Equivalence principle

Remarks

This is the mathematical form, in general relativity, of the physical equivalence principle. In a small enough freely falling laboratory, gravity cannot be detected, and the laws of special relativity hold. The weak equivalence principle says that the world line of a freely falling test body depends only on its initial position and velocity, not on its mass or composition. In Newtonian terms, inertial mass equals gravitational mass. Einstein's equivalence principle adds that all non-gravitational laws of physics take their special-relativistic form in a locally inertial frame. The proof: take an orthonormal basis $e_0, \ldots, e_3$ of $T_pM$ and use normal coordinates $x \mapsto \exp_p(x^\mu e_\mu)$; in normal coordinates the first derivatives of the metric vanish at $p$. The second derivatives of $g_{\mu\nu}$ cannot in general be removed: they vanish at $p$ in some coordinates only if the Riemann curvature tensor vanishes at $p$. So tidal effects, which are measured by curvature, remain detectable in a freely falling laboratory of finite size.

Going deeper

For a student

In the mathematics of general relativity the principle is built in: near any event, spacetime looks like Minkowski space to first order. This is why the laws of special relativity, written with tensors, can be carried to curved spacetime by replacing partial derivatives with covariant derivatives, the rule of minimal coupling. The curvature appears only at second order, and it governs the relative motion of nearby free bodies through the geodesic deviation equation.