Hi Ray,
I've made some progress on the hyper-drive. If it's possible, then some assumptions are necessary. First, hyperspace and standard space overlap. In other words, anywhere in the universe, I can cross. Second assumption, gravity works the same in both spaces, so G is the same; this makes the math easier. Third, momentum is conserved in both standard space and hyperspace. Fourth, energy is exchanged 1:1 between these two spaces. Fifth, hyperspace has a c'>>c; Planck constant for hyperspace is unknown. Sixth, hyperspace also has a 4D space-time; I don't know if anything scales.
I have a starship that can cross into hyperpsace. When it crosses, it either uses or else creates a spatial anomally, a space-time bubble. Inside of the bubble, standard space -time exists. At the boundary, all the interesting stuff happens. The standard space inside is not considered to be curved or compressed in spite of the bubble. Artificial gravity will come later. Conditions for creating such a bubble are still vague.
Since conservation of energy is upheld, I want to solve the Einstein equation for the stress-energy tensor. To keep it simple, I'm neglecting complicated curvatures in space, so Rmu,nu=0. I'm neglecting the Cosmological constant, for now, so NU=0. That leaves me with (-1/2)g_mu,nu(space-metric)R = (8piG/c^4)T_mu,nu(stress energy tensor). Solving for the stress energy tensor, I get,
T_mu,nu = -c^g_mu,nuR/2*8piG.
If energy exchanges 1:1 at the boundary between standard space and regular space, then the energy stress tensor can be the same at the boundary. So,
T_mu,nu = T'_mu,nu.
So after I drop the minus sign, 8,pi,2,and the gravitational constant, I assume is the same, here is what I get, R*c^4*g_mu,nu = R'*c'^4*g'_mu,nu.
At first, I thought that R and R' are the same for a sphere. But now I'm not sure.
Does it look like I'm approaching the problem in the right way? At this point, I'm not sure what to do with the metric-tensor. I'm assuming a zero gravity environment for right now. I'm also thinking about bringing back a G and G' if I can't get the spatial anomally to behave itself.
Any thoughts?
