Black hole shadows in nonminimally coupled Weyl connection gravity

This paper investigates black hole shadows in nonminimally coupled Weyl connection gravity, deriving modified Schwarzschild- and Reissner-Nordström-like solutions and using Event Horizon Telescope observations of Sgr A* to place meaningful observational bounds on the theory's Weyl parameter, thereby demonstrating the potential of horizon-scale imaging to constrain spacetime non-metricity.

The Gist

Imagine the universe as a giant, stretchy trampoline. In Einstein's famous theory of General Relativity, this trampoline is made of a single, perfect fabric called "spacetime." Massive objects like stars and black holes sit on it, creating dents that guide how everything else moves. This fabric is "metric-compatible," meaning if you slide a ruler along the fabric, its length never changes, no matter where you go.

But what if the fabric wasn't quite so perfect? What if, as you slid your ruler across the trampoline, it subtly stretched or shrank depending on where you were? This is the idea behind Weyl geometry, a concept proposed over a century ago that adds a hidden "tension field" (called a Weyl vector) to the fabric of the universe.

This paper asks a simple but profound question: If the universe has this extra "stretchiness," how would a black hole look?

The Cosmic Camera: Black Hole Shadows

To answer this, the authors use the Event Horizon Telescope (EHT), which is like a giant cosmic camera that took the first-ever picture of a black hole's "shadow."

Think of a black hole's shadow not as a hole in space, but as a silhouette. Imagine a streetlamp (the glowing gas around the black hole) shining behind a solid ball (the black hole). The ball blocks the light, creating a dark circle in the middle. The size of this dark circle depends entirely on the shape of the space around it.

In Einstein's world, we know exactly how big this shadow should be. But in this new "Weyl" world, the extra stretchiness of space might make the shadow look slightly different—either bigger, smaller, or shaped differently.

The Three Models: Testing the Fabric

The authors built three different "versions" of a black hole in this new theory to see how they would cast a shadow:

1. The Simple Stretch (Model I): Imagine a black hole where the "stretchiness" is constant but gets weaker the further you get from the center. The authors found that if this stretchiness is too strong, the shadow would look weirdly large or small. By comparing their math to the actual photo of the black hole at the center of our galaxy (Sagittarius A*), they realized the "stretchiness" must be incredibly weak. In fact, it has to be so weak that it's almost invisible, like trying to feel a single atom of wind on a hurricane.
2. The Complex Twist (Model II): This version allows for a more complicated arrangement of the stretchiness. Here, the rules are a bit more forgiving, but the conclusion is the same: the universe must be very close to Einstein's perfect fabric for the black hole to look the way we see it.
3. The Charged Stretch (Model III): This model adds an electric charge to the mix, like a black hole that is also a giant static electricity ball. They found that the "stretchiness" interacts with this charge. Interestingly, this stretchiness could hide the effects of a very strong electric charge. It's like wearing a pair of sunglasses that make a bright light look dimmer; the Weyl geometry "dims" the effect of the charge, allowing the black hole to have more charge than we thought possible without breaking the laws of physics as we see them.

The "Second Clock" Problem

You might wonder, "Why didn't Einstein just use this idea?" The paper mentions a historical problem called the "second clock effect." Imagine two identical clocks. If you take one on a walk around a park and the other stays home, in a Weyl universe, they might not agree on the time when they meet again, even if they were perfectly synchronized before. This sounds weird and unphysical.

However, the authors show that their specific version of the theory avoids this problem. The "stretchiness" is non-dynamical, meaning it's a fixed background rule rather than a wiggly, changing force. It's like the trampoline has a fixed pattern of stretchiness baked into it, rather than someone randomly pulling on the fabric. This keeps the math stable and the clocks happy.

The Big Takeaway

The authors used the "cosmic camera" (the EHT) to take a ruler to the fabric of space near a black hole. They found that:

• The fabric is mostly smooth: The universe is very close to Einstein's description. Any "stretchiness" (non-metricity) must be incredibly tiny.
• The shadow is a powerful tool: Just by looking at the size of the black hole's shadow, we can rule out wild theories about how the universe works.
• New possibilities exist: Even though the stretchiness must be tiny, it opens a door. It suggests that if we look closely enough, or if we find black holes with strong electric charges, we might see subtle hints that the universe is a bit more complex than Einstein imagined.

In short, this paper is like checking the seams of a suit. The suit (General Relativity) fits perfectly, but the authors checked to see if there were any hidden stitches (Weyl geometry) that might change the fit. They found that if those stitches exist, they are so fine they are practically invisible, but their presence is now mathematically constrained by the most beautiful photo we have of the cosmos.

Technical Summary

1. Problem Statement

The paper addresses the need to test extensions of General Relativity (GR) using high-precision astrophysical observations, specifically the black hole shadow images provided by the Event Horizon Telescope (EHT). While GR has passed many tests, alternative theories involving non-metricity (where the metric is not preserved under parallel transport) remain viable candidates for explaining cosmological phenomena like dark energy or galactic rotation curves without dark matter.

Specifically, the authors investigate Nonminimally Coupled Weyl Connection Gravity. This is a metric-affine theory where spacetime is described by a metric tensor $g_{\mu\nu}$ and a Weyl vector field $W_\mu$ that encodes non-metricity. Unlike many higher-order gravity theories, this framework maintains second-order field equations (avoiding Ostrogradsky instabilities) because the Weyl vector is non-dynamical. The central problem is to determine how the presence of the Weyl vector modifies the geometry of black holes and, consequently, their observable shadows, and to constrain the theory's parameters using EHT data for Sagittarius A* (Sgr A*).

