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Lunar laser ranging gives a more discriminating orbit readout. An observatory directs short laser pulses at a retroreflector array on the Moon, and a detector records returning photons. Round-trip time, statistically combined over several returns, constrains distance; pressure and temperature accompany the readout. The fitted model includes Newtonian gravity, terrestrial and lunar rotation, geophysical effects, and corrections from general relativity. Comparing orbit determination residuals across stations and model choices checks consistency. The inspected study reports $1.8\mathrm{cm}$ weighted root mean square residuals for its 2005–2011 ranges, rather than for a bare isolated inverse-square potential. It tests the equivalence principle using the Earth and Moon's response to the Sun; it is not a direct escape velocity measurement. Uneven coverage near new and full Moon, radiation pressure, and thermal expansion of reflector mounts limit that inference.^[ref]

The weak equivalence principle says that freely falling test bodies with identical initial conditions follow the same trajectory independently of composition, when nongravitational forces and appreciable self-gravity are negligible. In Newtonian language it implies universal gravitational-to-inertial mass ratio. The Einstein equivalence principle additionally requires local Lorentz invariance and local position invariance of nongravitational experiments: a sufficiently small freely falling laboratory realizes special relativity. The strong equivalence principle extends universality to gravitational experiments and self-gravitating bodies.

A falling elevator makes the local content clear. Uniform gravitational acceleration can be removed along a freely falling worldline by a suitable frame, but curvature cannot generally be removed over an extended region. Nearby geodesics have relative accelerationfor the displayed Riemann-sign convention. tidal forces therefore remain detectable even in free fall. Equivalence motivates a metric description and gravitational clock shifts, but does not uniquely select Einstein's field equations.

Tests compare accelerations of different materials, frequencies of clocks at different gravitational potentials, or the response of self-gravitating bodies. lunar laser ranging times pulses reflected by lunar retroreflectors and fits Earth-Moon motion in the Sun's field; a differential free-fall violation would add a characteristic periodic range perturbation. The fit must also model tides, station and reflector geometry and propagation delays, with solar-radiation pressure and reflector thermal effects checked as possible mimics. Williams and collaborators' 2012 analysis is a concrete example of this method, published after this examination.^[ref]

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.