Weak, Einstein, and Strong Equivalence Principles

The weak, Einstein, and strong equivalence principles are not three names for one proposition. They extend universality in different directions: from composition-independent motion of test bodies, to local nongravitational physics, and finally to experiments involving appreciable gravitational self-energy. A useful gravity test must identify which extension its observable actually probes.

Required background. Equivalence Principles and Gravitational Measurement supplies local inertial frames, free fall, clocks, and the limits of pointwise equivalence. Probability and Statistical Inference supplies likelihoods, nuisance parameters, and interval interpretation; the compact statistical model below makes the needed conventions explicit.

Helpful background. Weak-Field Gravity connects metric perturbations to measurable accelerations, while Tests of Gravity locates these distinctions in the wider experimental program.

Three operational statements

The weak equivalence principle (WEP), in its test-body form, says that the trajectory of an electrically neutral, negligibly self-gravitating body released with fixed initial position and velocity is independent of its internal composition. For two bodies $A$ and $B$ falling toward the same source, the standard differential observable is

$$\eta_{AB}=\frac{2(a_A-a_B)}{a_A+a_B}.$$

The sign depends on the declared acceleration direction and material order; the null statement is invariant: $\eta_{AB}=0$. To see what this measures, write $a_A=(1+\delta_A)g$ and $a_B=(1+\delta_B)g$ in one source field. Then $\eta_{AB}=2(\delta_A-\delta_B)/(2+\delta_A+\delta_B)\simeq\delta_A-\delta_B$ for small deviations. A universal change common to both bodies cancels: a differential experiment cannot determine an absolute rescaling of gravitational strength. A torsion balance, a differential accelerometer in orbit, or a dual-species atom interferometer can measure this ratio. Each comparison still needs an interaction range, source composition, test-body composition, and environmental model.

The Einstein equivalence principle (EEP) combines WEP with two further claims. Local Lorentz invariance says that the outcome of a local nongravitational experiment is independent of the freely falling laboratory’s velocity and orientation. Local position invariance says that the outcome is independent of where and when the experiment is performed. Clock redshift comparisons test position dependence; resonator, clock-anisotropy, and spin-precession searches test boost or orientation dependence. Will gives the operational separation and its connection to metric coupling in Will 2014, §2.1, pp. 6–25, PDF.

The strong equivalence principle (SEP) extends the local universality claim to gravitational experiments and to bodies whose gravitational binding energy is not negligible. One schematic sensitivity is

$$\frac{m_G}{m_I}=1+\eta_{\rm N}\frac{E_g}{Mc^2}+\cdots,$$

where $E_g\lt0$ is binding energy in the usual Newtonian convention and $\eta_{\rm N}=0$ in general relativity. For a uniform Newtonian sphere, assembling shells gives $E_g=-3GM^2/(5R)$, so the dimensionless expansion parameter is $E_g/(Mc^2)=-3GM/(5Rc^2)$. This explains why large compactness, rather than large mass alone, matters. The displayed mass-ratio equation is a weak-self-gravity parameterization, not a universal strong-field formula. Neutron stars require theory-dependent sensitivities because their masses and internal structures can respond nonlinearly to additional gravitational fields. Will uses the positive binding-energy magnitude; translating it to the negative $E_g$ here changes the displayed sign Will 2014, §4.3.1, Eq. (66), PDF.

Experiments classify the principle

Experiment	Compared outcome	Principle primarily tested	Essential boundary
Composition dipole or two test masses	$a_A-a_B$ in one external field	WEP	Negligible self-gravity; specified compositions and force range
Identical clocks at different potentials	Common fractional frequency shift	LPI component of EEP	Clock transfer, velocity, and potential model
Rotated clock, resonator, or spin system	Sidereal, orientation, or boost harmonics	LLI component of EEP	Coefficient basis and environmental harmonics
Earth and Moon falling toward the Sun	Differential motion including binding energy	Weak-self-gravity SEP	Lunar interior, tides, and planetary ephemeris
Pulsar and white dwarf in an external field	Differential compact-body acceleration	Strong-field SEP	Equation of state, timing model, and chosen gravity theory

The logical arrows are not reversible without additional hypotheses. EEP motivates universal coupling of nongravitational matter to a metric, but it does not select Einstein’s field equation. A scalar–tensor theory can preserve test-body universality while allowing a neutron star to acquire a scalar charge. That is a concrete counterexample to the inference “WEP therefore SEP.” Conversely, a clock-dependent redshift would violate LPI even if two ordinary test masses fell identically.

From observable to conclusion

For any of these tests, write the data as

$$\mathbf y=\mathbf m(\epsilon,\boldsymbol\nu)+\boldsymbol n, \qquad \mathcal L(\epsilon,\boldsymbol\nu) \propto \exp\!\left[-\frac12\mathbf r^{\mathsf T}C^{-1}\mathbf r\right],$$

where $\epsilon$ is the principle-violating parameter, $\boldsymbol\nu$ collects calibration and environmental nuisance parameters, $\mathbf r=\mathbf y-\mathbf m$, and $C$ is the declared covariance. A confidence interval is a property of a repeated-sampling construction; a credible interval additionally depends on a prior measure. Neither turns a phenomenological null result into proof that every alternative theory is absent.

The inference also cannot skip the source-to-observable map. A composition-dependent scalar force may couple to baryon number, neutron excess, electromagnetic binding energy, or another charge. Two materials with nearly parallel charge vectors give little discrimination even if their acceleration difference is measured precisely. Likewise, a strong-field theory can reduce exactly to the weak-field null while differing for compact bodies. The tested principle is therefore always qualified by composition, compactness, environment, range, and model space.

Limits of the hierarchy

“Local” does not mean an arbitrarily large laboratory. Connection coefficients can be removed along one freely falling worldline at one event, whereas tidal effects scale schematically as $R_{\alpha\beta\gamma\delta}L^2$ across a laboratory of size $L$. Detecting curvature is not an equivalence-principle violation. Nor does universality of free fall imply that every object follows a geodesic: spin, finite size, charge, radiation reaction, and tidal multipoles can produce known body-dependent motion outside the test-particle approximation.

The clean conclusion is thus conditional. A WEP experiment constrains differential response for the chosen bodies and source. An LLI or LPI experiment constrains specified coefficients or clock sensitivities. An SEP experiment constrains how self-gravity enters within a chosen compact-body theory. Keeping those statements separate prevents both underclaiming and the much more common error of promoting one null test into a universal verdict.

Exercise

A torsion balance finds $\eta_{AB}=0$ for two laboratory materials, while a scalar–tensor model predicts identical scalar charge-to-mass ratios for those materials but a nonzero scalar charge for neutron stars. Which principles and which sector does the experiment test?

Solution

The torsion balance tests test-body WEP for the selected material pair, source, interaction range, and environment. Because the model assigns the two materials equal charge-to-mass ratios, its predicted differential signal vanishes, so the null result does not distinguish it from general relativity. The neutron-star charge concerns strong self-gravity and therefore SEP. A pulsar–white-dwarf comparison, together with the theory-dependent compact-body sensitivities, is needed because laboratory WEP does not imply SEP.

References

• Will, Clifford M. “The Confrontation between General Relativity and Experiment.” Living Reviews in Relativity 17 (2014): 4. DOI; Open PDF.

Further reading

• Damour, Thibault. “Theoretical Aspects of the Equivalence Principle.” Classical and Quantum Gravity 29 (2012): 184001. DOI.
• Di Casola, Eolo, Stefano Liberati, and Sebastiano Sonego. “Nonequivalence of Equivalence Principles.” American Journal of Physics 83 (2015): 39–46. DOI.