Equivalence Principle Tests

The equivalence principle is the conceptual seed from which general relativity grows — and, uniquely among the pillars of the theory, it is a claim you can falsify in a laboratory. Galileo's observation that heavy and light bodies fall together, sharpened by Newton into the empirical identity of inertial and gravitational mass and elevated by Einstein into the local indistinguishability of gravity and acceleration, is not one proposition but a hierarchy of increasingly strong, separately testable claims. The Weak Equivalence Principle (WEP) — the universality of free fall — is the weakest and most precisely tested: the trajectory of an uncharged test body depends only on where it starts and how fast, never on what it is made of. The Einstein Equivalence Principle (EEP) adds two demands on all local non-gravitational physics: local Lorentz invariance (LLI), that a freely-falling laboratory has no preferred velocity or direction, and local position invariance (LPI), that it gets the same answers everywhere and everywhen. The Strong Equivalence Principle (SEP) extends this even to a body's gravitational self-energy — satisfied, to the order tested, only by general relativity itself. Schiff's conjecture ties these together: any complete, self-consistent theory obeying WEP must also obey LLI and LPI, so EEP is effectively the statement gravity is a metric theory — matter couples universally to one spacetime metric g_μν, and free bodies follow its geodesics. This article is the experimental spine of that claim: torsion balances and MICROSCOPE for WEP, Hughes–Drever clocks for LLI, redshift networks for LPI, and lunar ranging for SEP.

Key takeaways

The Einstein Equivalence Principle rests on three separately testable pillars: universality of free fall (WEP), local Lorentz invariance (LLI), and local position invariance (LPI).
WEP is quantified by the Eötvös parameter $\eta = 2(a_1-a_2)/(a_1+a_2)$ , now measured by the MICROSCOPE satellite as $\eta(\mathrm{Ti},\mathrm{Pt})=(-1.5\pm2.3_{\mathrm{stat}}\pm1.5_{\mathrm{syst}})\times10^{-15}$ — consistent with zero, testing WEP at the $10^{-15}$ level.
LPI predicts the gravitational redshift $\Delta\nu/\nu=\Delta\Phi/c^2$ , which follows from the equivalence principle alone — no field equations — and is now read by optical clocks over centimetre heights.
LLI demands no preferred frame; Hughes–Drever and clock-comparison experiments bound spatial anisotropy to $\lesssim 10^{-22}$ .
The Strong EP adds gravitational binding energy; lunar laser ranging bounds the Nordtvedt parameter to $|\eta_0|\lesssim 4\times10^{-4}$ , confirming that gravitational energy gravitates like every other kind.
Key idea: the equivalence principle is not one statement but a testable hierarchy — WEP (universal free fall), LLI (no preferred frame), and LPI (no preferred place or time); together they are empirically equivalent to the assertion that gravity is a metric theory, confirmed from the $10^{-15}$ (WEP) to $10^{-22}$ (LLI) level.

The core insight

The equivalence principle is not one statement but a testable hierarchy — WEP (universal free fall), LLI (no preferred frame), and LPI (no preferred place or time). Together (EEP) they are empirically equivalent to the assertion that gravity is a metric theory in which matter couples universally to a single spacetime metric, and they are confirmed from the $10^{-15}$ (WEP) to $10^{-22}$ (LLI) level.

Where this lives today

Equivalence-principle tests are among the most active precision frontiers in physics because a single violation would rewrite gravity. WEP is led by the MICROSCOPE satellite (CNES, 2016–2018), which compared the orbital free fall of concentric platinum and titanium cylinders to test WEP at the $10^{-15}$ level, with atom interferometers dropping rubidium and potassium clouds to extend the test into the quantum regime and aim below $10^{-17}$ . LPI is now a laboratory science: transportable optical-lattice clocks resolve the redshift over centimetres, while the eccentric-orbit Galileo satellites 5 and 6 improved the redshift test to $\sim10^{-5}$ . LLI feeds the Standard-Model Extension, where Hughes–Drever and comagnetometer experiments bound Lorentz violation to $\lesssim10^{-22}$ . And SEP is watched by lunar laser ranging and by binary pulsars — including one in a triple stellar system, whose free fall confirms SEP for a strongly self-gravitating body to parts in $10^{6}$ .

