Past exam of the relativity course of the University of Cambridge / 2013 / Paper 1 / 2 / a / Solution

Universality of free fall equates the passive gravitational response to inertial mass up to a universal choice of units. The Einstein equivalence principle further states that local nongravitational experiments in a freely falling laboratory obey special relativity independently of the laboratory's location and velocity. A supported laboratory instead has proper acceleration, just as an accelerating laboratory in flat spacetime does.

Around an event, freely falling coordinates can be chosen so that $g_{\mu \nu }={\eta }_{\mu \nu }$ and ${\Gamma }^{\alpha }{}_{\mu \nu }=0$ there. The first metric tensor derivatives vanish at that event, and nearby corrections begin at order curvature times squared separation. The metric tensor's second derivatives retain the Riemann tensor. Thus a local inertial frame removes Christoffel symbols at an event, while tidal gravity remains measurable over a finite region. Relative motion of separated freely falling test particles detects this curvature through geodesic deviation. Freely falling axes carried along an observer's world line extend the local construction, but do not make an extended curved region globally Minkowskian.