The Weyl Geometric Gravity black hole in light of the Solar System tests

TL;DR

This paper tests the Weyl Geometric Gravity (WGG) black hole against precision Solar System data by analyzing six classical tests of General Relativity. It derives η-dependent corrections for gravitational redshift, WEP/SEP, perihelion precession, light deflection, and radar time delay from the exact static, spherically symmetric WGG solution and confronts them with GP‑A, MICROSCOPE, LLR, and Cassini observations. The combined analysis yields a stringent bound |η| < 10^−10, ensuring compatibility with Solar System tests while allowing η to be much larger on galactic scales (η ~ 10^−15 to 10^−23 from rotation curve fits). This scale‑dependent constraint highlights that Weyl nonmetricity could be relevant cosmologically while remaining tightly suppressed locally, offering a clear pathway to test Weyl geometry across different astrophysical regimes.

Abstract

The Weyl geometric gravity theory, in which the gravitational action is constructed from the square of the Weyl curvature scalar and the strength of the Weyl vector, has been intensively investigated recently. The theory admits a scalar-vector-tensor representation, obtained by introducing an auxiliary scalar field, and can therefore be reformulated as a scalar-vector-tensor theory in a Riemann space, in the presence of a nonminimal coupling between the Ricci scalar and the scalar field. By assuming that the Weyl vector has only a radial component, an exact spherically symmetric vacuum solution of the field equations can be obtained, which depends on three integration constants. As compared to the Schwarzschild solution, the Weyl geometric gravity solution contains two new terms, linear and quadratic in the radial coordinate, respectively. In the present work we consider the possibility of testing and obtaining observational restrictions on the Weyl geometric gravity black hole at the scale of the Solar System, by considering six classical tests of general relativity (gravitational redshift, the Eötvös parameter and the universality of free fall, the Nortvedt effect, the planetary perihelion precession, the deflection of light by a compact object, and the radar echo delay effect, respectively) for the exact spherically symmetric black hole solution of the Weyl geometric gravity. All these gravitational effects can be fully explained and are consistent with the vacuum solution of the Weyl geometric gravity. Moreover, the study of the classical general relativistic tests also allows to constrain the free parameter of the solution.