What Do Precision Tests of General Relativity Actually Measure?

1. Introduction: The Experimental Canon

General relativity stands as one of the most precisely tested theories in physics. From the gravitational redshift measurements of Pound and Rebka (1960) to the multi-messenger observations of GW170817 (2017), a century of experiments has confirmed GR's predictions with extraordinary precision. Modern optical lattice clocks achieve fractional frequency comparisons at the 10⁻¹⁸ level and below, detecting gravitational time dilation across millimeter height differences. The Cassini spacecraft measured the Shapiro delay to constrain the PPN parameter γ to within 2 × 10⁻⁵ of unity. The Global Positioning System, which incorporates relativistic corrections as a matter of routine engineering, is often cited as the most practical demonstration that "GR works."

The standard narrative concludes that these tests leave no room for alternatives to GR. Any modification of gravity, the argument goes, would have been detected by now. The experimental foundations are secure.

This paper challenges that conclusion—not by disputing the experimental results, but by examining what those results actually constrain. A systematic methodological analysis reveals that the canonical precision tests share structural features that render them, as currently configured, unable to directly distinguish between general relativity and certain classes of alternatives, including the Temporal Equivalence Principle (TEP).

1.1 The Textbook Narrative

The experimental confirmation of GR is typically presented as a progression of increasingly precise tests:

Each entry represents a genuine experimental achievement. The question is not whether these experiments were performed correctly, but what they actually test.

1.2 The Central Question

This paper asks: What do precision tests of relativity actually constrain?

The analysis identifies five structural limitations shared by the experimental canon:

1. Two-Way Measurement Dominance: Nearly all tests use round-trip or reciprocity-even measurements, which are mathematically blind to direction-dependent effects.
2. Local/Global Conflation: Tests of local Lorentz invariance are extrapolated to claims about global spacetime structure.
3. Model-Dependent Calibration: Systems like GPS use GR-derived corrections to validate GR— demonstrating self-consistency within the framework, not independent confirmation.
4. The Conformal Loophole: Multi-messenger constraints bound the disformal sector of scalar-tensor theories while leaving the conformal sector unconstrained.
5. Theory-Laden Data Reduction: Systematic corrections assume the framework being tested, making independent falsification practically difficult within standard pipelines.

These limitations do not invalidate the experiments. They indicate that, in many cases, the tests primarily constrain parameter space within an assumed framework rather than systematically discriminating between alternatives.

The analysis throughout assumes a regime where disformal cone tilts are already constrained by multi-messenger bounds, and focuses on clock-sector structure that remains unconstrained within those limits.

1.3 The Temporal Equivalence Principle: Theory Definition

The Temporal Equivalence Principle (TEP) is defined formally by a single manifold endowed with two metrics and a universal matter coupling. It is not an ad hoc modification but a systematic framework for dynamical time.

The key distinguishing predictions are therefore not local redshift or two-way light-time relations (where TEP can agree with GR), but global and statistical observables that probe synchronization integrability and spatial structure:

• Local physics is identical to GR in freely falling laboratories (EEP; locally invariant c)
• Clock residuals can exhibit spatially structured correlations (e.g., exponential decay with distance) if the scalar field has a finite correlation length
• A convention-independent residual synchronization holonomy H_resid can be defined for direction-reversing, one-way closed loops after subtracting modeled GR loop effects
• Light and gravitational-wave propagation remain consistent with existing constraints: conformal rescaling preserves null cones, and any disformal cone tilt is bounded at the 10⁻¹⁵ level

The existing experimental canon strongly constrains local Lorentz violation, PPN departures in the gravitational/light-propagation sector, and disformal cone tilts. It does not yet directly probe spatial clock-correlation structure or residual holonomy in genuinely one-way, direction-reversing closed loops.

1.4 Structure of This Paper

Section 2 develops the methodological framework for analyzing precision tests, introducing key distinctions between gauge-invariant and convention-dependent observables, two-way and one-way measurements, and local and global tests.

Sections 3–7 examine the major categories of precision tests:

• Section 3: Gravitational redshift tests (Pound-Rebka, optical clocks)
• Section 4: Time dilation tests (Hafele-Keating, GPS)
• Section 5: Light propagation tests (Shapiro delay, VLBI)
• Section 6: Multi-messenger constraints (GW170817)
• Section 7: Resonator tests (Michelson-Morley, cavity experiments)

Section 8 proposes experiments that could genuinely discriminate between GR and TEP: triangle holonomy tests, interplanetary closed-loop timing, and GNSS correlation replication.

Section 9 discusses the philosophical implications, situating the findings within the broader context of underdetermination in physics. Section 10 concludes.

1.5 Related Work

This analysis builds upon and extends several distinct strands of literature in gravitational physics, metrology, and the philosophy of science.

Test Theories of Relativity: Frameworks for testing Lorentz invariance, such as the Mansouri-Sexl (1977) formalism and the Standard Model Extension (SME) (Colladay & Kostelecký 1998), have long parameterized deviations from relativity. These frameworks primarily focus on local Lorentz violation (anisotropy, boost dependence) in single-metric contexts. TEP differs by accepting exact local Lorentz invariance (consistent with the tightest SME bounds) while introducing non-integrable synchronization structure via a second metric.

Relativistic Metrology: The distinction between one-way and two-way time transfer is foundational in metrology (Ashby 2003). While two-way satellite time and frequency transfer (TWSTFT) and GPS Common-View are standard techniques for clock comparison, they are explicitly designed to cancel path delays. Recent proposals for "relativistic geodesy" with optical clocks (Mehlstäubler et al. 2018) focus on measuring the geopotential $W$, assuming GR. This work explores the inverse problem: using clocks to test the framework that defines $W$.

Philosophy of Simultaneity: The conventionality of simultaneity (Reichenbach 1928; Grünbaum 1973; Salmon 1977) establishes that one-way light speed is a convention in special relativity. Malament (1977) argued that causal connectivity makes standard synchronization unique, but this uniqueness holds only if the causal structure is singular (one metric). This paper operationalizes these philosophical insights in the context of two-metric theories, where the interplay between causal (light) and temporal (clock) structures makes the "convention" physically testable via holonomy.

