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abstract: |
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    <span>**Abstract**</span>

    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.
author:
- 'Yan Peng$^{1}$[^1]'
title: '<span>**Analytical studies on the hoop conjecture in charged curved spacetimes**</span>'
---

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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 $C\leqslant 4\pi \mathcal{M}$ @hc1 [@hc2]. 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 @hc3-@ahc9 and references therein.

Intriguingly, a few counterexamples against hoop conjecture were also presented @hc19 [@hc20]. In particular, Ref. @hc19 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 spcetime, the author claimed that Thorne hoop conjecture can be violated in horizonless charged fluid sphere spacetimes @hc19.

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 @hc21. 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 @hc19, 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 @hc21. 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.

Validity of the hoop conjecture for charged fluid spheres
=========================================================

It was widely believed that Thorne hoop conjecture is a fundamental property of classical general relativity. A lot of works indeed support the hoop conjecture @vknight@example.com. However, counterexamples against hoop conjecture were also presented in @hc19 [@hc20]. In particular, Ref. @hc19 constructed a gravity model of horizonless static charged fluid spheres. And the charged fluid sphere reads @metric1 [@metric2; @metric3; @metric4] $$\begin{aligned}
\label{AdSBH}
ds^{2}&=&e^{\nu}dt^{2}-e^{\lambda}dr^{2}-r^{2}(d\theta^2+sin^{2}\theta d\phi^{2}).\end{aligned}$$ The interior metric solution with uniform charge density was obtained in @hc19 as $$\begin{aligned}
\label{AdSBH}
e^{-\lambda}=[1-(1-f_{0})\upsilon^2]^2,\end{aligned}$$ $$\begin{aligned}
\label{AdSBH}
e^{\nu}=\frac{1}{4f_{0}}[(1-\varepsilon-f_{0})\upsilon^2+(1-3\varepsilon+f_{0}+\frac{2Q^2}{r_{0}^2})]^2e^{\lambda/2}.\end{aligned}$$ We define $\upsilon=\frac{r}{r_{0}}\leqslant 1$, $\varepsilon=\frac{M}{r_{0}}$ and $f_{0}=\sqrt{1-\frac{2M}{r_{0}}+\frac{Q^2}{r_{0}^2}}$, where $M$ is the total mass of the spacetime, Q is the sphere charge and $r_{0}$ is the sphere radius.

In the exterior region $r\geqslant r_{0}$, the background is the Reissner-Nordsr$\ddot{o}$m solution given by @RN1 [@RN2; @RN3; @RN4; @RN5; @RN6] $$\begin{aligned}
\label{AdSBH}
e^{\nu}=e^{-\lambda}=1-\frac{2M}{r}+\frac{Q^2}{r^2}.\end{aligned}$$ At the radius $r_{0}$, interior metric (2-3) and exterior metric (4) coincide with each other.

Thorne hoop conjecture states that horizons appear when and only when a mass $\mathcal{M}$ gets compacted into a region whose circumference C in every direction is $C\leqslant 4\pi \mathcal{M}$ @hc1 [@hc2]. The author in @hc19 claimed that Thorne hoop conjecture was violated by a relation $$\begin{aligned}
\label{AdSBH}
\frac{C(r_{0})}{4\pi M}\backsimeq 0.769<1\end{aligned}$$ with $\frac{M}{r_{0}}=0.65$ and $\frac{Q^2}{r_{0}^2}=0.39$. However, in the charged background, as stated in @hc21, 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. 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 $$\begin{aligned}
\label{AdSBH}
\frac{C(r_{0})}{4\pi M(r_{0})}\backsimeq 1.099>1,\end{aligned}$$ which is in accordance with the hoop conjecture in charged spacetime @hc21.

In the following, we provide an analytical proof on Thorne hoop conjecture for generic parameters. According to the weak energy condition (WEC), the energy density is nonnegative @metric1 [@metric2]. In this work, we can take a more general condition that the energy density is real (negative or nonnegative). The energy density of interior region is @hc19 $$\begin{aligned}
\label{AdSBH}
\rho_{m}=\frac{1-f_{0}}{8\pi r_{0}^2}\{6-\upsilon^2[5(1-f_{0})+\frac{Q^2}{r_{0}^2(1-f_{0})}]\}.\end{aligned}$$ It can be transformed into $$\begin{aligned}
\label{AdSBH}
\rho_{m}=-\frac{5\upsilon^2}{8\pi r_{0}^2}(1-f_{0})^2+\frac{3}{4\pi r_{0}^2}(1-f_{0})+\frac{Q^2\upsilon^2}{8\pi r_{0}^4}.\end{aligned}$$ We put $1-f_{0}$ in the general form $$\begin{aligned}
\label{AdSBH}
1-f_{0}=a+bi,\end{aligned}$$ where $a,b$ are real numbers.

