Einstein Equivalence Principle (EEP) and the Strong Equivalence Principle (SEP) both state that the trajectory of a point mass in a gravitational field depends only on its initial position and velocity, and is independent of its composition and structure.
On top of that, the EEP says that the outcome of any local non-gravitational experiment in a freely falling laboratory is independent of the velocity of the laboratory and its location in spacetime.
While the SEP claims that the outcome of any local experiment (gravitational or not) in a freely falling laboratory is independent of the velocity of the laboratory and its location in spacetime.
According to all sources that I have read, it is Einstein Equivalence Principle that we use as one of the hypothesis to derive General Relativity (for example, in page 50 in the book by Sean Carroll. Weinberg also talks about EP in page 70 of his book, but it was not clear to me which of the two formulations he was choosing).
I know that EEP and SEP are only distinguished by local gravitational effects, like for example gravitational self-interaction energy or maybe some torsion experienced by particle in a gravitational field.
But from my understanding of General Relativity, the assumption that lets us describe the theory using pseudo-Riemannian manifolds is that spacetime must look exactly the same for a free-falling observer than it does for an accelerated observer. If we allow gravitational effects, like maybe gravitational radiation, to change the physics for an observer in a gravitational field, are manifolds still a good description of the physics?
So in sumary my questions are:
Is EEP enough for General Relativity or should we impose SEP?
If we included a term in our theory that causes object to rotate as they fall in a gravitational field, it violates the SEP but not the EEP, right? Would such a term be allowed in General Relativity?
Am I wrong about objects that rotate while they fall satisfying the EEP? I guess you could detect such a rotation with non-gravitational experiments even if the cause was indeed of purely gravitational origin! Does that mean that a theory with EEP rules out any possible observable effect except maybe affecting how other particles feel gravitational interactions? As in for example making the gravitational force between two bodies weaker when they are in free-fall in a third body's gravitational field than they would be in an accelerated frame.
But wouldn't that also be measurable with non-gravitational experiments like, for example, measuring the resulting velocity?