Testing the Spacetime Geometry of Sgr A* with the Relativistic Orbit of S2 star

TL;DR

This work tests the spacetime geometry at the Galactic Center by propagating the S2 star's timelike geodesics through a suite of static, spherically symmetric spacetimes—ranging from Schwarzschild to Reissner–Nordström, regular black holes (Bardeen, Hayward), Simpson–Visser, and JNW naked singularities—and comparing predicted astrometric, spectroscopic, and relativistic redshift observables with VLT data while enforcing independent EHT shadow constraints. Fully relativistic orbit integration with Roemer and Einstein delays is combined with Bayesian MCMC to perform model comparison via AIC/BIC and to constrain the generalized charge-like parameter $q/M$. The results show that RN and BD spacetimes remain statistically indistinguishable from Schwarzschild with current S2 data, while HY, SV, and JNW are disfavored; the data place tight bounds on $q/M$ but do not decisively determine the central object’s causal structure. The findings emphasize the complementary role of stellar dynamics and horizon-scale imaging in probing strong-field gravity and highlight the need for spin-aware, fully relativistic projections and longer, higher-precision astrometric campaigns to break degeneracies and test the Kerr paradigm at the GC.

Abstract

In this work, we perform a relativistic test of the spacetime geometry of Sagittarius A* (Sgr A*) using the orbit of the S2 star. We consider a broad class of compact object models, including Schwarzschild, Reissner-Nordström, Bardeen, Hayward, and Simpson-Visser black holes, as well as the Janis-Newman-Winicour naked singularity spacetime. For each geometry, we integrate the timelike geodesic equations and consistently project the resulting trajectories onto astrometric and spectroscopic observables, incorporating Rømer time delay and relativistic redshift effects. The theoretical predictions are tested with current Very Large Telescope (VLT) observations of the S2 star, while simultaneously imposing constraints from the Event Horizon Telescope shadow size. We find that several spacetimes that are degenerate at the level of shadow imaging, most notably Schwarzschild, Reissner-Nordström, and Bardeen regular black hole geometries, remain statistically indistinguishable when tested against present S2 data. We further carry out a statistical model comparison based on the Akaike and Bayesian information criteria (AIC and BIC) to evaluate the relative performance of the alternative spacetime models. Our analysis also constrains the generalized charge like parameter $q/M$ in non-Schwarzschild spacetimes based on current S2 star observations, and identifies specific black hole and horizonless geometries that can be further tested with forthcoming high precision astrometric observations from the VLT and Keck telescopes.

Figures (6)

• Figure 1: Constraints on the generalized charge-like parameter $q$ inferred from the periastron precession of the S2 star. The shaded bands correspond to the $1\sigma$ and $2\sigma$ confidence intervals derived from the observed precession factor $f_{\rm sp}$. Each panel shows the allowed parameter range for a specific non-Schwarzschild spacetime model.
• Figure 2: Figure shows the marginalized posterior distributions scaled to 1 for $q/M$ for all models obtained by MCMC analysis. The dotted line represents the 2$\sigma$ upper limit for $q/M$ for all spacetimes. The corresponding $1 \sigma$ and $2\sigma$ upper limits on $q/M$ are given in Table \ref{['tab:q_constraints']}.
• Figure 3: This figure shows the combined astrometric and spectroscopic orbital fits of the S2 star using 24 years of VLT observations. The data are fitted within the Reissner--Nordström (RN) and Bardeen (BD) spacetimes and compared with the Schwarzschild (SCH) spacetime, which is shown by the dotted curve. For each spacetime, the left panel displays the best-fit sky-projected orbit, while the right panel shows the corresponding astrometric offsets and line-of-sight radial velocity measurements together with the model predictions.
• Figure 4: This figure shows the corner plot of the Schwarzschild model obtained from an MCMC analysis of the combined astrometric and spectroscopic S2 data. The contours indicate the $68\%$ and $95\%$ credible intervals. Best-fit values correspond to the posterior medians. The reduced chi-square is $\chi^2_\nu = 0.943$.
• Figure 5: This figure shows the corner plot of the RN model obtained from an MCMC analysis of the combined astrometric and spectroscopic S2 data. The contours indicate the $68\%$ and $95\%$ credible intervals. Best-fit values correspond to the posterior medians. The reduced chi-square is $\chi^2_\nu = 0.946$.