Past exam of the relativistic Astrophysics and Cosmology course of the University of Cambridge / 2016 / Paper 1 / 3 / e

The equivalence principle connects the universality of freely falling motion to the local description of gravity. In Newtonian language a test body's response to a given field is proportional to $m_g/m_i$. Equality of that ratio for different non-gravitating test bodies gives the weak equivalence principle: identical initial conditions yield identical free-fall trajectories, independent of composition. Negligible nongravitational forces and negligible disturbance of the background are part of the test-body limit.

The Einstein equivalence principle adds local Lorentz invariance and local position invariance for nongravitational experiments. A sufficiently small freely falling laboratory obeys the local laws of special relativity independently of its spacetime location and freely falling velocity. Uniform acceleration of a small laboratory can imitate the leading local effects attributed to a gravitational field. An observer held at rest in a gravitational field generally needs proper acceleration; a freely falling observer does not.

Mathematically, local inertial coordinates can set the metric to Minkowski form and the Christoffel symbols to zero at an event. The geodesic equation there has no connection term at that point:This coordinate choice does not remove curvature. With a standard curvature convention, nearby free-fall trajectories have relative accelerationThe tidal signal survives in an extended laboratory. Thus the principle is local and does not identify every gravitational field with acceleration globally. Finite-size spin and tidal couplings also qualify the ideal point-test-body statement.

The strong equivalence principle extends the universality to gravitational experiments and bodies whose gravitational self-energy is important. It is stronger than the Einstein principle and need not hold in every metric theory that satisfies local nongravitational equivalence. Comparing the Earth and Moon as they fall toward the Sun probes this distinction as well as composition effects. Lunar laser-ranging analyses test differential acceleration by fitting the lunar orbit and measured ranges; an interpretation in terms of self-energy requires separating ordinary composition-dependent violations.^[ref]

Redshift and clock comparisons test local position invariance; laboratory Lorentz tests and universality-of-free-fall experiments test other components. These are complementary tests, not deductions from a single numerical equality of inertial and gravitational mass. The principle motivates geometric gravity and minimal coupling, but does not by itself derive the complete Einstein field equations or determine the value of the gravitational constant.