Testing non-circular black hole spacetime with X-ray reflection

X-ray reflection spectroscopy is a powerful tool for testing the Kerr hypothesis and probing the strong gravity regime around accreting black holes. Most tests of general relativity (GR) assume that the spacetime around a black hole is circular, meaning the metric possesses a specific symmetry structure common to the Kerr solution. However, deviations from circularity are predicted by various modified gravity theories and non-vacuum general relativity solutions. In this work, we test a specific non-circular metric constructed based on a locality principle, where the deviation from the Kerr spacetime is driven by the local spacetime curvature. To accurately model the reflection spectrum in this background, we implement a relativistic ray-tracing code in horizon-penetrating (ingoing Kerr) coordinates, which are favored for their ability to avoid introducing curvature singularities at the horizon in non-circular spacetimes. We apply this model to the high-quality NuSTAR spectrum of the Galactic black hole binary EXO 1846-031. Our spectral analysis reveals a source with a high inclination angle ( ι ≈ 76∘ ι ≈ 76 ∘ ) and a near-extremal spin parameter ( a_* ≈ 0.98 a ∗ ≈ 0.98 ). While we identify a global minimum in the parameter space suggesting a non-zero deformation ( NP ≈ 0.12 ℓ NP ≈ 0.12 ), the 99% confidence interval fully encompasses the Kerr limit ( NP=0 ℓ NP = 0 ). We conclude that the current X-ray reflection data for EXO 1846-031 are consistent with the Kerr hypothesis. This work demonstrates the feasibility of using X-ray reflection spectroscopy to constrain non-circular metrics and establishes a framework for future tests.