Non-minimally coupled Weyl connection gravity in the Solar System and at the Galactic Center
We explore the phenomenological viability of non-minimally coupled Weyl connection gravity by confronting its static, spherically symmetric black hole solutions with classical weak-field Solar System tests and stellar-orbit observations near Sgr A*. In this geometric framework, non-metricity is encoded via a Weyl vector field, giving rise to two distinct families of Schwarzschild-like vacuum solutions characterized by a free parameter \(\omega\) with dimensions of length. We compute the corrections to four classical observables---gravitational redshift, Mercury's perihelion advance, light deflection, and radar echo delay---and derive stringent lower bounds on \(\omega\) using current observational data. For Solution I (purely radial Weyl vector), the leading metric corrections scale as \(1/\omega\), yielding bounds as strong as \(\omega \gtrsim 10^{30}\) m from perihelion precession. For Solution II (time-radial Weyl vector), the corrections scale as \(1/\omega^2\), resulting in weaker constraints, with \(\omega \gtrsim 10^{20}\) m from the same test. This marked difference arises from the distinct behavior of the linear corrections in each solution: while Solution I exhibits an unsuppressed linear term \(2r/\omega\) that dominates in the Solar System regime, Solution II features a linear term suppressed by an additional factor of \(M/\omega\), making the quadratic term \(-r^2/4\omega^2\) dominant throughout. We then analyze stellar orbits near the Galactic Center, finding that current observations of the S2 star provide complementary constraints on both solutions. (...)