Fluid Black Holes with Electric Field

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-Norstr\"om 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.


I. 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 Ω k = 0.000 ± 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 pure closed/open (S 3 /H 3 ) spatial topology is achieved by a constant matter field with the equation of state, where ρ > 0 for S 3 and ρ < 0 for H 3 . The resulting metric is well known as where k = +1/ − 1 represents S 3 /H 3 , and ρ = ±3/(8πR 2 0 ). For S 3 , the ranges of the radial coordinate, 0 ≤ r ≤ R 0 and r ≥ R 0 , are considered separately. (We shall call the former S 3 -I and the latter S 3 -II.) For S 3 -II, we take g 00 = +1 to consider only one time coordinate.
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) = −ρ(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) = −ρ(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 gravitating fluids in Refs. [8][9][10][11][12][13][14].
This paper consists as following. In Sec. II, we introduce the model and field equations. In Sec. III, we classify the solutions and discuss the spacetime structure. In Sec. IV, we discuss the geodesic motions. In Sec. V, we study the stability of the solutions. In Sec. VI, we conclude.

II. MODEL AND FIELD EQUATIONS
We consider the electric field and the perfect fluid in static state. The static metric ansatz for spherical symmetry is given by The energy-momentum tensor for the fluid is given by and we consider the equation of state which meets the S 3 /H 3 boundary condition, The field-strength tensor for the electric field is given by We consider the static electric field only, then the vector potential is given by Then the nonvanishing components of F µν in Eq. (6) are where 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 With the metric (3) and the energy-momentum tensors (4) and (9), the nonvanishing components of the Einstein's equation, G µ ν = 8π(T µ ν + T µ ν ), are .
Since the fluid and the electric field are minimally coupled only thorough gravity, the conservation of the energymomentum tensor is satisfied individually, ∇ µ T µν = 0 and ∇ µ T µν = 0, which provide the field equations, These field equations give solutions for fluid and electric field in terms of the gravitational field, Class

III. CLASSIFICATION OF SOLUTIONS
With the solutions in Eq. (14) and the equation of state (5), the Einstein equations (10)- (12) are solved, where, Q is the electric charge, α and β are integration constants, and ρ c = −9β/(8π). 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 α → ∞ and β → 0 with αβ = finite = M 2/3 . In order to catch the idea of the spatial topology, we transform the radial coordinate r to χ, and use the metric where b(χ) is introduced in the subsections below. We introduced a new parameter R 0 ≡ |α| which is related with the curvature. In addition, we introduce another parameter K ≡ 2R 2 0 |β| 3/2 interpreted as a mass parameter analogous to the fluid-only black hole investigated Ref. [7]. Depending on the signatures of α and β, the solutions are classified into three categories. Two of them meet the S 3 boundary condition, and the other does the H 3 condition. The classes are summarized in Table I. When both of the parameter K and the charge Q are turned off, the metric reduces to that of the pure S 3 /H 3 in Eq. (2) A. S3-I This is the case of α < 0 and β < 0. The transformation is performed by and the metric becomes Here, ρ c > 0. Both of the Ricci scalar and the Kretschmann scalar diverge at χ = 0 and π, i.e., there exist curvature singularities at both poles. For the pure fluid case (Q = 0) investigated in Ref. [7], the background S 3 topology is exposed at the boundary around the equator (χ ≈ π/2, i.e., r ≈ R 0 ), With the electric field, however, there is a charge correction, The location of the horizon is found from g −1 χχ = 0, Depending on the existence of the horizon, there are two types of solutions. (See Fig. 1 for the graphical view of the metric function.) (i) RN black-hole type solution: If J 1 > 0, there exist two horizons at χ h = χ ± , which coalesce when J 1 = 0. This solution mimics the Reisner-Nordström geometry of the charge black hole. The spacetime is regular at χ < χ − and χ > χ + . The singularity at the north pole (χ = 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 (χ = π) 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.
(ii) Naked singular solution: If J 1 < 0, there is no horizon. Both singularities are naked, but neither of them are accessible.

