⚛️ general relativity

Comments on the non-existence of unified hoop conjecture

While acknowledging Hod's conclusion that a unified Thorne's hoop conjecture fails for Kerr-Newman black holes, this paper identifies a notable exception where the conjecture holds in a unified manner for both static and spinning tidal charged braneworld black holes when using the same quasi-local mass.

Original authors: A. Bhattacharya, R. N. Izmailov, R. Kh. Karimov

Published 2026-07-02

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CC BY 4.0

Original authors: A. Bhattacharya, R. N. Izmailov, R. Kh. Karimov

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

Imagine you are trying to figure out when a giant, invisible net (a black hole) forms around a collapsing star. Physicists have a famous rule for this called the "Hoop Conjecture." Think of it like a hula hoop. The rule says: If you can wrap a hula hoop of a specific size around an object, and that object is heavy enough, it will collapse into a black hole.

For a long time, scientists thought this rule worked perfectly for simple, non-spinning objects. But a recent study by a researcher named Hod discovered a glitch: the rule works for static (still) objects, but it breaks when the object is spinning and has an electric charge (like a spinning, charged top). It's as if the hula hoop rule suddenly stops working the moment the object starts twirling.

The Paper's Discovery: A "Magic" Twist

The authors of this paper, Amrita Bhattacharya, Ramil N. Izmailov, and Ramis Kh. Karimov, say, "Wait a minute! There is a special case where the rule does work, even for spinning objects."

They found a solution in a specific type of universe described by string theory (a theory that tries to explain how the universe is built from tiny vibrating strings). In this universe, there are "braneworld" black holes.

Here is the trick they used:

The Standard World: In our normal universe, spinning black holes have an electric charge (like static electricity). When you plug this into the hoop rule, the math breaks.
The Braneworld Twist: In this special string theory universe, the black holes have something called a "tidal charge." It's not electric charge; it's a gravitational effect coming from extra dimensions (imagine gravity leaking in from a higher floor of a building).
The Magic Switch: The authors realized that if you take the math for the spinning, electrically charged black hole and simply swap the "electric charge" for this "tidal charge" (a mathematical move called a "Wick rotation"), the broken hoop rule suddenly fixes itself.

The Analogy: The Stretchy Rubber Band

Think of the "Hoop Conjecture" as a rubber band that needs to be tight enough to snap a balloon (forming a black hole).

In the normal spinning case: The balloon is spinning so fast and has so much electric charge that the rubber band stretches out too far and fails to snap. The rule says "No black hole," even though physics says there should be one.
In the braneworld case: The "tidal charge" acts like a special lubricant or a different material for the rubber band. Even though the balloon is spinning, this special rubber band stays tight enough to snap. The rule holds true!

What They Found in the Data

The authors did the math and created a table (Table 1) and some graphs (Figure 1) to prove this.

They tested different speeds of spin and different amounts of "charge."
The Result: For the braneworld black holes, the "Hoop Ratio" (a number that tells you if the rule works) is always greater than or equal to 1. This means the rule works perfectly every time.
The Contrast: For the normal spinning black holes (Kerr-Newman), that number drops below 1, meaning the rule fails.

The Catch (The "Price")

The paper points out a significant trade-off. To make the math work and save the Hoop Conjecture, you have to accept that the black hole isn't charged with electricity anymore. Instead, it's charged with this mysterious "tidal" effect from extra dimensions.

Summary
The paper argues that while the famous "Hoop Conjecture" fails for normal spinning, charged black holes, it is saved in the specific context of string theory's "braneworld" black holes. By swapping electric charge for a gravitational "tidal charge," the rule becomes unified again, working for both still and spinning black holes. However, this only works in this specific theoretical model, not in our everyday understanding of electrically charged black holes.

A Note on Energy
The authors also checked the "energy conditions" (rules about how energy behaves in the universe). They found that for these special braneworld black holes, some of these standard energy rules are broken (they take negative values). This is a known feature of these theoretical models, but it's part of the package deal for making the Hoop Conjecture work.

