Accretion Disks in Schwarzschild-MOG and Kerr-MOG Backgrounds: MOG Parameter in terms of Observational Quantities

This paper derives exact analytic formulas that express the mass, distance, rotation parameter, and the modified gravity coupling parameter $\alpha$ of Schwarzschild-MOG and Kerr-MOG black holes directly in terms of observable accretion disk quantities, thereby providing a method to empirically test deviations from standard general relativity.

The Gist

Imagine the universe as a giant, cosmic dance floor. For decades, physicists have believed that the most massive dancers—Black Holes—follow a very strict, simple choreography defined by Einstein's General Relativity. According to this "standard dance," a black hole is completely described by just two things: how heavy it is (Mass) and how fast it's spinning (Spin). It's like saying a dancer is only defined by their weight and their speed; everything else is just background noise.

But what if the music is slightly different? What if there's a hidden rhythm we haven't heard yet?

This is where Modified Gravity (MOG) comes in. It's a theory suggesting that gravity isn't just about mass; there's an extra "flavor" or "seasoning" to the universe, represented by a parameter called $\alpha$ (alpha). If $\alpha$ is zero, we are back to Einstein's standard dance. If $\alpha$ is not zero, the dance moves are slightly altered, and we might be able to detect this new rhythm.

The Problem: The Black Hole is a Mystery Box

The problem is that black holes are far away, dark, and incredibly dense. We can't just walk up to one and put a ruler against it or weigh it on a scale. We can only see the light from the hot gas (the accretion disk) swirling around them.

For a long time, scientists could only guess the black hole's mass and distance by making a lot of assumptions. It was like trying to figure out the speed of a car in the dark just by hearing the engine, without knowing the road conditions.

The Solution: Listening to the "Song" of the Light

This paper introduces a clever new way to "weigh" and "measure" these black holes by listening to the song of the light coming from the swirling gas.

Here is the analogy:
Imagine a lighthouse spinning in the fog.

1. The Frequency Shift (The Pitch): As the lighthouse spins toward you, the light waves get squashed (higher pitch/blue). As it spins away, they stretch out (lower pitch/red). This is the Doppler effect.
2. The Redshift Rapidity (The Change in Pitch): This is the paper's secret weapon. It's not just about the pitch; it's about how fast the pitch is changing as the lighthouse moves. It's like listening to a siren not just for its tone, but for how quickly that tone is sliding up or down.
3. The Redshift Acceleration (The Jerk): In the spinning black hole scenario, they add a third layer: how quickly that change in pitch is itself changing. It's the "jerk" in the motion.

The Magic Trick: Cracking the Code

The authors (Rojas and Momennia) did something brilliant. They took the complex math of Einstein's equations (and the new MOG equations) and turned them into a decoder ring.

They showed that if you measure three specific things from the light of the gas disk:

1. How much the light is shifted (The total pitch change).
2. How fast that shift is changing (The Rapidity).
3. The angle of the telescope (How wide your view is).

...you can solve a set of equations to find four hidden secrets simultaneously:

• Mass ($M$): How heavy the black hole is.
• Distance ($D$): How far away it is.
• Spin ($a$): How fast it's rotating (for spinning black holes).
• The MOG Parameter ($\alpha$): The "secret seasoning" that tells us if gravity works differently than Einstein predicted.

Why This Matters

Think of it like a forensic investigation.

• Old Way: "We think the black hole is 10 million miles away and weighs 1 billion suns, but we aren't sure because we have to guess the angle."
• New Way (This Paper): "We measured the light's song. The math says: It is exactly 10.04 million miles away, weighs exactly 1.02 billion suns, and the gravity here has a tiny bit of extra 'flavor' ($\alpha$)."

If they measure $\alpha$ and it's zero, Einstein wins again. If they measure $\alpha$ and it's not zero, then we have discovered a new law of physics!

The "Midline" and the "Line of Sight"

The paper gets very specific about where to look.

