# Fluid black holes with electric field

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## Abstract

We investigate the gravitational field of static perfect-fluid in the presence of electric field. We adopt the equation of state \(p(r)=-\rho (r)/3\) for the fluid in order to consider the closed (\(S_3\)) or the open (\(H_3\)) background spatial topology. Depending on the scales of the mass, spatial-curvature and charge parameters (*K*, \(R_0\), *Q*), there are several types of solutions in \(S_3\) and \(H_3\) classes. Out of them, the most interesting solution is the Reisner–Norström type of black hole. Due to the electric field, there are two horizons in the geometry. There exists a curvature singularity inside the inner horizon as usual. In addition, there exists a naked singularity at the antipodal point in \(S_3\) outside the outer horizon due to the fluid. Both of the singularities can be accessed only by radial null rays.

## 1 Introduction

The spatial topology of the Universe is one of the unresolved problems in cosmology. From the recent cosmic microwave background radiation data, the density fraction of the curvature is estimated as \(\Omega _k = 0.000 \pm 0.005\) (\(95\%\), *Planck* TT + lowP + lensing + BAO) [1]. Because of the observational error, it is not possible to determine the spatial topology from the data at the current stage. Some other efforts have been made in the inflation models in the closed/open universe [2, 3, 4, 5, 6]. The investigation of primordial density perturbation shows that the peculiar predictions of those models are beyond the resolution of the current observational data. Therefore, one needs to consider other ways in order to catch an idea of the background spatial topology, for example, the investigation of the gravitating localized objects in different topologies.

The metric (2) is the only solution to the Einstein’s equation with the matter of Eq. (1). There is no additional mass term unlike in vacuum which admits the flat Minkowski space as the massless limit of the Schwarzschild spacetime. In order to achieve a nontrivial structure such as a black hole in \(S_3/H_3\), other type of matter than Eq. (1) needs to be introduced. Then, the \(S_3/H_3\) nature will be exposed only at some place of space while a nontrivial geometry is formed elsewhere.

For the nontrivial geometrical structure that admits the inherent \(S_3\)/\(H_3\) topology, the static fluid configuration with the equation of state \(p(r) = -\rho (r)/3\) was recently studied in Ref. [7]. It was found that there are a black-hole solution (\(S_3\)-I, \(S_3\)-II, \(H_3\)), a nonstatic cosmological solution (\(S_3\)-II, \(H_3\)), and a singular static solution (\(H_3\)). The nontrivial geometries of these three types of solutions are sourced by fluid. At some region of space, the signature of the \(S_3/H_3\) topology appears (near the equator for \(S_3\)-I, near the center for \(S_3\)-II, and at the asymptotic region for \(H_3\)). In this sense, we interpret the nontrivial geometrical configuration as a gravitating object formed in the \(S_3/H_3\) background spatial topology. This object can be considered as a large fluid object which is produced in a global universe, or a local compact object which is produced in a local \(S_3\)/\(H_3\) space.

In this paper, we consider the same static fluid in Ref. [7] with the electric field in spherical symmetry. If there is only the electric field, the spacetime is described by the Reisner–Norström solution. If we add the constant matter of Eq. (1) to the electric field, there is no consistent static solution to the Einstein’s equation. Therefore, as in Ref. [7] we consider the fluid of \(p(r) = -\rho (r)/3\). The mixture of electric field and fluid form the geometry, and we expect that the \(S_3/H_3\) topology due to fluid unveils at some region of space. When the electric field is turned off, the system reduces to the fluid-only case investigated in Ref. [7]. There are some other works on the gravitating solutions for static fluids (see e.g., Refs. [8, 9, 10, 11, 12, 13, 14]).

