# On the status of the hoop conjecture in charged curved spacetimes

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## Abstract

The status and regime of validity of the famous Thorne hoop conjecture in spatially regular *charged* curved spacetimes are clarified.

## 1 Introduction

*M*will form an engulfing horizon if its circumference radius \(R=C/2\pi \) is equal to (or less than) the corresponding Schwarzschild radius 2

*M*[3]. That is, the hoop conjecture states that [1]

*charged*curved spacetimes [7, 8].

The main goal of the present compact paper is to clarify the status of the Thorne hoop conjecture in spatially regular charged spacetimes. In particular, below we shall explicitly demonstrate that the hoop conjecture is valid in charged curved spacetimes provided the mass parameter on the r.h.s of the hoop relation (1) is appropriately interpreted as the gravitational mass *M*(*R*) contained within the engulfing hoop of radius *R* and *not* as the total (asymptotically measured) mass \(M_{\infty }\) of the entire spacetime.

## 2 Validity of the hoop conjecture in spatially regular charged spacetimes

*total*mass \(M_{\infty }\) of the system (as measured by asymptotic observers), the hoop conjecture can be violated. As an illustrative example, Ref. [7] constructed a uniformly charged horizonless ball which is characterized by the dimensionless relations

*horizonless*charged matter configurations is characterized by the dimensionless ratio

However, we believe that in the Thorne hoop conjecture (1), which relates the mass parameter of the system to its circumference radius \(R=C/2\pi \), it is physically more appropriate to interpreted *M* as the gravitational mass contained *within* the engulfing hoop and not as the total mass of the entire curved spacetime.

*R*and electric charge

*Q*is \(T^0_0(r>R)=Q^2/8\pi r^4\) [9], the electromagnetic energy \(E_{\text {elec}}(r>R)=\int ^{\infty }_{R}T^0_0 4\pi r^2dr\)

*outside*the charged ball is given by the simple expression

*R*, electric charge

*Q*, and

*total*mass (energy) \(M_{\infty }\) as measured by asymptotic observers, the gravitational mass contained

*within*\((r\le R)\) the ball is given by

*not*violate the Thorne hoop conjecture (1).

## 3 Summary

In this compact paper we have explored the (in)validity of the Thorne hoop conjecture [1] in spatially regular *charged* curved spacetimes. Our analysis is motivated by the intriguing claims made in the physics literature (see e.g. [7, 8]) according to which this famous conjecture, which is widely believed to reflect a fundamental aspect of classical general relativity, can be violated by horizonless charged matter configurations.

The present analysis clearly demonstrates the fact that, as opposed to the claims made in [7, 8], the Thorne hoop conjecture is valid in charged spacetimes provided that, for a given radius *R* of the engulfing hoop, the mass parameter in the hoop relation (1) is appropriately interpreted as the gravitational mass *M*(*R*) contained *within* the hoop (sphere) of radius *R* and not as the total mass \(M_{\infty }\) of the entire spacetime.

## Notes

### Acknowledgements

This research is supported by the Carmel Science Foundation. I would like to thank Yael Oren, Arbel M. Ongo, Ayelet B. Lata, and Alona B. Tea for stimulating discussions.

## References

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