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The Einstein equivalence principle combines the weak equivalence principle with local Lorentz invariance and local position invariance of nongravitational experiments. In a sufficiently small freely falling laboratory, their results agree with special relativity independently of its location and velocity. The strong equivalence principle extends this claim to local gravitational experiments.

A torsion-balance comparison of the gravitational acceleration of bodies of different composition, testing equality of inertial and passive gravitational mass up to a universal proportionality factor. A universal acceleration law independent of test-body mass and composition agrees with this weak equivalence principle test. It does not by itself establish the strong equivalence principle for self-gravitating bodies.

Local Lorentz invariance requires the outcome of a local experiment, expressed in its freely falling apparatus, to be independent of the apparatus's velocity and orientation. In the Einstein equivalence principle it applies to nongravitational experiments; the strong equivalence principle also includes local gravitational experiments. It is distinct from local position invariance.

Local position invariance requires identical local experiments to have the same outcome independently of where and when they are performed. For nongravitational experiments it is part of the Einstein equivalence principle. Comparisons of clocks through gravitational redshift test this principle once the gravitational potential difference is independently specified; they do not by themselves test the whole strong equivalence principle.

A satellite experiment testing the universality of free fall. Electrostatic control keeps test masses made from titanium and platinum alloys on matched orbital motion; the required differential forces constrain composition dependence. A reference sensor uses masses of the same composition. The experiment tests the weak equivalence principle, rather than the full strong equivalence principle.^[ref]

Assume a metric theory in four dimensions, a Levi-Civita connection, covariant stress-energy conservation and field equations linear in second metric derivatives. The contracted Bianchi identity selects $G_{ab}=R_{ab}-g_{ab}R/2$. Its trace-reversed equation is $R_{ab}=\kappa (T_{ab}-g_{ab}T/2)+\Lambda g_{ab}$. In the convention $R_{00}\simeq -{\nabla }^2\phi$ and signature $(+---)$, nonrelativistic matter gives $R_{00}\simeq \kappa \rho /2$. The Poisson equation ${\nabla }^2\phi =4\pi G\rho$ fixes $\kappa =-8\pi G$. Reversing the curvature convention reverses this displayed coupling sign without changing the physical theory. The strong equivalence principle alone does not establish uniqueness without the stated additional assumptions.

A violation of the strong equivalence principle can make the gravitational-to-inertial mass ratio depend on a body's negative gravitational self-energy $E_G$. Bodies with different self-binding fractions would then fall differently in the same external field, even after composition-dependent weak-equivalence effects are removed. For the Earth and Moon this produces a polarization of the lunar orbit toward the Sun, measurable by lunar laser ranging. General relativity predicts ${\eta }_N=0$.

The strong equivalence principle extends universality of free fall to bodies with appreciable gravitational self-energy, and states that the outcome of any local experiment, including a gravitational experiment, in a freely falling laboratory is independent of its location and velocity. In a sufficiently small freely falling laboratory the external gravitational influence can be removed to first order by choosing a local inertial frame.

This does not remove tidal forces or the Riemann curvature tensor. The laboratory must be small compared with the length scale over which the external gravitational field varies. These qualifications distinguish a local equivalence statement from the incorrect assertion that gravity can be transformed away throughout an extended curved region.

The Pound-Rebka experiment measures a small gravitational redshift by comparing nuclear emission and resonant absorption at different heights in Earth's field. The $14.4\mathrm{keV}$ transition of iron-57 supplies the gamma-ray frequency standard. The Mössbauer effect permits recoil-free emission and absorption by nuclei bound in solids, giving a sufficiently narrow resonance. A source and absorber separated vertically by about $22.6\mathrm{m}$ have a predicted fractional frequency difference of order $2.5\times 10^{-15}$. A calibrated Doppler effect from slow source motion measures the shift through its effect on resonant absorption. Interchanging the upper and lower arrangements reverses the gravitational contribution and helps separate it from intrinsic source-absorber offsets; temperature-dependent frequency shifts must also be controlled. The observed change agrees with the weak-field prediction.