Classification of solutions. The signature of \(\rho _c\) is chosen so that \(f(\chi )g(\chi )>0\)

Class | \(\rho (\chi )\) | \(f(\chi )\) | \(g(\chi )\) |
---|---|---|---|

\(S_3\)-I | \(\frac{3}{8\pi R_0^2} \left[ 1- K \cot \chi -\frac{Q^2}{6R_0^2} (1-\cot ^2\chi ) \right] \) | \(\frac{\rho (\chi )}{\rho _c}, (\rho _c>0) \) | \(\frac{3}{8\pi \rho (\chi )} \) |

\(S_3\)-II | \(\frac{3}{8\pi R_0^2} \left[ 1 \mp K \tanh \chi -\frac{Q^2}{6R_0^2}(1+\tanh ^2\chi ) \right] \) | \(\frac{\rho (\chi )}{\rho _c},(\rho _c<0)\) | \(-\frac{3}{8\pi \rho (\chi )}\) |

\(H_3\) | \(-\frac{3}{8\pi R_0^2} \left[ 1 \mp K \coth \chi +\frac{Q^2}{6R_0^2}(1+\coth ^2\chi ) \right] \) | \(\frac{\rho (\chi )}{\rho _c},(\rho _c<0)\) | \(-\frac{3}{8\pi \rho (\chi )}\) |

## 2 Model and field equations

*E*(

*r*) is the electric field, and the prime denotes the derivative with respect to

*r*. The energy-momentum tensor for the electric field is given by

## 3 Classification of solutions

*Q*is the electric charge, \(\alpha \) and \(\beta \) are integration constants, and \(\rho _c = -9\beta /(8\pi )\). The above solutions reduce to those of the fluid-only solutions in Ref. [7] when \(Q=0\), and to the Reisner–Norström (RN) solution when \(\alpha \rightarrow \infty \) and \(\beta \rightarrow 0\) with \(\alpha \beta = \mathrm{finite} = M^{2/3}\).

*r*to \(\chi \), and use the metric

*K*and the charge

*Q*are turned off, the metric reduces to that of the pure \(S_3\)/\(H_3\) in Eq. (2)

### 3.1 \(S_3\)-I

*r*, \(\chi \) is double valued. The metric becomes

(i) *RN black-hole type solution* If \(J_1>0\), there exist two horizons at \(\chi _h=\chi _\pm \), which coalesce when \(J_1=0\). This solution mimics the Reisner–Nordström geometry of the charge black hole. The spacetime is regular at \(\chi <\chi _-\) and \(\chi >\chi _+\). The singularity at the north pole (\(\chi =0\)) is inside the inner horizon, and is not accessible by the timelike observers as in the RN black hole. The singularity at the south pole (\(\chi =\pi \)) is naked, but is not accessible either by the timelike observers as in the fluid black hole investigated in Ref. [7]. The geodesics are studied in the next section.

*Naked singular solution*If \(J_1<0\), there is no horizon. Both singularities are naked, but neither of them are accessible.

### 3.2 \(S_3\)-II

*r*. The metric becomes

Let us consider the \(\ominus \) solution.

- (i)
*RN black-hole type solution*For \(Q^2 \ge 3(1+K)R_0^2\), there are two horizons at \(\chi _\pm \) and this is the RN black-hole type. - (ii)
*Schwarzschild black-hole type solution*For \(3(1-K)R_0^2< Q^2 < 3(1+K)R_0^2\), there exists only one horizon. Inside the horizon (the trapped region), \(f(\chi ), g(\chi ) <0\) and \(\rho >0\). The spacetime is nonstatic in the trapped region, and static outside. The structure is similar to that of the Schwarzschild black hole. - (iii)
*Nonstatic solution*For \(Q^2 \le 3(1-K)R_0^2\), there is no horizon and the spacetime is nonstatic everywhere. This type of solution is special for \(S_3\)-II. This is analogous to the solution in Eq. (2) describing the region \(r\ge R_0\) in which the roles of the temporal and the radial coordinates are exchanged. If \(J_2<0\), there is one type of solution. - (iv)
*Regular solution*The spacetime is regular everywhere while \(\rho <0\). For the \(\oplus \) solution, the situation is the same with the \(\ominus \) solution with \(\chi \rightarrow {{-}}\chi \). Therefore, out of four types (i)-(iv), the only change is in (ii). Now, the region of \(\chi < \chi _h\) is static, and the region of \(\chi > \chi _h\) is nonstatic.