Theory-Laden Experimentation: The challenge of "epistemological circularity" in precision tests—where data reduction relies on the theory being tested—has been discussed by Franklin (1986) and others. This analysis extends that discussion to modern precision cosmology and GNSS data processing, identifying specific pipelines (e.g., common-mode filtering) that enforce theoretical priors.

To forestall misunderstanding, this paper does not claim:

• That GR is wrong
• That precision tests are flawed or incompetent
• That TEP is correct
• That all alternatives to GR are viable

The claim is narrower and more specific: the existing experimental canon does not directly probe the observables that distinguish GR from TEP. This underdetermination is resolvable through new experimental configurations that have not yet been performed.

2. Methodological Framework

Before examining individual experiments, it is essential to establish the conceptual tools required to distinguish what precision tests actually measure from what they are commonly claimed to measure. This section develops four key distinctions that structure the subsequent analysis.

2.1 Gauge-Invariant vs. Convention-Dependent Observables

Not all physical quantities carry the same epistemological weight. A fundamental distinction exists between gauge-invariant observables—quantities that remain unchanged under coordinate transformations—and convention-dependent quantities that depend on arbitrary choices.

Examples of gauge-invariant observables include:

• Proper time elapsed along a worldline: τ = ∫ ds
• Round-trip light travel time between two events
• Frequency ratio of co-located clocks
• Interference fringe count in closed optical paths

Examples of convention-dependent quantities include:

• One-way speed of light (depends on synchronization)
• Coordinate time difference between spatially separated events
• Simultaneity of distant events

The critical insight is that most precision tests measure gauge-invariant quantities, which by construction do not directly probe the specific loop and correlation observables that differ between GR and TEP in the regime considered here.

The central claim of this paper is that most high-precision tests to date access only the gauge-invariant subset of observables on which GR and TEP explicitly agree.

2.2 Two-Way vs. One-Way Measurements

The distinction between two-way (round-trip) and one-way measurements is not merely technical but fundamental to what can be tested.

Two-way measurements avoid this circularity by using a single clock:

However, this gauge-invariance comes at a cost: two-way measurements are inherently insensitive to direction-dependent effects. If light travels at c + v in one direction and c − v in the return direction, the round-trip time is:

The first-order direction-dependent term cancels exactly. This is not a limitation of experimental precision but a mathematical necessity: round-trip measurements are reciprocity-even and cannot probe reciprocity-odd effects.

2.3 Local vs. Global Tests

A second crucial distinction separates local tests—performed within a single freely falling frame—from global tests that probe the structure of spacetime across extended regions.

The TEP framework satisfies EEP exactly: in local freely falling frames, physics reduces to special relativity with invariant c. This means local tests—no matter how precise—cannot distinguish TEP from GR. The distinction arises in global and statistical observables: spatial structure in clock residuals and a GR-subtracted residual holonomy H_resid defined by genuinely one-way, direction-reversing closed loops.

The operational criterion for "local" is whether the measurement region is small compared to the curvature scale. Modern optical clock experiments operate at millimeter scales where spacetime curvature is negligible—they are quintessentially local tests.

2.4 Single-Path vs. Multipath Configurations

The final distinction concerns whether signals traverse a single path or multiple paths through spacetime.

TEP phenomenology arises from:

• Multipath configurations (gravitational lensing, where rays sample different field values)
• Statistical correlations (GNSS, where spatial structure ⟨φ(x)φ(x')⟩ is probed)
• Closed loops with direction reversal (residual holonomy H_resid after subtracting modeled GR loop effects)

Single-path, single-direction measurements—the dominant mode of precision tests—are structurally incapable of detecting these signatures.

2.5 Summary: The Measurement Taxonomy

The final row—one-way closed-loop measurements—represents the only configuration capable of testing synchronization integrability. No high-precision relativistic-gravity test has directly targeted a convention-independent, direction-reversing, one-way loop observable designed to detect non-integrable synchronization beyond modeled GR terms.

3. The Gravitational Redshift Tests

Gravitational redshift experiments—from Pound-Rebka (1960) to modern optical lattice clocks—are often cited as the most precise confirmations of general relativity. This section examines what these experiments actually test and why they do not directly probe the observables that distinguish GR from TEP.

3.1 The Canonical Experiments

3.1.1 Pound-Rebka-Snider (1960-1965)

The Pound-Rebka experiment measured the gravitational redshift of 14.4 keV gamma rays traveling 22.5 meters vertically in the Jefferson Tower at Harvard. The observed fractional frequency shift:

agreed with the GR prediction to within 1% (later improved to 0.1% by Pound and Snider).

3.1.2 Gravity Probe A (1976)

Gravity Probe A flew a hydrogen maser to 10,000 km altitude, comparing its frequency with ground-based masers. The experiment confirmed gravitational time dilation to 7 × 10⁻⁵ relative precision.

3.1.3 Modern Optical Lattice Clocks (2010-present)

Jun Ye's group at JILA has achieved gravitational redshift measurements at the 10⁻¹⁸ level and below, detecting time dilation across millimeter height differences. These represent the most precise measurements of gravitational effects ever performed.

3.2 What Redshift Tests Actually Measure

All gravitational redshift experiments measure the same fundamental quantity: the ratio of proper times at different gravitational potentials.

This ratio is:

• Gauge-invariant (independent of coordinate choice)
• Local (compares clocks at specific spacetime points)
• Common to all metric theories satisfying EEP

The experiments do not measure:

• Absolute synchronization between distant clocks
• Path-dependence of clock transport
• Global integrability of the synchronization field

3.3 The ACES Mission: A Missed Opportunity?

The Atomic Clock Ensemble in Space (ACES), scheduled for the International Space Station, will compare space-based and ground-based atomic clocks with unprecedented precision. The mission aims to test gravitational redshift at the 2–3 × 10⁻⁶ level (Savalle et al. 2019).

3.4 Summary: Local Brilliance, Global Blindness

The gravitational redshift tests are triumphs of experimental physics. They confirm that gravity affects time—a revolutionary insight that transformed our understanding of the universe. These experiments were not designed to probe global synchronization structure, and it would be unreasonable to criticize them for not testing something outside their scope. The point is simply that theories agreeing on local physics (like GR and TEP) do not directly probe the specific loop and correlation observables that differ between them in the regime considered here.

