Accretion of matter of a new bumblebee black hole

TL;DR

The paper investigates a static black hole in bumblebee gravity, introducing a Lorentz-violating parameter $χ$ that modifies photon geodesics and enlarges the shadow size, with the photon sphere remaining at $r_p = 3M$ but the critical impact parameter scaling as $b_p = 3\sqrt{3(1+χ)}\,M$.Using ray-tracing, it analyzes light deflection, direct emission, lensing rings, and photon rings for three thin-disk emission models (starting at the ISCO, the photon sphere, and the event horizon) and for static and infalling spherical accretions.The results show that larger $χ$ shifts all optical features outward and reduces observed brightness due to a redshift factor $g = \sqrt{\dfrac{1}{1+χ}\left(1 - \dfrac{2M}{r}\right)}$, with brightness further diminished by Doppler effects in infalling cases; intensities scale roughly as $I_{obs} \propto g^4$.These findings provide concrete observational signatures of Lorentz violation in strong gravity and offer guidance for interpreting horizon-scale images of accreting black holes in the presence of a fixed-norm bumblebee field.

Abstract

We investigate how the newly obtained static black hole in bumblebee gravity affects the behavior of accreting matter and its observable signatures. The Lorentz-violating parameter that characterizes this geometry modifies photon trajectories and shifts the location of the critical curve that defines the shadow. Using ray tracing, we examine light deflection, the structure of direct emission, lensing rings, and photon rings, and we explore three thin-disk emission models--starting at the ISCO, at the photon sphere, and at the event horizon--together with static and infalling spherical accretions. Larger values of this parameter enlarge the shadow, move all optical features outward, and suppress the observed intensity through gravitational redshift, with additional dimming produced by Doppler effects for infalling matter

Figures (10)

• Figure 1: Schematic comparison between two distinct operations starting from the original bumblebee black hole solution. Left: a genuine coordinate transformation rescales $t$ and transforms both the metric and the covector $b_\mu$, preserving the invariant norm $b_\mu b^\mu$ and describing the same physical configuration in a different coordinate chart. Right: absorbing the prefactor in $g_{tt}$ while keeping $b_\mu$ fixed in the coordinate basis breaks the VEV constraint, producing a configuration that does not satisfy the bumblebee field equations. The bottom box summarizes the tension: in the construction of the bumblebee black hole, $b_\mu$ is fixed by the VEV ansatz and the equations of motion; changing only the metric is inconsistent, while transforming $b_\mu$ covariantly preserves covariance but corresponds to a different vacuum configuration and therefore to a different black hole solution.
• Figure 2: The effective potential $V_{\mathrm{eff}}$ and the impact parameter $b_p$ of the new bumblebee black holes for different Lorentz violation parameters $\chi = 0.1$ and $\chi = 0.3$.
• Figure 3: Photon trajectories around the new bumblebee black hole in polar coordinates $(r, \phi)$ for different Lorentz violation parameters $\chi = 0.1$ and $\chi = 0.3$.
• Figure 4: Optical characteristics of the bumblebee black hole with Lorentz violation parameters $\chi = 0.1$ (left column) and $\chi = 0.3$ (right column).
• Figure 5: The first three transfer functions $r_{n}(b)$ for the bumblebee black hole with $\chi=0.1$ and $\chi=0.3$. The yellow, blue, and red curves correspond to the direct emission ($n=1$), lensing ring ($n=2$), and photon ring ($n=3$), respectively.