Weyl Geometric Black Holes

• Weyl Geometric Gravity Black Hole is defined as a solution derived from a gravitational theory that extends Riemannian geometry with a Weyl vector and a scalar degree of freedom.
• The metric exhibits additional constant, linear, and quadratic terms compared to Schwarzschild solutions, leading to modified horizon temperature, entropy, and stability properties.
• This framework provides insights into gravitational wave dynamics, astrophysical tests, and even laboratory analogues in Weyl semimetals for simulating black hole phenomena.

A Weyl Geometric Gravity Black Hole is a solution to gravitational field equations in a geometric framework in which spacetime is endowed not only with a metric but also with a Weyl vector (defining nonmetricity), often supplemented by a scalar degree of freedom. This broader geometry generalizes Riemannian geometry by permitting scale changes under parallel transport. Black holes in such theories exhibit metrics that deviate fundamentally from those in standard general relativity, typically manifesting additional terms (linear and/or quadratic in the radial coordinate) and supporting richer thermodynamic and dynamical phenomena. The paper, classification, and phenomenology of Weyl geometric black holes connect deep issues in classical and quantum gravity, gravitational wave astronomy, and modified gravity phenomenology.

1. Weyl Geometry and the Gravitational Action

Weyl geometry extends the concept of metric compatibility by introducing a nonmetricity condition,

$$\tilde{\nabla}_\lambda g_{\mu\nu} = -\alpha\,\omega_\lambda\, g_{\mu\nu},$$

where $\omega_\mu$ is the Weyl gauge field and %%%%1%%%% is a coupling constant. The basic geometric data are thus $(g_{\mu\nu},\,\omega_\mu)$.

The gravitational action in the canonical (“minimal” conformal) Weyl geometric gravity framework is typically constructed from the square of the Weyl scalar curvature (which itself contains both Ricci and Weyl vector contributions) and a Maxwell-type kinetic term for the Weyl vector: $$S = \int d^4x \sqrt{-g} \left[ \frac{1}{4!\,\xi^2}\tilde{R}^2 - \frac{1}{4} F_{\mu\nu}F^{\mu\nu} \right],$$ with $$F_{\mu\nu} = \partial_\mu \omega_\nu - \partial_\nu \omega_\mu$$. The action is often linearized by introducing an auxiliary scalar field $\phi$ via $\tilde{R}^2 \rightarrow 2\phi^2\tilde{R} - \phi^4$, resulting in a scalar–vector–tensor theory with nonminimal coupling between the scalar and the Ricci scalar: $$S = \int d^4x\,\sqrt{-g}\left[\frac{1}{12}\frac{\phi^2}{\xi^2}\left(R - 3\alpha \nabla_\mu \omega^\mu - \frac{3}{2}\alpha^2\omega_\mu\omega^\mu\right) - \frac{1}{4!}\frac{\phi^4}{\xi^2} - \frac{1}{4}F_{\mu\nu}F^{\mu\nu}\right]$$ (Yang et al., 2022).

This structure leads to field equations with effective nonminimal coupling between curvature, the Weyl vector, and the scalar, with $\omega_\mu$ encoding the nonmetricity and $\phi$ (or its square $\Phi\equiv\phi^2$) acting as a “dilaton”-like field.

2. Exact and Numerical Black Hole Solutions: Metric Structure

Spherically Symmetric Solutions

For static, spherically symmetric configurations, the general metric ansatz is: $$ds^2 = e^{\nu(r)}dt^2 - e^{\lambda(r)}dr^2 - r^2(d\theta^2 + \sin^2\theta\, d\varphi^2),$$ where the form of $\nu(r)$ and $\lambda(r)$ is determined by the coupled scalar–vector–tensor field equations. When the Weyl vector has only a radial component, and with the simplifying gauge $\nu+\lambda=0$, the scalar field and metric functions admit closed-form solutions: $$\Phi(r) = \frac{C_1}{(r+C_2)^2}, \qquad \omega_1(r) = \frac{1}{\alpha}\frac{\Phi'(r)}{\Phi(r)} = -\frac{2}{\alpha(r+C_2)},$$ and the metric function is: $$e^{-\lambda} = e^{\nu} = 1-\delta + \frac{\delta(2-\delta)}{3} \frac{r}{r_g} - \frac{r_g}{r} + C_3 r^2,$$ where $r_g$, $\delta$, and $C_3$ are integration constants (Yang et al., 2022, Sakti et al., 18 Jan 2024, Oancea et al., 2023, Khodadi et al., 19 Sep 2025). The parameter $C_3$ is often written as $-1/l^2$ with $l$ interpreted as a cosmological scale.