B. S3-II
This is the case of α < 0, β > 0. The transformation is performed by and the metric becomes (25) Here, ρ c < 0. The curvature is finite everywhere. The background S 3 topology is realized at the boundary around the center (χ ≈ 0, i.e., r ≈ R 0 ) with the charge correction. (For S 3 , ρ > 0 is necessary.) The location of the horizon is (The ± roots are valid for both ⊖ and ⊕.) The solutions are classified as below. (See also Fig. 2.) There are black-hole type solutions (both Schwarzschild and Reisner-Nordström types), but remember that they are not singular anywhere.
For the ⊖ solution, there are two types of solutions.
(U-i) Schwarzschild black-hole type solution: If Q 2 < 6R 2 0 (J 2 > 0 automatically), there exists only one horizon χ + . Inside the horizon, f (χ), g(χ) < 0 and ρ > 0. The spacetime is nonstatic inside the horizon, and static outside. The structure is similar to that of the Schwarzschild black hole.
(U-ii) Regular solution: If Q 2 > 6R 2 0 , there is no horizon and the spacetime is regular everywhere while ρ < 0.
For the ⊕ solution, the solutions are as following.
When J 2 > 0 and 0 < K < 1, there are tree types of solutions.
(L1-i) RN black-hole type solution: For 6R 2 0 < Q 2 , there are two horizons at χ ± and this is the RN black-hole type.
(L1-ii) Schwarzschild black-hole type solution: For 3(1 + K)R 2 0 < Q 2 ≤ 6R 2 0 , there is only one horizon at χ + . (L1-iii) Nonstatic solution: For Q 2 ≤ 3(1 + K)R 2 0 , there is no horizon and the spacetime is nonstatic everywhere. This type of solution is special for S 3 -II because this corresponds to the region r ≥ R 0 in pure S 3 in which the roles of the temporal and the radial coordinates are exchanged.
When J 2 > 0 and K > 1, there are tree types of solutions. (L2-i) RN black-hole type solution: , there is only one horizon at χ − . The spacetime is static inside the horizon, and nonstatic outside. This is a de Sitter-like solution which will be discussed in the next subsection.
(L3) Regular solution: When J 2 < 0, there is no horizon and the spacetime is regular everywhere.

C. H3
This is the case of α > 0, β > 0. The transformation is performed by and the metric becomes Here, ρ c < 0. The curvature diverges at χ = 0. The location of the horizon is The solutions are classified as below. (See also Fig. 2

.)
For the ⊖ solution in Eq. (28), there are three types of solutions for J 3 > 0.
(ii) dS-type solution: For Q 2 ≤ 3(K − 1)R 2 0 , there is only one horizon at χ + . 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 3KR 2 0 ≤ Q 2 , the solution is static everywhere, but with a singularity at the center.
For the ⊕ solution in Eq. (28), or for J 3 < 0, there is no horizon, and the solution is singular static like (iii).

D. Gauss' Law
Let us discuss the Gauss' law in the χ coordinate. The field-strength tensor F µν in Eq. (6) in the r coordinate with the components (8) is transformed to F ′ µν in the χ coordinate with the nonzero components, The electric flux is then where we used the relation ρ c = −9β/(8π). Compared with the Gauss' law in flat space, there is a correction due to fluid by the factor 8π|ρ c |.

