Imprints of the Lorentz-symmetry breaking on the precessing jet nozzle of M87*

Abstract

The approximately 11-year jet precession period observed in M87* strongly suggests that the supermassive rotating black hole with a tilted accretion disk, which could provide a powerful constraint for confining the parameters of black hole. In this paper, our aim is to utilize the observations of M87* to preliminarily constrain the parameters of the rotating black hole in Bumblebee gravity by modeling the motion of the tilted accretion disk particle with the spherical orbits. We compute spherical orbits and ISSOs, demonstrating that the conserved quantities energy $\mathcal{E}$, angular momentum $\mathcal{L}$, and Carter constan $\mathcal{K}$ depend on $(r,a,\ell,ζ)$, exhibiting distinct behaviors for prograde and retrograde orbits. For prograde orbits, the ISSO radius $r_{ISSO}$ decreases with spin parameter $a$ and LSB parameter $\ell$ and increases with the tilt angular $ζ$, whereas the opposite trends occur for retrograde orbits. Angular analysis shows that $θ$ oscillates within $(π/2-ζ, π/2+ζ)$, while $φ$ increases approximately linearly, enabling the determination of the oscillation period $T_θ$, azimuthal accumulation $φ(T_θ)/π$, and precession angular velocity $ω_t$. Using the observed jet precession period $T=11.24 \pm 0.47$ years with a fixed tilt $ζ=1.25^\circ$, the warp radius $r/M$ ranges from $(5.73,25.15)$ for prograde and $(6.16,26.46)$ for retrograde orbits, increasing with $a$ or $\ell$. Comparisons with Kerr limits ($r/M=14.12$ prograde, $16.1$ retrograde) suggest that $r/M>16$ may indicate a non-vacuum Bumblebee vector field. Incorporating the EHT shadow $θ_{sh}=42\pm3μ$as further constrains $r/M$ to $(5.82,22.61)$ and $(6.17,24.74)$, with discrepancies of $0.05\sim1.96$.

Figures (7)

• Figure 1: Parameters space of spin and LSB $(a,\ell)$. The colored region represents the allowed range of black holes.
• Figure 2: Angular momentum $\mathcal{L}$, energy $\mathcal{E}$ and Carter constant $\mathcal{K}$ for the spherical orbits with different parameters. Figures (a)-(c) show the prograde orbits and (d)-(f) describe the retrograde orbits. The dashed and solid red curves correspond to $\ell=0.2~\text{and}~\ell=0.3$, the dashed and solid blue curves correspond to $\zeta=\pi/6~\text{and}~\zeta=\pi/5$, and the dashed and solid green curves correspond to $a=0.5~\text{and}~a=0.7$, respectively.
• Figure 3: Characteristic quantities of the ISSO, including the radius $r_{ISSO}$, energy $\mathcal{E}_{ISSO}$, angular momentum $\mathcal{L}_{ISSO}$, and Carter constat $\mathcal{K}_{ISSO}$. Panels (a)–(d) depict the prograde orbits, while panels (e)–(h) present the retrograde orbits. The dashed and solid red curves correspond to $\ell=0.2~\text{and}~\ell=0.3$, the dashed and solid green curves correspond to $\zeta=\pi/6~\text{and}~\zeta=\pi/4$, respectively.
• Figure 4: Evolution of the angular coordinates $\theta$ and $\phi$ as functions of the coordinate time $t$. Panels (a)–(d) correspond to prograde spherical orbits, while panels (e)–(h) illustrate retrograde spherical orbits.
• Figure 5: Precession angular velocity $\omega_{t}$ as a function of the spin parameter $a$ for various values of ($\ell,\zeta,r/M$). The left panel corresponds to prograde spherical orbits, while the right panel shows retrograde ones. The dashed and solid red curves represent $\ell=0.2~\text{and}~\ell=0.3$, respectively; the dashed and solid blue curves denote $\zeta=\pi/4~\text{and}~\zeta=\pi/3$; and the dashed and solid green curves correspond to $r/M=9~\text{and}~r/M=11$, respectively.