If we want to solve a problem where a particle is forced off a geodesic, can we obtain identical results if we instead solve the problem where the particle is in a different spacetime but stays on a geodesic?
Long Version (The Thought Experiment):
(See Twin Paradox for background)
Twin A is floating stationary in space. Twin B is in a high-speed rocket which passes by twin A. At that moment, twin A and twin B synchronise their clocks.
Then, consider two different situations:
Space is globally flat Minkowski. Twin B coasts through flat space, using their thrusters to slowly accelerate before using them to turn around and travel back home to twin A.
Space is globally Schwarzschild, but twin A is situated very far away from the central black hole where space is, to good approximation, flat. Twin B immediately turns off their thrusters and coasts along their geodesic with no noticable acceleration in their frame. The Geodesic happens to take them close to the black hole, completely curving around the black hole and turning back on itself, back towards twin A (eventually passing by them again).
When the twins pass by the second time, we find that twin A's clock has avanced more time than twin B's clock. That is, twin A has aged more than twin B.
I know this is true in case (1) because it is the typical description of the twin paradox, which has a well-known answer (although I don't know how to explicitly calculate it).
I know this is true in case (2) because twin A and twin B are both just geodesics in a schwarzchild spacetime, so I was able to numerically integrate them and sure enough, I found that $\tau_B < \tau_A$.
My question is: Are situation (1) and (2) equivalent when viewed from twin A's frame?
Why I think they are equivalent:
- The Einstein Equivalence Principle states that acceleration and curvature are equivalent. A description that uses curvature (geodesics) to describe a particle's trajectory should be the same as a description that uses acceleration.
Why I'm not sure:
- Twin B feels a force of acceleration in case (1) but doesn't feel anything in case (2).
- In case (2), both twins A and B stay on geodesics forever. This is more akin to time dialation in Special Relativity which is happens when two particles are on different geodesics - Perhaps the time dialation we're seeing in (2) is more related to the SR type of time dialation which is presumably different from the type in (1)?