Non-Inertial Rigid Frames and the Principle of Equivalence

One version of the principle of equivalence, as originally formulated by Einstein, states that ``gravity" can be mimicked locally by going to an ``accelerated frame of reference". As highlighted by Synge, the physical content of this principle remains obscure in so far as it does not refer to the Riemann tensor Rabcd, which encodes the true effects of gravity. We here give the acceleration profile of a $Born$ rigid, Rindler$esque$, frame that can mimic a gravitational field corresponding to a given Rabcd. The generalised deviation equation that yields this result also has Centrifugal and Coriolis terms appearing in a purely relational context, yielding a similar connection between angular velocity of rotating, rigid inertial frames and the Riemann tensor. We comment briefly on implications for Mach principle.