The physical picture

Stand in a sealed elevator. On Earth's surface, drop two balls of different material and they strike the floor together (WEP). Now cut the cable: the elevator falls freely, the balls hover beside you, and — locally — gravity has vanished. Einstein's insight was that no experiment confined to the box can tell "freely falling in gravity" from "floating in deep space," nor "standing on Earth" from "accelerating at g in a rocket." That indistinguishability is the equivalence principle, in its sharpest form: the falling elevator and the drifting spacecraft are the same laboratory.

Figure

A freely falling elevator near Earth is locally indistinguishable from a spacecraft drifting in deep space: in both, platinum and titanium test bodies hover together because free fall is universal — any residual separation would register a nonzero Eötvös parameter, which MICROSCOPE finds consistent with zero at the 10⁻¹⁵ level

The word "local" is everything. Drop two balls separated horizontally and they slightly converge, both aiming at Earth's centre; separated vertically, the lower one falls faster. These tidal accelerations survive in every frame — the signature of true curvature, the residual the equivalence principle cannot erase. The three pillars ask three independent questions of the falling laboratory: does the trajectory care what the body is made of (WEP)? does the physics care how fast or which way the lab moves (LLI)? does it care where and when in the cosmos it sits (LPI)? A "yes" to any betrays matter coupling to something beyond one universal metric. The second figure shows how emphatically the answer for WEP has been "no" — the bound on $\eta$ falling six orders of magnitude across a century.

Figure

A century of tightening bounds on the Eötvös parameter η, the figure of merit for universality of free fall, on a logarithmic axis: Eötvös 1922 near 5×10⁻⁹, Dicke 1964 near 1×10⁻¹¹, Braginsky 1972 near 1×10⁻¹², Eöt-Wash near 2×10⁻¹³, and MICROSCOPE 2022 at the 10⁻¹⁵ level — each null result a sharper confirmation that all bodies fall alike

The core equations

The whole subject turns a null experiment into a number: each principle carries a figure of merit that is exactly zero if it holds, and whose measured bound is the strength of the test. The Weak Equivalence Principle compares the free-fall accelerations of two compositions in the same field — the Eötvös parameter is their fractional difference.

\eta \equiv 2\,\frac{a_1 - a_2}{a_1 + a_2} \qquad (1)

where: $\eta$ — dimensionless WEP-violation parameter, zero if free fall is universal; $a_1, a_2$ — free-fall accelerations of two bodies of different composition in the same field. Reading: as a normalized difference $\eta$ is insensitive to the absolute value of $g$ and reports only composition dependence — precisely what WEP forbids. Splitting the acceleration into gravitational and inertial mass makes explicit what $\eta$ captures.

m_I\,a = m_G\,g \;\Longrightarrow\; a=\frac{m_G}{m_I}\,g,\qquad \eta = 2\,\frac{(m_G/m_I)_1-(m_G/m_I)_2}{(m_G/m_I)_1+(m_G/m_I)_2} \qquad (2)

where: $m_I$ — inertial mass; $m_G$ — passive gravitational mass; $g$ — local field. Reading: WEP is exactly the statement that $m_G/m_I$ is one universal constant for every body, so it cancels and $\eta=0$ . Any species-dependent piece — from nuclear binding energy or a new fifth-force charge — shows up directly in $\eta$ . Gravity can be removed because at any point one can build a freely-falling frame in which the metric is Minkowski with vanishing first derivatives — the mathematical content of the equivalence principle.

