Strong gravitational lensing and Quasiperiodic oscillations as a probe for an electrically charged Lorentz symmetry-violating black hole

This study investigates how electric charge and Lorentz symmetry breaking jointly influence strong gravitational lensing and quasiperiodic oscillations around a charged black hole, utilizing observational data from supermassive black holes and microquasars to establish constraints on the Lorentz symmetry-violating parameter while revealing a compensatory effect between charge and symmetry breaking.

The Gist

Imagine the universe as a giant, invisible trampoline. In our everyday understanding of gravity (thanks to Einstein), heavy objects like black holes sit on this trampoline and create deep dips. When a marble (a photon of light) rolls past, it follows the curve of the dip. This is how gravity works in the standard model.

But what if the trampoline fabric itself has a weird, invisible texture or a slight tear in it? What if the marble also carries a tiny electric charge? This is the question physicist Sohan Kumar Jha is asking in his paper. He is investigating a very specific, exotic type of black hole that has two special features:

1. Electric Charge: Like a static shock, but on a cosmic scale.
2. Lorentz Symmetry Violation (LSB): Imagine the rules of the universe are slightly "broken" or tilted in one direction. In normal physics, the laws look the same no matter which way you spin or move. In this model, the fabric of spacetime has a preferred direction, like a grain in a piece of wood.

Jha wants to know: If such a black hole exists, how would it look to us, and how would it affect the things swirling around it?

To answer this, he uses two cosmic "flashlights" to shine a light on the black hole: Strong Gravitational Lensing and Quasiperiodic Oscillations (QPOs).

1. The Cosmic Shadow (Strong Gravitational Lensing)

Imagine you are standing far away from a black hole. You can't see the hole itself, but you can see its "shadow" against the bright background of the universe. This shadow is a dark circle surrounded by a bright ring of light.

• The Analogy: Think of the black hole as a heavy bowling ball on a trampoline. If you roll marbles (light) near it, they curve. Some get trapped, and some curve so much they form a bright ring around the dark center.
• The Twist: Jha found that the electric charge and the broken symmetry (LSB) act like two people pushing a swing from opposite sides.
  • The charge tries to make the shadow smaller.
  • The broken symmetry (if it's negative) tries to make the shadow bigger.
  • The Magic: Sometimes, these two forces cancel each other out perfectly! The black hole looks exactly like a "normal" black hole (a Schwarzschild black hole) with no charge and no broken symmetry, even though it's actually a complex, exotic object. It's like a magician's trick where the audience sees a normal rabbit, but the rabbit is actually wearing a hidden costume.

Jha looked at real photos of two famous black holes, M87* (the giant one in a distant galaxy) and Sgr A* (the one in the center of our Milky Way). By measuring the size of their shadows, he tried to figure out how much "broken symmetry" or "charge" they might have.

• The Result: The shadow sizes matched the predictions for normal black holes very well. He couldn't pin down the exact amount of electric charge, but he did put limits on how much the "broken symmetry" could be happening.

2. The Cosmic Heartbeat (Quasiperiodic Oscillations)

Now, imagine a black hole isn't just sitting there; it's eating a meal. It has a swirling disk of gas and dust (an accretion disk) spinning around it. As this material gets close to the edge of no-return, it vibrates and pulses, creating a rhythmic "heartbeat" in X-rays.

• The Analogy: Think of a child on a swing. If you push them at just the right time (resonance), they go higher. In these microquasars (small black holes eating stars), the gas vibrates at two specific frequencies that are locked in a 3:2 ratio (like a musical interval).
• The Test: Jha used the "Forced Resonance Model." He calculated how the charge and the broken symmetry would change the speed of these vibrations.
  • If the black hole has more charge, the vibrations speed up.
  • If the symmetry is broken, the vibrations slow down or speed up depending on the direction.
• The Result: By looking at the actual "heartbeats" of two specific microquasars (GRO J1655-40 and XTE J1550-564), Jha was able to get a much clearer picture. Unlike the shadow test, the heartbeat test allowed him to estimate both the charge and the broken symmetry parameter.

The Big Picture: Why Does This Matter?

This paper is like a detective story. We have a suspect (a new theory of gravity) that predicts black holes might have these weird properties.

1. The Shadow Test: Told us the suspect could be hiding in plain sight because the weird effects cancel each other out.
2. The Heartbeat Test: Caught the suspect in the act, giving us specific numbers for how "charged" and how "symmetry-broken" these black holes might be.

The Takeaway:
Jha's work shows that nature is tricky. Two different weird effects can combine to make a black hole look "normal" to our eyes (the shadow), but if we listen closely to its "heartbeat" (the X-ray oscillations), we can hear the secret. This helps scientists understand if our current laws of physics are perfect, or if there are tiny cracks in the foundation of the universe that we are just starting to notice.

In short: Black holes might be wearing masks, but their X-ray heartbeats are giving them away.

Technical Summary

1. Problem Statement

The study addresses the need to test modifications to General Relativity (GR) in the strong-field regime. While GR has been successful, various theoretical frameworks (string theory, non-commutative field theory) predict Lorentz Symmetry Breaking (LSB) at fundamental scales. Specifically, this paper investigates an electrically charged, Lorentz symmetry-violating (LV) black hole (referred to as a QKR BH), derived from a model involving a Kalb-Ramond (KR) field with a non-zero vacuum expectation value (VEV).

The core problem is to determine how the interplay between electric charge ($Q$) and the LSB parameter ($\alpha$) alters observable astrophysical phenomena compared to standard Schwarzschild or Reissner-Nordström (RN) black holes. The authors aim to constrain these free parameters using observational data from Supermassive Black Holes (SMBHs) and microquasars.

