⚛️ general relativity

Can a regular black hole be observationally distinguished from singular black holes as spinning lens partner in PSR-BH binaries?

Although the study proposes a novel time-of-arrival diagnostic based on frame dragging to distinguish regular Ayón-Beato and García black holes from singular Kerr and Kerr-Newman black holes in pulsar-binary systems, it concludes that under the thin-lens approximation, the observable differences are negligible (appearing only at third order or higher), rendering the two types of black holes indistinguishable despite their qualitative theoretical differences.

Original authors: G. Y. Tuleganova, R. Kh. Karimov, R. N. Izmailov, K. K. Nandi

Published 2026-06-29

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: G. Y. Tuleganova, R. Kh. Karimov, R. N. Izmailov, K. K. Nandi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ 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

The Big Question: Can We Spot a "Smooth" Black Hole?

Imagine you are a detective trying to solve a mystery. You have two suspects:

The "Singular" Black Hole: The classic version from standard physics. It has a center so dense and crushed that it breaks the laws of math (a "singularity"). Think of it like a hole in a trampoline that goes down forever.
The "Regular" Black Hole (AGBH): A newer, theoretical version. It also has a lot of mass, but its center is "smooth" and doesn't break math. Think of it like a hole in a trampoline that bottoms out on a soft, solid cushion instead of falling forever.

The authors of this paper asked: If we look at these two objects through a telescope, can we tell them apart?

The Detective Tool: The "Time-of-Arrival" Race

To solve this, the authors didn't look at how the black holes look (like their shadows). Instead, they looked at how they affect time.

Imagine a spinning carousel (the black hole) in the middle of a field.

The Setup: Two runners (light rays) start at the same time from a point behind the carousel. They have to run around the carousel to reach a finish line (the observer on Earth).
The Twist: Because the carousel is spinning, it drags the air (or space) around it.
- The runner going with the spin (co-rotating) gets a little push, like running on a moving walkway. They finish faster.
- The runner going against the spin (counter-rotating) has to fight the wind. They finish slower.

This difference in finish times is called the Time-of-Arrival (TOA) difference. It's like a cosmic race where the spinning object creates a "tailwind" for one side and a "headwind" for the other.

The Experiment: The Race Results

The authors ran this "race" theoretically for three different types of black holes:

The smooth, regular one (AGBH).
The classic spinning one with charge (Kerr-Newman).
The classic spinning one without charge (Kerr).

They calculated the time difference for two real-world scenarios where a pulsar (a flashing star) orbits a black hole:

Case A: A system similar to Cygnus X-1.
Case B: A system near the supermassive black hole at the center of our galaxy, Sagittarius A*.

The Verdict: They Look the Same

Here is the surprising conclusion: You cannot tell them apart.

Even though the "smooth" black hole and the "singular" black hole are fundamentally different inside (one has a soft cushion, the other has a bottomless pit), the race times were almost identical.

The Main Effect: The difference in arrival time is caused mostly by the spin of the black hole. This effect is huge enough to be measured (in microseconds).
The "Secret" Difference: The only thing that makes the smooth black hole different is a tiny mathematical tweak related to its electric charge. However, this difference only shows up in the third decimal place of the calculation.

The Analogy:
Imagine two identical-looking cars racing on a track. One car has a standard engine, and the other has a slightly modified engine.

The standard engine makes the car go 100 mph.
The modified engine makes it go 100.0000001 mph.
Even if you have a stopwatch, you can't tell the difference between the two cars because the speed difference is too small to measure.

Why This Matters (According to the Paper)

The authors emphasize that this is a weak field effect. This means they are looking at the black hole from far away, where gravity is relatively gentle.

What they found: At this distance, the "smoothness" of the black hole's center is hidden. The spin dominates the signal, and the subtle differences between the smooth and singular centers are drowned out.
The Limit: To see the difference, you would need to measure time with picosecond accuracy (trillionths of a second). The paper states that current technology is nowhere near that level. We can measure microseconds, but not picoseconds.

Summary

The paper concludes that while "Regular" black holes (which fix the math problems of the center) are theoretically possible, we cannot currently distinguish them from "Singular" black holes by watching light race around them in a binary star system. The "smoothness" is too subtle to be seen with our current tools, even though the two types of black holes are very different in their internal structure.

Key Takeaway: Nature is tricky. Two objects can be built very differently on the inside, but if you only look at them from the outside while they spin, they look exactly the same.

