The principle of equivalence states that it is possible to choose a locally "inertial" coordinate at every space-time point, in presence of an arbitrary gravitational field, where the explicit effects of gravity disappear.
If we take the simple example of a body ($m$) falling freely under gravity ($mg$)- let $x^{\mu}$ be the lab frame, which is of course inertia, and let $\xi^{\mu}$ be the frame locally chosen using priciple of equivalence. The z-components of the two frames can be connected as $\xi_z= x_z – gt^2$. Using this transformation, the effects of gravitations ( the $mg$ term) disappears from the equation of motion.
Clearly, the frame $\xi^{\mu}$ is accelerated in comparison to $x^{\mu}$. My question is, how can the frame $\xi^{\mu}$ be inertial when it is in acceleration wrt another inertial frame $x^{\mu}$?
Best Answer
In the context of General Relativity, the "lab frame" is not inertial, because it is not freely falling. This is the point of Einstein's "elevator" thought experiments: a freely falling frame in a gravitational field is equivalent to an inertial frame in deep space, and a stationary frame in a gravitational field is equivalent to an accelerated frame in deep space. Only the first category of frames is seen as "inertial" in the context of GR.