g_{\mu\nu}(\mathcal{P}) = \eta_{\mu\nu}, \qquad \partial_\alpha g_{\mu\nu}(\mathcal{P}) = 0 \qquad (3)

where: $g_{\mu\nu}$ — spacetime metric; $\eta_{\mu\nu}=\mathrm{diag}(-1,1,1,1)$ — Minkowski metric; $\mathcal{P}$ — the point at which the local inertial frame is erected; $\partial_\alpha$ — coordinate partial derivative. Reading: with the metric flat and its first derivatives zero at $\mathcal{P}$ , every Christoffel symbol vanishes there — the freely-falling observer feels no gravity. What the equivalence principle cannot remove is the second derivative of the metric: in Riemann normal coordinates the leading correction is quadratic, with the Riemann tensor as coefficient.

g_{\mu\nu}(x) = \eta_{\mu\nu} - \tfrac{1}{3}\, R_{\mu\alpha\nu\beta}(\mathcal{P})\, x^\alpha x^\beta + \mathcal{O}(x^3) \qquad (4)

where: $R_{\mu\alpha\nu\beta}$ — Riemann curvature tensor at $\mathcal{P}$ ; $x^\alpha$ — normal coordinates from $\mathcal{P}$ . Reading: the $-\tfrac13 R\,x x$ term is the tidal field; being a coordinate invariant it cannot be transformed away — which is why the equivalence principle is exactly local and why curvature, not the metric or connection, is the true measure of gravity. Local Position Invariance predicts the gravitational redshift, the workhorse LPI test: for a small height difference the fractional frequency shift equals the potential difference over $c^2$ .

\frac{\Delta \nu}{\nu} = \frac{\Delta \Phi}{c^2} = \frac{g\,h}{c^2} \qquad (5)

where: $\Delta\nu/\nu$ — magnitude of the fractional frequency shift between two clocks separated by $\Delta\Phi=gh>0$ , written positive: a signal climbing the well is redshifted by this fraction, one descending is blueshifted by it; $\Delta\Phi$ — Newtonian potential difference between emitter and receiver; $g$ — local acceleration; $h$ — height difference. Reading: a photon climbing a potential well loses energy and reddens; the deviation of the measured shift from this prediction defines the LPI anomaly $\delta=(\Delta\nu/\nu-\Delta\Phi/c^2)/(\Delta\Phi/c^2)$ , zero if LPI holds. The Strong Equivalence Principle is probed through gravitational self-energy: a self-gravitating body would fall anomalously if its binding energy did not gravitate like ordinary mass. The Nordtvedt effect quantifies this.

\eta_{\mathrm{SEP}} = \eta_0\,\frac{E_{\mathrm{grav}}}{m c^2},\qquad \eta_0 = 4\beta-\gamma-3 \qquad (6)

where: $\eta_{\mathrm{SEP}}$ — differential free-fall of a self-gravitating body; $E_{\mathrm{grav}}$ — gravitational self-binding energy; $mc^2$ — total rest energy; $\eta_0$ — Nordtvedt parameter, built from the post-Newtonian $\beta,\gamma$ . Reading: general relativity has $\beta=\gamma=1$ , so $\eta_0=0$ exactly — a nonzero $\eta_0$ would make Earth and Moon fall toward the Sun at different rates.

Derivation AGravitational redshift from the equivalence principle alone

The goal here is to derive the gravitational redshift $\Delta\nu/\nu=\Delta\Phi/c^2$ using nothing but local position invariance and the special-relativistic Doppler effect — Einstein's 1907 argument, made two years before he had any field equations. It shows the redshift is a consequence of the equivalence principle as such, not of the dynamics: any metric theory predicts exactly this leading shift, so a measured deviation attacks the principle itself. We work in a freely-falling frame, where gravity is absent and only kinematics remains.

Step 1. Set up a tower of height $h$ in a static gravitational field $g$ . At the instant light of frequency $\nu_e$ leaves the floor, release a freely-falling frame from rest coincident with the emitter. Why: in the falling frame gravity has been transformed away, so the photon travels with no gravitational shift and any frequency change must come from ordinary relative motion.

