General Relativity describes gravity as the curvature of spacetime. Mass and energy determine this curvature, and physical phenomena such as orbital motion, gravitational radiation, and the propagation of light are governed by the resulting geometry. In this framework gravity is not a force but a geometric property of spacetime itself.
Over the past century, General Relativity has been tested in many independent regimes. Planetary motion agrees with relativistic predictions, gravitational waves from compact binary mergers have been observed, and direct images of black hole shadows have been produced. Each of these observations probes a different physical manifestation of spacetime curvature. However, these tests are typically analyzed in isolation. Orbital dynamics, gravitational-wave signals, and black hole imaging are treated as separate probes, even when they concern the same astrophysical object. As a result, current tests do not directly verify whether all observations of a single black hole are mutually consistent with one and the same spacetime geometry.
The aim of this project is to address this missing consistency check. The guiding question is whether independent observations of the same black hole all describe the same spacetime, as predicted by General Relativity.
An uncharged, rotating black hole is described in General Relativity by the Kerr solution. This solution specifies the spacetime geometry outside the black hole and depends on only two physical parameters: the mass $M$ and the dimensionless spin $\chi$. The mass sets the overall curvature scale, while the spin controls the rotation and associated frame-dragging effects. The dimensionless spin is defined by
$$
\chi = \frac{J}{M^2}, \qquad |\chi| \le 1.
$$
A black hole cannot be observed directly. Instead, its spacetime geometry is inferred through its influence on matter, radiation, and light. In this work we focus on three observational sectors, each sensitive to different physical processes but governed by the same underlying Kerr spacetime.
In the dynamical sector, the curvature of spacetime determines the motion of massive bodies. In compact binary systems this motion produces gravitational waves whose phase evolution depends on the mass and spin of the black hole. Analysis of the inspiral signal yields an estimate $(M_{\mathrm{dyn}}, \chi_{\mathrm{dyn}})$.
In the ringdown sector, a perturbed black hole relaxes to equilibrium through a set of damped oscillations known as quasinormal modes. The frequencies and decay times of these modes depend only on the black hole mass and spin, independent of the details of the perturbation. Measurements of the ringdown signal provide a second estimate $(M_{\mathrm{rd}}, \chi_{\mathrm{rd}})$.
In the imaging sector, strong gravitational lensing near the black hole bends light into characteristic structures such as the photon ring and shadow. High-resolution images constrain the spacetime geometry through the size and shape of these features, leading to a third estimate $(M_{\mathrm{img}}, \chi_{\mathrm{img}})$.
The central hypothesis of this project is that if General Relativity correctly describes gravity, then these three independently inferred parameter pairs must be statistically consistent with a single Kerr spacetime. Differences between the measurements are expected due to experimental uncertainty, but they should not exceed what is allowed by those uncertainties. A statistically significant disagreement would indicate that the observations cannot be described by one common spacetime geometry.
To formalize this test, we describe the spacetime by the parameter vector $\Theta = (M, \chi)$. Each observational sector produces an estimate $\hat{\Theta}_{\mathrm{dyn}}$, $\hat{\Theta}_{\mathrm{rd}}$, and $\hat{\Theta}_{\mathrm{img}}$, with corresponding covariance matrices $\Sigma_{\mathrm{dyn}}$, $\Sigma_{\mathrm{rd}}$, and $\Sigma_{\mathrm{img}}$ encoding their uncertainties.
If all sectors are consistent, there should exist a common best-fit parameter vector $\bar{\Theta} = (\bar{M}, \bar{\chi})$ that minimizes the weighted discrepancy
$$
\chi^2(\Theta) = \sum_k
(\hat{\Theta}_k – \Theta)^{\mathrm{T}} \Sigma_k^{-1} (\hat{\Theta}_k – \Theta),
\qquad
k \in \{\mathrm{dyn}, \mathrm{rd}, \mathrm{img}\}.
$$
The minimizing value is given by
$$
\bar{\Theta}
=
\left(\sum_k \Sigma_k^{-1}\right)^{-1}
\left(\sum_k \Sigma_k^{-1} \hat{\Theta}_k\right).
$$
We then define the deviation of each sector from the common spacetime as $\delta \Theta_k = \hat{\Theta}_k – \bar{\Theta}$. The Kerr Trisector Closure statistic is
$$
T^2
=
\sum_k
\delta \Theta_k^{\mathrm{T}} \Sigma_k^{-1} \delta \Theta_k.
$$
This quantity measures the total inconsistency between the three sector estimates while accounting for their uncertainties. Since three independent measurements of two parameters are compared, the statistic follows a $\chi^2$ distribution with four degrees of freedom under the null hypothesis of consistency.
To validate the method, we test it using simulated data. We choose a true spacetime $\Theta^\ast = (M^\ast, \chi^\ast)$ and generate synthetic measurements $\hat{\Theta}_k = \Theta^\ast + \varepsilon_k$, where the errors $\varepsilon_k$ are drawn from Gaussian distributions with covariance $\Sigma_k$. In the consistent case, the resulting $T^2$ values follow the expected $\chi^2_4$ distribution. In Monte Carlo simulations, the mean value and rejection rate match theoretical expectations.
We then introduce a controlled bias in one sector, for example $\hat{\Theta}_{\mathrm{img}} = \Theta^\ast + \Delta + \varepsilon_{\mathrm{img}}$, while leaving the other sectors unchanged. In this case the $T^2$ distribution shifts to larger values and exceeds the consistency threshold with high probability. The individual contributions
$$
T_k^2 = \delta \Theta_k^{\mathrm{T}} \Sigma_k^{-1} \delta \Theta_k
$$
identify the sector responsible for the inconsistency.
These tests show that the Kerr Trisector Closure behaves as intended. It remains statistically quiet when all observations are consistent and responds strongly when a genuine inconsistency is present. The method provides a model-independent way to test the internal consistency of black hole spacetime measurements. Current observations do not yet allow a full experimental implementation, but future gravitational-wave detectors and improved black hole imaging may make such tests possible.