Observational Properties of Near-Maximally Spinning Supermassive Black Holes

Imagine the universe's most extreme objects: Supermassive Black Holes. These aren't just empty pits; they are cosmic engines spinning so fast they warp space and time itself. Scientists have long debated a specific question: How fast can these black holes spin?

There are two main theories about the "speed limit" for these cosmic giants:

The "Thorne Limit" (0.998): Named after physicist Kip Thorne, this suggests a black hole can spin almost at the absolute maximum speed allowed by physics, like a figure skater pulling in their arms to spin as fast as humanly possible.
The "Gammie Limit" (0.9375): Named after another physicist, this suggests that the friction and radiation from the swirling gas around the black hole act like a brake, preventing it from ever reaching that absolute top speed.

The Big Question: Does it actually matter which limit is true? If we look at a black hole, can we tell the difference between a "0.9375" spin and a "0.998" spin?

The Experiment: A Cosmic Movie Studio

To answer this, the authors (Tegan Thomas and her team) didn't just look at real black holes; they built virtual ones in a supercomputer.

Think of their computer code as a high-end movie studio. They created two identical sets of conditions:

Movie A: A black hole spinning at the "Gammie Limit" (0.9375).
Movie B: A black hole spinning at the "Thorne Limit" (0.998).

They simulated the "accretion disk"—the swirling soup of hot gas and magnetic fields that feeds the black hole—using the laws of General Relativity (Einstein's rules for gravity). Then, they used a "virtual camera" to take pictures and make movies of what these spinning monsters would look like to the Event Horizon Telescope (EHT), the real-world telescope array that famously took the first picture of a black hole.

The Surprise: They Look Identical

You might expect that a black hole spinning almost at the speed of light (0.998) would look wildly different from one spinning slightly slower (0.9375). You might expect the faster one to look brighter, flatter, or more chaotic.

But the results were shocking.

The authors found that the two movies looked almost exactly the same.

The Gas Flow: The way the gas swirled, the strength of the magnetic fields, and the power of the jets shooting out of the poles were nearly indistinguishable.
The Pictures: When they took "photos" of the black hole's shadow (the dark circle surrounded by a bright ring of light), the shapes, sizes, and brightness patterns were so similar that current telescopes couldn't tell them apart.

The Analogy: Imagine two race cars. One is going 199 mph, and the other is going 198 mph. If you take a blurry photo from a mile away, they look like the same car. Even if you zoom in a little, the difference is so tiny that your eyes (or current telescopes) can't spot it.

Why Does This Matter?

This is actually good news for scientists.

Simpler Models: Since the "slower" spin (0.9375) produces the same results as the "faster" spin (0.998), scientists can stop worrying about the extreme edge cases. They can use the simpler, slightly slower model to represent almost any fast-spinning black hole. It saves them from doing unnecessary complex math.
Current Telescopes are Blurry: The Event Horizon Telescope (EHT) is amazing, but it's like looking at a distant mountain through a slightly foggy window. It can see the mountain, but it can't see the tiny pebbles on the peak. The difference between these two spin speeds is like those pebbles.

The Only Way to Tell Them Apart: The "Photon Ring"

If the main picture looks the same, is there any way to tell the difference?

Yes, but it requires a much sharper lens. The authors point to a feature called the Photon Ring.

What is it? Imagine light particles (photons) getting trapped in a loop around the black hole, circling it like a race car on a track before escaping. This creates a very thin, sharp ring inside the main bright ring we usually see.
The Catch: This ring is incredibly thin and hard to see. Current telescopes can't resolve it clearly.
The Future: The authors suggest that a future mission called BHEX (Black Hole Explorer), which would put telescopes in space to create a giant virtual telescope, might be able to see this ring.

The Final Analogy:
Think of the black hole's shadow as a coin.

