstein postulated, gravity is caused by the curvature of space rather than a physical force with physical force carriers (i.e., gravity is acceleration), then we could indeed have a model of 4D space in which planetary acceleration follows an inverse square law rather than the inverse cube law predicted by the flux model(!!). I.e., the effective acceleration of a body under gravity would be determined by the property of the space -- how its curvature corresponds to the mass that occupies it -- rather than its dimensionality, which corresponds with the density of hypothetical force-carrying particles in N dimensions.

IOW, there may be a 4D analog of General Relativity in which planetary orbits are stable!!

[quickfur]

Hmm, further research turned up this paper: https://arxiv.org/abs/gr-qc/0609060 that seems to show that Gauss's Law still holds in general relativity generalized to d dimensions (i.e., (d-1) spatial dimensions + 1 temporal): the strength of gravity / spacetime curvature resulting from a massive body is proportional to 1/r^(d-2).

So the conclusion is that if we assume GR holds in a (4+1)D space-time (4D space + 1D time), then gravity will be an inverse cube law.

Which in turn means that if we want our 4D space to have an inverse square law, then GR does not hold, but a different kind of space-time geometry must hold.