Technical Summary: Comments on the non-existence of unified hoop conjecture

Problem Statement
The paper addresses a recent conclusion by Hod [1] regarding the non-existence of a unified version of Thorne's hoop conjecture. Thorne's conjecture posits that a black hole horizon forms if and only if a mass $M$ can be surrounded by a hoop of critical circumference $C = 4\pi M$ in all azimuthal directions, expressed as $H \equiv 4\pi M / C \geq 1$ . Hod demonstrated that while this conjecture holds for static Reissner-Nordström (RN) black holes when using the quasi-local mass $M(r \leq R_+)$ , it fails for spinning, electrically charged Kerr-Newman (KN) black holes under the same mass definition. Specifically, for KN black holes, the ratio $H$ drops below unity, violating the conjecture. The authors seek to investigate whether a unified formulation exists for a specific class of black holes derived from string theory braneworld scenarios.

Methodology
The authors analyze the validity of the hoop conjecture for "tidal charged" braneworld black holes. These solutions arise from the projection of 5D bulk gravity onto a 3-brane, described by effective 4D Einstein equations with a modified source term involving the projected Weyl tensor.

Metric Construction: The study utilizes the static braneworld metric (Eq. 4) and its spinning generalization (Eq. 7). A key feature of these metrics is the presence of a tidal charge parameter $q^2$ (where $q^2 > 0$ ) which enters the metric with a negative sign relative to the standard electric charge term in the RN or KN metrics. This is formally achieved via the transcription $Q \to iq$ (a Wick rotation) applied to the standard charged black hole solutions.
Quasi-local Mass Calculation: Following Hod's methodology, the authors calculate the quasi-local mass $M(R_+)$ enclosed within the event horizon radius $R_+$ . For the spinning tidal charged case, this involves integrating the energy-momentum components derived from the metric.
Hoop Ratio Evaluation: The dimensionless ratio $H = 4\pi M(R_+)/C$ is computed, where $C$ is the equatorial circumference of the black hole. The authors evaluate this ratio for various values of the spin parameter $a$ and the tidal charge $q$ .
Energy Condition Analysis: The paper examines the energy-momentum tensor components ( $\rho, p_R, p_\theta, p_\phi$ ) to determine the status of the Null (NEC), Weak (WEC), and Strong (SEC) energy conditions for these spacetimes.

Key Contributions and Results

Unified Hoop Conjecture for Braneworld Black Holes: The primary finding is that the hoop conjecture holds in a unified manner for both static and spinning tidal charged braneworld black holes. Unlike the Kerr-Newman case where $H < 1$ , the authors demonstrate that for braneworld black holes, the ratio satisfies $H \geq 1$ for all values of spin $a$ and tidal charge $q$ .
Quantitative Verification: Table 1 and Figure 1 provide numerical and graphical evidence showing that $H(a, q)$ increases monotonically with both $a$ and $q$ , remaining strictly greater than or equal to 1. In contrast, the corresponding plots for Kerr-Newman black holes show $H \leq 1$ .
Role of Tidal Charge: The negative correction to the Schwarzschild potential (arising from the $-q^2/R^2$ term) is identified as the crucial factor ensuring the validity of the conjecture. This term effectively counteracts the destabilizing effects of spin and charge that cause the conjecture to fail in standard KN black holes.
Energy Condition Violation: The analysis reveals that the tidal charged braneworld black holes violate the NEC, WEC, and SEC. The energy density and pressure components result in negative values for the sums required by these conditions (e.g., $\rho + p_\theta < 0$ ).

Significance and Claims
The paper claims to identify a "curious exception" to the non-existence of a unified hoop conjecture. While acknowledging Hod's correct conclusion that no unified conjecture exists for standard Kerr-Newman black holes using quasi-local mass, the authors assert that a unified formulation does exist for tidal charged braneworld black holes.

The significance of this work lies in the observation that a formal Wick rotation ( $Q \to iq$ ) applied to the Kerr-Newman metric transforms it into a braneworld solution that satisfies the hoop conjecture. However, the authors modestly note the "price" of this unification: the physical interpretation shifts from familiar electric charge to a tidal charge arising from higher-dimensional gravitational effects. Furthermore, this unification comes at the cost of violating standard energy conditions, a feature inherent to these specific braneworld solutions. The paper does not propose new experimental applications but highlights a theoretical consistency in string-theory-inspired black hole models that is absent in standard general relativity with electric charge.

Technical Summary: Comments on the non-existence of unified hoop conjecture

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