### 3.3 \(H_3\)

- (i)
*RN black-hole type solution*For \(3(K-1)R_0^2< Q^2 < 3KR_0^2\), there are two horizons at \(\chi _\pm \) and this is the RN black-hole type. - (ii)
*dS-type solution*For \(Q^2 \le 3(K-1)R_0^2 \), there is only one horizon at \(\chi _+\). The spacetime is static inside the horizon, and nonstatic outside. This is a de Sitter-like solution. This solution is achieved when the electric charge*Q*is small. When \(Q=0\), this corresponds to the cosmological solution of the fluid-only case in Ref. [7] for which the spacetime is nonstatic everywhere. It was interpreted as a universe expanding from an initial singularity. For the present case, however, the horizon is formed due to the electric field inside which the spacetime is static. - (iii)
*Naked singular solution*For \(Q^2 \ge 3KR_0^2\), the solution is static everywhere, but with a singularity at the center.

### 3.4 Gauss’ law

*r*coordinate with the components (8) is transformed to \(\mathcal{F}'_{\mu \nu }\) in the \(\chi \) coordinate with the nonzero components,

### 3.5 Mass

*M*as

Effective potential \(V(\chi )\)

Class | \(F(\chi )\) | \(V(\chi )\) |
---|---|---|

\(S_3\)-I | \(1- K \cot \chi -(Q^2/6R_0^2) (1-\cot ^2\chi ) \) | \(\frac{1}{2} [1- K \cot \chi -(Q^2/6R_0^2) (1-\cot ^2\chi )]\left( \frac{L^2}{\sin ^2\chi } +\frac{\varepsilon }{R_0^2} \right) \) |

\(S_3\)-II | \(-1 \pm K \tanh \chi +(Q^2/6R_0^2)(1+\tanh ^2\chi ) \) | \(\frac{1}{2} [-1 \pm K \tanh \chi +(Q^2/6R_0^2)(1+\tanh ^2\chi )] \left( \frac{L^2}{\cosh ^2\chi }+\frac{\varepsilon }{R_0^2} \right) \) |

\(H_3\) | \(1 \mp K \coth \chi +(Q^2/6R_0^2) (1+\coth ^2\chi )\) | \(\frac{1}{2} [1 \mp K \coth \chi +(Q^2/6R_0^2) (1+\coth ^2\chi )] \left( \frac{L^2}{\sinh ^2\chi } +\frac{\varepsilon }{R_0^2} \right) \) |

*s*is the signature of \(\rho _c\) (\(s=+1\) for \(S_3\)-I, and \(s=-1\) for the others). The mass depends on the radial coordinate \(\chi \).

For the fluid-only case (\(Q=0\)), the mass is still \(\chi \) dependent, while one has \(\mathcal{M}_{\mathrm{Sch}} = M\) for the ordinary Schwarzschild black hole. For the fluid black-hole solutions, one can show with the aid of Eq. (32) that the Misner–Sharp mass evaluated on the horizon coincides with the Schwarzschild mass, \(\mathcal{M}(\chi _h) = M\). This indicates that the horizon structure of the fluid black hole is the same with that of the Schwarzschild black hole.

For the ordinary RN black hole, the Misner–Sharp mass is given by \(\mathcal{M}_{\mathrm{RN}} =M-Q^2/(2r) = M-Q^2/[2R_0b(\chi )]\). For the RN black-hole type solutions obtained in this work (\(Q\ne 0\)), keeping the mass relation of *K* in Eq. (32), the Misner–Sharp mass evaluated on the horizons does not coincide with that of the ordinary RN black hole, \(\mathcal{M}(\chi _\pm ) \ne \mathcal{M}_\mathrm{RN}(\chi _\pm )\).

Although the horizon structure of the fluid black hole (\(Q=0\)) is the same with that of the ordinary one, the thermodynamics must be very different because the off-horizon structure is very different. We shall study the thermodynamics using the Misner–Sharp mass in a separate work including the charged case.