2. Methodology

Theoretical Framework

The study utilizes an action functional where the Ricci scalar is replaced by a generalized curvature scalar $\bar{R}$, coupled non-minimally to matter:
$$S = \int \left( \kappa f_1(\bar{R}) + f_2(\bar{R})\mathcal{L} \right) \sqrt{-g} \, d^4x$$
Here, $\bar{R} = R + \bar{\bar{R}}$, where $R$ is the standard Ricci scalar and $\bar{\bar{R}}$ arises from the non-metricity part of the connection. The affine connection is decomposed into the Levi-Civita part and a Weyl contribution defined by the vector $W_\mu$.

Black Hole Solutions

The authors analyze three distinct static, spherically symmetric black hole solutions derived from this framework:

1. Model I (Schwarzschild-like): A vacuum solution with a radial Weyl vector $W_\mu = (0, W(r), 0, 0)$. The metric components depend on mass $M$ and a Weyl integration constant $\omega$.
2. Model II (Schwarzschild-like variant): A vacuum solution with a more general Weyl vector configuration $W_\mu = (W_0(r), W_1(r), 0, 0)$.
3. Model III (Reissner–Nordström-like): A solution including a cosmological constant and electric charge $Q$. The charge is "dressed" ($\tilde{Q}$) due to the Weyl coupling, introducing an additional parameter $\zeta$.

Shadow Calculation

The authors compute the shadow radius ($r_{sh}$) for these geometries. The methodology involves:

• Deriving the equation of motion for null geodesics in the equatorial plane.
• Identifying the unstable photon sphere radius ($r_{ph}$) by solving $C'(r)/C(r) = A'(r)/A(r)$.
• Calculating the angular shadow radius for an observer at distance $r_O$ using the impact parameter $b_{cr}$.
• Comparing the theoretical shadow radius against the EHT observational constraints for Sgr A*. The constraints are expressed as a fractional deviation $\delta$ from the Schwarzschild prediction, with allowed ranges at $1\sigma$ and $2\sigma$ confidence levels:
  • $1\sigma$: $4.55 \lesssim r_s/M \lesssim 5.22$
  • $2\sigma$: $4.21 \lesssim r_s/M \lesssim 5.56$

Numerical Simulation

To visualize the effects, the authors simulated optical images of charged Weyl black holes surrounded by thin accretion disks. They used the Gralla-Lupsasca-Marrone (GLM) emission profiles and a relativistic ray-tracing code (GRAVITYp) to generate images for different values of the correction factor $\zeta$ and charge $Q$.

3. Key Contributions

• Derivation of Modified Shadows: The paper provides the first analytical derivation of black hole shadow radii specifically for nonminimally coupled Weyl connection gravity, linking the abstract Weyl integration constant $\omega$ to observable quantities.
• Parameter Constraints: It establishes quantitative lower bounds on the Weyl parameter $\omega$ required for the theory to remain consistent with current EHT data.
• Analysis of the "Dressed" Charge: The study elucidates the role of the parameter $\zeta$ in the Reissner–Nordström-like solution. It demonstrates how $\zeta$ acts as a correction factor that can either suppress or enhance the effective charge, thereby altering the shadow size and allowing for higher physical charges than standard GR would permit before violating observational bounds.
• Visualization of Non-Metricity Effects: By generating synthetic images, the authors show how variations in $\zeta$ and charge affect the photon ring structure and the overall shadow diameter, offering a visual signature for future high-resolution observations.

4. Results

• Constraints on $\omega$:
  • Model I: To satisfy the $2\sigma$ EHT constraint, the Weyl constant must be $\omega \gtrsim 10^{11.7} M$.
  • Model II: The constraint is slightly looser: $\omega \gtrsim 10^{10.5} M$.
  • Model III: The bound starts at $\omega \sim 10^{11.7} M$ for zero charge and becomes more relaxed as the dressed charge $\tilde{Q}$ increases.
• Behavior of Shadow Radius:
  • In Models I and III, the shadow radius is inversely proportional to $\omega$. As $\omega \to \infty$, the geometry approaches the standard Schwarzschild (or RN) limit, and the shadow converges to $3\sqrt{3}M$.
  • For small $\omega$, the shadow radius grows significantly, eventually exceeding the EHT observational bounds.
• Role of $\zeta$:
  • The parameter $\zeta$ modifies the relationship between the physical charge $Q$ and the dressed charge $\tilde{Q}$.
  • If $\zeta < 1$, the shadow size is suppressed (smaller), allowing for tighter constraints on charge.
  • If $\zeta > 1$, the shadow size is enhanced (larger), allowing the physical charge to exceed the standard RN limit (where a naked singularity would form in GR) while still remaining compatible with EHT data.
• Visual Differences: The simulated images reveal that while the accretion disk profile (GLM model) dictates the brightness distribution, the value of $\zeta$ and $Q$ significantly alters the size of the photon ring and the shadow diameter.

5. Significance

This work establishes a direct link between Weyl-based geometric theories and current astrophysical data. It demonstrates that:

1. Horizon-scale imaging is a powerful probe: Current EHT data is already sensitive enough to place meaningful limits on spacetime non-metricity, ruling out large deviations from GR in the strong-field regime.
2. Viability of Weyl Gravity: The theory remains viable provided the Weyl integration constant $\omega$ is sufficiently large (effectively pushing the non-metricity effects to scales far beyond the event horizon for Sgr A*).
3. Future Directions: The paper highlights the necessity of extending these analyses to rotating (Kerr-like) black holes, as spin effects could amplify deviations. It also calls for dedicated ray-tracing simulations within the Weyl framework to remove theoretical biases inherent in using GR-calibrated constraints.

In summary, the paper successfully uses the black hole shadow of Sgr A* to constrain a specific class of metric-affine gravity theories, showing that while non-metricity is not ruled out, it must be "weak" at the scale of the event horizon to be consistent with observations.