Exact solutions for charged spheres and their stability. I. Perfect Fluids.

The Reissner-Nordström metric (Reissner [1], Weyl [2] and Nordström [3]) is a solution of the Einstein-Maxwell equations that represents the exterior gravitational field of a spherically symmetric charged body. In fact, it is the unique asymptotically flat vacuum solution around any charged spherically symmetric object, irrespective of how that body may be composed or how it may evolve in time (Carter [4], Ruback [5] and Chruściel [6]). It contains no information about its source other than its total mass and total charge, and probably most interesting it imposes no constraints on its internal structure.

The internal structure of a charged sphere is established by solving the coupled Einstein-Maxwell equations. A solution of these equations describes the gravitational field inside the sphere as a function of the distribution of matter and energy within the sphere. The Einstein-Maxwell equations for a charged static spherically distribution of matter reduce to a set of three independent non-linear second order differential equations that connect the metric coefficients $g_{tt}$ and $g_{rr}$ with the physical quantities that represent the matter content of the sphere. A description of the matter content of a charged perfect fluid sphere includes functions that defines its inertial mass density $\rho_{fl}$, its pressure $p$ and the electric charge distribution $q$ or the electromagnetic energy density $\rho_{em}$. Thus, the complete system of equations to be solved consists of three linearly independent equations with five unknowns. This situation gives total freedom in choosing two of the five functions and then solving for the remaining three. It is therefore hardly surprising that there exists very many exact solutions of the Einstein-Maxwell equations for charged spheres. A comprehensive review of the various methods of solving the Einstein-Maxwell equations for charged spheres was done by Ivanov [7].

An important question that arises in the study of the structure of spherically symmetric compact objects is the following: what is the maximum of value $M/R$ (the total mass of the body divided its radius) that is allowed before gravitational collapse occurs? It is well known that for neutral perfect fluid spheres that the stability limit is given by the Buchdahl limit [8]:

$$\frac{M}{R}\leq\frac{4}{9}{~{}~{}~{}\rm for~{}uncharged~{}perfect~{}fluid~{}spheres}.$$

(1.1)

In case of charged compact objects this quantity becomes dependent on the total charge $Q$, since the addition of charge to the system increases its total energy and hence its total mass. Andréasson [9], has published a remarkable result that claims that for any compact spherically symmetric charged distribution the following relationship between $M/R$ and $Q/R$ holds:

$$\frac{M}{R}\leq\left(\frac{1}{3}+\sqrt{\frac{1}{9}+\frac{1}{3}\frac{Q^{2}}{R^{2}}}~{}\right)^{2}{~{}~{}~{}\rm for~{}all~{}charged~{}spheres}.$$

(1.2)

This formula generalizes the Buchdahl result for uncharged perfect fluid spheres. It is worth noting that in his derivation, Buchdahl placed the following constraints on the physical properties of the fluid: (i) $d\rho/dr<0$ i.e., the density decreases outward from the center of the sphere and (ii) the pressure is isotropic. In the derivation of his formula, Andréasson made the following assumptions about the matter content of the sphere: (i) $p_{r}+2p_{t}<\rho$ (in a spherically symmetric system it is possible to have the radial pressure, $p_{r}$ different from the tangential pressure, $p_{t}$) and (ii) $\rho>0$. We note that unlike the neutral case Andréasson did not require the condition $d\rho/dr<0$. Numerical investigations ([10] and [11]), of the upper bound of $M/R$ as a function of $Q/R$ have verified that the Andréasson formula provides a valid upper bound of $M/R$ for charged spheres. One of our aims in this study is to continue the investigation of the validity of the Andréasson limit using a mixture of analytical and numerical techniques.

A class of charge spheres that have received considerable attention in theoretical physics are extremal black holes. Extremal non-rotating charged black holes are interior solutions of the Reissner-Nordstrom metric in which the mass is equal to the charge in geometric units. These objects have zero surface gravity and their horizon structure is different from that of regular black holes. In general relativity extremal black holes are important in the study and proof of the third law of black hole thermodynamics. In string theory they are ubiquitous, in particular they were used to derived the Bekenstein-Hawking entropy formula and in general they are easier to describe than regular black holes quantum mechanically because of their vanishing surface gravity and consequently vanishing temperature for Hawking radiation [12]. In many solutions for charged spheres extremality is only achieved when $M/R$ = $Q/R$ = 1. In our study we will look for cases where extremality can occur before the upper bound on $Q/R$ and $M/R$ is reached.