Problem Statement
The paper addresses the question of whether a regular black hole (specifically the Ayón-Beato and García black hole, or AGBH) can be observationally distinguished from singular black holes (the Kerr-Newman and Kerr black holes) when acting as a spinning lens in Pulsar-Black Hole (PSR-BH) binary systems. While the AGBH is a regular solution of General Relativity coupled to nonlinear electrodynamics (removing the central singularity), and singular black holes possess a central singularity, the authors seek to determine if these qualitative geometric differences manifest in observable quantities. The study focuses on the "Time of Arrival" (TOA) difference, a diagnostic proposed to sample the frame-dragging effect of a spinning lens, distinct from the standard Shapiro gravitational time delay.

Methodology
The authors employ a theoretical framework based on the thin-lens approximation within the weak-field limit.

Metric Formulation: They utilize the spinning AGBH metric (derived via the Newman-Janis algorithm from the static AGBH solution) and compare it against the standard spinning Kerr-Newman (KNBH) and Kerr (KBH) metrics.
TOA Derivation: The study derives a generic formula for the difference in arrival times ( $\Delta t$ ) between two light rays emanating simultaneously from a source behind a spinning lens, passing on opposite sides (co-rotating and counter-rotating), and reaching an observer. This difference is attributed to frame dragging.
Expansion: The time delay is expanded in powers of the impact parameter ( $b$ ) and the mass-to-distance ratio ( $M/r$ ). The authors calculate the delay components up to the third order of smallness ( $\Delta t_1, \Delta t_2, \Delta t_3$ ).
Numerical Estimation: The derived analytic expressions are applied to two illustrative binary systems: a PSR-Cygnus X-1 system (stellar-mass black hole) and a hypothetical PSR-SgrA* system (supermassive black hole). The calculations assume specific parameters for mass ( $M$ ), spin ( $a$ ), charge ( $Q$ ), and distances ( $d_{OL}, d_{LS}$ ), ensuring the validity of the thin-lens approximation (small angles and impact parameters far exceeding the photon sphere radius).

Key Contributions and Results

Analytic Formulation: The paper provides explicit analytic expressions for the TOA difference components ( $\Delta t_1, \Delta t_2, \Delta t_3$ ) for the spinning AGBH, KNBH, and KBH spacetimes under the thin-lens approximation.
Order-by-Order Comparison:
- First Order ( $\Delta t_1$ ): The leading order term is found to be identical for all three black hole types and is independent of the charge-to-mass ratio ( $Q/M$ ). It depends primarily on the spin ( $a$ ) and mass ( $M$ ).
- Second Order ( $\Delta t_2$ ): The second-order terms for AGBH and KNBH are also found to be identical ( $\Delta t^{AGBH}_2 = \Delta t^{KNBH}_2$ ).
- Third Order ( $\Delta t_3$ ): The differences between the regular AGBH and the singular KNBH/KBH appear only at the third order of smallness. These terms contain the deviation parameters dependent on the gyromagnetic ratio ( $Q/M$ ).
Numerical Estimates:
- For the PSR-Cygnus X-1 system, the zeroth/first-order delay ( $\Delta t_1$ ) is estimated at approximately $0.028 \, \mu\text{sec}$ .
- For the PSR-SgrA* system, $\Delta t_1$ is estimated at approximately $4.2 \, \mu\text{sec}$ .
- The higher-order terms ( $\Delta t_2$ and $\Delta t_3$ ), which contain the distinguishing $Q/M$ factors, are calculated to be on the order of $10^{-6}$ to $10^{-15} \, \mu\text{sec}$ .

Significance and Claims
The paper concludes that while the AGBH and singular black holes exhibit significant qualitative differences in their spacetime structure (regularity vs. singularity), these differences are not observationally distinguishable via the TOA effect under the thin-lens approximation.

Indistinguishability: The numerical estimates indicate that the differences in TOA predictions between regular and singular black holes only manifest at the third order or higher. These higher-order corrections are so minute (requiring pico-second or sub-pico-second accuracy) that they are currently unmeasurable and likely unmeasurable even with future technology.
Diagnostic Limitation: Consequently, the TOA difference, while a valid probe of frame dragging, is not a sufficient diagnostic to distinguish between regular and singular black holes in the weak-field, thin-lens regime.
Future Work: The authors suggest that distinguishing these objects might require moving beyond the weak-field approximation to the strong-field limit (grazing the photon spheres), though this introduces mathematical complexities such as logarithmic divergences. They also note that studying TOA in "effective" spacetimes (as opposed to exact solutions of Einstein's equations) remains an open area for future investigation.

The study emphasizes that the TOA effect is fundamentally different from the Shapiro time delay and clarifies previous interpretations of the effect in the literature, but ultimately finds that current observational capabilities cannot resolve the subtle differences between regular and singular spinning black holes using this specific metric.