The proposed loop and correlation experiments should therefore be seen as complementary additions to, not replacements for, the existing experimental canon.

4. The Time Dilation Tests

Time dilation experiments—from Hafele-Keating's circumnavigating clocks to GPS satellite corrections—are frequently cited as strong evidence that "time runs differently" in different reference frames. This section examines the logical structure of these claims.

4.1 The Hafele-Keating Experiment (1971)

In October 1971, Joseph Hafele and Richard Keating flew cesium atomic clocks on commercial aircraft around the world—eastward and westward—comparing them with reference clocks at the U.S. Naval Observatory. The observed time differences:

The results are celebrated as confirming both special relativistic time dilation (velocity-dependent) and general relativistic gravitational time dilation (altitude-dependent).

4.2 Muon Lifetime Experiments

Cosmic ray muons, traveling at relativistic speeds, exhibit extended lifetimes compared to muons at rest. The observed lifetime dilation factor γ = 1/√(1 − v²/c²) agrees with special relativity to high precision.

4.3 The Galileo Eccentric Orbit Test (2018)

In 2014, two Galileo satellites (GSAT-0201 and GSAT-0202) were accidentally launched into highly eccentric orbits instead of circular ones. This "fortunate accident" enabled the most precise test of gravitational redshift since Gravity Probe A in 1976.

The eccentric orbits (perigee ~17,180 km, apogee ~26,020 km) produce a periodic modulation of the gravitational potential experienced by the onboard hydrogen masers. Delva et al. (2018) and Herrmann et al. (2018) independently analyzed 1008 days of data, measuring:

    α = (+0.19 ± 2.48) × 10⁻⁵

where α parameterizes deviations from the GR prediction (α = 0 in GR). This represents a factor 5.6 improvement over Gravity Probe A.

4.4 Gravity Probe B (2011)

Gravity Probe B measured two predictions of general relativity: the geodetic effect (spacetime curvature caused by Earth's mass) and frame-dragging (spacetime being "dragged" by Earth's rotation). Four ultra-precise gyroscopes orbited Earth for 18 months, measuring their precession relative to a distant guide star.

The results confirmed GR predictions:

4.5 The GPS System: Model-Dependent Calibration

The Global Positioning System is frequently cited as the most practical demonstration of relativistic effects. GPS satellites carry atomic clocks that are pre-corrected for both special relativistic (velocity) and general relativistic (gravitational) time dilation. This section examines the logical structure of GPS as a test of GR.

4.5.1 The Standard Narrative

GPS satellites orbit at ~20,200 km altitude with velocity ~3.9 km/s. The combined relativistic correction is approximately +38 μs/day (gravitational) minus 7 μs/day (velocity), yielding a net +31 μs/day correction applied to satellite clocks.

The argument proceeds: GPS works at meter-level accuracy; therefore, the relativistic corrections must be correct; therefore, GR is confirmed.

4.5.2 Model-Dependent Calibration

The phrase "assuming that general relativity is correct" is not a criticism of GPS engineering—it is a precise statement of the logical structure. GPS demonstrates that GR provides a self-consistent framework for navigation. It does not demonstrate that GR is the unique framework capable of doing so.

4.5.3 What a TEP Signal Would Look Like in GPS Residuals

If TEP effects exist, they would appear in GPS data as specific systematic patterns. Understanding these signatures is essential for determining whether GPS could detect or has already filtered out such effects.

4.5.4 Reported Evidence from GNSS Analysis

Exploratory analysis of GNSS clock products from multiple analysis centers (CODE, IGS, ESA) within the TEP research program has suggested systematic patterns consistent with TEP predictions:

• Distance-structured correlations: Clock residuals reportedly show exponential decay with inter-station distance, with a characteristic scale of order 10³–10⁴ km
• Cross-center consistency: The same pattern reportedly appears in independent processing pipelines (R² = 0.92-0.97 between centers)
• Temporal persistence: The correlation structure reportedly persists over 25 years of data (2000-2025)
• Ionospheric independence: The signal is reportedly invariant under geomagnetic storm stratification (Kp < 3 vs Kp ≥ 3)

These reported observations require independent verification. If confirmed, they would demonstrate that GPS/GNSS data contains systematic structure that standard processing treats as "error" rather than "signal." If refuted, they would constrain the TEP parameter space. Either outcome advances scientific understanding.

4.5.5 The Filtering Problem

4.5.6 The TEP Perspective

TEP predicts identical local clock rates to GR. The frequency corrections applied to GPS satellites would be the same under TEP. The difference arises in synchronization structure—whether the coordinate time t is globally integrable.

GPS uses predominantly two-way time transfer and common-view methods for synchronization. These methods are insensitive to residual holonomy H_resid. A nonzero H_resid beyond GR subtraction would manifest as systematic position errors with specific geometric signatures—but such signatures would be absorbed into the navigation solution as "common-mode errors" and filtered out.

4.6 Lunar Laser Ranging: The Nordtvedt Effect

Lunar Laser Ranging (LLR) has measured the Earth-Moon distance to millimeter precision since 1969, using retroreflectors placed by Apollo astronauts. LLR provides the most precise test of the Strong Equivalence Principle through the Nordtvedt effect.

4.6.1 The Nordtvedt Effect

If gravitational self-energy contributes differently to inertial and gravitational mass, the Earth and Moon would fall toward the Sun at slightly different rates, causing a polarization of the lunar orbit. The Nordtvedt parameter η quantifies this violation:

Current LLR solutions give (m_G/m_I)_E − (m_G/m_I)_M = (−0.8 ± 1.3) × 10⁻¹³, implying |η| ≲ few × 10⁻⁴ (conversion depends on the Earth–Moon self-energy difference; Williams et al. 2012).

4.7 Summary: Closed Loops That Don't Test Closure

The time dilation tests confirm that time is relative—a profound insight. But they confirm it through proper time comparisons along reuniting worldlines, not through synchronization of permanently separated clocks. The distinction is subtle but fundamental: proper time path-dependence is universal; synchronization path-dependence is what distinguishes theories.

