^{d-1}, which in 4d and higher would not produce stable orbits as in order for a planet to stay in orbit its star would need to be the only gravitational influence and it would need to move exactly in a circle, which tends to never happen in nature. For the hydrogen atom the electric potential would be 1/r

^{d-2}, which either doesn't have a ground state or doesn't have discrete energy levels.

As I understand it in Higher Dimensions in the case of massive photons ∇

^{2}V

_{E}-m

_{γ}

^{2}V

_{E}=0 https://physics.stackexchange.com/quest ... ic-charges in the case of a universe with a (++++...+-) metric, but in the case of a (++++...++) metric the equation would instead be ∇

^{2}V

_{E}+m

_{γ}

^{2}V

_{E}=0 with m

_{γ}being the rest mass of the photon. I have a blindspot in terms of knowing exactly how to derive these equations, however I have noticed that in 3d ∇

^{2}V

_{E}-m

_{γ}

^{2}V

_{E}=0 implies the Yukawa Potential https://en.wikipedia.org/wiki/Yukawa_potential and similarly ∇

^{2}V

_{E}+m

_{γ}

^{2}V

_{E}=0 implies the potential mentioned here https://www.gregegan.net/ORTHOGONAL/04/EM.html and massive photons plus it has the property that even in the case of massive photons the force from a line charge in d dimensions falls off at the same rate as the force from a point charge in d-1 dimensions, also imagining the photons as having imaginary mass

-(im

_{γ})

^{2}V

_{E}=m

_{γ}

^{2}V

_{E}while e

^{-ir}does not equal cos(r) or any other function with only real numbers.

Knowing that in a universe with a Euclidean metric kinetic energy has the opposite sign of mass energy and so it makes sense that if a system of two electric charges was more massive than if the objects had no electric charge then the electric charges would experience an attractive force as they would increase speed as they get closer to each other in order to balance out the increase in mass from potential energy. This should also mean that something similar to an electric force should also be able to hold planets together and hold planets in orbit in place of gravity, as electric forces being repulsive for like charges is one of the barriers for them holding planets or solar systems together in our universe.

In 5d Euclidean spacetime, assuming that the laplacian is assumed to only be applied to 4 of the five dimensions, as in not in the direction of spacetime velocity Y

_{1}(m

_{γ}r)/r is a solution to ∇

^{2}V

_{E}+m

_{γ}

^{2}V

_{E}=0 for the 4 dimensions it's applied to. In the interval of approximately 2 to 16.4 Y

_{1}(m

_{γ}r)/r is a curve that looks like this

with valeys and hills, which has valeys and hills with the energy change in the valeys and peaks being less steep than it would even be with r

^{-1}meaning that stable orbits should be possible in the valeys in the case of like charges and hills in the case of opposite charges.

The derivative of Y

_{1}(m

_{γ}r)/r is Y

_{2}(m

_{γ}r)/r and simulating two bodies with the force law Y

_{2}(m

_{γ}r)/r, with the initial velocities intentionally deviated so that they don't match sqrt(|a

_{c}r|) I get this for the smaller mass as it's motion

with it tracing out a star shaped orbit. Technically I could have made the orbit closer to a circular orbit, however I think having the distances repeat even after changing a lot can help illustrate how easy stable orbits are with this potential.

I found that in 4d there are two different versions of hyper-spherical coordinates I could use for the time independent Schrodinger Equation, so I chose both. In the one defined by x=rcostcosu y=rcostsinu z=rsintcosv w=rsintsinv there are basically two quantum number ms, and the sum of both ms must be odd if l is odd and even if l is even and the absolute values of both must sum to a number less than l. Using the shooting method to locate energy levels I find that it seems to be impossible to change l without changing the energy level. The energy levels I found between the first case of l=0 and the second case of l=0 are

and emiting a photon would cause an atom to get into a more excited state, rather than a less excited state like in our universe, and so it looks like it should be possible for an electron to emit a photon to get to a scattered state. I'm thinking one solution to this is for the energy of a photon at rest to be less than the largest difference between energy levels so that at some point the kinetic energy that would need to be added to get to a more excited state would be less than what could be gained from emitting a photon.

Simulating where an electron would likely be found using different configurations I get

with the electron being unlikely to be found where the electric interaction is repulsive, and more likely to be found in the first potential energy valley. When r is negligible compared to m

_{γ}then Y

_{1}(m

_{γ}r)/r reduces to being ar

^{-2}and so the inverse of the mass of the photon needs to be similar to the scale of an atom for massive photons to allow for stable atoms in higher dimensional Euclidean Spacetime. For massive photons to help keep planets in stable non circular orbits the inverse of the mass of the photon would need to be similar to the scale of a planet orbit. As a solution to this problem I would propose having a different electric force holding atoms together from the one keeping planets in orbit. Also I found that the energy levels using the other spherical coordinate system are the same as the first.