Putting (9) into (8), we arrive at the relation $$\begin{aligned}
\label{AdSBH}
\rho_{m}=(-\frac{5\upsilon^2a^2}{8\pi r_{0}^2}-\frac{5\upsilon^2b^2}{8\pi r_{0}^2}+\frac{3a}{4\pi r_{0}^2}+\frac{Q^2\upsilon^2}{8\pi r_{0}^4})
+(-\frac{5\upsilon^2a}{4\pi r_{0}^2}+\frac{3}{4\pi r_{0}^2})bi.\end{aligned}$$

At the center $\upsilon=\frac{r}{r_{0}}=0$, (10) yields $$\begin{aligned}
\label{AdSBH}
\rho_{m}(0)=\frac{3a}{4\pi r_{0}^2}
+\frac{3b}{4\pi r_{0}^2}i.\end{aligned}$$

Since $\rho_{m}(0)$ is real, there is $$\begin{aligned}
\label{AdSBH}
b=0.\end{aligned}$$ That is to say $f_{0}=\sqrt{1-\frac{2M}{r_{0}}+\frac{Q^2}{r_{0}^2}}$ is real according to (9), which yields $$\begin{aligned}
\label{AdSBH}
\frac{2M}{r_{0}}-\frac{Q^2}{r_{0}^2}\leqslant 1.\end{aligned}$$ In the case of $\frac{2M}{r_{0}}-\frac{Q^2}{r_{0}^2}= 1$, there is $1-\frac{2M}{r_{0}}+\frac{Q^2}{r_{0}^2}=0$ and $r_{0}$ is horizon according to (4). Since we study horizonless sphere, the spacetime should satisfies $$\begin{aligned}
\label{AdSBH}
\frac{2M}{r_{0}}-\frac{Q^2}{r_{0}^2}< 1.\end{aligned}$$

There is electric energy outside the charged sphere. In order to calculate $M(r_{0})$, we should subtract the exterior electric energy $E(r>r_{0})$ from the total energy M. Outside the sphere, one has the Maxwell field energy density $\rho(r)=\frac{Q^2}{8\pi r^4}$. The mass $E(r>r_{0})$ of the Maxwell field above the radius $r_{0}$ is given by $$\begin{aligned}
\label{AdSBH}
E(r>r_{0})=\int_{r_{0}}^{+\infty}4\pi r'^{2}\rho(r')dr'=\frac{Q^2}{2r_{0}}.\end{aligned}$$ So the mass $M(r_{0})$ within the radius $r_{0}$ is @hc21 [@mr1; @mr2] $$\begin{aligned}
\label{AdSBH}
M(r_{0})=M-E(r>r_{0})=M-\frac{Q^2}{2r_{0}}.\end{aligned}$$ With (14) and (16), we arrive at the relation $$\begin{aligned}
\label{AdSBH}
\frac{C(r_{0})}{4\pi M(r_{0})}=\frac{2\pi r_{0}}{4\pi (M-\frac{Q^2}{2r_{0}})}=\frac{1}{\frac{2M}{r_{0}}-\frac{Q^2}{r_{0}^2}}>1.\end{aligned}$$ It yields the inequality for the horizonless spacetime as $$\begin{aligned}
\label{AdSBH}
C(r_{0})> 4\pi M(r_{0})\end{aligned}$$ in accordance with Thorne hoop conjecture, which states that horizons appear when and only when the mass $M(r_{0})$ and circumference $C(r_{0})$ satisfy the relation $C(r_{0})\leqslant 4\pi M(r_{0})$ @hc21. We point out that relation (18) holds for generic parameters. Here we analytically prove Thorne hoop conjecture in spatial regular charged fluid sphere spacetimes.

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.

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.

<span>99</span>

K.S. Thorne, in Magic Without Magic: John Archibald Wheeler, ed. by J. Klauder (Freeman, San Francisco, 1972).

C.W.Misner,K.S.Thorne,J.A.Wheeler,Gravitation(Freeman,San Francisco, 1973).

I.H. Redmount, Phys. Rev. D 27, 699 (1983).

A.M. Abrahams, K.R. Heiderich, S.L. Shapiro, S.A. Teukolsky, Phys. Rev. D 46, 2452 (1992).

S. Hod, Phys. Lett. B 751, 241 (2015).

P. Bizon, E. Malec, and N. ó Murchadha, Trapped surfaces in spherical stars, Phys. Rev. Lett. 61, 1147 (1988).

P. Bizon, E. Malec, and N. ó Murchadha, Class. Quantum Grav. 6, 961 (1989).

D. Eardley, Gravitational collapse of vacuum gravitational field configurations, J. Math. Phys. 36,3004(1995).

J. Guven and N. ó Murchadha,Sufficient conditions for apparent horizons in spherically symmetric initial data, Phys. Rev. D 56,7658(1997).

J. Guven and N. ó Murchadha,Necessary conditions for apparent horizons and singularities in spherically symmetric initial data, Phys. Rev. D 56, 7666(1997).