5. The Light Propagation Tests

Light propagation experiments—Shapiro delay measurements, VLBI observations, and gravitational lensing studies—probe how electromagnetic radiation traverses curved spacetime. These tests have achieved remarkable precision in constraining the Parameterized Post-Newtonian (PPN) parameter γ. This section examines what these constraints actually imply.

5.1 The Shapiro Delay

In 1964, Irwin Shapiro predicted that radar signals passing near the Sun would experience a time delay due to spacetime curvature. The "fourth test of GR" has since been measured with increasing precision.

5.1.1 The Cassini Experiment (2003)

The most precise Shapiro delay measurement used radio signals between Earth and the Cassini spacecraft during solar conjunction. The result:

This constrains the PPN parameter γ to agree with GR (γ = 1) at the 10⁻⁵ level.

5.1.2 Data Reduction Concerns

The Cassini analysis required extensive corrections for systematic effects:

The Cassini inference is conditional on an end-to-end modeling stack: a spacetime model for signal propagation in the solar system, together with tropospheric, plasma, and spacecraft dynamics models. This does not weaken the measurement; it clarifies what is being tested. The result tightly constrains γ within the assumed signal model class.

5.2 Very Long Baseline Interferometry (VLBI)

VLBI observations of quasars near the Sun measure the apparent deflection of radio waves by solar gravity. The deflection angle:

where b is the impact parameter, has been measured to constrain γ at the 10⁻⁴ level.

5.3 The PPN Framework and Its Limitations

The Parameterized Post-Newtonian formalism provides a systematic way to compare metric theories of gravity. The PPN metric in isotropic coordinates:

where U = GM/rc² is the Newtonian potential. GR predicts β = γ = 1.

5.3.1 What PPN Tests Constrain

• γ: How much space curvature is produced by unit rest mass
• β: How much nonlinearity exists in the superposition of gravity
• Other parameters: Preferred-frame effects, conservation law violations

5.3.2 What PPN Tests Cannot Constrain

The PPN framework parameterizes the post-Newtonian limit under the assumption that the same effective metric governs the sector being tested (typically solar-system dynamics and light propagation). It is therefore not a complete language for theories in which additional structure appears in the matter and clock sector while remaining effectively GR-like in the photon and gravitational sectors.

• Clock transport and synchronization are governed by an effective metric distinct from the one constrained by light-propagation observables
• Convention-independent loop observables (residual holonomy after GR subtraction) are the primary targets
• Non-local statistical structure appears in clock residuals (spatial correlations) rather than in PPN light-bending parameters

In TEP, the matter metric is related to the gravitational metric by a disformal map g̃_μν = A(φ)g_μν + B(φ)∇_μφ∇_νφ. The conformal factor A(φ) rescales the matter sector while preserving null cones; the disformal term B(φ) can tilt null cones and is strongly constrained by multi-messenger observations. Consequently, PPN light-propagation tests can leave room for clock-sector effects encoded in A(φ) and for loop/correlation observables that are not representable as a single γ parameter.

5.4 The Conformal vs. Disformal Distinction

In scalar-tensor theories, the physical metric can differ from the gravitational metric through conformal and disformal transformations:

where A(φ) is the conformal factor and B(φ) is the disformal factor.

5.5 Summary: Constraining a Different Sector

Light propagation tests are precision triumphs that tightly constrain the post-Newtonian light-propagation sector (e.g., γ ≈ 1) and place strong limits on any disformal cone tilt (encoded in B(φ)). These constraints are largely orthogonal to clock-sector observables such as spatially structured correlations in clock residuals and GR-subtracted residual holonomy H_resid, which require genuinely one-way, direction-reversing configurations.

6. The Multi-Messenger Constraints

The event GW170817 detected gravitational waves from a binary neutron star merger, followed (+1.74 ± 0.05) s later by a gamma-ray burst (GRB 170817A). This timing coincidence yields an assumption-dependent constraint on the fractional speed difference between gravitational waves and light.

6.1 The GW170817 Event

On August 17, 2017, the LIGO-Virgo collaboration detected gravitational waves from a binary neutron star inspiral in NGC 4993, with luminosity distance of order 40 Mpc (130 million light-years; conservative bounds use the lower end of the distance interval) distant. The Fermi Gamma-ray Burst Monitor detected GRB 170817A approximately (+1.74 ± 0.05) s after the gravitational wave signal.

6.1.1 The Speed Constraint

Interpreting the observed delay as (at least partially) a propagation-time difference yields a bound on the fractional speed difference during the trip.

The numerical bounds depend on conservative assumptions about intrinsic emission-time offsets between the gravitational-wave and gamma-ray signals. In particular, the observable directly constrains differential propagation, while the emission-time difference is astrophysically model dependent (Abbott et al. 2017).

6.1.2 The Standard Interpretation

The constraint has been used to rule out or severely constrain numerous modified gravity theories, including:

• Scalar-tensor theories with disformal coupling
• Massive gravity theories
• Lorentz-violating gravity theories
• Certain dark energy models

6.2 What GW170817 Actually Constrains

6.2.1 The Conformal Loophole: Explicit Formulation

Understanding why GW170817 does not directly constrain TEP's conformal sector requires examining the two-metric structure explicitly. This section provides the mathematical derivation that shows conformal coupling cancels in arrival-time comparisons.

6.2.2 The Flight Time Calculation

Consider a signal (gravitational wave or photon) traveling from source S to detector D along a null geodesic. The coordinate flight time is:

where c_eff is the effective propagation speed in the relevant metric.

6.2.3 The Disformal Contribution

The disformal term B(φ)∇_μφ∇_νφ does modify the relative propagation:

This introduces a genuine speed difference:

GW170817's observed coincidence (after accounting for astrophysical emission-time uncertainties) implies that any net differential propagation delay accumulated over the propagation distance must be small. Operationally, the observation bounds the fractional speed difference at the few × 10⁻¹⁵ level under standard emission-time assumptions.

This is a real constraint on the disformal sector. It does not directly constrain the conformal factor A(φ) alone.

6.2.4 The Common-Mode Cancellation

GW170817-type constraints compare two signals that traverse essentially the same spacetime path and are time-stamped by clocks in the matter sector at the detector. Any common-mode rescaling of the local proper-time standard cannot be isolated by an EM–GW arrival-time comparison along a single path. The observable is therefore primarily sensitive to differential propagation effects, such as a disformal cone tilt.