The EHT (current telescope) can see the coin is round and shiny. It can't tell if the coin is made of gold or silver, or if the engraving on the edge is slightly different.
The Photon Ring is the tiny, intricate engraving on the coin's edge.
The BHEX (future space telescope) is a microscope that can finally read that engraving. Only then will we know if the black hole is spinning at 0.9375 or 0.998.

The Bottom Line

For now, if you are a scientist trying to understand a black hole, you can assume it's spinning at the "Gammie Limit" (0.9375). It's a safe bet that covers the "Thorne Limit" (0.998) too. The universe is hiding the subtle differences between these two speeds, and we will need to build better "eyes" (like the Black Hole Explorer) to finally catch a glimpse of the truth.

1. Problem Statement

Supermassive Black Holes (SMBHs) are theoretically described by the Kerr metric, characterized by mass ( $M_\bullet$ ) and dimensionless spin ( $a_\bullet$ ). While the theoretical maximum spin is $a_\bullet = 1$ , astrophysical processes likely impose lower limits. Two prominent theoretical limits exist in the literature:

$a_\bullet \approx 0.998$ : Proposed by Thorne (1974), based on the negative torque from radiation in a thin disk.
$a_\bullet \approx 0.9375$ : Proposed by Gammie et al. (2004) and others, suggesting relativistic jets and disk dynamics may cap spin at lower values.

Current observational methods (e.g., X-ray reflection spectroscopy) struggle to distinguish between these near-maximal spin values, often yielding distributions peaking at maximum spin. With the advent of the Event Horizon Telescope (EHT) and future missions like the Black Hole Explorer (BHEX), there is a need to determine if current simulations using $a_\bullet \approx 0.9375$ are representative of models with $a_\bullet \approx 0.998$ , and to identify which observables can effectively discriminate between them.

2. Methodology

The authors employed a multi-stage numerical pipeline to simulate and image accretion flows around SMBHs with spins of $a_\bullet = 0.9375$ and $a_\bullet = 0.998$ .

GRMHD Simulations:
- Code: Used KHARMA (General Relativistic Magnetohydrodynamics).
- Setup: Initialized with magnetized tori (Fishbone & Moncrief) with an inner radius of $20 r_g $and pressure maximum at$ 41 r_g$.
- Physics: Assumed a Magnetically Arrested Disk (MAD) state with plasma $\beta = 100$ and an adiabatic index $\hat{\gamma} = 13/9$ .
- Targets: Simulated flows for Sgr A* ($4.14 \times 10^6 M_\odot $) and **M87*** ($ 6.2 \times 10^9 M_\odot$).
- Resolution: Spherical static mesh (modified Kerr-Schild) with $288 \times 128 \times 128 $grid points. Simulations ran to$ t \approx 30,000 t_g $, with analysis focused on$ t \ge 15,000 t_g$.
Radiative Transfer (GRRT):
- Code: Used IPOLE to generate full-Stokes (I, Q, U, V) images.
- Parameters: Calculated at 230 GHz using the "fast-light" approximation.
- Thermal Models: Defined electron-ion temperature ratios ( $T_p/T_e$ ) based on gas vs. magnetic pressure dominance, using parameters $R_{high}$ and $R_{low}$ calibrated to reproduce EHT observables.
- Inclinations: Tested $150^\circ $and$ 90^\circ $for Sgr A*, and$ 163^\circ $and$ 90^\circ$ for M87*.

3. Key Contributions

Direct Comparison of Near-Maximal Spins: This is one of the first studies to directly compare GRMHD fluid properties and full-Stokes images between $a_\bullet = 0.9375$ and $a_\bullet = 0.998$ to test the sensitivity of current and near-future observables.
Validation of Simulation Representativeness: The study provides evidence that simulations using $a_\bullet \approx 0.9375$ are sufficient proxies for models with $a_\bullet \gtrsim 0.9375$ in most practical observational contexts.
Identification of the Photon Ring as the Primary Discriminator: The paper isolates the photon ring (specifically its shape and size) as the only viable observable to distinguish between these spin values, highlighting the necessity of space-based interferometry (BHEX).

