⚛️ general relativity

Terrestrial Sagnac delay in scalar-tensor-vector-gravity

This paper demonstrates that the dimensionless parameter $\alpha$ in Scalar-Tensor-Vector-Gravity (STVG) can be tightly constrained to the range $0 < \alpha < 10^{-5}$ by requiring that the theory's corrections to the terrestrial Sagnac delay remain consistent with observed GPS fluctuations.

Original authors: R. Kh. Karimov, R. N. Izmailov, K. K. Nandi

Published 2026-06-26

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CC BY 4.0

Original authors: R. Kh. Karimov, R. N. Izmailov, K. K. Nandi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: A New Kind of Gravity?

Imagine you are trying to figure out how heavy a suitcase is. In our current understanding of physics (General Relativity), when we look at galaxies spinning, they seem too light to hold themselves together. They should fly apart! To fix this, scientists invented "Dark Matter"—an invisible, ghostly substance that adds extra weight to hold the galaxies together.

But nobody has ever seen or touched this "ghost." It's like trying to find a specific person in a crowd by assuming they are invisible.

Enter STVG (Scalar-Tensor-Vector-Gravity). This is a theory proposed by physicist John Moffat. Instead of adding invisible ghosts, STVG suggests that the rules of gravity itself are slightly different. It says gravity is a bit stronger than we thought, but it also has a "repulsive" kick (like a spring) that pushes things apart at certain distances. This theory can explain why galaxies spin the way they do without needing any invisible dark matter.

The paper's authors want to test this theory. They are looking for a "smoking gun" that proves STVG is real or, more likely, proves it's wrong.

The Experiment: The "Sagnac Effect" (The Spinning Turntable)

To test this, the authors use a phenomenon called the Sagnac effect.

The Analogy:
Imagine a giant, spinning record player (a turntable). You stand on the edge with a flashlight. You shine two beams of light:

One beam goes with the spin (clockwise).
One beam goes against the spin (counter-clockwise).

Because the turntable is moving, the beam going against the spin has a shorter path to get back to you (you are moving toward it). The beam going with the spin has a longer path (you are running away from it).

When they meet back at your flashlight, they arrive at slightly different times. This time difference is the Sagnac delay.

On Earth, this happens all the time. The Earth is the turntable, and satellites or airplanes are the light beams. We use this effect to keep our GPS clocks synchronized. If we didn't account for this delay, your phone's map would be off by miles within minutes.

The Test: Measuring the "Ghost" in the Machine

The authors asked: If STVG is true, does it change the Sagnac delay?

In the STVG theory, there is a special number called $\alpha$ (alpha). This number represents how much the "new" gravity rules differ from Einstein's old rules.

If $\alpha = 0$ , STVG is just Einstein's General Relativity (no new physics).
If $\alpha > 0$ , the new gravity rules are active.

The authors calculated exactly how much the Sagnac delay would change if $\alpha$ had a value. They then compared this theoretical change to real-world data from two sources:

Old Data (Hafele-Keating & Allan, Weiss, Ashby): These were experiments using atomic clocks on airplanes and early GPS satellites. The measurements were good, but they had a "fuzziness" (error margin) of about 5 nanoseconds (a billionth of a second).
- The Result: Because the data was so fuzzy, the new gravity theory could hide in the noise. The authors found that $\alpha$ could be as high as 48 million and still fit the data. This wasn't very helpful; it's like trying to find a needle in a haystack when the haystack is the size of a mountain.
New Data (Updated GPS): The authors looked at much more precise data from modern GPS satellites. These satellites orbit high above the Earth and move in a "free-fall" path (geodesic). The measurements here are incredibly sharp. The "fuzziness" or fluctuation in the timing is now only about 0.25 nanoseconds.
- The Result: With this super-precise ruler, the "new gravity" theory can't hide anymore. If $\alpha$ were even slightly large, the GPS clocks would be off by more than the tiny 0.25 nanoseconds allowed.

The Conclusion: Tightening the Noose

The paper concludes that for the STVG theory to survive this test, the value of $\alpha$ must be incredibly small.

Old Limit: $\alpha$ could be up to 48,000,000.
New Limit: $\alpha$ must be less than 0.00001 ( $10^{-5}$ ).

The Metaphor:
Imagine you are trying to hear a whisper in a noisy room.

In the old experiment, the room was so loud (5 ns error) that you couldn't tell if the whisper (the new gravity effect) was there or not.
In the new experiment, the room is silent (0.25 ns error). If the whisper were even a little bit loud, you would hear it immediately. Since we don't hear it, the whisper must be almost non-existent.

Summary

The paper doesn't prove that STVG is wrong, but it severely restricts how "weird" it can be. It tells us that if this new theory of gravity exists, its effects on Earth are so tiny that they are practically invisible to our current best clocks. The "new gravity" parameter ( $\alpha$ ) must be vanishingly small, pushing the theory very close to Einstein's original General Relativity for objects like Earth.

