⚛️ general relativity

Generalized Eddington--Finkelstein Coordinates and Exact Vaidya-Type Solutions in Weyl Conformal Gravity

This paper investigates generalized Eddington–Finkelstein coordinates to derive, classify, and analyze both vacuum and non-vacuum Vaidya-type solutions with spherical, hyperbolic, and planar symmetries in Weyl conformal gravity, highlighting their structural non-triviality and key properties such as singularities, horizons, and gauge equivalence to Einstein spaces.

Original authors: Petr Jizba, Tereza Lehečková

Published 2026-09-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Petr Jizba, Tereza Lehečková

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Gravity is the force that holds our world together, keeping our feet on the ground and the planets in their orbits. For over a century, our best description of this force has been Albert Einstein's theory of general relativity. This theory works with stunning precision when we look at our solar system or even the space around our galaxy. However, when astronomers look out at the vast scale of the universe, at the swirling clusters of galaxies and the cosmic web that stretches across billions of light-years, Einstein's equations begin to struggle. To make the math match the observations, scientists have had to invent invisible ingredients called dark matter and dark energy, which we cannot see or touch but must assume exist to explain how the universe moves. Because these ingredients remain a mystery, some physicists have begun to ask if the problem lies not with the universe, but with the theory of gravity itself. They have turned their attention to alternative theories, including one called Weyl conformal gravity, which treats the size of space and time as flexible rather than fixed.

In a recent study, researchers Petr Jizba and Tereza Lehečková explored how this alternative theory of gravity behaves when stars and black holes change over time. They focused on a specific mathematical setup known as Vaidya-type solutions, which describe objects that are not static but are either gaining mass, losing mass, or radiating energy. To understand these dynamic processes, the team used a special way of mapping space and time called Eddington–Finkelstein coordinates. Imagine trying to describe a river that is flowing; standard maps might freeze the water in place, making it hard to see how the current moves. These special coordinates, however, are designed to flow along with the path of light rays, allowing scientists to track what happens as matter falls into a black hole or as a star radiates energy away. This approach is particularly useful for Weyl conformal gravity because, unlike Einstein's theory, this alternative framework does not have a fixed scale for mass, making the flow of light the most natural way to measure events.

The researchers set out to find all possible empty-space solutions for this theory, meaning they looked for how space and time curve when there is no matter present, only the gravitational field itself. They examined three different shapes of space: spherical, like a ball; planar, like a flat sheet; and hyperbolic, which curves like a saddle. In Einstein's theory of general relativity, the empty space around a spherical object is always static and unchanging; it does not evolve on its own. The team discovered that in Weyl conformal gravity, the situation is fundamentally different. They found that empty space can be dynamic, changing its shape and structure over time without any matter or energy to drive it. This is a profound departure from what we are used to, suggesting that in this theory, the geometry of the universe can shift and evolve even in a complete vacuum.

The study revealed that these dynamic solutions are not just simple variations of known black holes but belong to a vast family of possibilities. The researchers identified two main branches of solutions. One branch describes objects that behave like familiar black holes or white holes, but with a mass that changes over time. The other branch describes a background that can generate a constant acceleration, a kind of push or pull, without any physical source to cause it. This is a striking result because it implies that the theory can create effects that usually require a cosmological constant or dark energy, but does so naturally through the mathematics of the theory itself. The team also analyzed the boundaries of these objects, known as horizons, which are the points of no return for light. They found that in this theory, these horizons can move and change position over time, even though no matter is falling in or out. This is impossible in Einstein's theory, where the horizon of a black hole can only grow if matter is added.

To make sense of these complex shapes, the researchers compared their findings to a famous solution in Weyl gravity known as the Mannheim-Kazanas solution, which is the theory's version of the Schwarzschild black hole. They showed that their new, time-changing solutions are mathematically related to this older, static solution, but the connection is tricky. While they can be transformed into one another using a specific mathematical scaling, this transformation is not perfect everywhere; it works well in some places but breaks down at others. This means that while the local physics might look similar, the global structure of the universe in these new solutions is genuinely different. The researchers also confirmed that these solutions do not emit radiation in the traditional sense, yet they still evolve, which challenges the idea that change in gravity must always be driven by the movement of matter.

The paper concludes by addressing how these findings fit with other known results in the field. There is a theorem that suggests all spherical solutions in this theory should be equivalent to a single, static type of black hole. The researchers explain that this theorem holds true only in a limited, local sense. Because the mathematical tools used to connect these solutions involve singularities—points where the math becomes infinite—the connection does not hold up when looking at the entire universe. Therefore, the dynamic solutions they found are not just disguised versions of static black holes; they represent a distinct and richer set of possibilities. By mapping out these solutions, the study provides a clearer picture of how Weyl conformal gravity handles time, change, and the structure of space, offering a fresh perspective on whether the mysteries of the dark universe might be solved by changing our understanding of gravity itself.

