⚛️ general relativity

Low-mass X-ray binaries as a probe of Kerr-MOG black hole spacetime

This paper investigates rotating black holes in modified gravity (Kerr-MOG spacetime) by analyzing six stellar-mass black hole candidates, demonstrating that the modified gravity parameter $\alpha$ significantly influences accretion disk efficiency and jet power, thereby creating a degeneracy with spin that can be constrained using observational data to identify viable parameter regions for interpreting black hole binary properties.

Original authors: Bakhodirkhon Saidov, Bakhtiyor Narzilloev, Ibrar Hussain, Bobomurat Ahmedov

Published 2026-06-24

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CC0 1.0

Original authors: Bakhodirkhon Saidov, Bakhtiyor Narzilloev, Ibrar Hussain, Bobomurat Ahmedov

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, cosmic dance floor. For decades, physicists have believed the rules of this dance were set by Albert Einstein's General Relativity. In this standard view, black holes are like the ultimate dance partners: they have a specific mass and a specific spin (how fast they twirl), and they drag the fabric of space around them like a whirlpool.

But what if there's a hidden rulebook we haven't found yet? This paper asks that very question. The authors investigate a theory called MOG (Modified Gravity), which suggests that gravity behaves slightly differently than Einstein predicted, especially near massive objects like black holes. They want to see if this "MOG rulebook" fits the data better than the standard one.

Here is a simple breakdown of their investigation:

1. The New Dance Floor: Kerr-MOG

In the standard theory, a spinning black hole is described by just two numbers: how heavy it is and how fast it spins. In the Kerr-MOG theory (the new idea), there is a third number: a "gravity tweak" parameter (called $\alpha$ ).

Think of it like tuning a guitar.

Standard Gravity (Kerr): You have the strings (mass) and the tension (spin).
MOG Gravity: You have the strings, the tension, plus a special tuning peg ( $\alpha$ ) that changes how the strings vibrate.

The authors built a mathematical model of this new "tuned" black hole to see how it changes the space around it.

2. The Inner Edge of the Dance (The ISCO)

Around a black hole, matter swirls in a disk (like water going down a drain). There is a critical point called the ISCO (Innermost Stable Circular Orbit). Inside this point, matter can't hold its orbit and must plunge into the black hole.

The Finding: The authors found that the "MOG tuning peg" ( $\alpha$ ) moves this inner edge.
The Analogy: Imagine the inner edge of the dance floor. In standard gravity, the edge is at a fixed distance based on how fast the partner spins. In MOG, if you turn the tuning peg ( $\alpha$ ), the edge of the dance floor moves closer or further away, even if the spin stays the same.

3. The Two Clues: Light and Jets

To figure out which rulebook is correct, the authors looked at real black holes in our galaxy. They used two main clues:

Clue A: The Glow (Radiative Efficiency)
As matter falls into the black hole, it gets hot and glows brightly (mostly in X-rays). How bright it glows depends on how close the inner edge of the disk is.

The Problem: A fast-spinning black hole in standard gravity looks very similar to a slow-spinning black hole with the "MOG tuning peg" turned up. It's a degeneracy—like two different keys on a piano producing the same note. You can't tell them apart just by looking at the light.

Clue B: The Blast (Relativistic Jets)
Sometimes, black holes shoot out powerful beams of particles (jets) from their poles, like a cosmic firehose. The strength of this jet depends on how fast the black hole spins and the strength of its magnetic field.

The Strategy: The authors calculated how strong these jets should be under the MOG rules and compared them to what telescopes actually see.

4. The Detective Work: Testing 6 Black Holes

The team picked six famous stellar-mass black holes (like A0620-00 and GRS 1915+105) and tried to solve the puzzle: Can we find a combination of Spin and MOG-Tuning that explains both the Glow AND the Jet?

The "Sweet Spot": For some black holes (like A0620-00 and XTE J1550-564), they found a "sweet spot." There was a specific range of Spin and MOG-Tuning where the math matched both the light and the jets perfectly. It's like finding the perfect combination of ingredients that makes a cake taste just right.
The "Tight Squeeze": For the fastest-spinning black hole in their list (GRS 1915+105), the solution was very narrow. The MOG model only worked if the black hole was spinning at almost the maximum possible speed and the "tuning peg" was set to a very specific value. It was a tight squeeze, but it still fit.
The "Partial Match": For others, the two clues (light and jets) didn't perfectly line up unless you allowed for some measurement errors, but the MOG model still offered a plausible explanation.

5. The Conclusion

The paper concludes that the Kerr-MOG model is a viable candidate. It suggests that the universe could be running on these modified gravity rules, and that this model can successfully explain the behavior of several real black holes.

However, there is a catch: because the "Spin" and the "MOG-Tuning" can mimic each other (the degeneracy), we can't be 100% sure yet which one is the real answer. The authors argue that by combining different types of observations (like the glow and the jets), we can start to untangle this knot.

In short: The authors took a new theory of gravity, applied it to real black holes, and found that it fits the data just as well as, and in some cases better than, the standard theory. It's a strong hint that the "tuning peg" of gravity might exist, but we need more precise measurements to turn it all the way.

