Non-minimally coupled Weyl connection gravity in the Solar System and at the Galactic Center

Technical Summary: Non-minimally coupled Weyl connection gravity in the Solar System and at the Galactic Center

Problem Statement
General Relativity (GR) remains the standard model of gravity but faces unresolved issues, including the nature of dark matter and dark energy, the cosmological constant problem, and the lack of a consistent quantum extension. Alternative theories often introduce higher-order curvature invariants or additional fields, which can lead to Ostrogradsky instabilities. This paper investigates the phenomenological viability of non-minimally coupled Weyl connection gravity, a theory that incorporates Weyl geometry (characterized by non-metricity) into a non-minimal matter-curvature coupling framework. While this theory avoids Ostrogradsky instabilities and possesses a viable minimum energy space, its specific vacuum black hole solutions must be rigorously tested against observational data to constrain its free parameters and assess its physical consistency.

Methodology
The authors analyze two distinct families of static, spherically symmetric vacuum solutions derived from the field equations of non-minimally coupled Weyl connection gravity. These solutions are characterized by a free parameter $\omega$ (the Weyl constant) with dimensions of length. The study proceeds through the following steps:

1. Metric Derivation and Approximation: The authors examine two ansätze for the Weyl vector field $A_\mu$:

   • Solution I: A purely radial Weyl vector ($A_\mu = (0, A(r), 0, 0)$).
   • Solution II: A time-radial Weyl vector ($A_\mu = (A_0(r), A_1(r), 0, 0)$).
     For both cases, the metric functions are expanded in the weak-field regime relevant to the Solar System, retaining terms up to first order in small parameters ($\epsilon_1 = M/r$ and $\epsilon_2 = r/\omega$ or $r^2/\omega^2$).

2. Solar System Tests: The authors compute corrections to four classical observables for both solutions:

   • Gravitational redshift.
   • Perihelion advance of Mercury.
   • Light deflection by the Sun.
   • Radar echo delay (Shapiro delay).
     These corrections are compared against current observational uncertainties to derive lower bounds on $\omega$. The analysis explicitly addresses the normalization of test particle trajectories, comparing proper time parametrization (Levi-Civita geodesics) with affine parametrization (Weyl autoparallels) to ensure robustness.

3. Galactic Center Analysis: The study extends the analysis to the strong-field regime by modeling the orbits of the S2 star near the supermassive black hole Sagittarius A* (Sgr A*). The authors calculate corrections to the periastron precession and gravitational redshift of the S2 star, comparing them with data from the GRAVITY instrument and the Keck Observatory.

4. Asymptotic Analysis: The asymptotic behavior of the metric solutions is analyzed to define an effective cosmological constant ($\Lambda_{\text{eff}}$) and compare it with the observed value ($\Lambda_{\text{obs}}$).

Key Contributions and Results

• Distinct Scaling Behaviors: A primary finding is the qualitative difference in how the two solutions deviate from GR.

  • Solution I exhibits a leading linear correction term ($2r/\omega$) in the metric. Consequently, observables scale as $1/\omega$.
  • Solution II features a linear term suppressed by a factor of $M/\omega$, making the quadratic term ($-r^2/4\omega^2$) dominant in the Solar System regime. Consequently, observables scale as $1/\omega^2$.

• Solar System Constraints:

  • Solution I: The most stringent constraint arises from Mercury's perihelion precession, yielding $\omega \gtrsim 10^{30}$ m. Radar echo delay provides a strong bound of $\omega \gtrsim 10^{25}$ m. Gravitational redshift and light deflection yield weaker but still significant bounds ($\omega \gtrsim 10^{18}$–$10^{19}$ m).
  • Solution II: Constraints are generally weaker due to the $1/\omega^2$ scaling. Mercury's perihelion precession yields $\omega \gtrsim 10^{20}$ m, while light deflection provides the weakest bound ($\omega \gtrsim 10^{10}$ m).
  • Qualitative Differences: Solution I produces a retrograde correction (reducing the precession/redshift), while Solution II produces a prograde correction (enhancing them).

• S2 Star Constraints (Sgr A):*

  • Observations of the S2 star provide complementary constraints in a stronger gravitational field ($\Phi/c^2 \sim 10^{-3}$).
  • For Solution I, the S2 gravitational redshift yields $\omega \gtrsim 10^{24}$ m, and periastron precession yields $\omega \gtrsim 10^{21}$ m.
  • For Solution II, the S2 gravitational redshift yields $\omega \gtrsim 10^{22}$ m, and periastron precession yields $\omega \gtrsim 10^{17}$ m.
  • The S2 data confirms the qualitative distinction: Solution I reduces the precession, while Solution II enhances it.

• Effective Cosmological Constant:

  • Both solutions exhibit asymptotic quadratic terms resembling Schwarzschild-(anti-)de Sitter spacetimes.
  • Solution I corresponds to an anti-de Sitter-like behavior with $\Lambda_{\text{eff}}^{(I)} \simeq -3/\omega^2$. Given the tight Solar System constraints, $|\Lambda_{\text{eff}}^{(I)}|$ is constrained to be $\lesssim 10^{-60}$ m$^{-2}$, which is eight orders of magnitude smaller than $\Lambda_{\text{obs}}$.
  • Solution II corresponds to a de Sitter-like behavior with $\Lambda_{\text{eff}}^{(II)} \simeq 3/4\omega^2$. The weaker constraints on Solution II allow $\Lambda_{\text{eff}}^{(II)}$ to be as large as $\sim 10^{-41}$ m$^{-2}$, which is compatible with the order of magnitude of the observed cosmological constant.

Significance and Claims
The paper claims that non-minimally coupled Weyl connection gravity is phenomenologically viable but highly constrained. The results indicate that while the theory is not ruled out, the Weyl parameter $\omega$ must be extremely large, effectively suppressing Weyl-induced deviations from GR within the Solar System.

The authors highlight a crucial distinction between the two solution branches:

1. Solution I is tightly constrained by Solar System tests and predicts an anti-de Sitter asymptotic structure that cannot account for the observed cosmic acceleration.
2. Solution II is less constrained, allowing for a de Sitter-like asymptotic structure. This branch can naturally accommodate an effective cosmological constant of the same order as $\Lambda_{\text{obs}}$ without introducing an explicit cosmological constant term in the action.

The study concludes that while current Solar System tests confine the theory to a regime where its effects are negligible, stellar orbit observations near Sgr A* provide a complementary probe into the strong-field regime. The opposite signatures (retrograde vs. prograde corrections) in periastron precession offer a clear phenomenological distinction that could be tested with future high-precision observations (e.g., GRAVITY+). Ultimately, Solution II emerges as the more cosmologically promising branch, capable of reproducing cosmic acceleration geometrically while remaining consistent with all classical Solar System tests.