6.3 What Would Constrain Clock-Sector Structure?

To constrain clock-sector structure (including spatial correlations) and to test for non-exact time transport beyond modeled GR loop effects, one needs measurements that do not reduce to single-path, common-mode comparisons:

6.3.1 Multipath Configurations

Gravitational lensing produces multiple images that traverse different paths through the scalar field. If A(φ) varies spatially, different images experience different time delays—the "Shapiro delay" generalized to scalar fields.

This is the origin of TEP's "phantom mass" prediction: lensing time delays probe the integrated scalar field along different paths, potentially mimicking dark matter.

6.3.2 Statistical Correlations

If the scalar field has spatial structure ⟨φ(x)φ(x')⟩ ~ exp(−r/λ), clocks at different locations will exhibit correlated fluctuations. This is the signature suggested by exploratory GNSS analysis—distance-structured correlations in clock behavior.

Single-path measurements like GW170817 do not directly probe this structure because they sample only one realization of the field along one path. The proposed spatial-correlation and loop tests should be seen as complementary additions to, not replacements for, multi-messenger constraints.

6.3.3 Closed-Loop Holonomy

The experimentally relevant quantity is a GR-subtracted residual holonomy H_resid, defined from one-way time transfer around a direction-reversing closed loop after subtracting modeled GR loop effects (Sagnac, Shapiro, and gravito-magnetic contributions). In the conformal-only limit (B = 0), the A(φ) contribution is an exact gradient and yields H_resid = 0 on simply connected domains; a nonzero residual requires non-exact structure, for example a disformal coupling B(φ) ≠ 0 or more general non-metricity.

No single-path multi-messenger observation can probe H_resid because the signals travel from source to detector, not around direction-reversing closed loops.

6.4 The Broader Multi-Messenger Context

6.4.1 Future Events

Additional multi-messenger detections will improve statistics but not change the fundamental limitation: single-path measurements constrain differential propagation, not common-mode effects.

6.4.2 Strongly Lensed Events

A gravitationally lensed multi-messenger event would provide genuinely new information. If gravitational waves and electromagnetic signals from the same source arrive via different lensed paths, their relative timing probes the scalar field structure along different trajectories.

Such events are rare but would be transformative for constraining conformal coupling.

6.5 Summary: The Disformal Constraint

GW170817 is a landmark observation that opened the era of multi-messenger astronomy. Its constraints on modified gravity are real and important—for the disformal sector. Single-path EM–GW coincidence measurements primarily constrain differential propagation (for example, disformal cone tilts) and do not directly probe clock-sector correlation structure or GR-subtracted residual holonomy H_resid, which require multipath or direction-reversing closed-loop configurations. This is a statement about measurement geometry, not a criticism of the remarkable experimental achievement.

7. The Resonator Tests

Optical and microwave resonator experiments—descendants of the Michelson-Morley experiment—have achieved extraordinary precision in testing Lorentz invariance. Modern cavity experiments constrain anisotropy in the speed of light at the 10⁻¹⁸ level. This section examines the logical structure of these null results.

7.1 The Michelson-Morley Experiment (1887)

The Michelson-Morley experiment sought to detect Earth's motion through the luminiferous aether by measuring the difference in light travel time along perpendicular arms of an interferometer. The null result—no fringe shift as the apparatus rotated—is celebrated as disproving the aether and motivating special relativity.

7.1.1 What Michelson-Morley Actually Tests

The null result confirms that the two-way speed of light is isotropic to high precision. It does not test whether the one-way speed is isotropic—that would require synchronized clocks at the endpoints, introducing the synchronization convention problem.

7.2 The Kennedy-Thorndike Experiment (1932)

Kennedy and Thorndike modified the Michelson-Morley setup to have unequal arm lengths, testing whether the interference pattern changed as Earth's velocity varied over the year. The null result constrains time dilation effects.

7.3 Modern Cavity Experiments

Contemporary experiments use cryogenic optical or microwave cavities to achieve extraordinary precision. The resonance frequency of a cavity depends on its length and the speed of light:

where n is the mode number. By comparing cavities oriented in different directions, or monitoring a single cavity as Earth rotates, anisotropy in c can be constrained.

7.3.1 Precision Achievements

Modern experiments have achieved constraints on Lorentz violation at remarkable levels:

7.3.2 The Closed-Path Limitation

7.4 The Standard Model Extension (SME) Framework

Modern Lorentz violation tests are interpreted within the Standard Model Extension, which parameterizes all possible Lorentz-violating terms in the Standard Model Lagrangian. Cavity experiments constrain specific SME coefficients.

7.4.1 What SME Tests Constrain

• Anisotropy in the two-way speed of light
• Preferred-frame effects in electromagnetism
• CPT violation in photon propagation

7.4.2 What SME Tests Cannot Constrain

• One-way speed anisotropy (convention-dependent)
• Synchronization structure (requires one-way measurements)
• Effects that preserve two-way isotropy while violating one-way isotropy

7.5 The Sagnac Effect: A Partial Exception

The Sagnac effect—the phase shift in a rotating interferometer—is sometimes cited as measuring one-way light speed. In a rotating ring interferometer, counter-propagating beams accumulate different phases:

where A is the enclosed area and Ω is the rotation rate.

7.6 Summary: Isotropy of What, Exactly?

The resonator tests confirm that the laws of electromagnetism are the same in all directions—for round-trip measurements. This is a profound constraint on Lorentz violation. But it is not a constraint on synchronization structure, which requires genuinely one-way measurements that resonator experiments cannot provide.

TEP preserves local Lorentz invariance exactly. The scalar field φ modifies clock rates but not light propagation. Resonator experiments, which probe light propagation in closed paths, are structurally incapable of detecting TEP signatures.

8. What Would Actually Discriminate?

The preceding sections have established that canonical precision tests of GR share common methodological limitations: two-way measurements, local comparisons, single-path configurations, and theory-laden data reduction. This section proposes experiments that could genuinely discriminate between GR and TEP.

The following proposals are ordered by increasing baseline length and technological ambition, from terrestrial triangles to interplanetary loops.