The prediction follows either from an accelerating laboratory and the equivalence principle, or directly from a static metric. With $x^0=ct$, write $g_{00}=N^2\simeq 1+2\Phi /c^2$. The time-translation Killing vector field gives a conserved photon energy $E_{\xi }=p_a{\xi }^a$, while a static observer has four-velocity $u^a={\xi }^a/N$ and measures $h\nu =E_{\xi }/N$. ThusFor an upward ray over height $H$, $\Delta \nu /\nu \simeq -gH/c^2$; downward propagation has the opposite sign. The equivalent compensating Doppler speed has magnitude $gH/c$, about $7.4\times 10^{-7}\mathrm{m}{\mathrm{s}}^{-1}$ for this baseline. Identical clocks at rest have $d\tau =Ndt$, so a clock at higher potential runs faster relative to the common static coordinate time. This does not mean that the nuclear transition changes its locally measured frequency with height: the comparison involves transporting a photon between different local proper-time standards.

The weak equivalence principle asserts universal free fall for test bodies. The Einstein equivalence principle adds local Lorentz invariance and local position invariance for nongravitational experiments in a freely falling laboratory. The redshift comparison tests the clock aspect of local position invariance and supports a metric description of gravity. The strong equivalence principle extends these requirements to self-gravitating bodies and local gravitational experiments, including the contribution of gravitational binding energy. Their outcomes should be independent of the external location and motion of a freely falling laboratory when tidal effects are negligible. A nuclear redshift experiment is therefore not a test of the whole strong equivalence principle, and its success does not uniquely establish the gravitational field equations.

The geometrical realization uses the Levi-Civita connection of a spacetime metric. At any event, normal coordinates make $g_{ab}={\eta }_{ab}$ and the Christoffel symbols vanish. Freely falling test particles follow geodesics; local nongravitational equations take their special-relativistic form at that event. The Riemann curvature tensor need not vanish, so finite laboratories can measure tidal geodesic deviation. The strong principle concerns removing a uniform external gravitational field locally, not removing curvature throughout an extended region.

For a perfect fluid, local rest-frame isotropy and the absence of heat flux or viscosity give $T^{\hat{a}\hat{b}}=\mathrm{diag}(\rho ,p,p,p)$, where $\rho$ is rest-frame energy density. With the PDF's signature $(+---)$, $u^a{u}_a=1$ and $h^{ab}=g^{ab}-u^a{u}^b$ projects spatially. Its covariant form isThe pressure sign follows from the negative spatial metric; it gives positive diagonal stresses in the rest frame. Local stress-energy conservation, ${\nabla }_a{T}^{ab}=0$, produces the two projectionsThe first is the energy equation, including pressure work; the second is the relativistic Euler equation. Its rest-frame spatial pressure force is $-\nabla p$.

Further assumptions are needed to derive the Einstein field equations. Take a local generally covariant metric equation, linear in curvature and of second differential order, and demand compatibility with stress-energy conservation. The available rank-two tensors are $R_{ab}$, $R{g}_{ab}$ and a constant multiple of $g_{ab}$. Defining $R_{ab}=R^c{}_{acb}$, the contracted Bianchi identity gives ${\nabla }^a{R}_{ab}=\frac{1}{2}{\nabla }_{b}R$. Hence the combination $R_{ab}+\alpha R{g}_{ab}$ is identically divergence-free only when $\alpha =-\frac{1}{2}$. This determines the Einstein tensor $G_{ab}=R_{ab}-R{g}_{ab}/2$; a cosmological constant term remains allowed.

The coupling sign must be matched to the actual curvature convention. Here $R^a{}_{bcd}$ represents $[{\nabla }_d,{\nabla }_c]$, and a static weak field has $R_{00}\simeq -{\nabla }^2\Phi /c^2$. With negligible pressure and cosmological term, a trial equation $G_{ab}=C{T}_{ab}$ implies $R_{00}\simeq C{\rho }_{\mathrm{mass}}{c}^2/2$. Comparing with the Poisson equation ${\nabla }^2\Phi =4\pi G{\rho }_{\mathrm{mass}}$ fixes $C=-8\pi G/c^4$. Thus the Einstein equation in the sequential-derivative curvature convention isEquivalently, $R_{ab}=\Lambda g_{ab}-(8\pi G/c^4)(T_{ab}-T{g}_{ab}/2)$, with $T=\rho -3p$. Pressure therefore gravitates as well as supplying a material force. The cosmological constant is not fixed by the local Newtonian matching.