Step 2. The light takes time $t=h/c$ to reach the top. During that interval the falling frame — accelerating downward at $g$ relative to the lab — acquires a downward speed $v=gt=gh/c$ , so relative to that frame the receiver at the top is moving upward, away from the source. Why: the lab accelerates upward relative to the freely-falling frame, like the far wall of a rocket.

Step 3. Apply the first-order Doppler shift for a receiver receding at $v$ . With the signed convention $\Delta\nu\equiv\nu_{\rm rec}-\nu_{\rm em}$ this reads $\Delta\nu/\nu=-v/c=-gh/c^2$ — the minus sign is the statement that the climbing photon arrives reddened. Higher-order terms are of order $(gh/c^2)^2$ and utterly negligible. Why: to leading order the classical and relativistic Doppler formulae agree, and only this term is needed since $gh\ll c^2$ .

Step 4. Rewrite via the potential $\Delta\Phi=gh$ to reach the frame-independent statement $(\nu_{\rm rec}-\nu_{\rm em})/\nu=-\Delta\Phi/c^2$ : light climbing out of a well reddens. Quoted as a magnitude, as in equation (5) and the worked examples, this is $|\Delta\nu|/\nu=\Delta\Phi/c^2$ . For the Pound–Rebka tower, $g=9.80\ \mathrm{m/s^2}$ and $h=22.5\ \mathrm{m}$ give $gh/c^2\approx2.45\times10^{-15}$ . Why: the potential form frees the shift from the geometry and lets it be compared directly with the metric prediction and experiment.

Go deeper · Derivation B · 1 minThe local inertial frame and the curvature it cannot erase

Derivation BThe local inertial frame and the curvature it cannot erase

The aim is to make precise both halves of the equivalence principle: that gravity can always be removed to first order by choosing a freely-falling frame, and never at second order because the residue is the Riemann tensor. This is the quantitative content of "strictly local" — what separates a genuine gravitational field from a merely accelerated coordinate system. We build Riemann normal coordinates about a point and read off the metric expansion.

Step 1. At any point $\mathcal{P}$ choose coordinates so that $g_{\mu\nu}(\mathcal{P})=\eta_{\mu\nu}$ and $\partial_\alpha g_{\mu\nu}(\mathcal{P})=0$ . A coordinate transformation has just enough free parameters to impose both: ten to diagonalize the metric to Minkowski form and forty to kill its first derivatives. Why: with first derivatives gone every Christoffel symbol vanishes at $\mathcal{P}$ , so the geodesic equation reduces to $\ddot x^\mu=0$ — a freely-falling observer sees force-free straight-line motion.

Step 2. The second derivatives of the metric contain more components than a coordinate transformation can absorb; the irreducible remainder is precisely the twenty independent components of the Riemann tensor. Why: this is why curvature, not the metric or connection, is the coordinate-invariant carrier of gravitational information.

Step 3. Expanding with those constraints gives $g_{\mu\nu}(x)=\eta_{\mu\nu}-\tfrac13 R_{\mu\alpha\nu\beta}(\mathcal{P})\,x^\alpha x^\beta+\dots$ , the coefficient fixed by matching to the geodesic-deviation equation. Why: the same $R_{\mu\alpha\nu\beta}$ that governs tidal accelerations between neighbouring geodesics appears in the metric expansion, tying "tides" and "curvature" into one object.

Step 4. Estimate the scale at which the residue matters: near Earth the tidal gradient is $2g/R_\oplus\approx3.1\times10^{-6}\ \mathrm{s^{-2}}$ , so across a $1\ \mathrm{m}$ lab the differential acceleration is only $\sim3\times10^{-6}\ \mathrm{m/s^2}$ , one part in $3\times10^{6}$ of $g$ . Why: the equivalence principle is not approximate but exactly local — it holds to whatever accuracy you shrink the box, corrections scaling as curvature times size squared.