IV. GEODESICS
In this section, we discuss the geodesics of the solutions. We focus mainly on the black-hole solutions. For simplicity, we define a function, where s is the signature of ρ c (s = +1 for S 3 -I, and s = −1 for the others). The geodesic equations become t-eq. : φ-eq. : From Eqs. (33) and (34), we denote the conserved quantities E (energy) and L (angular momentum) as The χ-equation can be derived from the metric as where ε = 0, 1 for null and timelike geodesics, individually. On the θ = π/2 plane, Eq. (36) becomes where the effective potential is given by We summarize V (χ) in Table II. The effective potential V (χ) of the black-hole type solutions is plotted in Fig. 4-6.
For the RN black-hole type solution of S 3 -I, the singularities at both poles are not accessible except by the radial null geodesic. For the fluid-only case in Ref. [7], the one at the north pole inside the horizon was accessible since the inner geometry was similar to that of the Schwarzschild black hole. However, for the present case, it is not because the inner geometry is similar to that of the charged black hole. The nonaccessibility to the naked singularity at the south pole is similar to the fluid-only case. The geodesic observer starting from the outer static region falls into the inner static region passing the intermediate nonstatic region. Afterwards, the observer bounces back to the nonstatic region and then enters the outer static region. This later motion after the bounce proceeds in the other copy of the spacetime accompanied in the usual RN geometry. The geodesic as a whole is an oscillatory orbit in the infinite tower of the RN spacetime.
For S 3 -II, the Schwarzschild black-hole type solution has the similar geodesic structure to that of the usual Schwarzschild black hole. When the energy level (Ẽ) is low, all the geodesic motions fall into the black hole. For the RN black-hole type solution, when the energy level 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 center of the black hole where the curvature is finite. When the energy level is high enough, the geodesic observer can escape to the asymptotic infinity.
For H 3 , the singularity at the center is not accessible except by the radial null geodesic, which is different from the fluid-only case. Similarly to the S 3 -I, it is due the electric charge. When the energy level is low, the geodesic motion is oscillatory as in S 3 -I. When the energy level is high, the geodesic observer can reach the asymptotic infinity. Another interesting solution is dS-type. For this solution, the geodesics escape from the static region crossing the de Sitter-like horizon and reach asymptotic infinity. This is different from the pure de Sitter space in which there can be a stable geodesic motion inside the horizon.   There is no stable geodesic motion inside the de Sitter-like horizon. All the geodesics escape from the static region crossing the horizon.

V. STABILITY
In this section, we study the stability of the solutions. We introduce linear spherical scalar perturbations with the metric ansatz, The metric perturbations are introduced as where ǫ is a small parameter, and the subscript 0 stands for the background solutions obtained in Sec. III. Using the function F (χ) = 8πR 2 0 ρ 0 (χ)/3s defined in Eq. (32), where ρ 0 (χ) is the background solution in Table I, we have .
The contravariant form of the energy-momentum tensor for fluid is written as with the velocity four-vector u µ = u 0 (t, χ), u 1 (t, χ), 0, 0 .
For the fluid at hand, p = −ρ/3, the perturbations for the energy density and the four-velocity are introduced by We have u 1 0 (χ) = 0 for the comoving background fluid. From the normalization u µ u µ = −1, we have u 0 0 (χ) = 1/ f 0 (χ) and u 0 . For the electric field, we introduce the simplest perturbation along the radial direction only, by which there is no magnetic field induced by the perturbation, where E 0 (χ) is given in Eq.
Therefore, the perturbations of the four-vector, u 0 1 and u 1 1 in Eqs. (47) and (48), are expressed by the background functions and the metric perturbations. There are seven equations in total for four perturbations, f 1 , g 1 , ρ 1 and E 1 ; three from Einstein's equation, two from ∇ µ T µν = 0 , and two from ∇ µ T µν = 0. Four of them are independent equations. After manipulating equations with the equation for ϕ(χ) is decoupled as where σ ≡ 1/(8πR 4 0 ρ c s) = 1/(8πR 4 0 |ρ c |) > 0 for all classes. The coefficients of the above equation depend only on the background functions F (χ) and b(χ).
By transforming the radial coordinate and the amplitude function as where N is a normalization constant, we get the perturbation equation in the nonrelativistic Schrödinger-type, The potential is given by Since there always exists a positive eigenvalue Ω for any type of potential U , i.e., ω 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 [15]. The perturbation introduced in this work may develop such an instability in the RN black-hole type solution.

VI. 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) = −ρ(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 class S 3 -II corresponds to the r ≥ R 0 region of the S 3 space). 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. Depending on the classes of the background topology, the black-hole center and the asymptotic infinity can, or cannot be accessed by the geodesic observer.
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) [16][17][18][19][20]. 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.