Setting $c=1$ for the action normalization, the same metric equation follows from the Einstein-Hilbert action $S_g=(2\kappa {)}^{-1}\int \sqrt{-g}(R-2\Lambda )d^{4}x$, where $\kappa =8\pi G$, together with a matter action obeying $\delta S_m=\frac{1}{2}\int \sqrt{-g}{T}_{ab}\delta g^{ab}{d}^{4}x$. After the metric boundary term is removed or supplied separately, the bulk variation is proportional to $G_{ab}+\Lambda g_{ab}+\kappa T_{ab}$. This also makes the sign convention explicit. General covariance, the field equation and the contracted Bianchi identity are mutually consistent with the fluid conservation laws. The experiment motivates the local metric description; the curvature and conservation assumptions, plus the Newtonian limit, supply the additional route to the field equation.

The strong equivalence principle extends universality of free fall to bodies with appreciable gravitational binding energy and requires results of sufficiently local experiments, including gravitational experiments, in freely falling laboratories to be independent of their location and velocity. The Einstein equivalence principle makes the corresponding statement for nongravitational experiments; the weak principle concerns test-body free fall without significant self-gravity. In a locally inertial frame the nongravitational laws take their special-relativistic form, while internal gravitational experiments retain the same local gravitational laws and coupling constants. They are not declared to have no gravity.

In a metric description, one may set $g_{ab}={\eta }_{ab}$ and ${\partial }_c{g}_{ab}=0$ at a point by Riemann normal coordinates, so the Levi-Civita connection vanishes there. This removes a locally uniform external gravitational acceleration, not the Riemann curvature tensor. Tidal effects between separated bodies, or accumulated over a finite-duration experiment, remain. The local limit must control both the laboratory's size and duration. Gravitationally self-bound bodies are included in the strong version, so it is stronger than equality of inertial and passive gravitational mass for ordinary test particles.

A heuristic field-equation argument supplements this principle with further assumptions: gravity is described only by a Lorentzian metric, matter couples universally to it, the affine connection is Levi-Civita, and the leading local field equation is a symmetric covariant equation with at most second metric derivatives and is linear in those second derivatives. Matter obeys stress-energy conservation ${\nabla }^a{T}_{ab}=0$. Curvature therefore supplies the natural candidates $R_{ab}$ and $R{g}_{ab}$, with a constant multiple of $g_{ab}$ also allowed. Write the candidate as $R_{ab}+cR{g}_{ab}+\Lambda g_{ab}=\kappa T_{ab}$. The contracted Bianchi identity and metric compatibility giveTo permit varying scalar curvature consistently with matter conservation, take $c=-1/2$. This gives the Einstein tensor $G_{ab}=R_{ab}-g_{ab}R/2$ andThe cosmological constant is constant so that its divergence vanishes. These assumptions motivate the usual Einstein equation; the equivalence principle alone does not exclude theories with extra fields or higher curvature terms.

The sign and normalization are fixed by the supplied Newtonian limit. Use signature $(+---)$, the curvature commutator convention of Q1, and the Ricci contractionThis is the contraction for which $R_{00}\simeq -{\nabla }^2\phi$ in the question. In four dimensions, tracing the candidate equation gives $-R+4\Lambda =\kappa T$, and substituting back givesFor slowly moving pressureless matter, $T_{00}\simeq \rho$, $T\simeq \rho$ and $g_{00}\simeq 1$. On local scales where the cosmological term is negligible, $R_{00}\simeq \kappa \rho /2$. Compare this with $R_{00}\simeq -{\nabla }^2\phi$ and the Poisson equation ${\nabla }^2\phi =4\pi G\rho$. The Newtonian normalization of the Einstein field equations is thereforeUnits with light speed one are used, as in the paper; for conventional physical stress-energy units the coupling is $-8\pi G/c^4$. If the opposite Ricci/curvature sign convention is adopted, the displayed geometric coupling sign changes accordingly. With the cosmological term retained in the same weak-field calculation, the Newtonian equation becomes ${\nabla }^2\phi =4\pi G\rho -\Lambda$, consistent with this sign choice.

