# Analytical studies on the hoop conjecture in charged curved spacetimes

## Abstract

Recently, with numerical methods, Hod clarified the validity of Thorne hoop conjecture for spatially regular static charged fluid spheres, which were considered as counterexamples against the hoop conjecture. In this work, we provide an analytical proof on Thorne hoop conjecture in the spatially regular static charged fluid sphere spacetimes.

## 1 Introduction

One famous conjecture in general relativity is the Thorne hoop conjecture, which states that horizons appear when and only when a mass \(\mathcal {M}\) gets compacted into a region whose circumference C in every direction is \(C\leqslant 4\pi \mathcal {M}\) [1, 2]. This upper bound can be saturated in the case of Schwarzschild black hole with the horizon radius \(r_{0}=2\mathcal {M}\). If generically true, such conjecture would signify that black holes form if matter/energy is enclosed in a small enough region. At present, there are a lot of works addressing the hoop conjecture, see [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29] and references therein.

Intriguingly, a few counterexamples against hoop conjecture were also presented [30, 31]. In particular, Ref. [30] constructed the horizonless charged fluid sphere configurations with uniform charge densities. For \(\frac{M}{r_{0}}=0.65\) and \(\frac{Q^2}{r_{0}^2}=0.39\), the horizonless charged sphere satisfies a relation \(\frac{C(r_{0})}{4\pi M}\backsimeq 0.769<1\), where *M* is the total mass of the spacetime, Q is the sphere charge and \(r_{0}\) is the sphere radius. According to the relation \(\frac{C(r_{0})}{4\pi M}<1\) in the horizonless spacetime, the author claimed that Thorne hoop conjecture can be violated in horizonless charged fluid sphere spacetimes [30].

However, as stated by Hod, it is physically more appropriate to interpreted the mass term \(\mathcal {M}\) in Thorne hoop conjecture as the gravitational mass \(M(r_{0})\) contained within the radius \(r_{0}\) and not as the total mass *M* of the entire curved spacetime [32]. In fact, there is electric energy outside the charged sphere. For the same parameters \(\frac{M}{r_{0}}=0.65\) and \(\frac{Q^2}{r_{0}^2}=0.39\) as [30], Hod reexamined the validity of hoop conjecture for charged fluid spheres and numerically obtained the relation \(\frac{C(r_{0})}{4\pi M(r_{0})}\backsimeq 1.099>1\), which is in fact in accordance with the hoop conjecture in charged spacetime [32]. Along this line, it is still meaningful to analytically examine Thorne hoop conjecture for spatial regular charged fluid spheres with generic parameters.

The rest of the paper is organized as follows. We shall introduce the gravity model of spatial regular static charged fluid spheres. We provide an analytical proof on Thorne hoop conjecture for horizonless charged fluid spheres with generic parameters. Finally, we will briefly summarize our results.

## 2 Validity of the hoop conjecture for charged fluid spheres

*M*is the total mass of the spacetime, Q is the sphere charge and \(r_{0}\) is the sphere radius.

*M*of the entire curved spacetime. With the same parameters \(\frac{M}{r_{0}}=0.65\) and \(\frac{Q^2}{r_{0}^2}=0.39\), Hod reexamined the model and numerically obtained the relation for horizonless spheres as

*a*,

*b*are real numbers.

## 3 Conclusions

We analytically examined the validity of Thorne hoop conjecture in spatial regular charged curve spacetimes. We took the natural assumption that the matter energy density is real. On this real energy density assumption, with generic parameters, we found that horizonless charged fluid sphere should satisfy the relation (18) in accordance with Thorne hoop conjecture. In summary, we provided an analytical proof on Thorne hoop conjecture in the spatially regular static charged fluid sphere spacetimes.

## Notes

### Acknowledgements

This work was supported by the Shandong Provincial Natural Science Foundation of China under Grant No. ZR2018QA008. This work was also supported by a grant from Qufu Normal University of China under Grant No. xkjjc201906.

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