= Solution
\b[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:
$$
\frac{\mathrm d^2x^\mu}{\mathrm d\tau^2}
+\Gamma^\mu_{\alpha\beta}
\frac{\mathrm dx^\alpha}{\mathrm d\tau}
\frac{\mathrm dx^\beta}{\mathrm d\tau}=0.
$$
This coordinate choice does not remove curvature. With a standard curvature convention, nearby free-fall trajectories have relative acceleration
$$
\frac{D^2\xi^\mu}{\mathrm d\tau^2}
=-R^\mu{}_{\nu\alpha\beta}u^\nu\xi^\alpha u^\beta.
$$
The 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.<Lunar Laser Ranging Tests of the Equivalence Principle by James G. Williams and others>[2012 abstract, differential Earth-Moon free fall and dependence on composition and gravitational self-energy]{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>.
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