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Three mutually perpendicular mirrors reverse a ray's direction when it encounters all three. With orthonormal normals $a,b,c$, write the reflections as $I-2P_a$, $I-2P_b$, $I-2P_c$. Their orthogonal projection matrices have pairwise products zero and sum to $I$ by the resolution of the identity in an orthonormal basis. The product of the three reflections is therefore $-I$. It reverses every incident direction independently of the order in which the mirrors are met. Real apertures restrict which incoming rays actually reach all three reflecting faces. Such retroreflectors provide the returning-light geometry in lunar laser ranging.

A researcher identified with James G. Williams and Slava G. Turyshev on their 2012 lunar laser ranging study. Their analysis tests the equivalence principle by fitting precise Earth–Moon distance measurements to a model of the lunar orbit and checking whether different analysis choices change the inferred result.^[ref]

A researcher in lunar laser ranging and tests of gravity. With Slava G. Turyshev and Dale Boggs, he analyzed whether Earth and the Moon fall toward the Sun at the same rate. Timing laser pulses reflected by lunar retroreflectors turns the Earth–Moon system into an experiment on the equivalence principle.^[ref]

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]

For an arbitrary vector $\mathbf{x}$, expansion in the orthonormal basis gives $x_i=(a_i{a}_j+b_i{b}_j+c_i{c}_j)x_j$, with summation over $j$. Since this holds for every $\mathbf{x}$, the resolution of the identity in an orthonormal basis isLet $P_a=\mathbf{a}{\mathbf{a}}^T$, and similarly for $b,c$. The three orthogonal projection matrices sum to $I$ and have pairwise products zero. ThusAny ray meeting all three ideal perpendicular mirrors therefore returns in the opposite direction, independently of reflection order. This is the geometry of a corner reflector, a kind of retroreflector used in lunar laser ranging.

I would not assign the lunar mare an automatic terrestrial 100–125 km compensation level. For present-day mare basins, a local equal-column-mass model may have no physically valid compensation depth; first test whether lithospheric flexure supports the load. For ancient thermally driven subsidence that did relax, choose a common datum beneath the cooling layer in deformable lunar mantle appropriate to that epoch.

The Moon is smaller than Earth and has lost a larger fraction of its original heat. Its cold, mechanically strong outer shell can be much thicker relative to its radius than the terrestrial oceanic lithosphere. Lunar mare basalts filled pre-existing impact basins: their loading, basin excavation, mantle uplift, cooling and the evolving strength of the lithosphere all matter. A thick elastic lithosphere can support lateral stress differences, so equal column mass need not hold even after a long time. The strong gravity anomalies of lunar mascons, measured by GRAIL, are evidence that the present mare-basin system is not everywhere locally compensated.^[ref]

For an explicitly assumed ancient relaxed thermal model, estimate its thermal penetration depth from thermal diffusivity and the elapsed cooling time, infer the contemporaneous mechanical lid from gravity-topography admittance or basin deformation, and place $z_c$ below that layer where lateral pressure anomalies relax. For example, a hypothetical several-hundred-kilometre-thick lid requires a deeper datum than that lid, not the shallow crust-mantle boundary. The paper supplies neither an epoch-specific thermal model nor a load wavelength, so a unique kilometre value is not determined. If the chosen datum is a substantial fraction of the lunar radius, spherical geometry and radial variation in gravity must replace the flat, constant-gravity column approximation.

Does the Moon have an asthenosphere? Not a demonstrated shallow, globally connected Earth-like layer underlying mobile plates; but “no weak or partially molten mantle anywhere, ever” is too strong. Apollo lunar seismology indicates a cold outer interior and more attenuating deep regions. Reanalyses support a deep low-velocity or partially molten region near the lunar core, while lunar laser ranging and tidal Love numbers constrain deep deformability and tidal dissipation. Such a deep weak region is sometimes called a lunar asthenosphere; it is not automatically the correct local compensation surface for a mare load.^[ref]

The inferred melt fraction and distribution remain model dependent: Dissipation at tidal and seismic frequencies in a melt-free Moon by Francis Nimmo and others demonstrates that much of the dissipation can also be explained by hot solid-state anelasticity, although its model does not reproduce every frequency-dependent constraint. The Moon was hotter during mare emplacement and could relax loads differently then. Choose the datum from the relevant thermal and mechanical history, or use a flexural model when local isostasy fails.

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]

A retroreflector returns incident light approximately toward its source. Corner-cube arrays make the lunar laser ranging round trip possible.

A physicist working on experimental tests of gravity. His joint work with James G. Williams and Dale Boggs uses lunar laser ranging to compare the ratios of gravitational mass to inertial mass for Earth and the Moon. The comparison probes the equivalence principle using bodies with different compositions and gravitational binding energies.^[ref]