The strong equivalence principle asserts universality of free fall, including gravitational self-energy, and local independence of nongravitational and gravitational experiments from a freely falling laboratory's location and velocity. In a sufficiently small local inertial frame, uniform gravitational acceleration is removed and local physics takes its special-relativistic form. The Riemann curvature tensor remains measurable through tidal effects; a finite laboratory is not made exactly flat. Freely falling test bodies follow geodesics, clocks measure proper time, and minimally coupled light follows null geodesics in geometrical optics. Local nongravitational laws retain their special-relativistic form even when different laboratories observe gravitational redshift.

The comma-to-semicolon prescription replaces flat partial derivatives by the Levi-Civita connection, and the Minkowski metric by the curved metric tensor. It implements this local correspondence for minimally coupled matter. However, an identity obtained by commuting partial derivatives need not survive unchanged when they become covariant derivatives.

Raising the indices of the field-strength definition in flat spacetime gives $F^{ab}={\partial }^a{A}^b-{\partial }^b{A}^a$. Substitution into the sourced Maxwell equations yieldsso the first potential equation isPartial derivatives commute, giving the second formNo Lorenz gauge assumption was used.

Apply the comma-to-semicolon prescription before commuting derivatives. The two candidate left-hand sides becomeUse the cover curvature $C$ from Question 1, with $C_{ab}=C^c{}_{acb}$. The vector commutator is $[{\nabla }_c,{\nabla }_d]A^a=-C^a{}_{bcd}{A}^b$. HenceThis is the derivative-order ambiguity in covariant Maxwell potential equations. At a point in Riemann normal coordinates, the connection itself vanishes but its derivatives need not; the strong equivalence principle does not dispose of this curvature term by a coordinate choice.

Covariantizing the antisymmetric field strength and its Maxwell equations directly gives ${\mathcal{P}}_1^a={\mu }_0{J}^a$, orThis formulation has gauge invariance under $A_a\mapsto A_a+{\nabla }_a\chi$, since $F_{ab}$ is unchanged. In contrast, the uncorrected ${\mathcal{P}}_2$ acquires $-C^a{}_b{\nabla }^b\chi$ under that transformation. Thus preserving the original electromagnetic gauge symmetry singles out the directly covariantized Maxwell equations, rather than treating the two potential equations as interchangeable in matter. The equivalence principle alone is not a unique recipe for every possible higher-derivative or curvature coupling.

In a Ricci-flat vacuum spacetime, $C_{ab}=0$, so ${\mathcal{P}}_1={\mathcal{P}}_2$ and this particular ambiguity disappears even though the full Riemann curvature tensor may be nonzero. This uses vacuum with zero cosmological constant; an Einstein vacuum with nonzero cosmological constant is not Ricci flat.

For the terrestrial estimate, the relevant effect is Ricci curvature inside matter, not merely the tidal field outside the Earth. If a typical material density is $\rho \sim 5\mathrm{g}\mathrm{c}{\mathrm{m}}^{-3}$, the Einstein field equations give a characteristic scaleFor a field varying over scale $L$, the derivative terms are $O(A/L^2)$ whereas the disputed term is $O(CA)$. Their relative size is $O(C{L}^2)$: about $10^{-22}$ for a metre-scale experiment. Even substituting the Earth's radius gives onlyA curvature length comparable to $C^{-1/2}\sim 10^{13}\mathrm{cm}$ would be needed for an order-one effect. In principle non-vacuum measurements could distinguish different curvature couplings, but an ordinary terrestrial laboratory has an exceedingly small signal, overwhelmed by its ordinary electromagnetic response to matter. This is a practical scale estimate, not a theorem that an arbitrarily precise experiment is impossible.

