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

This paper utilizes the observed 11-year jet precession period of M87* to constrain the parameters of a rotating black hole in Bumblebee gravity, demonstrating how Lorentz-symmetry breaking affects spherical orbits and innermost stable spherical orbits, and suggesting that a warp radius exceeding 16 may indicate the presence of a non-vacuum Bumblebee vector field when combined with EHT shadow constraints.

The Gist

Imagine the universe as a giant, cosmic dance floor. Usually, the rules of physics say that no matter how you spin, slide, or tilt, the laws of gravity and motion stay the same. This is called Lorentz symmetry. It's like saying the dance floor looks and feels exactly the same whether you're dancing in the morning or the evening, or whether you're facing north or south.

But what if the dance floor isn't perfectly uniform? What if there's a subtle "ripple" or a hidden texture that changes how things move depending on your direction? This is the idea of Lorentz-symmetry breaking (LSB).

This paper is a detective story where the authors try to find evidence of these "ripples" by looking at the most famous dancer in the universe: M87*, a supermassive black hole at the center of a galaxy.

Here is the story broken down into simple concepts:

1. The Mystery: The Wobbly Jet

M87* is famous for shooting out a massive beam of energy (a jet) that looks like a giant lighthouse beam. Recently, astronomers noticed something strange: this beam isn't just pointing in one direction; it's wobbling or precessing (like a spinning top that is about to fall over).

This wobble happens in a perfect cycle, taking about 11.24 years to complete one full circle. The scientists think this wobble is caused by the black hole's spin dragging the space around it, combined with the fact that the disk of gas swirling into the black hole is tilted at an angle.

2. The Theory: The "Bumblebee" Gravity

The authors are testing a specific theory called Bumblebee Gravity.

• The Analogy: Imagine the vacuum of space isn't empty, but filled with a invisible "honey" (the Bumblebee field). Usually, this honey is calm. But in this theory, the honey has a preferred direction, like a river flowing one way.
• The Effect: If a black hole spins in this "flowing honey," the rules of gravity change slightly compared to Einstein's standard theory. The black hole might act a bit differently, like a skater spinning on ice that has a hidden current underneath.

3. The Experiment: Simulating the Dance

The authors built a mathematical model to see how particles (gas and dust) would move around this "Bumblebee" black hole. They focused on spherical orbits—particles that don't just circle the equator but bob up and down like a ball on a string, tracing a sphere.

They asked: If the "honey" (LSB) exists, how does it change the wobble of the jet?

They found three main "knobs" that control the dance:

1. Spin ($a$): How fast the black hole is spinning.
2. The Tilt ($\zeta$): How much the gas disk is leaning.
3. The "Honey" Strength ($\ell$): How strong the Lorentz-symmetry breaking effect is.

The Findings:

• Prograde vs. Retrograde: If the gas spins the same way as the black hole (prograde), the "honey" makes the orbits shrink and slow down. If it spins the opposite way (retrograde), the "honey" pushes the orbits out and speeds them up.
• The Wobble Speed: The stronger the black hole spins or the stronger the "honey" is, the faster the jet wobbles.

4. The Detective Work: Matching the Clues

The authors took the real-world data (the 11.24-year wobble period and the size of the black hole's "shadow" seen by the Event Horizon Telescope) and tried to fit their "Bumblebee" model to it.

They calculated: Where must the gas be located (the "warp radius") for the wobble to match the 11.24-year observation?

The Results:

• The "Sweet Spot": They found that for the math to work, the gas must be located at a specific distance from the black hole.
• The Smoking Gun: In standard Einstein gravity (no "honey"), there is a hard limit on how far out this gas can be. However, their calculations showed that if the "honey" (LSB) exists, the gas could be much further out.
• The Conclusion: If the actual gas disk is located further out than Einstein's theory allows (specifically, if the distance is greater than 16 times the black hole's mass), it would be strong evidence that Lorentz symmetry is broken and that the "Bumblebee" field is real.

5. Why This Matters

This paper doesn't prove the "honey" exists yet, but it sets up a very precise test.

• It tells astronomers: "If you measure the jet's wobble and the black hole's shadow, and you find the gas is in this specific 'forbidden' zone, then Einstein's theory needs a little update."
• It shows that the spin of the black hole is the most important factor in this dance, more so than the "honey" itself.

Summary in a Nutshell

Think of the black hole as a spinning top. The "Bumblebee" theory suggests the table it's spinning on has a hidden texture. The authors calculated how this texture would change the top's wobble. By comparing their calculations to the real wobble of M87*, they found a specific range of distances where the "hidden texture" would be necessary to explain what we see. If future observations confirm the gas is in that range, we might have to rewrite the rules of gravity!

Technical Summary

1. Problem Statement

The paper addresses the need to constrain parameters of rotating black holes within Bumblebee gravity, a theoretical framework incorporating spontaneous Lorentz-symmetry breaking (LSB). While the Event Horizon Telescope (EHT) has provided images of the M87* black hole shadow, and radio observations have detected a periodic precession in its relativistic jet, existing models often rely on standard General Relativity (Kerr metric). The authors aim to:

• Determine how LSB parameters (specifically the coupling constant $\ell$) and black hole spin ($a$) affect the dynamics of massive particles in tilted accretion disks.
• Utilize the observed 11.24-year jet precession period and the EHT shadow radius of M87* to constrain the LSB parameter and the black hole's spin.
• Investigate whether deviations from the Kerr metric (specifically $r/M > 16$) can serve as observational signatures of a non-vacuum Bumblebee vector field.

