⚛️ general relativity

Detecting Lorentz-violation induced by a tensor field with S-star's motion around Sgr A*

This paper utilizes Markov Chain Monte Carlo analysis of S2, S38, and S55 star orbital data around Sgr A* to constrain a Lorentz-violating parameter in Kalb-Ramond gravity, achieving bounds on the parameter $\ell$ that are three orders of magnitude tighter than previous Event Horizon Telescope imaging constraints.

Original authors: Qi Qi, Yu Sang, Xiao-Mei Kuang

Published 2026-08-19

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CC BY 4.0

Original authors: Qi Qi, Yu Sang, Xiao-Mei Kuang

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

Deep in the center of our Milky Way galaxy lies a monster of gravity: a supermassive black hole known as Sagittarius A*. For decades, astronomers have watched a small group of stars, called S-stars, zip around this invisible giant. These stars move so fast and get so close to the black hole that they experience extreme gravity, far stronger than anything we feel here on Earth. This environment acts as a natural laboratory for testing the rules of physics. The most famous set of rules is General Relativity, Albert Einstein's theory of how gravity works. While this theory has passed every test we have thrown at it so far, many physicists suspect it might not be the whole story. They believe that at the most fundamental level, the universe might have a slight preference for one direction over another, a concept known as Lorentz symmetry. If this symmetry is broken, it would mean the laws of physics change depending on which way you are facing or moving, a possibility that could help unify gravity with the other forces of nature.

A team of researchers from Yangzhou University in China has taken a fresh look at the motion of one specific star, S2, to see if they can find evidence of this symmetry breaking. Instead of looking at the black hole's shadow, as other teams have done, they analyzed the precise path S2 traces as it orbits the central black hole. They used a theoretical model that includes a specific type of invisible field, called a Kalb-Ramond field, which is predicted by some advanced theories of gravity. This field would act like a subtle, invisible wind that slightly alters the shape of spacetime around the black hole, changing how the star moves. By comparing the actual observations of the star's position and speed against the predictions of this new model, the researchers were able to test whether such a field exists.

The researchers gathered decades of data from powerful telescopes, tracking the star's position in the sky and measuring how fast it moves toward and away from us. They then ran a massive computer simulation to see how well the new model fit the data. This simulation involved adjusting fourteen different variables at once, including the mass of the black hole, the distance to it, and the specific shape of the star's orbit, to find the combination that matched the observations best. They tested two different assumptions about how these variables might be distributed, ensuring their results were robust. The goal was to find a specific number that represents the strength of the Lorentz-violating effect. If this number were zero, it would mean the laws of physics are perfectly symmetrical, just as Einstein predicted. If it were anything else, it would signal a crack in the foundation of our current understanding.

The results were remarkably precise. The team found that the data is consistent with the standard theory of gravity, meaning there is no strong evidence for this new field yet. However, they were able to set a very tight limit on how strong such a field could possibly be. They calculated that if this Lorentz-violating effect exists, it must be incredibly small, with a value around one hundred-thousandth of a percent. To put this in perspective, this new limit is about one thousand times more precise than the best constraints obtained from images of the black hole's shadow taken by the Event Horizon Telescope. This means that watching the star dance around the black hole provides a much sharper tool for testing these theories than simply taking a picture of the black hole itself.

The study also looked at other stars in the same region, such as S38 and S55, and even tried combining the data from all three stars to see if it would improve the results. While the other stars provided useful information, the data for S2 was so much more complete and precise that it remained the most powerful tool for the analysis. The researchers noted that while their limits are not as tight as those obtained from experiments within our own solar system, testing gravity near a supermassive black hole is a completely different and necessary challenge. The extreme environment near the black hole offers a unique place to look for effects that might be hidden in the calmer gravity of our solar neighborhood.

Ultimately, this work demonstrates that the orbital paths of stars around the galactic center are sensitive enough to detect even the tiniest deviations from Einstein's theory. The researchers found no sign of the Lorentz violation they were hunting for, but by narrowing the range of possibilities so significantly, they have cleared the path for future discoveries. As new telescopes come online with even greater precision, these stellar orbits will continue to serve as a rigorous test bed for the deepest laws of the universe, helping us understand whether the fabric of spacetime is truly uniform or if it holds hidden secrets waiting to be uncovered.

Technical Summary: Detecting Lorentz-violation induced by a tensor field with S-star's motion around Sgr A*

Problem Statement
While General Relativity (GR) has been rigorously tested in the Solar System, the extreme gravitational environment near supermassive black holes (SMBHs) offers a unique regime to search for deviations, such as Lorentz symmetry breaking (LSB). Theories like string theory and loop quantum gravity suggest LSB may occur at the Planck scale, manifesting as small corrections in gravitational fields. The Standard-Model Extension (SME) provides a framework for these corrections, specifically through the Kalb–Ramond field, an antisymmetric tensor field arising naturally in string theory effective actions. In this scenario, spontaneous LSB is triggered by a non-minimal coupling between the Kalb–Ramond field ( $B_{\mu\nu}$ ) and gravity, leading to a non-zero vacuum expectation value (VEV) that sets a preferred direction in spacetime. This paper aims to constrain the dimensionless Lorentz-violating parameter $\ell$ (defined as $\xi b^2$ , where $\xi$ is the coupling strength and $b^2$ is the norm of the VEV) by analyzing the orbital dynamics of S-stars around the SMBH Sgr A*.

