[pavel_lishin]

Isn't part of it the spacetime geometry, specifically your path through it?

It makes absolute sense for my gas tank to be lower if I take a long roundabout path from A to B while you take a short one, even though our cars have no idea about what the other one is doing.

[johncolanduoni]

Yes, general relativity is all about the "metric tensor" of space, which is just a fancy way of saying it's about assigning distances to paths. For paths you can actually physically take (i.e. movement slower than the speed of light relative to local space) this ends up corresponding to the time experienced along that path.