4. Key Results

A. GRMHD Fluid Properties

Accretion Rate: The average accretion rate for $a_\bullet = 0.998$ was slightly lower ( $\sim 11.94$ ) than for $a_\bullet = 0.9375$ ( $\sim 12.67$ ), but the variability ( $\sigma/\mu$ ) was also lower for the higher spin case.
Magnetization ( $\phi_{BH}$ ): Both models achieved MAD states with $\phi_{BH} \approx 52-54$ , consistent with theoretical predictions. Variability was slightly lower for the $a_\bullet = 0.998$ case.
Jet Power Efficiency: The jet power efficiency ( $\eta$ $η$ ) for both models agreed well with the Blandford-Znajek (BZ) mechanism predictions, including higher-order corrections for near-maximal spins.
- $a_\bullet = 0.9375$ : $\eta \approx 1.35$ (Predicted $\approx 1.49$ ).
- $a_\bullet = 0.998$ : $\eta \approx 1.84$ (Predicted $\approx 2.0$ ).
Spin Evolution: Both models exhibited negative spin-up parameters ( $s$ ), indicating spin-down due to jet extraction, though the $0.998$ case spun down slightly faster.

B. GRRT Imaging Properties

Total Intensity: Time-averaged images for both spins were remarkably similar. Ring asymmetry metrics showed no statistically significant difference between the two spin values, even at edge-on inclinations.
Polarization: Linear polarization fraction ( $m_{net}$ $m_{n e t}$ ), circular polarization ( $v_{net}$ $v_{n e t}$ ), and the phase/magnitude of the second mode ( $\beta_2$ $β_{2}$ ) showed no significant difference between the two spin models.
- For Sgr A*, both models fit observational constraints.
- For M87*, high-spin models generally failed to fit constraints (consistent with previous findings favoring lower/retrograde spins for M87*), but the difference between 0.9375 and 0.998 was negligible.
Variability: The light curve variability for Sgr A* remained high for both models ( $\sigma/\mu \approx 0.28$ ), failing to resolve the discrepancy between EHT models and observations ( $\sigma/\mu \sim 0.1$ ).

C. The Photon Ring

The Only Discriminator: The photon ring (sub-rings $n=1, 2$ ) showed distinct differences in radius and shape between the two spin values.
Edge-On Sensitivity: At $90^\circ $inclination, the photon ring radius and shape (specifically the flattening on the Doppler-beamed side) differed between$ a_\bullet = 0.9375 $and$ 0.998$.
Resolution Requirements: The difference in the $n=2$ photon ring for an edge-on Sgr A* was approximately 1.88 $\mu$ as. This scale is below current EHT resolution but within the capabilities of the proposed Black Hole Explorer (BHEX) mission (resolution $\sim 5 \mu$ as, potentially better with space baselines).

5. Significance and Conclusions

Model Robustness: For most practical purposes, including interpreting current EHT data for Sgr A*, simulations with $a_\bullet = 0.9375$ can be used to represent systems with spins up to $a_\bullet = 0.998$ . The fluid dynamics and standard image morphologies do not change significantly in this near-maximal regime.
Future Observations: Distinguishing between near-maximal spin values requires resolving the photon ring. Current ground-based VLBI cannot achieve the necessary precision; space-based extensions like BHEX are required to measure the subtle shape and size differences in the photon ring.
Calibration: The study provides necessary calibration data to interpret future BHEX measurements, specifically regarding the angle-dependent bias of photon sub-rings relative to the theoretical "critical curve" ( $n=\infty$ ).

In summary, while the spin of an SMBH fundamentally dictates its energy extraction efficiency, the observational signatures in standard accretion flow images (intensity and polarization) are nearly identical for $a_\bullet = 0.9375$ and $0.998$. The definitive test for these extreme spins lies in the high-resolution imaging of the photon ring.