Take $z$ positive outward and define the separation as upper minus lower position. The Newtonian potential of a point mass gives the acceleration $a(z)=-{\partial }_z\varphi =-GM/z^2$. Expanding its difference across a separation $\zeta \ll z$ gives the tidal force equationThe upper particle falls slightly less rapidly, so the separation increases. If the upper particle carries the electric charge and the electric field points outward, the extra relative acceleration is $qE/m$. Reversing the field or placing the charge on the lower particle changes its sign. Write this signed acceleration as $a_e$ and put $k=2GM/z^3$. With initial relative velocity zero and nearly constant $k$ and $a_e$, the ordinary differential equation has solutionFor $k{t}^2\ll 1$, the leading changes due to the tidal force and electric field are respectivelywith the stated sign convention. Thus, for a nonzero fixed electric acceleration,A laboratory of size $L$ has $|{\zeta }_0|\le L$, so this ratio tends to zero as $L\to 0$. This is local tidal acceleration scaling: gravitational relative acceleration is proportional to separation, whereas the electric difference persists even at coincident positions.

The duration also matters. At fixed time, $\Delta {\zeta }_g\to 0$ in absolute size, but $\Delta {\zeta }_g/{\zeta }_0\simeq (GM/z^3)t^2$ need not tend to zero. To keep the electrically displaced particles in the shrinking laboratory, take $|a_e|t^2\lesssim L$; then $|\Delta {\zeta }_g|/L\lesssim GML/(z^3|a_e|)\to 0$. Alternatively a stationary terrestrial laboratory has a free-fall time of order $\sqrt{L/g}$, giving $|\Delta {\zeta }_g|/|{\zeta }_0|=O(L/z)$. Both make precise the small spacetime-region limit. An extended, long-running experiment can still measure arbitrarily small tidal forces.

The weak equivalence principle says that freely falling test particles with the same initial data have composition-independent motion. In particular, a freely falling frame removes the common gravitational acceleration of both particles. The electric accelerations depend on charge and cannot be removed for charged and uncharged particles simultaneously by the same change of frame.

The Einstein equivalence principle extends the freely falling laboratory statement to all local nongravitational experiments: their outcomes obey special relativity, with local Lorentz invariance and local position invariance. The strong equivalence principle also includes local gravitational experiments and bodies with significant gravitational binding energy. It asserts universality of their free fall and independence of local experimental outcomes from the laboratory's location and velocity. The two-particle experiment illustrates the local distinction between gravity and a nongravitational force; by itself it does not establish this stronger claim about self-gravitating bodies.

In a metric theory, the common freely falling trajectories are timelike geodesics of a Lorentzian metric. Normal coordinates make $g_{ab}={\eta }_{ab}$ and the Christoffel symbols zero at one point. Fermi normal coordinates do the same along a reference timelike geodesic. The residual effects are controlled by the Riemann curvature tensor through geodesic deviation, and enter the local metric at quadratic order in distance. Curvature is a physical tidal field, so no coordinate choice removes it over a finite region.

A relativistic theory should consequently couple matter universally to the metric tensor, recover nongravitational laws in local inertial frames and recover the Newtonian limit. General relativity implements this with the Einstein field equations; the contracted Bianchi identity is consistent with stress-energy conservation. The equivalence principle motivates this geometric description but does not alone uniquely determine the field equations. Extra gravitational fields may satisfy the Einstein equivalence principle for ordinary matter while failing the strong equivalence principle because local gravitational measurements or self-gravitating trajectories depend on those fields.

The strong equivalence principle asserts universality of free fall even for bodies with gravitational self-binding, together with independence of local experimental outcomes, including gravitational experiments, from the location and velocity of a freely falling laboratory. “Local” means that external tidal variation across the apparatus can be neglected. It does not mean that all curvature, or a laboratory's own gravitational field, can be removed. The weak principle concerns freely falling test bodies without appreciable self-gravity; laboratory composition tests alone therefore do not establish the strong version.

