= Solution
The phrase <equivalence principle> covers several related claims. The <weak equivalence principle> concerns the universality of <free fall> for <test particles> of negligible self-gravity, with the same initial conditions and no relevant nongravitational <forces>. In a Newtonian notation,
$$
m_I\mathbf a=m_G\mathbf g
\quad\Longrightarrow\quad
\mathbf a=\frac{m_G}{m_I}\mathbf g.
$$
Composition-independent motion requires the ratio of passive <gravitational mass> to <inertial mass> to be universal. Its common value can then be absorbed into the normalization of $\mathbf g$; the empirical assertion is universality, not the numerical convention used to express the ratio.
The <Einstein equivalence principle> includes that universality and extends the claim to local nongravitational experiments through <local Lorentz invariance> and <local position invariance>. A sufficiently small freely falling laboratory should recover the nongravitational laws of <special relativity>. The <strong equivalence principle> extends the scope to local gravitational experiments and bodies with appreciable gravitational self-energy. An experiment establishing one part of the weak statement does not automatically establish these stronger claims.
The motivating elevator comparison makes both the power and the limitation clear. Release a small object in a freely falling cabin. The cabin and object share the local <free fall>, so the object initially floats relative to the cabin. In a cabin supported against <gravity>, a released object instead accelerates relative to the floor, as it does in a suitably accelerated cabin in flat <spacetime>. This suggested replacing a universal gravitational <force> by geometric <inertial motion>. It does not identify a finite planetary <gravitational field> with an accelerated frame everywhere: separated falling objects can have a measurable relative <acceleration>.
In <general relativity>, the precise local geometric statement is that one can choose <normal coordinates> at an event $p$ with
$$
g_{\mu\nu}(p)=\eta_{\mu\nu},\qquad
\partial_\rho g_{\mu\nu}(p)=0,\qquad
\Gamma^\mu{}_{\alpha\beta}(p)=0.
$$
This eliminates the connection contribution to the <geodesic equation> at that point. It does not generally set the <Riemann curvature tensor> to zero. In normal coordinates the first metric corrections away from $p$ are quadratic in the coordinate separation, and <geodesic deviation> measures the associated <tidal forces>. Equivalence is a local approximation for finite experiments whose extent and duration make these corrections negligible; it is not an assertion that curvature has been transformed away.<General Relativity by David Tong>[sections 3.3.2 and 3.3.4: normal coordinates and geodesic deviation]{ref}
There are direct experimental consequences beyond the elevator thought experiment. An <Eötvös experiment> compares bodies of different composition on a <torsion balance>, using a differential signal to test the universality of gravitational <acceleration>. <MICROSCOPE (satellite)> performed an analogous test in orbit using electrostatic control of titanium-alloy and platinum-alloy test masses. Its first published result in 2017 constrained their differential <free fall>; this was already available before the examination. It tested a specific weak-equivalence prediction, not every possible gravitational law.<The MICROSCOPE mission: first results of a space test of the Equivalence Principle by Pierre Touboul and others>[2017 abstract and experimental description: composition comparison and differential control forces]{ref} Comparisons of different <atomic clocks> can instead test <local position invariance>, including the expected <gravitational redshift>. These measurements constrain related but distinct parts of the framework.
The principle remains important as a restriction on how matter couples to gravity and as an organizing guide for experimental tests. It does not, on its own, derive the <Einstein field equations>. Universally coupling matter to a <metric tensor> still leaves choices for that metric's <action>; the <equivalence principle does not uniquely fix field equations>. For example, a metric gravitational action containing a <scalar curvature> squared term can retain the same minimal matter coupling while changing the gravitational dynamics. Selecting the <Einstein-Hilbert action> requires further dynamical assumptions and empirical success. The present role of the principle is consequently precise and limited: it supports local relativistic physics and universal matter coupling, while curvature, gravitational field equations and their quantitative predictions remain additional content.
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