8.1 The Triangle Holonomy Test

The most direct test of synchronization integrability is the measurement of holonomy around a closed loop using one-way signals. This test must explicitly distinguish TEP holonomy from the well-known Sagnac effect.

8.1.1 Formal Definition: The Loop Observable

The loop observable $H$ is defined operationally as a concrete estimator constructed from time-tagged data. Consider three stations $i \in \{A, B, C\}$ exchanging optical pulses.

The experimental challenge is to model or control the first five terms to isolate the sixth.

8.1.2 Distinguishing TEP Holonomy from Sagnac Effect

A skeptic will immediately note that any closed loop on a rotating Earth measures the Sagnac effect ($\Delta t = 4A\Omega/c^2$). The triangle holonomy test must explicitly distinguish TEP effects from Sagnac rotation.

The numerical values quoted below are order-of-magnitude forecasts intended to clarify the experimental scale. In practice, the effective Sagnac term and the achievable subtraction residual depend on loop geometry, ephemerides and Earth-orientation modeling, link non-reciprocity control, and the time-transfer calibration strategy.

8.1.3 The Large-Area Strategy

Contrary to "zero-area" approaches that eliminate signal along with noise, the original TEP framework (Smawfield 2025e) emphasizes that the holonomy $H$ scales with the loop area (flux of the time-transport curvature). To maximize the signal-to-noise ratio, a Large-Area MEO Strategy is proposed:

8.1.4 Explicit Error Budget

The experimental design relies on precise modeling to isolate the TEP residual. The Sagnac effect poses the most significant challenge:

8.1.5 Multiple Geometries for Cross-Validation

To further distinguish TEP from Sagnac effects, use multiple triangle configurations:

8.1.6 Technical Requirements

8.1.7 The Critical Innovation

The key requirement is genuinely one-way links. Current time transfer methods (two-way satellite time transfer, GPS common-view) use round-trip or common-mode techniques that cancel holonomy by construction.

One-way optical links between ground stations and satellites (or between satellites) could provide the required configuration. The European Space Agency's ACES mission includes a one-way link capability that could, in principle, be used for holonomy measurements—though this is not part of the planned science program.

8.1.8 Objections and Replies

8.1.9 Formal Note: Why Conformal-Only Models Give H_resid = 0

8.1.10 The Conformal-Only Limit and TEP Testability

The preceding analysis raises a critical question: if conformal-only TEP (B = 0) predicts H_resid = 0 after GR subtraction—identical to GR—how can the triangle holonomy test discriminate between the theories?

8.2 Interplanetary Closed-Loop Timing

Deep space baselines are attractive for probing disformal structure because they accumulate propagation effects over long paths. However, a two-point one-way asymmetry is not, by itself, an operational observable because it depends on unknown clock offsets and drifts between endpoints. An interplanetary discriminator therefore requires an explicitly offset-invariant, direction-reversing closed-loop construction.

8.2.1 The Observable: AU-Scale Loop Holonomy

8.2.2 Physical Content

The signal of interest is the GR-subtracted residual holonomy $H_{\text{AU,resid}} = H_{\text{AU,raw}} - H_{\text{AU,GR}}$. The modeled GR term includes Shapiro delay contributions from solar-system bodies, kinematic loop terms from platform motion, and plasma dispersion. A nonzero residual with the predicted geometry dependence would indicate non-exact time transport beyond the modeled GR loop effects.

8.2.3 Proposed Configuration

• Three-node geometry using one Earth terminal and two spacecraft (or one spacecraft plus a second Earth terminal), enabling both loop orientations.
• Optical time transfer with recorded emission and reception time tags for each directed link.
• Dual-frequency calibration (or independent plasma monitoring) to control dispersive delays.
• End-to-end modeling of ephemerides and Shapiro delays to construct $H_{\text{AU,GR}}$.

8.2.4 Forecast and Feasibility

8.3 GNSS Correlation Replication

Exploratory analyses within the TEP research program have suggested distance-structured correlations in GNSS clock data with correlation length on the order of 10³–10⁴ km. Independent, blinded replication of this analysis would provide strong evidence for or against the TEP interpretation.

8.3.1 The Existing Evidence

Exploratory analysis from the TEP research program, using 25 years of CODE clock products and cross-validated with IGS and ESA analysis centers, suggests:

• Exponential decay of clock correlations with distance
• Correlation length of order a few × 10³ km (reported example: λ ≈ few × 10³ km under one estimator)
• Consistency across three independent analysis centers (R² = 0.92–0.97)
• Orbital-dynamics coupling (reported correlation with Earth's orbital velocity)
• Approximate alignment with the CMB frame (reported within tens of degrees)

Status: These findings constitute preliminary evidence requiring independent replication. This paper treats them as a motivating hypothesis and does not assume their numerical values when specifying the discriminating measurement geometries.

8.3.2 The Reproducibility Mandate

For these claims to be scientifically accepted, future analysis must move beyond preliminary reports to a fully transparent Reproducibility Package. Any confirming study must provide:

8.3.3 Replication Requirements

Independent replication should:

1. Use raw GNSS data (not processed clock products) to avoid filtering artifacts
2. Apply minimal processing to preserve spatial correlations
3. Test multiple constellations (GPS, GLONASS, Galileo, BeiDou)
4. Verify the correlation length is consistent across datasets
5. Test for systematic artifacts (ionospheric, tropospheric, orbital)

8.3.4 Falsification Criteria

The TEP interpretation would be falsified if:

• Correlation length falls outside 500-20,000 km range
• Correlations disappear in raw (unprocessed) data
• Correlations are explained by known systematic effects
• Different constellations show inconsistent correlation lengths

8.4 Optical Clock Networks and Environmental Screening

The next generation of optical clock networks—connected by optical fiber or free-space links—could provide unprecedented sensitivity to synchronization structure and environmental screening mechanisms.

8.4.1 Current Developments

Several projects are developing continental-scale optical clock networks:

• European fiber network connecting national metrology institutes
• NIST-JILA optical link in Boulder, Colorado
• Proposed intercontinental optical links

8.4.2 TEP Sensitivity: Two Phases

Phase I: Distance Correlations. Optical clocks achieve $10^{-18}$ stability, potentially sensitive to distance-dependent correlations at the $\delta\tau/\tau \sim 10^{-18}$ level over 1000 km baselines.