Technical Summary: Generalized Eddington–Finkelstein Coordinates and Exact Vaidya-Type Solutions in Weyl Conformal Gravity

Problem Statement
While Einstein's General Relativity (GR) is highly successful at solar system scales, it faces challenges at cosmological scales, often requiring the introduction of dark matter and dark energy to reconcile observations with theory. Weyl Conformal Gravity (WCG) offers an alternative higher-derivative gravity theory with gauged scale invariance, capable of addressing these issues without exotic matter. However, the study of dynamical spacetimes in WCG, particularly those analogous to the Vaidya solution (which models radiating stars or black holes), remains underexplored. Existing literature has utilized Eddington–Finkelstein (E-F) coordinates to study general properties of spherically symmetric solutions and black hole dynamics in conformal gravity, but exact dynamical solutions in WCG using these coordinates had not been derived. Furthermore, the structural differences between WCG and GR vacuum solutions in dynamical settings, and their relationship to the Birkhoff–Riegert theorem, require clarification.

Methodology
The authors employ generalized Eddington–Finkelstein-like coordinates to analyze the field equations of Weyl Conformal Gravity. The metric ansatz (Eq. 1) incorporates a line element with a function $H(r, w)$ and allows for spherical ( $K=1$ ), planar ( $K=0$ ), and hyperbolic ( $K=-1$ ) symmetries. The time coordinate $w$ represents retarded or advanced time, facilitating the study of dynamical processes.

The core of the methodology involves solving the Bach vacuum equation, $B_{\mu\nu} = 0$ , which is the field equation derived from the Weyl action. The authors:

Solve the $rr$-component of the Bach equation first to determine the general form of the metric function $H(r, w)$ .
Substitute this general form back into the remaining components of the Bach tensor to derive constraints on the arbitrary functions of time.
Analyze the resulting solutions by computing Newman–Penrose Weyl scalars to determine algebraic types (Petrov classification).
Investigate the existence of Conformal Killing Vector Fields (CKVFs) and their relation to the Birkhoff–Riegert theorem, specifically examining whether these dynamical solutions can be mapped to static Einstein spaces via Weyl transformations.
Extend the analysis to non-vacuum cases involving Coulombic electric fields and null dust radiation.

Key Contributions and Results

Exact Vacuum Solutions: The paper derives all vacuum dynamical solutions for the generalized E-F metric in WCG. Unlike GR, where the vacuum solution for this metric is static (Schwarzschild–(anti)-de Sitter), WCG admits structurally non-trivial dynamical vacuum solutions.
- Branch 1: Solutions where $F_2(w) = 0$ , parameterized by arbitrary functions $F_3(w)$ and $F_4(w)$ . This branch includes de Sitter-like solutions and dynamical Mach-type backgrounds.
- Branch 2: Solutions where $F_2(w) \neq 0$ . Here, the functions $F_3(w)$ and $F_4(w)$ are determined by $F_1(w)$ and $F_2(w)$ via specific differential relations (Eq. 23). This branch generalizes the Mannheim–Kazanas (M-K) solution to a dynamical context.
Algebraic and Geometric Properties:
- Petrov Type: All derived vacuum solutions are of Petrov type D (if $\Psi_2 \neq 0$ ) or type O (conformally flat if $\Psi_2 = 0$ ). The Weyl scalar $\Psi_4$ vanishes, indicating the absence of a radiation field in these vacuum solutions.
- Singularities: The singularity at $r=0$ is confirmed as a genuine curvature singularity via the divergence of scalar invariants (e.g., $C^2$ ).
- Horizons: The authors identify trapping horizons defined by the condition $H(r, w) = 0$ . A significant finding is that in WCG, these horizons can change position over time even in the absence of matter radiation (since $\Psi_4=0$ ), a phenomenon impossible in GR vacuum solutions.
Birkhoff–Riegert Theorem and Weyl Equivalence:
- The study addresses the Birkhoff–Riegert theorem, which states that spherically symmetric electrovacuum solutions in WCG are Weyl equivalent to the charged M-K solution.
- The authors demonstrate that while their dynamical solutions are locally Weyl equivalent to static M-K (or topological black hole) solutions, the required Weyl transformations are often singular. Consequently, the equivalence is local; globally, the spacetimes differ in characteristics such as Noether–Wald charges and homotopy groups.
- The paper identifies a class of solutions (Linear Mass Case) that admit a Conformal Killing Vector Field, allowing a transformation to a static frame where thermodynamic and causal properties can be analyzed more easily.
Non-Vacuum Solutions: The authors derive and classify solutions incorporating a Coulombic electric field and null dust radiation, extending the standard Vaidya solution to the WCG framework.

Significance and Claims
The paper claims to provide the first derivation of exact Vaidya-type solutions in Weyl Conformal Gravity using Eddington–Finkelstein coordinates. The primary significance lies in demonstrating that WCG possesses a rich set of dynamical vacuum solutions that are structurally distinct from GR. Specifically, the existence of time-dependent horizons in vacuum (without radiating matter) highlights a fundamental difference in the causal structure of WCG compared to GR.

The authors emphasize that while many WCG solutions are locally Weyl-equivalent to Einstein spaces, the global properties of these spacetimes can differ significantly due to the singular nature of the required conformal factors. This distinction is crucial for understanding the physical interpretation of WCG solutions, particularly regarding global charges and topological features. The work establishes a framework for analyzing dynamical processes, such as black hole evaporation or accretion, within the context of conformal gravity, offering a tool to explore time-dependent scenarios that are difficult to treat in standard Schwarzschild coordinates.

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