Methodology
The authors model the spacetime around Sgr A* using a static, spherically symmetric Schwarzschild-like metric derived from Kalb–Ramond gravity. The metric differs from the standard Schwarzschild solution by the parameter $\ell$ , which modifies the radial and temporal components.

Theoretical Derivation:
- The geodesic equations for timelike test particles are derived in the equatorial plane.
- An analytical expression for the pericentre precession is obtained, keeping terms to the leading order in $\ell$ . The result shows that the Lorentz-violating parameter introduces a linear correction to the GR precession: $\Delta\phi \simeq \Delta\phi_{GR} - \pi\ell$ .
- The orbital trajectory is parameterized using the semi-major axis $a$ and eccentricity $e$ , with the motion described via the eccentric anomaly.
Data Analysis and Modeling:
- Dataset: The study utilizes public astrometric and spectroscopic data for the S2 star, collected over three decades by the Keck Observatory and the VLT (GRAVITY instrument). This includes 145 astrometric positions, 44 radial velocity measurements, and the measured pericentre advance ratio ( $f_{sp} = 1.10 \pm 0.19$ ).
- Orbital Model: The authors numerically integrate the geodesic equations in the Kalb–Ramond background. To compare with observations, they apply:
  - Coordinate transformations (Thiele-Innes constants) to project the orbital plane onto the sky plane.
  - Corrections for reference frame offsets and linear drifts ( $x_0, y_0, v_{x0}, v_{y0}$ ).
  - Rømer time delay corrections to account for the finite speed of light.
  - Relativistic frequency shifts (Doppler and gravitational redshift) to model radial velocities.
Statistical Analysis:
- A 14-dimensional Markov Chain Monte Carlo (MCMC) analysis is performed using the emcee package.
- The parameter space includes: central mass ( $M$ ) and distance ( $R_0$ ); six Keplerian orbital elements ( $a, e, \iota, \omega, \Omega, t_{apo}$ ); five reference-frame terms ( $x_0, y_0, v_{x0}, v_{y0}, v_{z0}$ ); and the Lorentz-violating parameter $\ell$ .
- Two prior distributions are tested: a uniform prior (to avoid bias) and a Gaussian prior (centered on GR best-fit values from the Gravity Collaboration).
- The likelihood function combines astrometric positions, radial velocities, and the precession constraint.

Key Contributions and Results

Constraints on $\ell$ : The MCMC analysis yields constraints on the Lorentz-violating parameter $\ell$ $ℓ$ at the $1\sigma$ $1 σ$ confidence level:
- Uniform Prior: $\ell = 1.60 \times 10^{-5} {}^{+1.38\times10^{-4}}_{-1.76\times10^{-4}}$
- Gaussian Prior: $\ell = -1.02 \times 10^{-5} {}^{+1.26\times10^{-4}}_{-1.19\times10^{-4}}$
- In both cases, the best-fit values are of the order $10^{-5}$ , and the results are consistent with GR (i.e., $\ell=0$ ) within current observational uncertainties.
Comparison with Other Probes: The constraints obtained from S2 orbital dynamics are approximately three orders of magnitude tighter than those derived from Event Horizon Telescope (EHT) imaging of Sgr A* (which constrain $\ell$ to $\sim O(10^{-2})$ ). While Solar System tests provide tighter bounds, this study probes a distinct physical regime characterized by strong gravitational fields near an SMBH.
Multi-Star Analysis: The authors also performed MCMC simulations using data from S38 and S55 stars, as well as joint analyses (S2+S38, S2+S55, and S2+S38+S55).
- The best-fit values for $\ell$ in all simulations remain of the order $10^{-5}$ .
- However, S2 provides the most stringent constraints on $M$ , $R_0$ , and $\ell$ . The joint fits do not significantly improve the constraints on $\ell$ compared to S2 alone, primarily because the public data for S38 and S55 are fewer and less precise.

Significance
The paper demonstrates that the orbital dynamics of S-stars, particularly S2, serve as a sensitive probe for Lorentz violation induced by the Kalb–Ramond field in the strong-gravity regime of the Galactic Center. The study establishes that stellar dynamics can provide constraints on LSB parameters that are significantly more precise than current black hole shadow imaging. Although current data are consistent with GR, the methodology provides a robust framework for testing gravity extensions. The authors note that future improvements in data quality from next-generation facilities (e.g., GRAVITY+, ELT) and the inclusion of more S-stars will further refine these tests. Additionally, the authors suggest that future work should incorporate the spin of Sgr A* (Kerr-like solutions in Kalb–Ramond gravity) for a more realistic description, as the current study assumes a static, non-rotating black hole.

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