Go deeper · Derivation C · 1 minSchiff's conjecture — why weak forces Einstein

Derivation CSchiff's conjecture — why weak forces Einstein

The aim is the heuristic proof of Schiff's conjecture: that a theory obeying the Weak Equivalence Principle, together with local energy conservation, is forced to obey Local Position Invariance too, so the three seemingly independent pillars collapse into one. The payoff is enormous — a single ultra-precise WEP experiment indirectly constrains LLI and LPI, and EEP becomes tantamount to "gravity is a metric theory," the licence for the whole PPN programme.

Step 1. Suppose WEP holds but LPI fails, so that two atomic species A and B redshift by different fractions when raised through a potential difference $\Delta\Phi$ . Take an atom of species A at height $h$ , let it emit a photon, and drop the photon to the ground. Why: assuming a differential redshift is the hypothesis we intend to destroy by contradiction.

Step 2. Track the energy. The photon gains $E\,\Delta\Phi/c^2$ falling through the potential, exactly the work needed to have raised the equivalent mass $E/c^2$ — provided the redshift matches the potential universally. Why: energy conservation for one species fixes the factor $\Delta\Phi/c^2$ ; the question is whether a second species may differ.

Step 3. Close a two-species loop: emit with A, absorb with B, convert, lower, repeat. If A and B redshifted by different fractions, each cycle would return more energy than it cost, a perpetuum mobile. Local energy conservation forbids it, so the redshift must be species-independent: $\Delta\nu_A/\nu_A=\Delta\nu_B/\nu_B=\Delta\Phi/c^2$ . Why: demanding no free energy per cycle is exactly the demand that LPI hold, derived from WEP plus conservation.

Step 4. A parallel construction links WEP to LLI, and the chain is made rigorous in the THεμ formalism, where a single non-metric coupling generates WEP, LLI, and LPI violations of comparable size. Why: this is why EEP is effectively the single statement that all matter couples to one metric — "gravity is geometry."

Limiting cases

The strength of an equivalence-principle claim depends entirely on which class of physics the local laboratory is permitted to contain — test bodies only, all non-gravitational fields, or self-gravitating matter as well.

Case	Result	Meaning
Weak EP (WEP)	$a_1=a_2$ for all test bodies	Only trajectories of uncharged, negligibly self-gravitating bodies are constrained; tested at the $10^{-15}$ level by MICROSCOPE.
Einstein EP (EEP)	$\text{WEP}+\text{LLI}+\text{LPI}$	All local non-gravitational physics is universal and frame/position-independent; implies gravity is a metric theory.
Strong EP (SEP)	EEP extended to self-gravity and local $G$	Gravitational binding energy falls like all energy; satisfied by GR, generically violated by scalar-tensor theories. Tested by lunar ranging.
Local inertial limit	$g_{\mu\nu}(\mathcal{P})=\eta_{\mu\nu},\ \partial_\alpha g_{\mu\nu}(\mathcal{P})=0$	At a point the metric is Minkowski with vanishing first derivatives; curvature enters only at second order via $R_{\mu\alpha\nu\beta}$ .
Extended body / long baseline	tidal term $-\tfrac13 R_{\mu\alpha\nu\beta}x^\alpha x^\beta$	Over finite size the equivalence principle fails at order (curvature)×(size)²; geodesic deviation is physical and frame-independent.

Common trapthinking the freely-falling elevator is exactly Minkowski

The equivalence principle is strictly local, and the most common error is to forget it. There is no global frame in which gravity disappears: a freely-falling elevator is Minkowski only to first order in its size, with corrections of order $R_{\mu\alpha\nu\beta}$ times the squared box dimensions. Two balls released side by side slowly converge — real tidal motion, governed by the Riemann tensor, that no coordinate change can remove. A uniformly accelerated rocket in flat spacetime has zero Riemann tensor and no such convergence; that difference is how the occupant tells true gravity from acceleration. Equally, WEP applies only to bodies of negligible self-gravity: a neutron star or planet, with significant binding energy, probes the logically distinct Strong equivalence principle, which can be violated even where WEP holds perfectly.

