Ok to keep the inverse square law

[Vector_Graphics]

Is it okay to keep the inverse-square law from 3d when making a 4d universe? Would this have any unintended side effects (as well as the main effect of keep orbits stable)?

[PatrickPowers]

I don't know. I've long wondered whether having an inverse-square law would allow you to have a perpetual motion machine or some other non-obvious problem.

Here in 3D an inverse square law means any complete surface around the source of gravity has the same potential energy. In 4D that's no longer true. The potential energy around the surface increases dramatically the further away you go. That seems like you are getting something for nothing, but any specific problem is beyond me. It makes gravitons impossible but no one has proved they exist.

[steelpillow]

Assuming a basic Newtonian model, a 4D universe would have an inverse-cube law.

Not even an inverse square law could keep it stable; the 4th dimension means that orbital energy can shuffle between pairs of dimensions (planes), so that no stable orbital plane can exist. So you'd need to postulate some other, unknown law of 4D physics to keep things stable.

[Vector_Graphics]

I don't know. I've long wondered whether having an inverse-square law would allow you to have a perpetual motion machine or some other non-obvious problem.

Here in 3D an inverse square law means any complete surface around the source of gravity has the same potential energy. In 4D that's no longer true. The potential energy around the surface increases dramatically the further away you go. That seems like you are getting something for nothing, but any specific problem is beyond me. It makes gravitons impossible but no one has proved they exist.

Well, let's see if an inverse linear law in 3d would cause problems.

[Vector_Graphics]

Assuming a basic Newtonian model, a 4D universe would have an inverse-cube law.

Not even an inverse square law could keep it stable; the 4th dimension means that orbital energy can shuffle between pairs of dimensions (planes), so that no stable orbital plane can exist. So you'd need to postulate some other, unknown law of 4D physics to keep things stable.

...no? At least with an inverse square law, any two-body problem happens on a 2d plane relative to one of the objects.

[quickfur]

Yeah my understanding was that an inverse square law would work in any dimension as far as stable orbits are concerned, because as far as a 2-body problem is concerned, the results would be independent of the dimension of the ambient space; the two orbiting bodies only "know" the plane that they are orbiting each other in. However, there may be other unintended consequences.

[Vector_Graphics]

Yeah my understanding was that an inverse square law would work in any dimension as far as stable orbits are concerned, because as far as a 2-body problem is concerned, the results would be independent of the dimension of the ambient space; the two orbiting bodies only "know" the plane that they are orbiting each other in. However, there may be other unintended consequences.

What do you even think steelpillow means by "orbital energy"/

[quickfur]

Not sure, but usually when we speak of energy wrt orbits it refers to the total energy in the orbital system, i.e., the kinetic energy (the orbital motion itself) + potential energy (the gravitational potential). It may also include the overall angular momentum of the system and its conservation.