Can Gravity Deviate from 1/r² at Short Distances?

[Doctor Luz]

The physics books show as that gravity deppends on distance at 1/r^2. But how we know that that is true at very short distances? Is there any proof? Could gravity deppend on 1/r^n whit n>2 at very short distances? What implications could have?

 

[HallsofIvy]

There have, in fact, been many experiments on this. All laboratory experiments (using relatively small masses) show F= (GMm)/r^2. On the other hand, the precession of Mercury's orbit indicates that that is not exactly true: Newton's law has to be replaced by the theory of General Relativity which does not give a formula anything like that but, rather, asserts that a certain tensor must be 0, thus setting the geometry of the space near a massive body.

 

[Javier]

As stated above, the 1/r form of the potential has been tested for weak fields to mm scales or so. On the large scale, dealing with stronger fields, this 1/r form is correct to a great degree, for deviations too large would lead to strange orbits of the planets that we *don't* observe.

However, as stated above, the exact theory of gravitation has some small corrections for things like planets and GPS satellites orbiting Earth, which has been verified by observations.

So for mm to astronomical distances, Newton's law of gravitation works well for weak fields. For "mediumly strong" fields, the corrections due to general relativity can't be ignored, like in the example of the precession of Mercury mentioned above. So GR works well for mm to astronomical distances for weak to mediumly strong spacetime curvatures (if you want, "field strengths").

But for very strong curvatures of spacetime, GR does not make reasonable predictions, the most extreme case is when a singularity is found in the theory. Of course, observation has not yet told us what the more precise theory would be. Furthermore, there is nothing to tell us that at smaller scales than we've tested, GR is not modified. In fact, in both of the major theories with quantum gravity (string theory and non-perturbative quantum gravity [i.e. "quantum geometry"]) GR is modified at smaller scales than we have tested to date.

 

[Creator]

Originally posted by Doctor Luz
The physics books show as that gravity deppends on distance at 1/r^2. But how we know that that is true at very short distances? Is there any proof? Could gravity deppend on 1/r^n whit n>2 at very short distances? What implications could have?

Strictly speaking, Doc, and put simply, in classical physics 1/r^2 dependence was derived for spherically symmetric matter only; (deviations from spherical symmetry requires corrective factors). However this 1/r^2 gravity dependence applies only for points outside the the sphere.

For points inside a sphere (of uniform density), gravity will vary as 1/r.

Creator

 

[elas]

Strictly speaking, Doc, and put simply, in classical physics 1/r^2 dependence was derived for spherically symmetric matter only; (deviations from spherical symmetry requires corrective factors). However this 1/r^2 gravity dependence applies only for points outside the the sphere.

This is correct up to a point, but in the case of particle, atomic and surprisingly, galactic fields; the 1/r rule applies. It is not simply a question of the presence or absence of mass but the relationship between force and the elasticity of the force carrier. I am aware that my statement does not agree with current thinking but if you examine the Galactic Gravity Problem you will observe that my statement solves the problem.
It all depends on how close (relatively speaking) the field nuclei are to each other. Particle and galactic fields are relatively close stretching the fields to their maximum hence 1/r. Planetary and stellar fields are relativly far apart and their influence on the fields between bodies is relatively weak hence 1/r^2.