The Einstein Equivalence Principle

The Einstein equivalence principle (EEP) strengthens universal free fall by making two further claims about local nongravitational experiments. In a freely falling laboratory, their outcomes are independent of the laboratory’s constant velocity and orientation, and independent of where and when the experiment is performed. Together these statements motivate a single Lorentzian metric that governs clocks, rods, freely falling test bodies, and minimally coupled nongravitational fields.

Required background. The formulation uses special-relativistic local laws, measurements made by relativistic observers, and the Newtonian gravitational field as a limiting comparison.

Helpful background. Tensor fields and local frames make the covariant statement economical, while accelerated-observer measurements help identify when a laboratory is not freely falling.

The three empirical components

EEP is the conjunction of three distinct propositions:

1. Weak equivalence principle: the trajectories of suitable freely falling test bodies are independent of composition and internal structure.
2. Local Lorentz invariance: the outcome of any local nongravitational experiment is independent of the velocity and orientation of the freely falling apparatus.
3. Local position invariance: the outcome of any local nongravitational experiment is independent of the spacetime location and epoch at which it is performed.

“Local” and “nongravitational” do substantial work. The apparatus must be small enough that tidal changes across it are below the retained accuracy. The experiment may involve electromagnetic, weak, strong, or mechanical processes, but not the gravitational field’s own dynamics or a comparison of gravitational self-energy. Will gives this operational decomposition and explains how its three components are tested by different classes of experiments Will 2014, § 2.1, pp. 6–15, PDF.

The components ask different experimental questions, and one should not substitute a successful test of one for tests of the others. At a phenomenological level, common trajectories can be postulated while a clock response depends on a preferred velocity; a varying scalar coupling can instead make local frequency ratios depend on position. Such examples distinguish the claims, but specifying one anomalous response is not yet a complete, self-consistent theory. Conservation laws and the internal energies of composite bodies can connect clock anomalies to free-fall anomalies. These connections, and the limitations of the conjecture that WEP alone implies EEP, are discussed in Will 2014, § 2.2.1, pp. 16–17, PDF.

Local special relativity and tidal remainders

Choose a freely falling timelike worldline and an orthonormal tetrad at an event $p$. In suitable local coordinates,

$$g_{ab}(p)=\eta_{ab}, \qquad \partial_c g_{ab}(p)=0, \qquad \Gamma^a{}_{bc}(p)=0.$$

Minimally coupled nongravitational equations then take their special-relativistic form at $p$. For example, the covariant Maxwell equations reduce at the central event to their inertial-coordinate form because the connection terms vanish there. This is a pointwise statement; it does not say that the metric is constant throughout the room.

If the laboratory has characteristic size $L$ and the curvature scale is represented by $|R|$, normal-coordinate corrections begin schematically at

$$g_{ab}=\eta_{ab}+O(|R|L^2),$$

with curvature gradients entering at higher order. Therefore the EEP limit requires $|R|L^2\ll1$ and corresponding control over the experiment’s duration. Reall’s normal-coordinate construction shows explicitly that the metric can be made Minkowskian with vanishing first derivatives at the central event Reall 2013, § 4.4, pp. 47–48, PDF. Along a freely falling central geodesic, Manasse and Misner compute the surviving quadratic curvature terms and their validity scale Manasse and Misner 1963, p. 743, Eqs. (65)–(66), and p. 745, Eqs. (84)–(86). A sufficiently sensitive extended experiment can detect tides without contradicting EEP; it has simply gone beyond the pointlike limit.

Why EEP motivates metric coupling

If all local nongravitational physics uses the same freely falling frames, a universal metric provides their common light cones, proper times, and local Lorentz transformations. Matter equations are then constructed covariantly from that metric and its Levi-Civita connection. Dimensionless local readings—frequency ratios, scattering angles, or clock-rate comparisons—are the appropriate observables; coordinate components alone are not.

This reasoning motivates the class often called metric theories of gravity: spacetime carries a Lorentzian metric, freely falling test bodies follow its timelike geodesics, and local nongravitational laws reduce to special relativity in freely falling frames. It does not choose the equation determining the metric. Einstein gravity, scalar–tensor models with universal matter coupling in an appropriate frame, and higher-curvature metric theories can share an EEP limit while having different gravitational field equations.

Minimal coupling is also not a mechanical uniqueness recipe. Flat-space actions differing by a divergence can yield curvature-dependent differences after naïve covariantization, so a proposed curved-spacetime matter action still requires physical input. But that ambiguity does not make every curvature coupling compatible with EEP. A pointwise ordinary-matter term such as $\xi R\phi^2$ leaves a curvature term in the scalar field equation even in normal coordinates at $p$, because $R(p)$ cannot be transformed away. The local law therefore does not reduce to its special-relativistic form; giving every species the same value of $\xi$ would not cure that failure. Okon and Callender 2011, § 5, p. 142, Eq. (6) use this example to expose the distinction between removing the connection and removing curvature.

Two other cases must be kept separate. Higher-curvature terms in the gravitational action can change the equation that determines $g_{\mu\nu}$ while leaving a universal minimally coupled matter action intact. Curvature-dependent finite-size operators can also describe tidal response of an extended body and disappear in the point-particle, point-laboratory limit. EEP constrains the pointwise nongravitational law; it neither selects the gravitational field equation nor forbids controlled tidal corrections outside its local test-body domain.

Domain and common overstatements

EEP does not cover strongly self-gravitating bodies, whose gravitational binding energy can affect motion. It does not imply global Minkowski coordinates, fix spacetime topology, eliminate curvature, or establish Einstein’s equation. Nor does “uniform acceleration is equivalent to a uniform gravitational field” hold globally: a finite accelerated frame can possess horizons, and a genuinely uniform Newtonian field cannot reproduce arbitrary tidal curvature.

An independent check is to compare scales. Shrinking the apparatus with the central tetrad fixed sends $|R|L^2$ to zero and recovers a special-relativistic nongravitational law. Doubling $L$ increases the leading tidal correction by a factor of four. That scaling cleanly separates a failure of locality control from an intrinsic violation of local Lorentz or position invariance.

Exercises

Classify each hypothetical observation: (a) a spectral ratio depends on the freely falling apparatus’s orientation, (b) the same ratio varies with epoch after all local environmental variables are controlled, and (c) two neutral test bodies with different compositions acquire different accelerations.

Solution

(a) violates local Lorentz invariance because orientation is part of the laboratory’s local Lorentz state. (b) violates local position invariance because the dimensionless outcome depends on epoch. (c) violates WEP within the stated test-body domain. Each observation therefore attacks a different component of EEP; none should be relabeled as all three without further evidence.

References

• Manasse, F. K., and Charles W. Misner. “Fermi Normal Coordinates and Some Basic Concepts in Differential Geometry.” Journal of Mathematical Physics 4 (1963): 735–745. DOI.
• Okon, Elias, and Craig Callender. “Does Quantum Mechanics Clash with the Equivalence Principle—and Does It Matter?” European Journal for Philosophy of Science 1 (2011): 133–145. DOI.
• Reall, Harvey S. General Relativity. Cambridge Part III lecture notes (2013). PDF.
• Will, Clifford M. “The Confrontation between General Relativity and Experiment.” Living Reviews in Relativity 17, 4 (2014). DOI; arXiv.