Worked examples

1 · The Pound–Rebka redshift (fully worked)

A gamma-ray photon is emitted at the base of the Jefferson tower at Harvard and detected $h=22.5\ \mathrm{m}$ up, with $g=9.80\ \mathrm{m/s^2}$ . What fractional frequency shift does LPI predict, and what emitter speed would mimic it?

Step 1. Apply equation (5): $\dfrac{\Delta\nu}{\nu}=\dfrac{gh}{c^2}=\dfrac{220.5}{8.99\times10^{16}}$ . The photon climbs the well, so it redshifts.

Step 2. Evaluate: $\dfrac{\Delta\nu}{\nu}\approx\mathbf{2.45\times10^{-15}}$ — the celebrated "one part in $10^{15}$ " that Pound and Rebka set out to measure in 1959.

Step 3. The equivalent Doppler speed is $v=c\,(\Delta\nu/\nu)=gh/c\approx\mathbf{7.4\times10^{-7}\ m/s}$ — under a micron per second. Pound and Rebka moved the source on a loudspeaker cone at exactly such speeds to sweep the Mössbauer line, agreeing with the prediction to $\sim1\%$ , the first terrestrial confirmation of the gravitational redshift. A 22.5 m tower shifts a gamma ray by 2.45×10⁻¹⁵, mimicked by a 0.74 μm/s source velocity.

2 · Optical clocks resolve LPI over a tabletop

Strontium optical-lattice clocks reach a fractional frequency uncertainty near $1\times10^{-18}$ . Over what height does the gravitational redshift become resolvable, and what does this let them test?

Step 1. Set $\Delta\nu/\nu = gh/c^2$ equal to the clock resolution: over $h=1\ \mathrm{m}$ , $gh/c^2=9.80/(8.99\times10^{16})=\mathbf{1.09\times10^{-16}}$ — a hundred times the clock floor, easily resolved.

Step 2. Push down to $h=1\ \mathrm{cm}$ : $gh/c^2=\mathbf{1.09\times10^{-18}}$ , right at the $10^{-18}$ frontier. Two such clocks stacked a centimetre apart tick at measurably different rates — the redshift as a benchtop effect, demonstrated by JILA and NIST.

Step 3. Comparing clocks of different atomic species over the same height, or over the annually varying solar potential as Earth orbits, bounds the LPI anomaly $\delta$ and searches for drifting constants. A 1 cm height gives a 1.09×10⁻¹⁸ redshift — resolvable today, turning LPI into a laboratory measurement.

3 · The Nordtvedt effect and the falling Earth

The Earth's gravitational self-binding energy is $E_{\mathrm{grav}}/m_\oplus c^2\approx-4.6\times10^{-10}$ of its rest energy, and lunar ranging bounds the Nordtvedt parameter to $|\eta_0|\lesssim4\times10^{-4}$ . What differential free-fall of Earth and Moon toward the Sun does this permit?

Step 1. A uniform-density sphere has $E_{\mathrm{grav}}=-\tfrac35 GM^2/R$ , so $|E_{\mathrm{grav}}/m c^2|=\tfrac35\,GM/(Rc^2)\approx\mathbf{4.2\times10^{-10}}$ ; the centrally condensed real Earth gives $\approx4.6\times10^{-10}$ .

Step 2. From equation (6), the maximum SEP violation is $\eta_{\mathrm{SEP}}=\eta_0\,(E_{\mathrm{grav}}/mc^2)\lesssim(4\times10^{-4})(4.6\times10^{-10})\approx\mathbf{1.8\times10^{-13}}$ ; the Moon's far smaller binding energy is negligible here.

