The images of Brans-Dicke-Kerr type naked singularities

TL;DR

The paper analyzes images of Brans-Dicke-Kerr type naked singularities in the Brans-Dicke theory, showing that the BD parameter $\omega$ qualitatively alters horizon structure and photon trajectories. By applying backward ray-tracing, it reveals distinct shadow morphologies: for $a \le M$ the shadow persists and becomes flatter as $\omega$ decreases, while for $a > M$ a pair of gray patches emerges, signaling photons that cross negative radii regions. Notably, when $a \le M$ and $\omega < 1/2$, the shadow takes a two-headed jellyfish shape with self-similar fractal features due to chaotic scattering, providing a unique observational signature. These results suggest that BD-Kerr naked singularities imprint observable differences from Kerr and Kerr-de Sitter spacetimes, offering a foundation for testing Brans-Dicke theory with future high-precision shadow measurements.

Abstract

We have studied the images of the Brans-Dicke-Kerr spacetime with a dimensionless Brans-Dicke parameter $ω$, which belongs to axisymmetric rotating solutions in the Brans-Dicke theory. Our results show that the Brans-Dicke-Kerr spacetime with the parameter $ω>-3/2$ represents naked singularities with distinct structures. For the case with $a \leq M$, the shadow in the Brans-Dicke-Kerr spacetime persists, gradually becomes flatter and smaller as $ω$ decreases. Especially when $ω<1/2$, the shadow in the image exhibit a very special ``jellyfish" shape and possesses a self-similar fractal structure. For the case with $a > M$, a distinct gray region consisting of two separate patches appears in the image observed by equatorial observers. This indicating that the Brans-Dicke-Kerr spacetime can be distinguished from the Kerr and Kerr-de Sitter cases based on its image. These effects of the Brans-Dicke parameter could help us to reveal the intrinsic structure of the Brans-Dicke-Kerr spacetimes and provide a foundation for testing Brans-Dicke theory through future high-precision observations.

Figures (8)

• Figure 1: The variation of images with the Brans-Dicke parameter $\omega$ for the Brans-Dicke-Kerr spacetime with fixed $a=0$. Here we set the mass parameter $M=1$, $r_{obs}=8M$ and $\theta_{obs}=\pi/2$. The figures from left to right correspond to $\omega=0.4$, $1$, $10$, and $500$, respectively.
• Figure 2: The variation of images with the Brans-Dicke parameter $\omega$ for the Brans-Dicke-Kerr spacetime with fixed $a=0.5$. Here we set the mass parameter $M=1$, $r_{obs}=8M$ and $\theta_{obs}=\pi/2$. The figures from left to right correspond to $\omega=0.4$, $1$, $10$, and $500$, respectively.
• Figure 3: The variation of shadows with the Brans-Dicke parameter $\omega$ for the Brans-Dicke-Kerr-type black holes and naked singularities with fixed $a=0.99$. Here we set the mass parameter $M=1$, $r_{obs}=8M$ and $\theta_{obs}=\pi/2$. The figures from left to right correspond to $\omega=0.4$, $1$, $10$, and $500$, respectively.
• Figure 4: The variation of images with the Brans-Dicke parameter $\omega$ for the Brans-Dicke-Kerr spacetime with fixed $a=1.05$. Here we set the mass parameter $M=1$, $r_{obs}=8M$ and $\theta_{obs}=\pi/2$. The figures from left to right correspond to $\omega=0.4$, $1$, $10$, and $500$, respectively.
• Figure 5: Photon trajectories in the $(r-\theta)$ plane for the Brans-Dicke-Kerr spacetime with $a=0.5$. The black half-ellipses denote surfaces of constant Boyer-Lindquist radius $r$, while the black dashed hyperbolas correspond to surfaces of constant latitude coordinate $\theta$. The black dot-dashed line represents the event horizon, the orange solid line indicates the secondary curvature singularity, and the other colored solid curves show photon trajectories for $x=-0.3$ with different values of $y$. In the left panel, the purple, red, green, and blue curves are associated with $y=4.5, 4.168, 3.5, 3.0$; in the right panel, they take values $y=1.6, 1.2, 0.8, 0.4$.