Phase II: Environmental Screening Maps. TEP predicts that the scalar field profile $\phi(x)$ is screened by local matter density. Comparing clocks at different altitudes (sea level vs mountain vs LEO) allows mapping the screening profile. After subtracting GR redshift, TEP predicts residual frequency shifts of $10^{-19}$–$10^{-18}$ over tens of kilometers for a screening length $\lambda_{scr} \sim 10$ km.

8.4.3 Critical Design Considerations

To test TEP, optical clock networks must:

• Use one-way comparisons (not just common-mode rejection)
• Span continental-to-global baselines (order 10³–10⁴ km)
• Operate continuously to detect temporal variations
• Deploy transportable clocks to varying environments (density/altitude) to test screening

8.5 Matter-Wave Interferometry

Atom interferometers provide a complementary probe sensitive to the gradients of the conformal factor $\nabla \ln A(\phi)$.

• Objective: Measure $\nabla\nabla \ln A(\phi)$ via differential phases in atom interferometers.
• Design: Long-baseline or tall fountain interferometers (Sr, Rb) in gradiometric configurations.
• Discrimination: Combine with mechanical gravimeters. GR affects both; TEP scalar gradients affect atoms (via $A(\phi)$) differently from bulk mass (if composition-dependent couplings exist, though A is universal at leading order).

8.6 Explicit TEP Predictions

To ensure falsifiability, TEP makes specific numerical predictions for each proposed test. These predictions are derived from the theoretical framework (Smawfield 2025a). Numerical ranges are order-of-magnitude targets; where they are informed by reported GNSS analyses, they are treated as provisional pending independent replication.

8.7 Summary: The Experimental Frontier

These experiments share a common feature: they break the two-way symmetry that characterizes existing precision tests. By measuring one-way propagation, closed-loop residual holonomy H_resid, or spatial correlations, they probe the synchronization structure that distinguishes GR from TEP.

None of these experiments has been performed with the explicit goal of testing synchronization integrability. The experimental frontier lies not in achieving greater precision within existing paradigms but in designing fundamentally new measurement configurations.

9. Discussion: The Underdetermination Problem

The analysis presented in Sections 3-7 reveals a systematic pattern: precision tests of general relativity constrain parameters within an assumed framework rather than discriminating between frameworks. This section examines the philosophical implications and situates the findings within the broader context of theory testing in physics.

9.1 The Duhem-Quine Thesis in Relativistic Metrology

The Duhem-Quine thesis holds that scientific hypotheses cannot be tested in isolation; any test involves auxiliary assumptions that could be revised instead of the primary hypothesis. This thesis finds stark expression in precision tests of GR.

9.1.1 Theory-Laden Observation

Every precision measurement requires corrections for systematic effects. These corrections are derived from theoretical models:

• Tropospheric delay corrections rely on refractivity models combined with a relativistic signal-path geometry
• Solar plasma corrections rely on electron-density models combined with spacecraft ephemerides and tracking geometry
• Orbital dynamics are expressed in relativistic reference systems and ephemerides
• Clock comparisons adopt a coordinate-time standard within an assumed relativistic reference frame

The framework being tested generates the corrections applied to test it. This is standard practice in precision metrology: systematic corrections require theoretical and empirical auxiliary models; the logical point is that the "test" is conditional on those auxiliaries. "Testing GR" is therefore more accurately described as "testing the self-consistency of a GR-anchored data-reduction pipeline within a specified model class."

9.1.2 The Auxiliary Hypothesis Problem

When a measurement agrees with GR, the conclusion is: "GR is confirmed." When a measurement disagrees, the response is typically to revise auxiliary assumptions (calibration errors, unmodeled systematics) rather than question GR.

This asymmetry is rational—GR has enormous empirical support—but it illustrates how discrepancies are often underdetermined between core theory and auxiliary models unless experiments are designed to vary the auxiliaries independently.

9.2 The Conventionality of Simultaneity

The philosophical literature on the conventionality of simultaneity, initiated by Reichenbach and developed by Grünbaum, Salmon, and others, directly addresses the issues raised in this paper. This section translates these philosophical insights into operational terms accessible to experimental physicists.

9.2.1 The Operational Problem: Measuring with the Thing You're Measuring

9.2.2 The Rubber Ruler Analogy

9.2.3 Reichenbach's ε-Synchronization

Reichenbach formalized this insight mathematically. Einstein's synchronization convention (ε = 1/2, meaning light takes equal time in each direction) is not empirically determined. Any value 0 < ε < 1 yields an empirically equivalent theory.

The one-way speed of light is:

Only the round-trip speed c is empirically determined. The choice ε = 1/2 is conventional, not empirical—unless you have an independent way to synchronize clocks.

9.2.4 Malament's Theorem: The Uniqueness Claim

Malament (1977) proved that Einstein synchronization is the unique synchronization definable from the causal structure of Minkowski spacetime. This is often cited as resolving the conventionality debate in favor of ε = 1/2.

9.2.5 The Two-Clock Thought Experiment

9.2.6 Implications for TEP

TEP is not a simple Reichenbach ε ≠ 1/2 anisotropic light-speed theory. This distinction is critical for understanding what TEP actually claims and how it differs from conventional alternatives to special relativity.

9.3 Underdetermination vs. Falsifiability

The underdetermination identified in this paper is not the radical underdetermination of Quine (where infinitely many theories fit any finite data). It is a specific, resolvable underdetermination: existing tests do not directly probe the observables that distinguish GR from TEP, but new tests could.

9.3.1 The Structure of the Underdetermination

9.3.2 Resolvable Underdetermination

The underdetermination is resolvable because GR and TEP make different predictions for observables that have not yet been measured with appropriate configurations. The triangle holonomy test, interplanetary closed-loop timing, and GNSS correlation replication could all distinguish the theories.

This is not a failure of falsifiability but a gap in the experimental program. The tests that would discriminate have simply not been performed.

9.4 The Sociology of Precision Tests

Why have discriminating tests not been performed? Several factors contribute:

9.4.1 Two-Way Convenience

Two-way measurements are experimentally simpler. They require only one clock, avoid synchronization problems, and provide gauge-invariant results. The experimental tradition naturally evolved toward two-way configurations.