The metric deviates from Schwarzschild–de Sitter by the appearance of a constant term and an $r$-linear term, direct signatures of the scalar and Weyl vector contributions. The Weyl vector profile modulates the scalar field which, via nonminimal coupling, alters both causal structure and thermodynamics.

Higher-Dimensional Generalizations

In higher-dimensional theories, the Weyl-Yang-Kaluza–Klein (WYKK) approach yields metrics of the form: $$f(r) = 1 - \frac{2M}{r} + \frac{Z^2}{r^2} - \frac{\Lambda}{3} r^2,$$ where $Z$ is a charge-like parameter intertwined with the geometry of extra-dimensional squashed $S^3$ and the warping function, whose constancy or variability dictates the emergence of new black hole classes analogous to Reissner–Nordström–de Sitter or Schwarzschild–de Sitter (Kuyrukcu, 2016).

3. Black Hole Thermodynamics in Weyl Geometric Gravity

Horizon and Temperature

The event horizon radius $r_H$ is defined by the outermost root of $e^{\nu(r_H)} = 0$. The surface gravity and Hawking temperature follow: $$\kappa_H = \left. \frac{1}{2} \nu'(r) e^{(\nu-\lambda)/2} \right|_{r_H}, \qquad T_H = \frac{\kappa_H}{2\pi}.$$ For the Weyl geometric black hole: $$T_H = \frac{1}{4\pi} \left(\frac{r_g}{r_H^2} - 2\frac{r_H}{l^2} - \frac{\delta(\delta-2)}{3r_g} \right)$$ (Sakti et al., 18 Jan 2024).

Entropy and Free Energy

The entropy is generically modified due to the nonminimal coupling: $S_{BH} = \pi r_H^2 \Phi_H,$ where $\Phi_H$ is the value of the scalar field on the horizon. This departs from the standard Bekenstein–Hawking area law, reflecting the variable effective Newton’s constant $G_{eff} = 1/\Phi$.

The Helmholtz free energy, crucial for global stability, is: $F = M_H - T_H S_{BH},$ with $M_H$ obtained from integrating $dM_H = T_H dS_{BH}$. Sign of the specific heat $C_H = dM_H/dT_H$ determines local stability; for small black holes $C_H<0$ (unstable), while for large $r_H$ approaching (A)dS scale, $C_H>0$ (stable) (Sakti et al., 18 Jan 2024).

Comparison with General Relativity

In Schwarzschild–de Sitter, the metric lacks the constant and linear-in-$r$ Weyl terms, and the entropy obeys $S = \pi r_H^2$ (in units with $G=1$). In Weyl geometric gravity, the entropy scaling and temperature are shifted, leading to distinguishable thermodynamic signatures.

4. Dynamical and Wave Phenomena: Quasinormal Modes and Stability

Weyl geometric black holes exhibit nontrivial dynamics for linear perturbations. In theories constructed with quadratic curvature corrections (Einstein–Weyl gravity, for example), the black hole spectrum includes both massless and massive spin-2 modes (Myung, 2013, Podolsky et al., 2018, Konoplya et al., 27 Jan 2025, Konoplya et al., 2 May 2025). The spherically symmetric (s-mode) perturbations may exhibit Gregory–Laflamme–type instabilities, especially when a nonzero graviton mass is present: $$( \bar{\square} - 2\Lambda/3 - 1/(3\beta) ) \delta G_{\mu\nu} + 2 \bar{R}_{\rho\mu\sigma\nu} \delta G^{\rho\sigma} = 0,$$ with instability for $0 < M < \mathcal{O}(1)/r_0$ (Myung, 2013). For higher-order and massive gravity terms, ringdown waveforms and quasinormal modes (QNMs) deviate from those in GR, with phenomena such as secondary QNM branches induced by “dark matter” (conformal) terms and late-time power-law tails (Konoplya et al., 27 Jan 2025, Konoplya et al., 2 May 2025). Wormholes in these geometries present distinct QNM and shadow signatures compared to black holes.

