Things I know already
1) Rubber sheet analogy of GR is yet another misleading piece of info
2) differential geometry makes sense
3) equivalence principle makes sense
4) special relativity makes sense
I am trying to develop a better understanding of curvature in space-time.
Current Dilema
Trying to refine understanding of the effect of curved space-time on a stationary object. We all know that stationary objects do fall. Curvature, I reasoned, can surely only be experienced, (and therefore only cause an effect, such as an apparent acceleration), if an object has a trajectory; if it is moving relative to the curve. Massive stationary objects have a trajectory only in time, so the answer must be that curvature of time alone can be responsible for the effects we call gravity.
Question I Think I Need to Ask
This is a theoretical scenario.
A point mass is;
a) stationary in
b) a spatially flat volume, which
c) is curved in time in a simple way
(a geometry that is purely theoretical, but allows the question to focus on the effects of time)
d) by what mechanism does this mass experience a change in velocity?
I know I haven't used many mainstream GR terms, but I hope the question makes sense.
Other Aspects
Curved space is not a massive conceptual challenge, but curvature in space-time is more difficult. There seem to be less familiar concepts to relate distortions of the temporal dimension to.
The answer I am not looking for is that GR can only be "understood" by directly using the maths. In this case I would initially think that would really just mean you don't know (It's not like the philosophy struggle people go through with, for example, the meaning of wave-functions). However, I'd be happy to be pointed to a mathematical treatment of this kind of scenario that I can scrutinise; (that will be easier for me if lower level constructs are used).
Thanks in advance for your help.
