Equivalence-Principle and Universality Tests

An equivalence-principle test is meaningful only after the principle is specified. Universality of free fall for weakly self-gravitating bodies, local Lorentz invariance, local position invariance, and the strong equivalence principle are related but not identical. Quantum test masses add state preparation and readout without turning every universality test into a direct probe of quantum gravity.

Required background. Quantum-Gravity Observables and Test Taxonomy fixes the observable-to-claim chain. Acceleration, Gravity, and the Limits of Equivalence-Principle Arguments separates local acceleration arguments from global field effects. Helpful background. Detector and Instrument Validation Ledger supplies instrument controls, and Matching Amplitudes to Background Observables connects low-energy operators to measurable forces.

Which universality is being tested?

For two test bodies $A$ and $B$ in the same external field, the Eötvös parameter is

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

A null result constrains composition-dependent acceleration in the experimental environment. If a light scalar $\varphi$ couples through

$$\mathcal{L}_{\mathrm{int}} =\varphi\sum_i d_i\mathcal{O}_i,$$

then nuclear binding, electromagnetic energy, quark masses, and spin content give different bodies distinct effective scalar charges. A measured or bounded $\eta_{AB}$ constrains combinations of $d_i$, the scalar range, the source composition, and screening assumptions. It is not automatically a bound on every ultraviolet completion.

The weak equivalence principle concerns freely falling test bodies. The Einstein equivalence principle adds local Lorentz and position invariance for nongravitational experiments. The strong form extends universality to gravitational self-energy and local gravitational experiments. Lunar ranging and pulsar systems test self-energy effects that laboratory masses cannot. One should not translate among these tests without a theory that relates their coefficients.

First application: a dual-species atom interferometer

In a light-pulse Mach–Zehnder atom interferometer, the leading phase for species $s$ is

$$\phi_s=\mathbf{k}_{\mathrm{eff},s}\!\cdot\!\mathbf{a}_s T_s^2 +\phi_{\mathrm{laser},s} +\phi_{\mathrm{sys},s}.$$

The differential phase does not directly equal $(a_A-a_B)T^2$ when effective wavevectors, pulse times, trajectories, or scale factors differ. Define calibrated scale factors $S_s=k_{\mathrm{eff},s}T_s^2$ and estimate

$$\widehat{\eta}_{AB} =\frac{2(\phi_A/S_A-\phi_B/S_B)} {\phi_A/S_A+\phi_B/S_B}.$$

The first application is to propagate gravity gradients, wavefront aberrations, Coriolis terms, magnetic and ac-Stark shifts, initial cloud offsets, and source-mass uncertainty into this estimator. Reversing the momentum transfer changes the inertial phase while many offsets do not; alternating species order and source configuration supplies further null channels.

Quantum coherence enables precise phase comparison, but the inferred parameter remains universality of acceleration for the chosen internal states and compositions. Delocalization can make tidal terms and wave-packet separation important. A spin superposition tests spin-dependent forces only if its magnetic environment and internal-energy contribution are controlled.

The MICROSCOPE satellite’s final titanium–platinum result,

$$\eta_{\mathrm{Ti,Pt}} =\big[-1.5\pm2.3\,({\rm stat}) \pm1.5\,({\rm syst})\big]\times10^{-15},$$

was consistent with zero MICROSCOPE Collaboration 2022. This is an exceptionally strong null test of composition-dependent free fall in that channel. Its correct quantum-gravity interpretation is a constraint on low-energy nonuniversal couplings, not a detection and not a universal comparison of microscopic programs.

Quantum states do not erase classical systematics

An atom interferometer compares phases of quantum paths, whereas a torsion balance or drag-free satellite compares classical trajectories. The ontological description of the probe differs, but both require a source model, calibration, and an operator map. Tests with atoms in different internal or spin states can probe couplings unavailable to bulk materials. They do not establish a separate “quantum equivalence principle” unless that phrase is tied to a precise transformation law or Hamiltonian.

For a Hamiltonian

$$H=mc^2+\frac{\mathbf{p}^2}{2m_i}+m_g U(\mathbf{x})+H_{\mathrm{int}},$$

free-fall universality constrains $m_g/m_i$. Redshift or clock tests constrain how $H_{\mathrm{int}}$ couples to $U$. A single phase can mix both; the pulse sequence and internal-state history determine which combination is measured.

Adversarial control: inject a false differential acceleration

Offset the two atomic clouds by $\Delta\mathbf{x}$ in a gravity gradient $\Gamma_{ij}=\partial_i\partial_j U$. The induced differential acceleration is

$$\delta a_i=\Gamma_{ij}\Delta x_j.$$

If the analysis omits the measured initial offset, it can return a nonzero $\eta_{AB}$. A credible pipeline must recover this injected effect, remove it with independently measured gradients and trajectories, and propagate the correction uncertainty. Wavefront curvature and platform rotation should receive analogous injections.

The strongest surviving claim after these controls is either a null bound on a specified coupling combination or an unexplained differential acceleration. Only after replication across compositions, source masses, trajectories, and instrument principles could the latter become evidence for a new long-range interaction. Connecting it to quantum gravity would require an additional matching calculation.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Asenbaum, P., C. Overstreet, M. Kim, J. Curti, and M. A. Kasevich. “Atom-Interferometric Test of the Equivalence Principle at the $10^{-12}$ Level.” Physical Review Letters 125, 191101 (2020). DOI.
• MICROSCOPE Collaboration. “MICROSCOPE Mission: Final Results of the Test of the Equivalence Principle.” Physical Review Letters 129, 121102 (2022). DOI.
• Will, C. M. Theory and Experiment in Gravitational Physics, 2nd ed. Cambridge University Press (2018). Publisher.