In the usual "windowless room" thought experiment it is stated that there is no experiment which can be performed which will determine if the room is undergoing "uniform accleration" far from any massive body or is in a "uniform gravitational field". The two situations are therefore said to be "equivalent". Also, since the spacetime in the uniform acceleration case is obviously flat the equivalent spacetime in the uniform gravitation must also be flat. So, my question is a pretty basic one. How do we construct the right coordinate systems for doing the "equivalent" windowless room thought experiments? Specifically: In the gravitational/accelerated non-inertial frame what is the form of the gravitational field (as a function of position)? If the field is a non-constant function of position then how is it considered either uniform or flat? How are simultaneity, distance, and time experimentally defined in the uniform field? What is the correct transformation between the accelerated and an inertial frame? Can you simply use the Lorentz transform to boost between observers with uniform (non-inertial) motion in the gravitational field? (i.e. since spactime is flat do four-vectors still transform like usual, especially the four-momentum) Any other hints would be appreciated.