Lunar laser ranging supplies evidence specifically sensitive to the strong version. The Earth and Moon have different fractions of gravitational self-energy. If those fractions affected gravitational and inertial mass differently, their accelerations toward the Sun would differ and the lunar orbit would acquire a periodically varying polarization, the Nordtvedt effect. Combining lunar ranging with laboratory bounds on composition-dependent differential acceleration separates this self-energy effect. A determination available before this examination found the self-energy contribution to the Earth–Moon difference in $m_G/m_I$ to be $(-2.0\pm 2.0)\times 10^{-13}$, consistent with zero; see the 2004 experimental analysis. Universality-of-free-fall experiments, gravitational redshift measurements and tests of local Lorentz invariance support the weaker constituent principles, but should not be confused with a direct self-gravity test.

In general relativity, all matter couples to one metric tensor, and freely falling test particles follow its geodesics. At any event, Riemann normal coordinates give $g_{ab}={\eta }_{ab}$ and ${\Gamma }^a{}_{bc}=0$. Nongravitational local equations reduce to their special-relativistic forms in that frame. Gravitational binding is governed by the same metric field equations rather than a separate composition-dependent force, and the local gravitational coupling is universal. These features implement the strong equivalence principle. Curvature still governs geodesic deviation, so this construction removes uniform acceleration, not tidal gravity.

To geometrize Newtonian mechanics, use a four-dimensional manifold with coordinates $(t,x^1,x^2,x^3)$, absolute time ${\tau }_a=(dt{)}_a$ and a degenerate spatial inverse metric $h^{ij}={\delta }^{ij}$, $h^{0a}=0$. For a gravitational potential $\Phi (t,x)$ define the torsion-free Newton–Cartan gravitational connection byIt satisfies ${\nabla }_a{\tau }_b=0$ and ${\nabla }_a{h}^{bc}=0$. An affine geodesic has $d^{2}t/d{\lambda }^2=0$; choosing $\lambda =t$ makes its spatial equationsThis is Newton's free-fall law. The geometry is Newton–Cartan theory, with an affine connection and degenerate metric data, rather than a nondegenerate four-dimensional Lorentzian metric. The spatial slices themselves remain Euclidean.

Use the curvature conventionIn these coordinates the quadratic connection terms vanish, and the only nonzero curvature components, apart from antisymmetry in $c,d$, areIn particular $R_{00}={\nabla }^2\Phi$ and all other components of the Ricci tensor vanish. This Newtonian tidal curvature gives relative acceleration $\delta {\ddot{x}}^i=-{\partial }_i{\partial }_j\Phi \delta x^j$. Spatially constant gravitational acceleration can be removed by an accelerating frame because its Hessian vanishes. The Poisson equation for Newtonian gravity becomes $R_{ab}=4\pi G\rho {\tau }_a{\tau }_b$. Fluid mechanics is compatible with the same connection: the covariant continuity equation is the usual mass-conservation equation, and the spatial Euler equation is $u^b{\nabla }_b{u}^i=-{\rho }^{-1}{\partial }_{i}p$, which includes the gravitational force through the connection.

Now replace this absolute-time geometry by the Lorentzian metric of general relativity, with signature $(-,+,+,+)$. A perfect fluid haswhere $\epsilon$ is rest-frame energy density. In units $c=1$, $\epsilon \simeq \rho$ for a nonrelativistic fluid. The Newtonian equation suggests a covariant curvature equation sourced by $T_{ab}$. However $R_{ab}=\kappa T_{ab}$ alone conflicts with generic stress-energy conservation, because the contracted Bianchi identity gives ${\nabla }^a{R}_{ab}=\frac{1}{2}{\nabla }_{b}R$. Among expressions linear in Ricci curvature, conservation therefore selectsA constant multiple of the metric is also divergence-free, giving the possible cosmological constant term. This is a heuristic selection using locality, universal metric coupling and second-order metric equations; Newtonian matching by itself does not prove uniqueness among every possible gravity theory.

