The Weak Equivalence Principle

The weak equivalence principle (WEP) is a universality statement about the motion of carefully idealized test bodies. It does not say that every object in every circumstance moves alike. It says that, after matching initial event and initial velocity and holding non-gravitational charges and structure outside the experiment, the trajectory in a given external gravitational field is independent of composition and mass.

Required background. We use special-relativistic worldlines, observer-measured motion, and the Newtonian field and potential as the operational comparison model.

Helpful background. Tensor fields provide the covariant language used below, and accelerometer readings distinguish free fall from supported or forced motion.

Operational statement of the principle

Consider two neutral, nonspinning bodies $A$ and $B$ released at the same event $p$ with the same future-directed timelike initial four-velocity. Treat both as probes: their size and gravitational self-field are negligible compared with the scale and strength of the prescribed external field. The WEP states that changing the bodies’ internal composition does not change the spacetime curve they follow, apart from a relabeling of the curve parameter.

“Same initial velocity” is indispensable. Two stones launched in different directions trace different paths without violating universality. “Same event” is equally important in an inhomogeneous field. The comparison must also control electric charge, spin, buoyancy, drag, thermal effects, and other forces. Will formulates these restrictions as universality of free fall and separates them from the stronger components of the Einstein equivalence principle Will 2014, § 2.1.1, pp. 7–9, PDF.

In the Newtonian test-body limit,

$$\mathbf a_A=-q_A\boldsymbol\nabla\Phi, \qquad \mathbf a_B=-q_B\boldsymbol\nabla\Phi, \qquad q_I=\left(\frac{m_{\mathrm P}}{m_{\mathrm I}}\right)_I.$$

Universality requires the dimensionless ratios $q_A$ and $q_B$ to agree for all tested compositions. A common value can be absorbed into the normalization of $\Phi$; only differences between ratios are exposed by differential free-fall measurements.

Trajectories, parameters, and local comparison

The relativistic formulation compares unparametrized curves because the WEP alone does not supply a universal clock parameter. If $x^\mu(\lambda)$ describes one trajectory, any smooth monotone relabeling $\lambda\mapsto\lambda'(\lambda)$ describes the same geometric path. Later, metric proper time will select a natural parameter for massive ideal clocks, but importing that conclusion into the definition would make the principle circular.

The statement is local in another sense. A sufficiently small probe samples one external field at one location. Extended bodies in a tidal field experience forces depending on multipole moments; spinning bodies can experience spin–curvature coupling; strongly self-gravitating bodies carry significant gravitational binding energy. None of these effects is a composition dependence within the elementary test-body WEP. Reall emphasizes the passage from universal local acceleration to common freely falling coordinates, while retaining tidal limitations Reall 2013, §§ 1.2–1.4, pp. 2–5, PDF.

One convenient observable is a signed differential-acceleration parameter along the applied field,

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

It is constructed to cancel the common field strength at leading order. Its interpretation still requires identical initial data, the same sampling region, and a model for nongravitational disturbances. The durable content needed here is the null hypothesis $\eta_{AB}=0$, not a time-sensitive numerical bound.

Controlled counterexamples

A uniform electromagnetic field illustrates why the qualifications are physical rather than verbal. For a particle with electric charge $q$, rest mass $m$, and four-velocity $u^\mu$,

$$m\,u^\nu\nabla_\nu u^\mu=qF^\mu{}_{\nu}u^\nu.$$

Two particles with different $q/m$ follow different trajectories despite identical initial data. This does not violate WEP, because the bodies were not differing only in composition under a universal gravitational interaction; they carried an uncontrolled nongravitational charge. Likewise, an accelerometer mounted on a supported laboratory reads nonzero proper acceleration, whereas an ideal freely falling accelerometer reads zero. Comparing supported and freely falling bodies is not a WEP test.

The counterexample also shows what WEP does not establish. Universal trajectories are compatible with many metric theories and do not determine the equation sourcing the metric. Nor does test-body universality constrain gravitational self-energy strongly enough to establish the strong equivalence principle.

From universality to geometric structure

A family of common free-fall paths suggests an affine or projective connection: through each event and admissible initial direction passes a preferred trajectory. Light propagation supplies a conformal structure, and compatibility conditions can motivate a Lorentzian metric. Those are additional constructive steps. WEP itself provides the empirical universality that makes a single geometry plausible; it does not yet prove that the connection is Levi-Civita, that light follows its null curves, or that curvature obeys Einstein’s equation.

The independent Newtonian check is immediate: if $q_A=q_B$, subtracting the two equations of motion gives $\Delta\mathbf a=0$. Conversely, if $\boldsymbol\nabla\Phi\ne0$ and differential acceleration vanishes for arbitrary compositions, then their passive-to-inertial ratios must agree within the idealization. This two-way implication is narrower—and more precise—than the slogan that “all masses are equal.”

Exercises

Two particles are released with the same four-velocity in an external electromagnetic field. Show that equality of their trajectories for arbitrary $F_{\mu\nu}$ requires equal charge-to-mass ratios, assuming both remain test particles.

Solution

At the common initial event their covariant accelerations are

$$a_A^\mu=\frac{q_A}{m_A}F^\mu{}_{\nu}u^\nu, \qquad a_B^\mu=\frac{q_B}{m_B}F^\mu{}_{\nu}u^\nu.$$

For arbitrary fields and velocities the contractions can be nonzero in any allowed spatial direction. Equality therefore requires $q_A/m_A=q_B/m_B$. Electromagnetic motion is universal only within a class sharing that ratio, unlike the WEP claim for neutral test bodies of differing composition.

References

• 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.