Step 3. Such a differential would polarize the lunar orbit toward the Sun by tens of centimetres. Decades of ranging to the Apollo retroreflectors see none, confirming that gravitational energy gravitates exactly like all other energy — a clean success of the Strong equivalence principle. General relativity predicts η₀ = 0 (β = γ = 1); ranging confirms it to |η₀| ≲ 4×10⁻⁴.

Physics of the portrait

The equivalence principle carries the name of a Hungarian baron who never set out to test relativity — he was mapping the Earth's gravity — yet built the instrument that still defines the field.

Origins · 1 minLoránd Eötvös and how this was discovered

Loránd Eötvös

1848–1919 · torsion-balance tests of the universality of free fall

Baron Loránd Eötvös of Vásárosnamény was a Hungarian physicist whose torsion balance turned Newton's assertion that inertial and gravitational mass are equal from a philosophical convenience into a laboratory number. From the 1880s until his death he suspended pairs of masses of different materials from a fine fibre and looked for the tiny differential twist a composition-dependent free fall would produce as the Earth's rotation modulated the centrifugal and gravitational pulls. His genius was to turn a static, unmeasurable difference into a signal at a known frequency by rotating the apparatus — the trick that still underlies every modern test. By 1922, in results published posthumously with Pekár and Fekete, he had shown $m_G/m_I$ universal to a few parts in $10^{9}$ , unrivalled for four decades. When Einstein built general relativity on the exact equality of the two masses, Eötvös's null result was the empirical ground beneath it. The lineage runs straight from his fibre to Dicke and Braginsky, to the Eöt-Wash balances, and to MICROSCOPE — each reporting the same parameter $\eta$ he defined.

History

1590

s — Galileo argues that bodies of different weight fall at the same rate

1687

— Newton posits and pendulum-tests the equality of inertial and gravitational mass

1907

— Einstein states the equivalence principle and derives the gravitational redshift

1922

— Eötvös, Pekár and Fekete publish η ≲ few×10⁻⁹ from the torsion balance

1960

— Pound and Rebka confirm the redshift in a 22.5 m tower

1976

— Gravity Probe A verifies the redshift to 1.4×10⁻⁴

2022

— MICROSCOPE measures η(Ti,Pt) = (−1.5 ± 2.3 ± 1.5)×10⁻¹⁵, null at the 10⁻¹⁵ level

Going further

Texts. Misner, Thorne & Wheeler, Gravitation (1973), §16, §38–40 · Wald, General Relativity (1984), §4.1 · Carroll, Spacetime and Geometry (2004), §4.1, §5.1 · Hartle, Gravity (2003), §6 · Schutz, A First Course in General Relativity, 2nd ed. (2009), §5 · Will, Theory and Experiment in Gravitational Physics, 2nd ed. (2018), Ch. 2. The key numerical benchmarks used throughout this article are: the MICROSCOPE WEP bound $|\eta|=(-1.5\pm2.3_{\mathrm{stat}}\pm1.5_{\mathrm{syst}})\times10^{-15}$ ; the Gravity Probe A redshift anomaly $\delta\sim1.4\times10^{-4}$ ; the Hughes–Drever LLI anisotropy $\lesssim10^{-22}$ ; and the lunar-ranging Nordtvedt bound $|\eta_0|\lesssim4\times10^{-4}$ . Spot-checks on the worked examples: the Pound–Rebka shift $gh/c^2=2.45\times10^{-15}$ ; the optical-clock redshift $1.09\times10^{-16}$ per metre and $1.09\times10^{-18}$ per centimetre; the Earth self-energy fraction $\tfrac35 GM/(Rc^2)\approx4.2\times10^{-10}$ ; $\eta_0=4\beta-\gamma-3=0$ for $\beta=\gamma=1$ ; and the coefficient $-\tfrac13$ in equation (4).

