The research team, led by M. Castillo Alarcón of the Center for Research and Advanced Studies of the National Polytechnic Institute in Mexico City, together with Leonel Bixano, I. A. Sarmiento-Alvarado, Claudio Salas-Pérez and the veteran dark matter theorist Tonatiuh Matos, set out to correct a persistent weakness in the literature. Most existing models of black holes wrapped in dark matter halos begin by assuming a relationship between the two fundamental functions that define a static, spherically symmetric spacetime: the temporal metric function A(r), which governs gravitational redshift and time dilation, and the radial metric function B(r), which governs spatial curvature along the radial direction. In the pure Schwarzschild vacuum solution these functions are exact reciprocals of one another, satisfying A(r)B(r) = 1. Many authors simply carry that reciprocal relation over to halo systems as a convenient ansatz, without checking whether the Einstein field equations actually permit it once matter is present.

Castillo Alarcón and colleagues refused to make that assumption. They started from the most general static, spherically symmetric line element, with A(r) and B(r) treated as fully independent functions, and modeled the dark matter halo as an anisotropic fluid, one whose radial pressure differs from its tangential pressure. This flexibility matters because dark matter need not behave like a perfect fluid; different microscopic candidates, from cold collisionless particles to ultralight scalar fields, can generate different stress-energy configurations. By feeding three widely used halo density profiles directly into the Einstein equations, the Burkert profile favored by observations of dwarf and low-surface-brightness galaxies, the Einasto profile that fits simulated halos so well, and the multistate scalar field dark matter model developed over decades by Matos and collaborators, the team solved for the metric functions, the radial pressure and the tangential pressure simultaneously, without imposing any of them by hand.

The construction works as follows. The radial function B(r) is determined by the total gravitational mass enclosed within a given radius, which combines the black hole mass with the accumulated halo mass. Crucially, the authors assume that dark matter vanishes inside a critical radius r_in, equal to three times the Schwarzschild radius, because any dark matter orbiting closer than the innermost stable circular orbit of a massive particle would be captured by the black hole. Inside that boundary the spacetime is exactly Schwarzschild, in accordance with Birkhoff’s theorem, though the surrounding halo imprints a uniform gravitational redshift on the inner vacuum region that cannot be removed globally. Outside the boundary, the temporal function A(r) is built from a deformation potential that measures precisely how the halo warps the clock rate relative to the isolated black hole case.

The radial pressure of the halo is reconstructed from the observed circular velocities of stars, using the exact relativistic relation between circular geodesics and the gravitational potential. The tangential pressure then follows from the conservation of the stress-energy tensor, the anisotropic generalization of the Tolman-Oppenheimer-Volkoff equation that governs hydrostatic equilibrium in relativistic stars. With the full system in hand, the authors verified that all three density profiles satisfy the strong energy condition, meaning the halo source is physically reasonable rather than exotic in a pathological sense. They also demonstrated that the transition between the inner vacuum region and the outer halo at r_in satisfies the Darmois-Israel junction conditions: the induced metric and the extrinsic curvature match perfectly on both sides, so no thin shell of exotic surface matter, no surface pressure and no surface tension appear at the boundary. The halo begins smoothly, not as a membrane.

The headline result emerged from both numerical computation and a semi-analytical proof. Even though no reciprocal relation was ever assumed, the product A(r)B(r) equals one plus corrections of order ten to the minus six throughout the halo. The reason is a small dimensionless parameter that the authors call the halo strength, defined as eight pi times the gravitational constant times the central halo density times the square of the scale radius, divided by the square of the speed of light. For Milky Way parameters this factor is roughly a millionth. The Einstein equations show that the variation of the product A(r)B(r) is controlled entirely by this parameter, with the radial pressure contributing only a second-order correction and the tangential pressure not entering the relation at all. In other words, the reciprocal structure so often assumed in the literature is not an arbitrary guess; it is a derived consequence of the sheer weakness of the halo’s relativistic pull.

This finding has immediate practical value. Shadow calculations, gravitational lensing studies, quasinormal mode analyses and accretion disk models that adopt A(r) = 1/B(r) for halo systems can now do so with a theoretical license rather than a leap of faith. The authors note that the result is robust even if the sharp inner cutoff of the halo is replaced by a smooth taper, and that the three density profiles, despite their very different origins as phenomenological fits, simulation-motivated shapes and microscopic field theory constructions, produce nearly identical metric functions. From the perspective of light rays skimming the black hole, the team’s null geodesic calculations show that one essentially cannot tell whether the black hole is embedded in a halo or floating in perfect vacuum; the differences only become visible at distances far from the hole.

To test whether the model survives contact with reality, the researchers confronted it with two of the most precise measurements in modern astrophysics. The first is the pericenter precession of the star S2, which whips around Sagittarius A* every sixteen years on a highly eccentric orbit, passing within about fourteen hundred Schwarzschild radii of the black hole. The GRAVITY Collaboration has measured the relativistic precession of this orbit to remarkable accuracy. Integrating the geodesic equations in the halo spacetime for all three density profiles, the team obtained a precession of 12.1687 arcminutes per orbit, with a fitted scaling factor of 1.11 plus or minus 0.21, comfortably inside the observational range reported by GRAVITY.

The second test is the shadow of Sagittarius A* itself. Using the general formula for the angular radius of a shadow in a static, spherically symmetric spacetime, and adopting the measured distance to the galactic center of 8178 parsecs, the authors computed an angular diameter of 50.1746 microarcseconds for all three halo profiles. The Event Horizon Telescope reports 51.8 plus or minus 2.3 microarcseconds, so the prediction sits squarely within the error bars. Two independent observables, one probing the orbital dynamics of matter far from the hole and the other probing the bending of light near the photon sphere, are simultaneously satisfied. The dark matter halo, at least with the densities inferred for the Milky Way, does not push the spacetime outside what astronomers have already measured.

The study does leave honest caveats. The construction delivers a self-consistent static equilibrium but does not prove dynamical stability under radial perturbations; because the radial pressure depends nonlocally on the enclosed halo mass, a proper stability analysis would require solving the linearized Einstein equations with specified constitutive relations, a task the authors explicitly defer to future work. The treatment of the region inside the innermost stable orbit, where captured dark matter would accumulate, is also acknowledged as a simplification. Nevertheless, the paper’s central contribution stands: the near-reciprocity of the metric functions in black hole-halo systems is now a proven property rooted in the feeble relativistic strength of galactic dark matter, quantified by a single small parameter. As horizon-scale interferometry sharpens its resolution and stellar orbit monitoring extends to fainter stars closer to the event horizon, the framework offers theorists a rigorous foundation for asking the next question: how small a deviation from Schwarzschild would it take for the Event Horizon Telescope, or its successors, to finally see the dark matter that surrounds the shadow?

Subject of Research: Exact general relativistic modeling of a Schwarzschild black hole surrounded by a dark matter halo

Article Title: Determination of the metric of a Schwarzschild black hole surrounded by a halo of dark matter

Article References: Determination of the metric of a Schwarzschild black hole surrounded by a halo of dark matter. (n.d.). https://doi.org/10.1140/epjc/s10052-026-16375-8

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16375-8

Keywords: black holes, dark matter, general relativity, Schwarzschild metric, dark matter halo, Sagittarius A*, Event Horizon Telescope, GRAVITY Collaboration, S2 star, anisotropic fluid, Einstein field equations, shadow radius