The breakdown of eikonal/geodesic correspondence for QNMs in quadratic gravity, due to nontrivial potential structure, highlights the limitations of WKB approaches in this context (Konoplya et al., 2 May 2025).

5. Astrophysical and Observational Implications

Solar System Tests and Constraints

The exact static spherically symmetric solution,

$$f(r) = 1 - \eta + \frac{\eta(2-\eta)}{3} \frac{r}{r_g} - \frac{r_g}{r} - \frac{\Lambda}{3} r^2,$$

must reduce to the Schwarzschild limit for Solar System consistency. Detailed analysis of gravitational redshift, perihelion precession, light deflection, radar echo delay, and the Nordtvedt effect constrains $|\eta| \lesssim 10^{-10}$, ensuring observational agreement with general relativity within current sensitivities (Khodadi et al., 19 Sep 2025).

Distinguishing Features

Non-Schwarzschild black holes and wormholes in Weyl geometric gravity can have smaller shadow radii and different QNM spectra compared to GR black holes of equal mass. These phenomena are potential targets for observation via black hole imaging and gravitational wave astronomy (Konoplya et al., 27 Jan 2025). The presence of linear and quadratic terms in the metric affects lensing, galaxy rotation curves, and potentially offers a mechanism for simulating “dark matter” or “dark energy” effects at astrophysical scales.

Laboratory Analogues

In condensed matter, “type-II” Weyl fermions in overtilted Weyl semimetals simulate black hole horizons and Hawking radiation, with experimental signatures tractable at room temperature—offering analog models to paper gravitational phenomena (Volovik, 2016).

6. Extensions, Generalizations, and Prospects

Various generalizations have been studied:

• Higher-dimensional Gauss–Bonnet–Weyl black holes: Solutions in $D=5$ with entropy modified by a term linear in the horizon radius, distinct from four-dimensional behavior (Bahamonde et al., 3 Apr 2025, Malik et al., 2019).
• Non-minimally coupled Weyl connection gravity: Vacuum black holes featuring additional $r$-linear terms and a nonzero curvature scalar, and otherwise forbidden Reissner–Nordström-like solutions appear with matter fields (Lima et al., 2 Oct 2024).
• Weyl-invariant extensions of Einstein equations: Yielding black holes with potentials that grow logarithmically at large radius and supporting non-decaying cosmological vector modes (Bañados, 24 Feb 2024).
• Rotating black holes in conformal gravity: The Weyl-Kerr solution introduces extra linear-in-$r$ and cosmological terms, leading to different ergoregion and horizon structures, and shifting extremal limits compared to the GR Kerr metric (Asuncion et al., 2 Jul 2025).

Observational signatures sensitive to these modifications—especially in gravitational waveforms, shadow radii, or deviations in orbital dynamics—form the basis of current and prospective tests of Weyl geometric gravity.

7. Summary Table: Key Features Across Representative Classes

Metric Terms	Gravitational Degrees	Thermodynamics	Instability	Distinguishing Signature(s)
$1/r$, const, $r$, $r^2$ (Weyl)	Massless + Weyl/dilaton	Modified $T$, $S$, $C$, $F$	GL (massive mode)	Solar System, galaxy rotation, QNMs
$1/r$, $r^2$, charge (Einstein-Maxwell-Weyl)	Massless + massive + EM	Entropy correction, 2 BH branches	QNM shifts	Absent Reissner–Nordström analog
Logarithmic potential (Weyl gauged Einstein)	Tensors, Vectors, Scalar	Not computed	Not established	Slow fall-off, vector cosmology
Schwarzschild/AdS-like w/ warping (WYKK)	Yang–Mills, graviton	Analogous to RNdS, quantized charge	Not discussed	“Squashing” parameter, new solutions

Modifications in metric structure, horizon and thermodynamic properties, and wave propagation—all tied to the extended geometric fields (Weyl vector, scalar, nonmetricity)—are haLLMarks of Weyl geometric black holes. The constraints from precision tests and distinctive signatures in strong gravity regimes remain the principal drivers for ongoing and future research in this domain.