Strong-Field Equivalence Tests

Strong-equivalence tests ask whether gravitational binding energy and strong internal fields fall, gravitate, and radiate universally. Neutron stars make this question nonperturbative: an alternative theory can agree with every weak-field test yet assign a compact star a large effective charge. The clean signatures are compactness-dependent differential acceleration, dipole radiation, and other body-dependent timing effects, interpreted through a specified stellar model and gravity theory.

Required background. Equivalence Principles and Gravitational Measurement supplies the distinction between EEP and SEP. Probability and Statistical Inference supplies likelihood and covariance concepts; the timing likelihood below states the local statistical bridge.

Helpful background. Weak-Field Gravity fixes the matching to Solar-System fields, while Tests of Gravity connects compact bodies to laboratory and wave tests.

Compact-body charges

For a body whose mass depends on an ambient scalar $\varphi_0$, define an effective scalar charge

$$\alpha_A=\left.\frac{\partial\ln m_A(\varphi_0)}{\partial\varphi_0}\right|_{N_A},$$

holding conserved particle number $N_A$ fixed along a cold, nonrotating stellar sequence with a fixed equation of state. Here $m_A$ is the Einstein-frame mass and $\varphi_0$ is a dimensionless scalar with a declared kinetic normalization; otherwise the numerical charge changes under a field redefinition. A weakly bound object typically has $\alpha_A$ close to the weak-field coupling; a neutron star can depend strongly on mass, equation of state (EoS), and coupling parameters. In spontaneous scalarization, $\alpha_A$ can grow nonperturbatively even when weak-field coupling is small Damour and Esposito-Farèse 1996, §§II–IV, PDF.

In an external scalar field sourced by body $C$, the differential acceleration of $A$ and $B$ has the schematic form

$$\Delta\mathbf a_{AB}\propto (\alpha_A-\alpha_B)\alpha_C\boldsymbol\nabla U_C.$$

The proportionality hides theory-dependent normalization; the robust null checks are $\Delta\mathbf a=0$ when the charges are equal and recovery of the GR result when all additional charges vanish. An asymmetric neutron-star–white-dwarf binary also has a time-varying scalar dipole. Its leading extra energy loss scales as $(\alpha_A-\alpha_B)^2$ and can enter earlier in orbital velocity than GR’s quadrupole flux. Direct free-fall and radiative tests therefore probe complementary products of charges. For instance, a very weakly charged external source $\alpha_C\simeq0$ suppresses a free-fall signal even if $\alpha_A-\alpha_B$ is appreciable; the same pair can still radiate a dipole if the field is light enough. Conversely, equal charges suppress both differential channels without removing a modification of the conservative force. The detailed observational maps are developed in triple timing and dipole damping.

Timing observables and nuisance physics

A hierarchical triple supplies an external field: compare the acceleration of an inner neutron star and white dwarf toward the outer white dwarf. A binary supplies radiation: compare the intrinsic orbital-period derivative with the theory prediction after subtracting kinematic, Galactic, mass-loss, and tidal contributions. Freire and Wex derive the timing and contamination structure and then map it to SEP and scalar-tensor tests in Freire and Wex 2024, §3.4.2, Eqs. (28)–(34), §3.5.1, Eq. (35), and §5.

For arrival-time residuals $\mathbf r$, a Gaussian approximation uses

$$\ln\mathcal L=-\frac12\mathbf r^{\mathsf T}C^{-1}\mathbf r -\frac12\ln\det C+\text{constant},$$

where $C$ includes radiometer noise, pulse jitter, dispersion variations, and red timing processes. The forward model also carries ephemeris, clock, distance, proper motion, Galactic acceleration, companion mass, and EoS nuisance parameters. Priors on neutron-star mass and EoS can determine whether a scalarization threshold lies inside the sampled region, so a bound on a phenomenological charge is not automatically a theory-independent exclusion.

Evidence and alternatives at 2026-08-19

Position or channel	Evidence/status at 2026-08-19	Limitation or counterevidence
GR strong equivalence	Triple-system and binary-pulsar timing remain consistent with universal compact-body fall and quadrupolar damping	Each result is conditional on timing, kinematic, Galactic, EoS, and population models; consistency does not prove uniqueness
Perturbative scalar charge	Asymmetric pulsar–white-dwarf systems strongly constrain dipole radiation across sampled masses	Equal-charge systems suppress the dipole; massive or screened fields and unobserved mass windows can evade a simple map
Nonperturbative scalarization	Stellar calculations predict charge thresholds in viable-looking weak-field parameter regions	Pulsar timing has closed important windows, but thresholds shift with EoS, scalar mass, rotation, and coupling function

The PSR J0337+1715 triple is a direct strong-field free-fall comparison because a neutron star and inner white dwarf fall in the outer companion’s field. Archibald et al. report the timing construction and its null result in Archibald et al. 2018, pp. 73–76 and Methods. Pulsar–white-dwarf orbital-decay measurements supply negative evidence for large dipole channels. Zhao et al. combine multiple systems to close a previously weakly tested spontaneous-scalarization mass window Zhao et al. 2022, §§2–4.

The conservative conclusion is that GR passes these tests and broad classes of massless scalar-tensor deviations are strongly restricted. The competing possibility is not “any modification survives,” but that theory-specific massive, screened, rotating, or differently coupled branches can map outside the tested phenomenological coordinate. Whether such branches are consistent and independently motivated belongs to their theory analysis. A universal strong-field verdict remains open and model-dependent.

Independent discriminator. The independent check first verifies analytically that both differential acceleration and dipole flux vanish in GR and equal-charge limits. It then computes one scalarization curve with an independent stellar-structure solver across at least three radial resolutions and several EoSs, and propagates a blinded point on that curve into synthetic timing residuals. No such numerical run was executed for this page; a claimed threshold requires retained code, EoS tables, tolerances, and convergence output.

What could change the conclusion. The conclusion could change after a superseding triple- or binary-timing dataset, an EoS measurement that shifts scalarization thresholds, discovery of a pulsar in an uncovered mass or external-field regime, or an independently reproduced theory branch that passes weak-field matching and the synthetic-residual discriminator.

Exercise

Why can a double-neutron-star binary be less sensitive to dipole radiation than a neutron-star–white-dwarf binary even though both neutron stars are more compact?

Solution

Dipole power depends on the difference of effective charges, schematically $(\alpha_A-\alpha_B)^2$, rather than on either compactness alone. Two neutron stars with similar masses and structures can acquire similar charges, so their dipole nearly cancels. A neutron star and weakly bound white dwarf can have a much larger charge contrast. Therefore the asymmetric system may give the stronger dipole test, while the double-neutron-star system remains valuable for quadrupolar dynamics and other strong-field effects.

References

• Archibald, Anne M., et al. “Universality of Free Fall from the Orbital Motion of a Pulsar in a Stellar Triple System.” Nature 559 (2018): 73–76. DOI.
• Damour, Thibault, and Gilles Esposito-Farèse. “Tensor-Scalar Gravity and Binary-Pulsar Experiments.” Physical Review D 54 (1996): 1474–1491. DOI; Open PDF.
• Freire, Paulo C. C., and Norbert Wex. “Gravity Experiments with Radio Pulsars.” Living Reviews in Relativity 27 (2024): 5. DOI; Open PDF.
• Zhao, Junjie, Paulo C. C. Freire, Michael Kramer, Lijing Shao, and Norbert Wex. “Closing a Spontaneous-Scalarization Window with Binary Pulsars.” Classical and Quantum Gravity 39 (2022): 11LT01. DOI; Open PDF.