Testing generalized spacetimes for black holes using the Hod function representation of the hoop conjecture

Problem Statement
The paper addresses the validity of Thorne's hoop conjecture within the context of generalized, regular, spherically symmetric static black hole solutions coupled to nonlinear electrodynamics. Specifically, it investigates a two-parameter family of solutions constructed by Fan and Wang. A central issue identified is that applying the hoop conjecture naively using the asymptotic ADM mass ($M_\infty$) can lead to apparent violations of the conjecture, even in spacetimes that genuinely possess event horizons. The authors aim to determine whether these generalized solutions, including a proposed "new class" of solutions, truly represent black holes by utilizing a corrected mass definition.

Methodology
The authors employ the "Hod function" approach, originally suggested by Hod, which redefines the mass term in the hoop conjecture. Instead of using the total asymptotic mass ($M_\infty$), the conjecture is restated using the gravitational mass $M(r \le R)$ contained within the hoop radius $R$.

1. Metric Analysis: The study analyzes the generic metric function $f(r)$ for the Fan and Wang solutions, characterized by arbitrary exponents $\mu$ and $\nu$, a Schwarzschild mass parameter $M_{Sch}$, and an asymptotic electromagnetic mass $M_e$.
2. Critical Value Determination: The authors determine critical values ($x_c, s_c$) where the metric function $f(r)$ transitions between having horizons and not having horizons. These are found by solving simultaneous equations $f=0$ and $\partial_x f = 0$.
3. Hod Function Construction:
   • For the case $M_{Sch} = 0$ (purely electromagnetic regular black holes), the authors integrate the energy density $T^0_0$ to derive the enclosed mass $M(r \le R)$ and construct the Hod function $H_f$.
   • For the case $M_{Sch} \neq 0$, they extend this derivation to include the Schwarzschild mass term, resulting in a generalized Hod function $H_f(w, \mu, \nu; x, s)$.
   • A similar procedure is applied to the "new class" of solutions proposed by Fan and Wang, which features a different metric structure $g(r)$.
4. Verification: The validity of the conjecture is tested by checking if the Hod function satisfies the inequality $0 < H \le 1$ for configurations with horizons ($s \le s_c$) and $H > 1$ for those without ($s > s_c$).
5. Mass Equivalence: The paper analytically demonstrates that the Hod mass $M(r \le R)$ is mathematically identical to the Misner-Sharp geometrical quasilocal mass $m(r \le R)$ for these spacetimes.

Key Results

• Validation of Generalized Solutions: For the standard Fan and Wang solutions (generalizing the Bardeen and Hayward metrics), the extended Hod function successfully restores the validity of the hoop conjecture. The inequality $0 < H_f \le 1$ holds for all cases where horizons exist ($s \le s_c$), confirming that these spacetimes genuinely represent black holes.
• Rejection of the "New Class": The proposed "new class" of solutions (where the metric function differs in the denominator term) fails the test.
  • The critical parameter $s_c$ is found to be negative for all tested values of $\mu$ and $w$, which contradicts the physical requirement that $s > 0$ (derived from charge and length parameters).
  • The curvature invariants (Ricci scalar and Kretschmann scalar) are finite but suggest a wormhole-like topology (negative Ricci scalar) rather than a black hole.
  • The Hod function for this new class violates the hoop conjecture inequalities (yielding negative values or values $>1$ in regimes where horizons are expected), indicating these solutions do not qualify as black holes under the conjecture.
• Mass Equivalence: The study confirms that the Hod mass is exactly the matter counterpart of the Misner-Sharp mass. Consequently, any conclusion drawn from the Hod function is strictly a conclusion of General Relativity.

Significance and Claims
The paper claims that the Hod function serves as a robust, non-trivial tool for testing whether generalized spacetime solutions represent black holes, particularly when asymptotic mass definitions lead to misleading results. By applying this method, the authors support the validity of the Fan and Wang solutions (generalizing Bardeen and Hayward) as true black holes while arguing against the proposed "new class" of solutions due to topological inconsistencies and violations of the hoop conjecture.

The authors emphasize that their analysis is methodological, providing a framework to test complex solutions in nonlinear electrodynamics and potentially in other modified gravity theories. They explicitly state that the adjudication of black hole status based on the Hod function carries the authority of General Relativity due to its equivalence with the Misner-Sharp mass. The paper does not propose new experimental applications but rather offers a theoretical consistency check for existing mathematical solutions.