To fix the coefficient, first set the cosmological constant to zero. The trace-reversed equation would read $R_{ab}=\kappa (T_{ab}-\frac{1}{2}T{g}_{ab})$. In the fluid rest frame, $T=-\epsilon +3p$, so $R_{00}=\frac{\kappa }{2}(\epsilon +3p)$. In the weak static Newtonian limit, $g_{00}=-(1+2\Phi )$ and $R_{00}\simeq {\nabla }^2\Phi$. For negligible pressure the Poisson equation for Newtonian gravity then requires $\kappa /2=4\pi G$. Restoring $c$ uses $x^0=ct$, $R_{00}\simeq {\nabla }^2\Phi /c^2$ and $T_{00}\simeq \rho c^2$, giving $\kappa =8\pi G/c^4$. ThusPressure contributes to relativistic active gravity, while the low-pressure weak-field limit recovers Newton's Poisson equation. The comparison fixes the Newtonian normalization but does not fix $\Lambda$, which is an independent parameter.

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,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 $p$ withThis 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.^[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 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.^[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.

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 weak equivalence principle is universality of free fall for sufficiently small test bodies, independent of composition, when non-gravitational forces and self-gravity are negligible. Equality of inertial and passive gravitational mass makes this possible in the Newtonian limit. The Einstein equivalence principle adds local Lorentz invariance and local position invariance for nongravitational experiments: a small freely falling laboratory has the local physics of special relativity.

This motivates a universal spacetime metric. Freely falling unforced test particles follow geodesics, $u^{\nu }{\nabla }_{\nu }{u}^{\mu }=0$. At a point one can choose local inertial coordinates with vanishing connection, but curvature and tidal relative acceleration remain measurable over a finite region. Gravity cannot be transformed away globally. A uniformly accelerating small laboratory and a local gravitational field can have the same nongravitational experimental behaviour without having identical global geometry.

Clock comparisons predict a fractional rate difference of order $g\Delta h/c^2$ in a weak static field. The Pound-Rebka experiment tested the associated photon redshift through resonant gamma-ray emission and absorption at different heights, using controlled Doppler shifts to match the tiny energy displacement. Such a result tests the metric redshift prediction, not uniquely every component of the gravitational field equations.^[ref]

general relativity adds the dynamical Einstein field equations,The equivalence principle alone does not derive this unique equation; assumptions about covariance, dynamics and the Newtonian limit are also needed. The strong equivalence principle extends the universality claim to self-gravitating systems and gravitational experiments. Alternative metric theories can satisfy the weaker local principles while differing in gravitational dynamics, so composition tests, clock/redshift tests and self-gravity tests probe distinct claims.

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.

Local equivalence and tides. In Newtonian mechanics, inertial mass $m_{\mathrm{i}}$ measures the response to a force, whereas passive gravitational mass $m_{\mathrm{g}}$ couples to an imposed Newtonian gravitational field. A freely falling body obeysThe observed composition independence of free fall says that this ratio is universal. With a common choice of units it is one. This is the Newtonian content of the weak equivalence principle for test particles; it does not by itself specify all nongravitational physics.

The Einstein equivalence principle further says that local nongravitational experiments in a sufficiently small freely falling laboratory have the form predicted by special relativity, independently of the laboratory's location and motion. In such a local inertial frame, the metric tensor is locally the Minkowski metric and the Christoffel symbols vanish at the selected event. The strong equivalence principle extends the universality claim to local gravitational experiments and bodies with significant gravitational self-energy. None of these statements removes measurable tidal forces across a finite apparatus.

In an ideal small freely falling lift, a released ball initially floats relative to the cabin, because both undergo the same free fall. Two separated balls, however, need not retain their separation: they sample different parts of the gravitational field. For a point source, linearizing $\mathbf{g}=-GM\mathbf{r}/r^3$ across a small separation $\mathbf{\xi }$ gives the tidal tensor predictionA radial separation stretches with coefficient $2GM/r^3$, whereas a transverse separation is compressed with coefficient $-GM/r^3$. Initially parallel velocities therefore develop relative motion even though the individual bodies experience zero proper acceleration.

The relativistic statement is geodesic deviation. With the Riemann curvature tensor conventiona connecting vector between neighboring freely falling worldlines satisfiesHere $D/d\tau$ is the covariant derivative along the reference geodesic, and $U^{\mu }$ its four-velocity. Curvature can be nonzero where the Christoffel symbols have been made zero. In the weak, static, slow-motion limit this equation reduces to ${\ddot{\xi }}^i=-{\partial }_i{\partial }_j\Phi {\xi }^j$, recovering the tidal tensor above. A finite falling lift can detect gravity through these relative accelerations; local equivalence does not imply a globally uniform field.

