Veiled Singularities in Einstein-Weyl Gravity: Stability and Physical Interpretation of Horizonless Solutions

TL;DR

The paper investigates a class of horizonless solutions in Einstein–Weyl gravity, focusing on attractive naked singularities of the $(-2,2)$ type. It develops a Frobenius-based classification of static, spherically symmetric solutions, isolates the veiled naked singularities with near-origin behavior $h(r)\sim h_2 r^2$ and $f(r)\sim f_{-2}/r^2$, and introduces the veil radius as a diagnostic for the region where the near-origin description holds. Through a time-domain analysis of tensor perturbations, the authors show linear stability across the physical parameter space, with oscillatory massive tails $\phi(t)\sim \sin(m_2 t-\delta) t^{-5/6}$, and they reveal photon-ring structure and shadows that resemble Schwarzschild black holes while remaining horizonless. The results position veiled singularities as viable horizonless analogs in quadratic gravity, offering new insights into singularity resolution, stability, and observational phenomenology beyond General Relativity.

Abstract

We investigate a class of horizonless solutions in Einstein-Weyl gravity, corresponding to the so-called attractive naked singularities of the (-2,2) type. In contrast to General Relativity, where naked singularities are generically unstable and excluded by the cosmic censorship conjecture, we show that these configurations are linearly stable under tensor perturbations. By numerically evolving the perturbation equations in the time domain, we find that all modes decay with characteristic oscillatory tails, a behavior consistent with the dynamics of massive field perturbations in quadratic gravity. This establishes that attractive naked singularities in Einstein-Weyl gravity are dynamically stable and can persist as stationary configurations. We argue that these horizonless configurations are observationally concealed, and therefore we refer to them as veiled singularities. Their stability and phenomenological similarity to black holes suggest that they may represent viable horizonless alternatives in higher-derivative theories of gravity, offering a novel perspective on the interplay between singularity resolution, stability, and effective field dynamics beyond Einstein's theory.

Figures (6)

• Figure 1: In the top panels we plot the metric functions $h(r)$ and $f(r)$ for three different parameter choices, with the blue solid line corresponding to $M=0.6$ and $S^-_2=0.05$, the red dashed line to $M=0.2$ and $S^-_2=-0.2$, and the green dotted line to $M=2.0$ and $S^-_2=0.01$. The bottom panels show the logarithmic derivatives $\frac{rh'(r)}{h(r)}$ and $\frac{rf'(r)}{f(r)}$, which approach $2$ for $h(r)$ and $-2$ for $f(r)$ as $r\to 0$, in agreement with the expected near-origin scaling.
• Figure 2: Null geodesics around a naked singularity in arbitrary units (for reference, we used $M=0.6$ and $S_2^-=0.07$). Left panel illustrates a ray-tracing simulation, with an observer at $x=-\infty$ and an illuminated screen assumed at $x=+\infty$. Orange lines are light rays that reach the observer, blue lines are light rays deflected back by strong lensing effects. and black lines are light rays that reach the singularity. The dashed and solid circles indicate the position of the external photon ring and the veil radius. Right panel illustrate the face-on view of the singularity from the observer position.
• Figure 3: Effective potential as a function of the tortoise coordinate for different values of $M$ and $S_2^-$. Blue solid line: $M=0.6$ and $S^-_2=0.05$; red dashed line: $M=0.2$ and $S^-_2=-0.2$; green dotted line: $M=2.0$ and $S^-_2=0.01$. In all cases, we have a potential well with a finite value in the origin.
• Figure 4: Two different examples of the time evolution of the perturbations with the corresponding linear fit, on the left with $M=0.6$ and $S_2^-=0.05$, on the right with $M=0.2$ and $S_2^-=-0.2$ showing characteristic beating patterns.
• Figure 5: Diagram of the stability as a function of the parameters. The colors scale represents the variation of the imaginary part of the perturbations frequencies. Solid red and blue lines correspond to non-Schwarzschild black holes, while the dashed black line represents Schwarzschild black holes.