Technical Summary: Terrestrial Sagnac Delay in Scalar-Tensor-Vector-Gravity

Problem Statement
Scalar-Tensor-Vector-Gravity (STVG), a modified gravity theory developed by Moffat, offers a potential alternative to General Relativity (GR) by explaining galactic rotation curves, cluster dynamics, and cosmological data without invoking dark matter. The theory introduces a dimensionless parameter, $\alpha$ , which modifies the gravitational coupling constant ( $G = G_N(1 + \alpha)$ ) and generates a repulsive Yukawa-like force. While STVG has shown success in various astrophysical contexts, the permissible range of values for $\alpha$ remains a subject of investigation. Previous literature suggests a range of $0.03 < \alpha < 2.47$ based on different central source masses. This paper aims to constrain $\alpha$ further by analyzing its impact on the terrestrial Sagnac delay, specifically examining whether STVG corrections exceed the observational uncertainties in modern Global Navigation Satellite System (GPS) data.

Methodology
The authors derive the Sagnac time delay for light beams traveling in opposite directions around a rotating massive body (modeled as the Earth) within the framework of the Kerr-STVG metric. The study proceeds through the following steps:

Theoretical Framework: The paper revisits the STVG action and field equations, focusing on the Kerr-MOG (Modified Gravity) black hole solution. The metric is derived assuming a constant gravitational coupling $G = G_N(1 + \alpha)$ and a massive vector field $\phi_\mu$ with a gravitational charge $Q = \sqrt{\alpha G_N M}$ .
Derivation of Delay Formulas:
- Non-Geodesic Motion: The authors calculate the Sagnac delay for a source/receiver undergoing forced circular motion (non-geodesic), analogous to the Hafele-Keating airborne clock experiment. They derive an exact expression for the delay (Eq. 38) and its post-Newtonian expansion (Eq. 43), isolating terms dependent on mass ( $M$ ), spin ( $a$ ), and the STVG parameter ( $\alpha$ ).
- Geodesic Motion: The calculation is extended to free-fall circular orbits (geodesic motion), relevant for GPS satellites. The authors derive the exact Sagnac delay for geodesic motion in the Kerr-STVG spacetime (Eq. 55) and its weak-field approximation (Eq. 56).
Application to Terrestrial Data: The derived formulas are applied to Earth's parameters ( $M_\oplus, a_\oplus, \omega_\oplus$ $M_{\oplus}, a_{\oplus}, ω_{\oplus}$ ).
- Case 1 (Non-Geodesic): The authors compare the STVG correction terms against the error residuals of the classic Hafele-Keating experiment and the more precise Allan, Weiss, and Ashby (AWA) GPS-based measurements (error $\sim 5$ ns).
- Case 2 (Geodesic): The authors utilize updated data from Two-Way Satellite Time and Frequency Transfer (TWSTFT) experiments. They focus on the diurnal fluctuation ( $\Delta(\delta\tau_S)$ ) of the Sagnac delay caused by satellite jitter, which has been measured to be approximately $\pm 0.25$ ns.

Key Contributions and Results
The paper provides the first derivation of the Sagnac delay specifically for the Kerr-STVG metric, separating the contributions of the STVG parameter $\alpha$ from standard GR effects (mass and spin).

Non-Geodesic Constraint: Applying the non-geodesic formula to the AWA data (error $\sim 5$ ns), the authors find that the $\alpha$ -dependent correction term is negligible compared to the error. This yields a loose upper bound of $\alpha \leq 4.80 \times 10^7$ . The authors note this bound is not highly informative.
Geodesic Constraint (Primary Result): By applying the geodesic formula to the updated TWSTFT data, which offers a much tighter fluctuation limit of $\Delta(\delta\tau_S) \approx 0.25$ ns, the authors derive a significantly stricter constraint. The $\alpha$ -dependent term in the geodesic delay is calculated to be $7.05 \times 10^3 \alpha$ ns. Requiring this correction to be less than or equal to the observed fluctuation ($0.25$ ns) leads to the constraint:
$\alpha < 3.54 \times 10^{-5}$
Consequently, the paper proposes the range $0 < \alpha < 10^{-5}$ for terrestrial applications.

Significance and Claims
The paper claims that the terrestrial Sagnac delay, when analyzed through the lens of updated GPS and TWSTFT precision, provides a powerful diagnostic tool for constraining modified gravity theories.

Refinement of Bounds: The study demonstrates that using geodesic motion data (GPS satellites) combined with high-precision fluctuation analysis reduces the upper bound on $\alpha$ by several orders of magnitude compared to previous non-geodesic analyses or literature values ( $0.03 < \alpha < 2.47$ ).
Consistency with Observations: The authors assert that their derived constraint ( $\alpha < 10^{-5}$ ) is consistent with the theory's ability to explain other phenomena (like neutron star masses) while suggesting that for Earth-mass sources, the deviation from GR must be extremely small.
Modest Scope: The authors explicitly state that this constraint is an upper bound. They do not claim to prove $\alpha$ is exactly zero, but rather that if STVG is correct for Earth-mass objects, $\alpha$ must be exceedingly small ( $< 10^{-5}$ ) to remain consistent with current terrestrial timing precision. The paper concludes that this result suggests a potential improvement over previous bounds proposed for stellar mass sources.

The Big Picture: A New Kind of Gravity?

The Experiment: The "Sagnac Effect" (The Spinning Turntable)

The Test: Measuring the "Ghost" in the Machine

The Conclusion: Tightening the Noose

Summary

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