Technical Summary: Low-mass X-ray Binaries as a Probe of Kerr-MOG Black Hole Spacetime

Problem Statement
General Relativity (GR) faces unresolved fundamental challenges, including the accelerated expansion of the Universe, spacetime singularities, and the quantization of gravity. Modified theories of gravity, such as Modified Gravity (MOG) or Scalar-Tensor-Vector Gravity (STVG) proposed by John Moffat, offer alternatives that address astrophysical issues like galaxy rotation curves and dark matter without invoking non-baryonic dark matter. While black hole solutions in MOG have been constructed, their astrophysical implications in the strong-field regime remain under-explored. Specifically, there is a need to determine whether the Kerr-MOG spacetime, characterized by mass ( $M$ ), spin ( $a$ ), and a modified gravity parameter ( $\alpha$ ), can consistently reproduce the observed properties of stellar-mass black hole candidates, particularly regarding accretion disk radiative efficiency and relativistic jet power.

Methodology
The authors investigate the Kerr-MOG spacetime metric and apply it to six well-known stellar-mass black hole candidates: A0620-00, H1743-322, XTE J1550-564, GRS1124-683, GRO J1655-40, and GRS1915+105. The study employs a two-pronged observational approach:

Accretion Disk Analysis: Using the Novikov–Thorne thin accretion disk model, the authors calculate the Innermost Stable Circular Orbit (ISCO) radius and the resulting radiative efficiency ( $\eta$ ). They analyze how the ISCO location and $\eta$ vary as functions of the spin parameter $a$ and the MOG parameter $\alpha$ . The study utilizes observational estimates of radiative efficiencies derived from the continuum-fitting method (CFM) to constrain the allowed regions in the $(a, \alpha)$ parameter space.
Relativistic Jet Analysis: The power of transient relativistic jets is modeled using the Blandford–Znajek (BZ) mechanism, which extracts rotational energy from the black hole via magnetic fields threading the event horizon. Theoretical jet power ( $P_{BZ}$ ) is calculated based on the horizon angular velocity ( $\Omega_H$ ) and compared against observed transient jet energetics. The analysis considers two representative jet Lorentz factors, $\Gamma = 2$ and $\Gamma = 5$ , to account for moderately and highly relativistic outflows.

The core of the methodology involves identifying regions in the $(a, \alpha)$ parameter space where the theoretical predictions for both radiative efficiency and jet power simultaneously match the observational data for each source.

Key Contributions and Results

Spacetime Structure and Degeneracy: The study demonstrates that the MOG parameter $\alpha$ significantly alters the spacetime structure, specifically the location of the ISCO. An increase in $\alpha$ generally expands the ISCO radius for fixed spin, while increasing spin reduces it. This creates a degeneracy where different combinations of $a$ and $\alpha$ can produce similar radiative efficiencies, complicating the isolation of modified gravity effects without additional constraints.
Radiative Efficiency Constraints: The radiative efficiency $\eta$ is shown to depend on both $a$ and $\alpha$ . For low spin values, $\eta$ increases with $\alpha$ , whereas for high spin values, $\eta$ decreases as $\alpha$ increases. By applying CFM-derived efficiency estimates, the authors constrain the allowed $(a, \alpha)$ regions for each source.
Jet Power and Lorentz Factor Dependence: The analysis reveals that the consistency between observed jet power and theoretical predictions is sensitive to the assumed jet Lorentz factor ( $\Gamma$ $Γ$ ).
- A0620-00: Significant overlap exists between efficiency and jet power constraints for both $\Gamma=2$ and $\Gamma=5$ , suggesting the Kerr-MOG model can consistently explain this system with parameters around $a \approx 0.04$ and $\alpha \approx 0.16$ .
- H1743-322 and XTE J1550-564: These sources show significant overlap regions, indicating compatibility with the Kerr-MOG framework, though the specific parameter ranges shift depending on $\Gamma$ .
- GRS 1124-683 and GRO J1655-40: Compatibility is more sensitive to the Lorentz factor. For GRS 1124-683, a clear overlap is found only for $\Gamma=5$ , while for GRO J1655-40, a lower Lorentz factor ( $\Gamma=2$ ) yields better simultaneous agreement.
- GRS 1915+105: As a highly spinning source ( $a > 0.98$ ), compatibility is achieved only within a very narrow range of Kerr-MOG parameters, specifically near-extremal spin configurations. The authors note that previous studies had not found such overlapping regions for this source in other spacetime models.

Significance and Claims
The paper claims that the Kerr-MOG spacetime provides a viable framework for interpreting the observed properties of several black hole X-ray binaries. The primary significance lies in demonstrating that modified gravity parameters can be constrained by combining independent observational probes: accretion disk radiative efficiency and relativistic jet power.

The authors emphasize that while the Kerr-MOG model can reproduce the data for multiple sources, it introduces a degeneracy between spin and the modified gravity parameter. Consequently, the study argues that combining constraints from both accretion and jet phenomena is essential to break this degeneracy. The results suggest that for some sources, the Kerr-MOG model offers a consistent explanation, while for others (like GRS 1915+105), the model restricts the viable parameter space to narrow, near-extremal configurations. The authors conclude that future observations with improved precision and larger samples of black hole systems are necessary to place stronger constraints on the MOG parameter and further test the viability of modified gravity in the strong-field regime.