New Exact Black Hole Solutions Emerge From Gravity With a Twist of Torsion

More than a century after Karl Schwarzschild found the first exact solution to Einstein’s field equations, physicists continue to ask whether the black holes described by general relativity are the whole story. A new theoretical study published in The European Physical Journal C by Lehel Csillag and Tiberiu Harko of Babes-Bolyai University and their collaborators has derived two exact vacuum black hole solutions in a little-known extension of Einstein’s theory called semi-symmetric metric gravity, a framework in which spacetime carries not only curvature but also a very special kind of torsion. The results offer a rare gift in modified gravity: closed-form, fully analytical metrics that can be tested directly against observations of the Solar System and probed for thermodynamic and dynamical stability.

To understand what makes this possible, it helps to recall what torsion is. In general relativity, the connection that tells parallel-transported vectors how to move is the Levi-Civita connection, which is torsion-free and metric-compatible. In Einstein-Cartan theory, torsion is sourced algebraically by the spin density of matter and effectively vanishes in vacuum. Poincare gauge theory allows several independent torsion modes with multiple coupling constants, while teleparallel and f(T) gravity abandon curvature altogether and describe gravity purely through torsion. Semi-symmetric metric gravity, rooted in a class of connections introduced by Friedmann and Schouten in 1924, sits in an elegant middle ground: the spacetime retains both curvature and torsion, but the torsion tensor is completely determined by a single vector field, the torsion vector. In four dimensions this reduces the 24 independent components of a general torsion tensor to just 4, making it arguably the simplest torsional extension of general relativity.

The field equations of the theory are the vacuum Einstein equations built from the semi-symmetric Ricci tensor and scalar, which acquire extra contributions from the torsion vector and its derivatives. Csillag and Harko assumed a static, spherically symmetric metric and a purely radial torsion vector, and imposed the familiar Schwarzschild-like condition that the two metric functions sum to zero. Under these assumptions the coupled differential equations collapse into a solvable system. The outcome is a bi-parametric family of exact solutions whose metric function can be written compactly as a product of a quadratic factor in the radial coordinate and the standard Schwarzschild factor, normalized by a term involving the ratio of the gravitational radius to a new length scale. One integration constant reproduces the usual gravitational radius, while the second, denoted R_g, is a genuinely new cosmological-type length scale with no counterpart in Einstein’s theory.

The structure of the solution depends on a discrete sign choice. For one branch, the black hole possesses a single horizon at the gravitational radius, closely resembling Schwarzschild. For the other branch, a second, outer horizon appears at the radius R_g itself, and the torsion vector vanishes exactly at the inner horizon. In the limit where the new length scale vastly exceeds the gravitational radius, the metric smoothly reduces to the Schwarzschild solution, so general relativity is recovered whenever torsional effects are negligible. The authors also showed that the torsional contribution can be reinterpreted as an effective gravitational radius and an effective geometric density, quantities that behave as if the vacuum were filled with a smooth, geometry-generated mass distribution.

The real power of an exact solution is that it can be confronted with data, and the authors applied the three classical tests of general relativity. For the perihelion precession of Mercury, they derived the correction to the standard 42.94 arcseconds per century predicted by general relativity. Observations give 43.11 plus or minus 0.21 arcseconds per century, leaving a residual of 0.17 plus or minus 0.21 arcseconds that could be attributed to torsion. Assuming the entire residual comes from the new physics, the test fixes the sign of the new parameter and yields a lower bound of roughly 5 times 10 to the 26 centimeters for R_g, a scale far exceeding the Solar System and approaching cosmological proportions.

The deflection of light by the Sun and the radar echo delay between Earth and Mars tell a consistent story. Long-baseline radio interferometry constrains the fractional deviation of the light-deflection angle to at most a few parts in ten thousand, which translates into a torsional length scale of about 1.7 times 10 to the 25 centimeters. The Shapiro time delay measured in the Earth-Mars-Sun system gives a remarkably close value of about 2.1 times 10 to the 25 centimeters. Intriguingly, these bounds correspond to inverse-square values in the same ballpark as, though somewhat larger than, the observed cosmological constant, suggesting that the new length scale may be related to the dark energy scale that drives the accelerated expansion of the Universe, perhaps through a relation in which R_g scales as the inverse square root of the cosmological constant.

The thermodynamic analysis reveals genuinely new behavior. For the single-horizon branch, the Hawking temperature is shifted by a term proportional to one over R_g, and the Bekenstein-Hawking entropy takes the standard area form. For the two-horizon branch, the inner black hole horizon has a temperature reduced by the same kind of correction, while the outer horizon at R_g turns out to be a double root of the metric function. As a consequence, its Hawking temperature vanishes exactly, making it a genuinely cold horizon: it carries finite area and finite entropy but does not radiate thermally in the usual Hawking sense, a behavior reminiscent of extremal and cosmological horizons but even more extreme. In both branches the heat capacity of the black hole horizon is negative whenever the temperature is positive, reproducing the familiar thermodynamic instability of asymptotically flat black holes, whereby losing energy makes the hole hotter.

Stability was examined through linear perturbations. Scalar perturbations lead to a Schrodinger-like equation with an effective potential whose peak is determined by a quintic equation, so the scalar quasinormal spectrum must be computed numerically and is deferred to future work. The vector, or electromagnetic, sector, however, can be solved analytically. The effective potential remains non-negative throughout the physical region between the horizons, and a first-order WKB analysis shows that the imaginary parts of the quasinormal frequencies are negative, meaning vector perturbations decay in time. The vector sector is therefore linearly stable, an essential consistency check for any proposed black hole metric. Depending on the sign of the new parameter, the torsional correction either raises or lowers the potential barrier that governs how gravitational waves ring down from the hole, offering a potential observational signature.

Perhaps the most provocative implication lies at galactic and cluster scales. Because the new terms in the metric grow with radius, they become relevant only over megaparsec distances, precisely the scales where galaxies and clusters appear to contain far more gravitating mass than their visible stars and gas can supply. The torsion-generated effective density could mimic the gravitational effect of dark matter, and the authors outline how stellar velocity dispersions in clusters, analyzed through the Jeans equation with a Plummer stellar profile, could constrain the effective mass profile and the torsion vector directly. In cosmology, the same torsion sector modifies the Friedmann equations and admits de Sitter-type accelerated expansion without extra scalar fields, hinting at a geometric origin for inflation and dark energy alike. From Mercury’s orbit to the edge of galaxy clusters, a single torsion vector may quietly reshape gravity’s story, and the new exact solutions give astronomers a concrete target to test it against.

Subject of Research: Exact vacuum black hole solutions in semi-symmetric metric gravity with vectorial torsion and their observational constraints

Article Title: Exact black hole solutions with semi-symmetric vectorial torsion

Article References: Exact black hole solutions with semi-symmetric vectorial torsion. (n.d.). https://doi.org/10.1140/epjc/s10052-026-16189-8

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16189-8

Keywords: black holes, torsion, semi-symmetric metric gravity, modified gravity, Schwarzschild solution, Solar System tests, perihelion precession, light deflection, radar echo delay, Hawking temperature, quasinormal modes, dark matter