• The Midline: Imagine looking at the black hole from the side, where the gas is moving directly toward or away from you. This is where the pitch shift is the most extreme.
• The Line of Sight: Imagine looking straight down the barrel of the gun, where the gas is moving across your view. This is where the "rapidity" (the speed of the pitch change) is strongest.

By combining data from these two specific angles, the math "unscrambles" the variables. It's like solving a puzzle where you need pieces from two different sides to see the whole picture.

The Bottom Line

This paper is a user manual for the universe. It provides a precise, mathematical recipe for astronomers to take raw data from telescopes (like the Event Horizon Telescope) and instantly calculate the true properties of black holes, including whether our understanding of gravity needs a tune-up.

It turns the mysterious, invisible black hole into a measurable object, using the light it steals from the stars as a messenger. And the best part? If the universe is playing by Einstein's rules, this recipe will give the standard answer. If the universe is playing by the new MOG rules, this recipe will reveal the secret.

Technical Summary

1. Problem Statement

The paper addresses the challenge of testing Modified Gravity (MOG), specifically Scalar–Tensor–Vector Gravity (STVG), against General Relativity (GR) using observational data from black hole accretion disks.

• Context: While GR has been confirmed by gravitational wave detections and Event Horizon Telescope imaging, alternative theories like MOG propose that the effective gravitational coupling is dynamical ($G = G_N(1+\alpha)$) and includes a massive vector field. This introduces a new coupling parameter, $\alpha$, which modifies the spacetime metric.
• The Gap: Previous studies have derived black hole parameters (mass $M$, spin $a$, distance $D$) in standard GR or specific modified gravity models using frequency shifts. However, there was a lack of exact, closed-form analytic relations that express the MOG parameter $\alpha$ directly in terms of measurable quantities from accretion disks without relying on auxiliary non-observable parameters.
• Objective: To derive exact analytic formulas that determine the black hole mass ($M$), MOG coupling parameter ($\alpha$), distance ($D$), and rotation parameter ($a$) solely from directly observable quantities (frequency shifts, aperture angles, and their time derivatives) for both static (Schwarzschild-MOG) and rotating (Kerr-MOG) black holes.

2. Methodology

The authors employ a general relativistic framework applied to the specific metrics of Schwarzschild-MOG and Kerr-MOG spacetimes. The methodology involves:

• Geodesic Analysis:
  • Deriving the four-velocity components ($U^\mu$) for massive test particles on circular equatorial orbits.
  • Deriving the four-momentum ($k^\mu$) and impact parameter ($b_\phi$) for null geodesics (photons).
  • Establishing the metric functions for Schwarzschild-MOG (static) and Kerr-MOG (rotating), which include the MOG parameter $\alpha$.
• Frequency Shift Formulation:
  • Calculating the total frequency shift ($1+z$) for photons emitted by orbiting matter and detected by a distant observer.
  • Decomposing the shift into gravitational ($z_g$) and kinematic ($z_{kin}$) components.
• Introduction of Higher-Order Kinematic Invariants:
  • Redshift Rapidity ($\dot{z}$): Defined as the proper-time derivative of the redshift. This acts as a relativistic invariant to help resolve parameter degeneracies.
  • Redshift Acceleration ($\ddot{z}$): Defined as the proper-time derivative of the redshift rapidity (analogous to "jerk" in Newtonian mechanics). This is introduced specifically for the rotating Kerr-MOG case to disentangle the spin parameter $a$ from other variables.
• Observational Constraints:
  • Evaluating these quantities at two critical locations: the midline ($\phi = \pm \pi/2$, where redshift/blueshift is maximal) and the line of sight (LOS) ($\phi \approx 0$, where rapidity is maximal).
  • Constructing a system of non-linear equations linking observables ($z, \dot{z}, \ddot{z}, \delta$) to unknowns ($M, \alpha, D, a, r_e$).
• Analytic Inversion:
  • Solving the system of equations to obtain closed-form expressions for the unknown parameters in terms of the observables.

