There seems to be a relationship between these two ...... (can't find the word that describes them both -"actors"? )

It is commonly said that the latter causes the former to bend and the former causes the latter to move.

 

 

Are there any theories that describe( or aim to describe) both as one thing rather than two? (emergent properties ,would that be the term?)

 

 

Is there any property of both Space and Energy (shorthand terms perhaps ) that is conserved,perhaps as the Universe evolves?

 

That would imply that at the moment close to the Big Bang that this quantity (whatever it might be) be be the same as it is at this moment and at all other moments

 

Would that be a good way of looking for this possible common property behind what to me look like very separate phenomena?

Space and energy are not directly related. Time and energy are. Basically, if the physics is time translationally invariant then energy is conserved.

Is there another property that is related to space in the same way? Momentum?

 

I'm just wondering if there is some sort of transformation between space-time coordinates and energy-something state space ...

Is there another property that is related to space in the same way? Momentum?

 

I'm just wondering if there is some sort of transformation between space-time coordinates and energy-something state space ...

People do deal with what we call extended phase spaces. As you know, space and momentum are canonical conjugates and you can also view time and energy in that way. So people develop a formalism where the phase space is (x,t, p E). This has a canonical symplectic structure and so I guess you can think about canonical transformations here. I have not looked at this properly for a while.