9.4.2 The "GR is Correct" Prior

GR has been spectacularly successful. The prior probability assigned to alternatives is low, reducing motivation to design experiments specifically to test them. Resources flow toward improving precision within the GR framework rather than testing the framework itself.

9.4.3 The PPN Paradigm

The PPN formalism provides a systematic way to parameterize deviations from GR. But PPN is a parameterization of the post-Newtonian limit under assumptions about which effective metric governs the sector being tested (typically solar-system dynamics and light propagation). This makes it an exceptionally powerful framework for constraining single-metric departures in those sectors, while leaving open the possibility of additional clock-sector structure that is not representable as a small set of PPN light-propagation parameters.

9.4.4 Scientific Conservatism

When experiments confirm GR, they are celebrated. When anomalies appear (Pioneer anomaly, flyby anomaly), enormous effort goes into finding conventional explanations. This is scientifically appropriate—extraordinary claims require extraordinary evidence, and GR has earned its status through a century of successful predictions. The point is not that this conservatism is wrong, but that it shapes which experiments get funded and performed.

9.5 The Broader Context

9.5.1 Dark Matter and Dark Energy

The standard cosmological model requires ~95% of the universe to consist of unknown dark matter and dark energy. These are inferred from gravitational effects assuming GR is correct on all scales.

If GR is not the complete description of gravity—if TEP or another modification is required—the inferred dark sector could be partially or wholly an artifact of applying the wrong theory. The stakes for testing GR alternatives are high.

9.5.2 The Hubble Tension

The discrepancy between early-universe (CMB) and late-universe (Cepheid/SNe) measurements of the Hubble constant has reached 5σ significance. This tension could indicate new physics—potentially related to modifications of GR.

9.5.3 The S₈ Tension

Similarly, measurements of matter clustering (S₈) show tension between CMB predictions and direct measurements. Modified gravity theories, including scalar-tensor theories like TEP, could potentially resolve these tensions.

9.6 Summary: The State of the Evidence

10. Conclusions

This paper has presented a systematic methodological analysis of the canonical precision tests of general relativity. The analysis reveals that these tests—while representing extraordinary achievements in experimental physics—share structural features that, as currently configured, do not directly probe the specific loop and correlation observables that distinguish single-metric theories (GR) from two-metric alternatives such as the Temporal Equivalence Principle.

10.1 Principal Findings

The analysis in Sections 3–8 demonstrates that the five structural limitations identified in §1.2—two-way measurement dominance, local/global conflation, model-dependent calibration, the conformal loophole, and theory-laden data reduction—are not isolated issues but interconnected features of the experimental tradition.

The underdetermination is structural, not accidental. Two-way measurements evolved because they are simpler and provide gauge-invariant results. The PPN formalism, which guides most precision tests, constrains the post-Newtonian limit in the sectors it parameterizes (solar-system dynamics and light propagation). This makes it exceptionally powerful for ruling out many alternatives to GR, while leaving open the possibility of additional clock-sector structure that is not reducible to a small set of PPN light-propagation parameters.

10.2 What the Tests Do Establish

The critique presented here should not be misread as claiming that precision tests are worthless. They establish important facts:

• GR provides a self-consistent framework for describing gravity
• Local Lorentz invariance holds to 10⁻¹⁸ precision
• PPN parameters match GR predictions (γ = β = 1)
• GW–EM propagation speeds are bounded to differ by at most a few × 10⁻¹⁵ under standard emission-time assumptions (constraining disformal coupling)
• No evidence for preferred-frame effects in two-way measurements

These are genuine constraints on the space of viable theories. Many alternatives to GR are ruled out by these tests. But TEP is not among them in the regime considered here: it can be formulated to satisfy the existing constraints while introducing additional clock-sector structure that is not directly targeted by the canonical measurement geometries.

10.3 The Path Forward

10.3.1 Discriminating Experiments

The underdetermination is resolvable. Experiments that could distinguish GR from TEP include:

• Triangle Holonomy Test: One-way timing around a large-area closed loop (e.g., ground-satellite-ground), measuring a GR-subtracted residual holonomy $H_{\text{resid}}$. GR predicts $H_{\text{resid}} = 0$; a nonzero residual indicates non-exact time geometry.
• Matter-Wave Interferometry: Atom interferometers in gradiometric configurations sensitive to gradients of the scalar field $\nabla\nabla \phi$ via the conformal factor, distinguishable from GR gravity by combining with mechanical gravimeters.
• Interplanetary Closed-Loop Timing: AU-scale, direction-reversed loop measurements constructed from one-way time-tagged links, targeting a GR-subtracted residual holonomy analogous to the terrestrial triangle observable.
• GNSS Correlation Replication: Independent, blinded analysis of raw GNSS data to verify or refute the distance-structured correlations suggested by exploratory analyses within the TEP research program.
• Optical Clock Networks: Continental-scale networks using one-way comparisons to probe synchronization structure at 10⁻¹⁸ precision.

10.3.2 The Scientific Imperative

The dark matter problem, the Hubble tension, and the S₈ discrepancy all suggest that current understanding of gravity may be incomplete. Testing alternatives to GR is not merely an academic exercise—it addresses fundamental questions about the nature of the universe.

Optical clocks, one-way optical links, and GNSS infrastructure already exist. What is needed is a shift in experimental philosophy: from improving precision within the GR framework to designing tests that could falsify it.

10.3.3 Experimental Priorities

The proposed discriminating experiments can be ranked by technical feasibility, discriminating power, and resource requirements:

10.4 Concluding Remarks

10.5 Summary Table: The Experimental Canon

Every entry in the "TEP Status" column reads "Compatible." This reflects the structural nature of the underdetermination. The experimental canon, for all its remarkable precision, was designed to test predictions that GR and TEP share. The experiments succeed brilliantly at their intended purpose; they simply were not designed to probe the specific observables where GR and TEP differ.

This is not a criticism of the experimentalists who performed these measurements—their work represents some of the greatest achievements in precision physics. It is an observation about the logical structure of theory testing: to distinguish between theories, one must measure observables where they make different predictions.

References