Builds on: Newtonian Limit of General Relativity · Weak-Field Limit and Linearized Gravity · Gravitational Redshift. Leads to: Parametrized Post-Newtonian (PPN) Formalism · Pound–Rebka Experiment · Geodetic Precession.

Ask the tutor: Ask me to walk through Einstein's 1907 redshift argument line by line, or to show why the $-\tfrac13$ curvature term makes the equivalence principle exactly local.

What is the difference between the weak, Einstein, and strong equivalence principles?

They form a nested hierarchy of ever-stronger claims. The Weak EP says uncharged test bodies fall identically regardless of composition — a statement only about trajectories. The Einstein EP adds two demands on all local non-gravitational physics: local Lorentz invariance (no preferred velocity or direction) and local position invariance (no preferred place or time), and it implies gravity is a metric theory. The Strong EP extends both to self-gravitating bodies and to locally measured $G$ . General relativity satisfies all three; most alternatives satisfy WEP and EEP but violate SEP — which is why SEP tests discriminate so sharply.

Why can't the equivalence principle hold globally?

Because tidal forces are real and cannot be transformed away. In a freely-falling frame you can make the metric Minkowski and its first derivatives vanish at one point, so gravity disappears locally. But the second derivatives survive as the Riemann curvature tensor, producing the geodesic deviation that pulls neighbouring free-fall trajectories together or apart — two balls released across a falling elevator slowly converge because both aim at Earth's centre, whereas a uniform acceleration in flat spacetime produces none. That residual tidal motion is the coordinate-invariant signature of true gravity, so the equivalence principle is exactly and only local.

How does a torsion balance turn a static effect into a clean signal?

A composition-dependent free fall would deflect two masses on a beam by slightly different amounts, twisting the suspending fibre by a fixed angle — hopeless to measure against slow drifts and thermal noise. Eötvös's insight was to rotate the apparatus, so the would-be constant torque becomes a periodic signal at a known frequency. Lock-in detection there rejects everything not modulated in step, pushing the sensitivity to $\eta$ down by orders of magnitude. Eöt-Wash balances rotate on a turntable; MICROSCOPE uses its orbital motion.

Does the gravitational redshift prove general relativity specifically?

No — and that is precisely why it is valuable. The redshift $\Delta\nu/\nu=\Delta\Phi/c^2$ follows from local position invariance and special relativity alone, so every metric theory of gravity predicts the same leading result. A redshift measurement therefore tests the equivalence principle — the foundation shared by all metric theories — not the specific field equations. To distinguish general relativity from competitors you need effects depending on the post-Newtonian $\beta$ and $\gamma$ : light deflection, Shapiro delay, perihelion precession. The redshift confirms gravity is geometric; the PPN tests pin down which geometry.

What would a violation of the equivalence principle actually look like?

It depends on which pillar breaks. A WEP violation would show up as a nonzero Eötvös parameter — two materials falling at measurably different rates, most likely from a new composition-dependent "fifth force" coupling to baryon or lepton number. An LPI violation would appear as a redshift anomaly $\delta\neq0$ or a drift of fundamental constants with the local potential. An LLI violation would make transition frequencies depend on the laboratory's orientation or velocity through the cosmos. Any of these would signal matter coupling to a field beyond a single universal metric, breaking the geometric picture and pointing to new physics such as scalar-tensor theories or Lorentz-violating extensions.

Why does testing WEP so precisely also constrain LLI and LPI?

Because of Schiff's conjecture: any complete, self-consistent theory that satisfies the Weak equivalence principle is forced, by energy conservation and internal consistency, to satisfy local Lorentz invariance and local position invariance too. The heuristic proof is a perpetual-motion argument — if two atomic species redshifted differently (an LPI violation) while both obeyed WEP, you could build a cycle that manufactures free energy; ruling that out forces universal redshift. The THεμ test-theory formalism makes this rigorous. So an ultra-precise WEP null result — like MICROSCOPE's — quietly bounds all three.