Weak-field connection and potential. Take $x^0=ct$ and the $+---$ metric signature. There is an index error in the printed formula for the Christoffel symbols: its first differentiated metric component must be $g_{\nu 0}$, not $g_{\mu 0}$. The correctly contracted expression isThe offending term vanishes under the stipulated time independence, so this correction does not change the intended calculation. Write $g_{\mu \nu }={\eta }_{\mu \nu }+h_{\mu \nu }$ and treat both $h$ and its derivatives as first order. To that order $g^{00}=1-h_{00}$, $g^{0i}=O(h)$, and $g^{ij}=-{\delta }^{ij}+O(h)$. A stationary metric tensor givesFor the temporal component the inverse-metric coefficient is already first order, soFor a spatial component the zeroth-order inverse metric contributes:The time component of the approximate geodesic equation says that $dt/d\tau$ is constant to the retained order. In its spatial component, convert the proper time derivatives to coordinate-time derivatives and cancel $(dt/d\tau {)}^2$:Comparison with the Newtonian equation $d^2\mathbf{x}/d{t}^2=-\nabla \Phi$ identifies $h_{00}=2\Phi /c^2+C$. For an isolated source, choose the gravitational potential to tend to zero at infinity and normalize the time coordinate there, giving $C=0$. ThusThe quantity $\Phi$ is the Newtonian gravitational potential, or gravitational potential energy per unit mass, in $\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}={\mathrm{m}}^2{\mathrm{s}}^{-2}$. For a point source, $\Phi =-GM/r$. A static clock has $d\tau =\sqrt{g_{00}}dt\approx (1+\Phi /c^2)dt$, connecting the Newtonian limit with gravitational time dilation.

Frequency of the orbiting source. Treat the gas blob as a test particle on a timelike circular Schwarzschild geodesic, with pressure and other nongravitational forces negligible. Treat our very distant location as infinity, so the observer's own gravitational potential is negligible. These are the assumptions that make a unique answer in terms of $R_S/r$ possible.

Put $f=1-R_S/r$ and use a local static observer at the emission point as an intermediate measuring frame. The remote observer lies on the symmetry axis of the orbit. A ray arriving on that axis has zero conserved axial angular momentum, so its local direction is perpendicular to the emitter's azimuthal velocity. Bending within a meridional plane does not alter this orthogonality. The transverse relativistic Doppler effect therefore giveswhere $v$ is the orbital speed measured by that local static observer. The gravitational redshift between static Schwarzschild observers then givesA gravitational circular orbit has $v^2=GM/r+O((GM/r{)}^2/c^2)$ in the weak-field limit, so $v^2/c^2=R_S/(2r)+O((R_S/r{)}^2)$. Expanding both factors,Solving for the emitted frequency givesThe gravitational part is $R_S/(2r)$ and the transverse kinematic time dilation contributes another $R_S/(4r)$; the absence of line-of-sight motion does not remove this second effect.

As an exact check, the local speed on a timelike circular Schwarzschild geodesic satisfies $v^2/c^2=\mu /(r-2\mu )$, where $\mu =GM/c^2=R_S/2$. Therefore the face-on frequency shift of a circular Schwarzschild source iswhose expansion agrees with the boxed answer. If the gas is supported or accelerated by pressure or magnetic stresses rather than following a geodesic, its orbital speed is not fixed by the mass and radius alone; the appropriate first-order result is instead ${\nu }_{\mathrm{em}}\approx {\nu }_{\mathrm{ob}}[1+GM/(r{c}^2)+v^2/(2c^2)]$, with $v$ supplied independently.

Freely falling test particles with identical initial position and velocity follow the same trajectory independently of their material composition. The principle neglects appreciable self-gravity and nongravitational forces. The Einstein equivalence principle extends the local statement to nongravitational experiments; the strong equivalence principle also includes gravitational experiments and self-gravitating bodies.