2. Methodology

The authors employ a multi-step theoretical and numerical approach:

• Theoretical Framework: They utilize the rotating black hole metric in Bumblebee gravity, derived via the modified Newman-Janis algorithm. The metric depends on mass ($M$), spin ($a$), and the LSB parameter ($\ell = \varrho b^2$).
• Particle Dynamics: They analyze the motion of massive test particles using the Hamilton-Jacobi equation. They derive the equations of motion for the coordinates $(t, r, \theta, \phi)$ and identify conserved quantities: Energy ($E$), Angular Momentum ($L$), and the Carter constant ($K$).
• Orbital Modeling:
  • Spherical Orbits: They focus on spherical orbits (constant radius $r$) to model particles in a tilted accretion disk. The tilt angle is denoted by $\zeta$.
  • Innermost Stable Spherical Orbits (ISSOs): They solve for the conditions $R(r) = R'(r) = R''(r) = 0$ to determine the stability limits of these orbits.
• Precession Analysis:
  • They numerically integrate the angular equations of motion to track the evolution of $\theta$ and $\phi$ over time.
  • They calculate the oscillation period $T_\theta$ in the polar direction and the accumulated azimuthal angle $\phi(T_\theta)$.
  • The precession angular velocity ($\omega_t$) is defined as $\omega_t = |\phi(T_\theta) - 2\pi| / T_\theta$.
• Observational Constraints:
  • They relate the theoretical precession frequency to the observed jet period ($T \approx 11.24$ years) using the formula $T = 2\pi/\omega_t \times (GM/c^3)$.
  • They fix the tilt angle $\zeta = 1.25^\circ$ based on observational data.
  • They cross-reference results with the EHT shadow constraint ($\theta_{sh} = 42 \pm 3 \mu as$) and the parameter ranges derived in a previous study (Ref. [78]) to narrow down the allowed $(a, \ell)$ parameter space.

3. Key Contributions

• Analytical Derivation: The paper provides explicit dependencies of conserved quantities ($E, L, K$) and ISSO properties on the LSB parameter $\ell$, spin $a$, and tilt $\zeta$ for both prograde and retrograde orbits.
• Distinction of Orbital Behaviors: It systematically contrasts prograde ($L>0$) and retrograde ($L<0$) orbits, revealing that the LSB parameter $\ell$ suppresses prograde dynamics but enhances retrograde angular momentum and radius.
• Jet Precession as a Probe: It establishes a direct link between the jet nozzle precession period and the "warp radius" ($r_{warp}$) of the accretion disk, treating the jet launch point as lying outside the ISSO.
• Combined Constraints: It demonstrates how combining jet precession data with shadow imaging provides tighter constraints on alternative gravity theories than either method alone.

4. Key Results

A. Orbital Dynamics and ISSOs

• Prograde Orbits: Increasing the spin $a$ or LSB parameter $\ell$ decreases the ISSO radius ($r_{ISSO}$), energy ($E$), and angular momentum ($L$). Increasing the tilt $\zeta$ increases $r_{ISSO}$ and $E$ but decreases $L$.
• Retrograde Orbits: The trends are largely reversed. Increasing $a$ increases $r_{ISSO}$, while increasing $\ell$ increases $r_{ISSO}$, $L$, and $K$ but decreases $E$.
• Precession Velocity ($\omega_t$):
  • $\omega_t$ increases with $a$ and $\ell$ for both orbit types.
  • For prograde orbits, $\omega_t$ increases with tilt $\zeta$; for retrograde, it decreases.
  • The influence of $\zeta$ on $\omega_t$ is relatively weak compared to $a$ and $\ell$.

B. Constraints on M87* Parameters

Using the observed precession period $T = 11.24 \pm 0.47$ years and fixed tilt $\zeta = 1.25^\circ$:

• Warp Radius ($r/M$) Ranges:
  • Prograde: $r/M \in (5.73, 25.15)$.
  • Retrograde: $r/M \in (6.16, 26.46)$.
• Kerr Limit Comparison: In standard Kerr gravity (where $\ell=0$), the maximum warp radii are $r/M \approx 14.12$ (prograde) and $16.1$ (retrograde).
  • Key Finding: If the derived warp radius $r/M > 16$, it strongly suggests the presence of a non-vacuum Bumblebee vector field (i.e., $\ell \neq 0$).
• Impact of Shadow Constraints: Incorporating the EHT shadow radius ($\theta_{sh} = 42 \pm 3 \mu as$) further restricts the parameter space:
  • Prograde: $r/M \in (5.82, 22.61)$.
  • Retrograde: $r/M \in (6.17, 24.74)$.
  • The discrepancy between prograde and retrograde radii remains significant (0.05–1.96), indicating that the spin parameter $a$ has a more dominant effect on the warp radius than the LSB parameter $\ell$.

5. Significance

• Testing Alternative Gravity: The study provides a robust framework for testing Lorentz-symmetry breaking using real astrophysical data (M87*), moving beyond theoretical speculation.
• Diagnostic Tool: It identifies a specific observational signature: a warp radius $r/M > 16$ would be a "smoking gun" for Bumblebee gravity effects, distinguishing it from standard General Relativity.
• Multi-Messenger Synergy: The paper highlights the power of combining jet precession timing (dynamic information) with shadow imaging (geometric information) to break degeneracies in black hole parameter estimation.
• Future Outlook: While the current model simplifies the accretion disk dynamics (ignoring viscosity and magnetic fields), the results offer a foundational baseline for future, more complex GRMHD simulations in modified gravity theories.

In conclusion, the paper successfully demonstrates that the precessing jet of M87* is a sensitive probe for the LSB parameter in Bumblebee gravity, offering a pathway to constrain the nature of spacetime at the Planck scale through macroscopic astrophysical observations.