2. Methodology

The authors employ a two-pronged observational approach to probe the QKR BH metric:

A. Theoretical Framework

• Metric: The study utilizes a static, spherically symmetric metric derived from the Einstein-Kalb-Ramond action with non-minimal coupling to electromagnetism. The metric function is given by:
  $$A(r) = \frac{1}{1-\alpha} - \frac{2M}{r} + \frac{Q^2}{(1-\alpha)^2 r^2}$$
  where $\alpha$ is the LSB parameter and $Q$ is the electric charge.
• Horizons: Conditions for the existence of event and Cauchy horizons are derived, establishing a parameter space where a black hole solution exists ($Q^2 \leq M^2(1-\alpha)^3$).

B. Strong Gravitational Lensing (SGL) Analysis

• Photon Trajectories: The authors analyze null geodesics in the equatorial plane to determine the photon sphere radius ($x_m$) and the critical impact parameter ($b_m$), which defines the shadow radius.
• Deflection Angle: Using the strong-field limit approach (Bozza's formalism), they calculate the deflection angle $\gamma_D(b)$ as a function of the impact parameter.
• Observables: Three key lensing observables are computed:
  1. Angular radius of the shadow ($\theta_\infty$).
  2. Angular separation between the outermost image and the rest ($s$).
  3. Relative magnification of the brightest image ($r_{mag}$).
• Data Comparison: These theoretical values are compared against Event Horizon Telescope (EHT) observations of M87* and Sgr A* to constrain $\alpha$ and $Q$.

C. Quasiperiodic Oscillations (QPOs) Analysis

• Particle Dynamics: The motion of test particles near the Innermost Stable Circular Orbit (ISCO) is analyzed.
• Epicyclic Frequencies: The authors derive expressions for radial ($\nu_r$) and latitudinal ($\nu_\theta$) epicyclic frequencies in the QKR background.
• Resonance Model: They employ the forced resonance model, assuming the observed twin-peak QPOs correspond to:
  • Lower frequency: $\nu_L = \nu_\theta$
  • Upper frequency: $\nu_U = \nu_r + \nu_\theta$
• Data Comparison: The model is tested against high-frequency QPO data from microquasars GRO J1655-40 and XTE J1550-564 to extract best-fit values for $\alpha$ and $Q$.

3. Key Contributions and Findings

A. Interplay of Charge and LSB

A significant finding is the cancellation effect between the electric charge and the LSB parameter.

• The study identifies specific combinations of negative $\alpha$ and positive $Q$ where the competing effects cancel each other out.
• In these specific cases, the event horizon ($r_h$), shadow radius ($b_m$), and ISCO radius ($r_{isco}$) become indistinguishable from those of a standard Schwarzschild black hole, despite the presence of charge and LSB.
• Negative $\alpha$ generally increases the shadow radius and ISCO radius compared to RN/Schwarzschild, while positive $\alpha$ decreases them.

B. Constraints from Strong Gravitational Lensing (SGL)

• Shadow Observations: Using angular diameter data for M87* and Sgr A*, the authors derived bounds on the LSB parameter $\alpha$:
  • Sgr A*: $\alpha \in [-0.0597, 0.1263]$ (within $1\sigma$).
  • M87*: $\alpha \in [-0.0897, 0.0095]$ (within $1\sigma$).
• Charge Constraint: SGL observations alone could not constrain the electric charge $Q$. The shadow size is insensitive to $Q$ in certain regimes or the degeneracy with $\alpha$ is too high. However, the theoretical condition for the existence of a black hole ($Q^2 \leq M^2(1-\alpha)^3$) was used to place an upper bound on $Q$ based on the allowed $\alpha$.

C. Constraints from Quasiperiodic Oscillations (QPOs)

• Charge Constraint: Unlike SGL, QPO observations successfully constrained both parameters.
• Best-Fit Values: Using the forced resonance model for the microquasars, the authors obtained the following best-fit values (with uncertainties):
  • For Upper Frequency ($\nu_U$): $\alpha = -0.3^{+0.012}_{-0.012}$ and $Q/M = 0.625^{+0.086}_{-0.096}$.
  • For Lower Frequency ($\nu_L$): $\alpha = -0.2^{+0.011}_{-0.011}$ and $Q/M = 0.625^{+0.060}_{-0.073}$.
• Physical Insight: The analysis revealed that increasing the black hole mass pushes the ISCO further out, lowering the QPO frequencies. Conversely, increasing the charge $Q$ tends to increase the frequencies, while increasing $\alpha$ (positive) decreases them.

4. Significance

• Complementary Probes: The study demonstrates that while SGL is effective for constraining the LSB parameter ($\alpha$), it fails to constrain the electric charge ($Q$) independently. In contrast, QPOs provide a complementary and powerful method to constrain both parameters simultaneously.
• Degeneracy Breaking: The research highlights how different observational channels (photon trajectories vs. matter dynamics) break parameter degeneracies in modified gravity models.
• Model Viability: The results confirm the viability of the QKR black hole model as a candidate for describing real astrophysical objects, showing that the combined effects of charge and Lorentz violation can mimic standard GR solutions under specific conditions.
• Future Outlook: The authors suggest that next-generation telescopes with higher precision will be crucial for tightening these constraints and distinguishing between QKR, Schwarzschild, and RN black holes definitively.

In summary, this paper provides a rigorous phenomenological test of a Lorentz-violating, charged black hole model, utilizing the synergy between strong gravitational lensing and high-frequency QPOs to place the first simultaneous bounds on both the LSB parameter and electric charge in this specific theoretical framework.