E. Malec, Event horizons and apparent horizons in spherically symmetric geometries, Phys. Rev. D 49, 6475 (1994).

E. Malec and ó Murchadha,The Jang equation, apparent horizons, and the Penrose inequality, Class. Quantum Grav. 21,5777(2004).

T. Zannias, Phys. Rev. D 45, 2998 (1992).

T. Zannias, Phys. Rev. D 47, 1448 (1993).

E. Malec,Isoperimetric inequalities in the physics of black holes, Acta Phys. Pol. B 22, 829 (1991).

M. Khuri,The Hoop Conjecture in Spherically Symmetric Spacetimes, Phys. Rev. D 80, 124025 (2009).

H. Bray and M. Khuri, Asian J. Math. 15, 557 (2011).

R. Schoen and S.-T. Yau, Commun. Math. Phys. 90, 575(1983).

Takeshi Chiba, Takashi Nakamura, Ken-ichi Nakao, Misao Sasaki, Hoop conjecture for apparent horizon formation, Class. Quant. Grav. 11(1994)431-441.

Takeshi Chiba, Apparent horizon formation and hoop concept in nonaxisymmetric space, Phys. Rev. D 60(1999)044003.

Ken-ichi Nakao, Kouji Nakamura, Takashi Mishima, Hoop conjecture and cosmic censorship in the brane world, Phys. Lett. B 564(2003)143-148.

G.W. Gibbons,Birkhoff's invariant and Thorne's Hoop Conjecture, arXiv:0903.1580\[gr-qc\].

M. Cvetic, G.W. Gibbons, C.N. Pope,More about Birkhoff's Invariant and Thorne's Hoop Conjecture for Horizons, Class. Quant. Grav. 28(2011)195001.

John D. Barrow, G. W. Gibbons, Maximum Tension: with and without a cosmological constant, Mon. Not. Roy. Astron. Soc. 446(2014)3874-3877.

John D. Barrow, G.W. Gibbons, A maximum magnetic moment to angular momentum conjecture, Phys. Rev. D 959313532382.

Edward Malec, Naqing Xie, Brown-York mass and the hoop conjecture in nonspherical massive systems, Phys. Rev. D 91(2015)no.8,081501.

Fabio Anz$\grave{a}$, Goffredo Chirco,Fate of the Hoop Conjecture in Quantum Gravity, Phys. Rev. Lett. 119(2017)no.23,231301.

Shahar Hod, Bekenstein's generalized second law of thermodynamics: The role of the hoop conjecture, Phys. Lett. B 7519313532382.

Shahar Hod, The gravitational two-body system: The role of the Thorne hoop conjecture, Eur. Phys. J. Plus 134(2019)no.3,106.

J.P. de Le¨®n, Gen. Relativ. Grav. 19, 289 (1987).

W.B. Bonnor, Phys. Lett. A 99, 424 (1983).

Shahar Hod, On the status of the hoop conjecture in charged curved spacetimes, Eur. Phys. J. C (2018)78:1013.

D. Núñez, H. Quevedo, and D. Sudarsky,Black Holes Have No Short Hair,Phys. Rev. Lett. 76, 571 (1996).

S. Hod, Hairy Black Holes and Null Circular Geodesics,Phys. Rev. D 84, 124030 (2011)

Yan Peng, Scalar condensation behaviors around regular Neumann reflecting stars, Nucl. Phys. B 9349313532382.

S. Chandrasekhar, The Mathematical Theory of Black Holes, (Oxford University Press, New York, 1983).

Shahar Hod, The Reissner-Nordstr?m black hole with the fastest relaxation rate, The European Physical Journal C 78,935(2018).

Pallab Basu, Chethan Krishnan, P. N. Bala Subramanian,Hairy Black Holes in a Box, JHEP 11(2016)041.

Nicolas Sanchis-Gual,Juan Carlos Degollado,Pedro J.Montero,Jos A. Font,Carlos Herdeiro,Explosion and final state of an unstable Reissner-Nordström black hole, Phys. Rev. Lett. 1169313532382.

Sam R Dolan,Supakchai Ponglertsakul,Elizabeth Winstanley, Stability of black holes in Einstein-charged scalar field theory in a cavity, Phys. Rev. D 929313532382.

Yan Peng, On instabilities of scalar hairy regular compact reflecting stars,JHEP 9313532382.

Yan Peng, Scalar field configurations supported by charged compact reflecting stars in a curved spacetime, Phys. Lett. B 7809313532382.

Yan Peng, Large regular reflecting stars have no scalar field hair, Eur. Phys. J. C 79(2019)no.4,309.

Yan Peng, Hair distributions in noncommutative Einstein-Born-Infeld black holes, Nucl. Phys. B 9313532382.

[^1]: vknight@example.com

{'source': 'arXiv_out/1906'}