3. Key Contributions

1. Exact Analytic Solutions for MOG: The paper provides the first exact, closed-form relations expressing the MOG coupling parameter $\alpha$, black hole mass $M$, and distance $D$ purely in terms of observables for the Schwarzschild-MOG background.
2. Extension to Rotating Spacetimes (Kerr-MOG): The formalism is extended to the rotating case. Crucially, the authors demonstrate that the redshift acceleration is necessary to uniquely determine the spin parameter $a$ alongside $M$, $\alpha$, and $D$ in the rotating MOG background.
3. Parameter Disentanglement: The derived formulas successfully break the degeneracy between the mass, distance, and the modified gravity parameter. In standard GR, $\alpha=0$, and the formulas naturally reduce to known GR results.
4. New Observables: The paper rigorously defines and utilizes redshift rapidity and redshift acceleration as measurable quantities in the context of accretion disks, showing how they encode higher-order spacetime curvature information.

4. Key Results

• Schwarzschild-MOG (Static Case):
  • The authors derived explicit formulas for $\alpha$, $M$, $D$, and the emitter radius $r_e$ using the total frequency shift ($z$), telescope aperture angle ($\delta$), and redshift rapidity ($\dot{z}$) measured at the midline.
  • Proposition 1: It is proven that for $\alpha > 0$, the absolute total frequency shift $|z_{MOG}|$ at the midline is strictly larger than the corresponding shift in standard Schwarzschild spacetime ($|z_{Schw}|$). This provides a direct observational signature of MOG.
  • Formula (57): An explicit expression for $\alpha$ is given solely in terms of $z_{MOG1}, z_{MOG2}, \delta_m$, and $\dot{z}_{MOG}$.
• Kerr-MOG (Rotating Case):
  • The inclusion of the spin parameter $a$ requires the redshift acceleration ($\ddot{z}$).
  • The paper derives a 5x5 system of equations solved analytically to find $M, a, \alpha, D, r_e$.
  • Opposing Effects: The analysis of Figure 2 reveals that increasing the spin parameter $a$ increases the frequency shift separation and rapidity amplitude, whereas increasing the MOG parameter $\alpha$ decreases these values. This opposing behavior is crucial for distinguishing between rotation and modified gravity effects.
  • Weak-Field Limits: In the Newtonian limit, the redshift rapidity reduces to the line-of-sight projection of Keplerian acceleration (modified by $1+\alpha$), and redshift acceleration reduces to the projection of the Keplerian jerk.
• Recovery of GR: In the limit $\alpha \to 0$, all derived expressions continuously reduce to the standard Schwarzschild and Kerr results found in previous literature, validating the consistency of the framework.

5. Significance

• Empirical Test of Modified Gravity: The work provides a concrete, model-independent method to test MOG against General Relativity. By measuring $\alpha$ directly from accretion disk data, astronomers can quantify deviations from GR without assuming a specific dark matter distribution.
• Black Hole Parameter Estimation: The derived formulas are "concise and suitable for incorporation into black hole parameter-estimation pipelines." They offer a way to determine the distance to a black hole and its mass simultaneously, which is often difficult using standard methods.
• Application to Real Systems: The formalism is designed for thin accretion disks around supermassive black holes, such as those found in Active Galactic Nuclei (AGNs) with H$_2$O megamaser disks (e.g., NGC 4258), where the geometric assumptions hold well.
• Theoretical Advancement: By introducing redshift acceleration as a tool for parameter decoupling in rotating spacetimes, the paper opens new avenues for analyzing high-order kinematic effects in strong gravitational fields.

In summary, this paper bridges the gap between theoretical modified gravity models and observational astrophysics by providing a rigorous mathematical toolkit to extract the MOG coupling parameter $\alpha$ directly from the spectral and temporal properties of photons emitted by accretion disks.