## 4 Geodesics

*E*(energy) and

*L*(angular momentum) as

For \(S_3\)-II, the RN black-hole type solution, when the energy level (\({{\tilde{E}}}\)) is low, the oscillatory orbit is similar to that of \(S_3\)-I. When the energy level is increased, the geodesic observer can reach the inner static region behind the inner horizon. When the energy level is high enough, the geodesic observer can escape to the asymptotic infinity in the static region. The Schwarzschild black-hole type solution has the similar geodesic structure to that of the usual Schwarzschild black hole. When the energy level is low, all the geodesic motions fall into the black hole. However, \(V(\chi )\) approaches a constant value as \(\chi \rightarrow -\infty \).

## 5 Stability

*N*is a normalization constant, we get the perturbation equation in the nonrelativistic Schrödinger-type,

*U*, i.e., \(\omega ^2 <0\), this system is

*unconditionally unstable*.

The stability story is very similar to the fluid-only case. When perturbations are introduced to the static fluid, the fluid becomes time dependent, which drives the Universe to undergo the Friedmann expansion. This type of instability does not necessarily mean that the black-hole structure is destroyed. Instead, the instability indicates that the background universe undergoes expansion while the black-hole structure sustains.

When the perturbation of the electric field is considered, the instability can be related with the destruction of the black-hole structure. It is known that the Cauchy (inner) horizon of the charged black hole is unstable to form a singularity [16]. The perturbation introduced in this work may develop such an instability in the RN black-hole type solution.

## 6 Conclusions

We investigated the gravitational field of static fluid plus electric field. Both of the fluid and the electric field are the sources of the gravitational field, but the way to curve the spacetime is a bit different from each other. By adopting the equation of state \(p(r) = -\rho (r)/3\), the fluid is responsible for the topology of the background space. The spatial topology can be either closed (\(S_3\)) or open (\(H_3\)). Such a nature of the spatial topology is not observed everywhere. Instead, the signature of the background spatial topology appears at some place of the spacetime.

Based on the background topology, there exist various types of solutions in three classes which we named as \(S_3\)-I, \(S_3\)-II, and \(H_3\). Interesting classes are \(S_3\)-I and \(H_3\) although the class \(S_3\)-II has most varieties in solution. The most interesting solutions are the black-hole solutions. Due to the presence of the electric field, the black-hole geometry mimics that of the Reisner–Norström spacetime. This type of black hole exists in both \(S_3\) and \(H_3\) spaces. (There exists also a Schwarzschild-type black hole in \(S_3\)-II.) The central singularity inside the black hole of this type of solution is due to the electric source as well as the fluid source. There is a naked singularity in \(S_3\)-I at the antipodal point which is not accessible except by the radial null rays. The formation of this singularity is caused by the fluid. The geodesics of the Reisner–Norström black-hole type solution exhibit the oscillatory orbit in the infinite tower of the spacetime encountered in the usual Reisner-Norström geometry.

All the solutions obtained in this paper are unconditionally unstable. This is not surprising because the stability story is similar to the fluid-only case in Ref. [7]. The reason of the instability is that the static fluid becomes unstable (time dependent) with small perturbations and drives the background geometry to the Friedmann expansion. In addition, there is an electric field for which it is well known that the pure charged black-hole solution (Reisner–Norström geometry) is unstable under perturbations.

The solutions investigated in this paper are useful in studying the magnetic monopole in the closed/open space, which is under investigation currently. Usually, the outside geometry of the magnetic monopole is the same with that of the charged black hole (Reisner–Norström geometry) [17, 18, 19, 20, 21]. Since we obtained the charged black-hole solution in \(S_3\)/\(H_3\) with the aid of fluid, it is very interesting to investigate the magnetic monopole in the presence of fluid. It may give rise to insight about the monopole in the closed/open space. The asymptotic geometry of this type of the gauge monopole is worth while to investigate and will be very interesting to compare with the usual monopole geometry. In addition, the removal of the singularity is also a very interesting issue. For the usual case, the monopole field removes the singularity of the charged solution. For this case, however, the formation of the singularity is caused not only by the electric charge, but also by the fluid. It is interesting to see if the monopole field can regularize the singular behavior of the fluid.

## Notes

### Acknowledgements

The author is grateful to Hyeong-Chan Kim and Gungwon Kang for useful discussions. This work was supported by the grant from the National Research Foundation funded by the Korean government, no. NRF-2017R1A2B4010738.

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