This paper is organized as follows in the next section we develop the full Einstein-Maxwell equations for the interior gravitational field of a charged sphere. In section 3 we solve the equations and consider their stability properties and the formation of extremal black holes in these solutions. In section 4 we discuss our results and formulate our conclusions. We note that a prime (^′) or a comma (,) denotes derivatives with respect $r$, and a semi-colon (;) is used to represent covariant derivatives. We will work in units where $c=G=1$.

The material presented in the next section is well known. However, for consistency and clarity we have chosen to show the full derivation of the electromagnetic energy-momentum tensor for a static radial electric field. A complete pedagogical description of the derivation of the general form of the perfect fluid and electromagnetic energy-momentum tensors is given in Alder, Bazin and Schiffer [13].

We are interested in studying the interior gravitational field of charged spheres, therefore we will assume a spherically symmetric metric of the form

$$ds^{2}=-e^{2\nu}dt^{2}+e^{2\lambda}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta d\phi^{2}.$$

(2.1)

In this paper we are concerned only with spherically symmetric static solutions of the coupled Einstein-Maxwell equations, therefore $\nu\,=\,\nu(r)$ and $\lambda\,=\,\lambda(r)$ are functions of $r$ only. The Einstein field equations are

$$G_{\alpha\beta}=R_{\alpha\beta}-\frac{1}{2}g_{\alpha\beta}R=8\pi T_{\alpha\beta}.$$

(2.2)

The spheres that we will study are composed of a perfect fluid with a charge distribution that creates a static radial electric field. The energy-momentum tensor $T_{\alpha\beta}$, will thus be written as

$$T_{\alpha\beta}=(T_{\alpha\beta})_{pf}+(T_{\alpha\beta})_{em},$$

(2.3)

with $(T_{\alpha\beta})_{pf}$ the energy-momentum tensor for a perfect fluid and $(T_{\alpha\beta})_{em}$ the energy-momentum tensor associated with the electric field.

The energy momentum tensor for a static spherically symmetric perfect fluid with a rest-frame energy-mass density $\rho$ and isotropic pressure $p$ [13] is

$$(T_{\alpha\beta})_{f}=(\rho+p)u_{\alpha}u_{\beta}+pg_{\alpha\beta}.$$

(2.4)

Here, $\rho$ and $p$ are functions of $r$ only and $u_{\alpha}$ is the 4-velocity of the fluid.

The four velocity $u_{\alpha}$ is defined such that

$$g_{\alpha\beta}u^{\alpha}u^{\beta}=-1.$$

(2.5)

Since we are considering a fluid at rest, we will take

$$u_{r}=u_{\theta}=u_{\phi}=0,~{}~{}~{}{\rm and}~{}~{}~{}u_{t}=-(-g^{tt})^{-\frac{1}{2}}=-e^{\nu(r)}$$

(2.6)

thus,

$$u_{\alpha}=-\delta_{\alpha}^{t}e^{\nu}$$

(2.7)

and the energy momentum tensor for a perfect fluid is

$$(T_{\alpha\beta})_{pf}={\rm{diag}}(-\rho e^{-2\nu},p\,e^{-2\lambda},p\,r^{2},p\,r^{2}\sin^{2}\theta).$$

(2.8)

The energy-momentum tensor of the electric field, $(T_{\alpha\beta})_{em}$ is constructed from the electromagnetic field tensor $F_{\alpha\beta}$:

$$(T_{\alpha\beta})_{em}=\frac{1}{4\pi}(F_{\alpha}^{~{}\tau}F_{\beta\tau}-\frac{1}{4}g_{\alpha\beta}F_{\tau\kappa}F^{\tau\kappa}).$$

(2.9)

We are interested in studying a fluid whose charge distribution gives rise to static radial electric field therefore the only non-zero components of $F_{\alpha\beta}$ are $F_{rt}\,=\,-F_{tr}=\,E(r)$, thus

$$F_{\alpha\beta}=\delta_{\alpha}^{r}\delta_{\beta}^{t}E(r).$$

(2.10)

Maxwell’s equations in curved space-time are:

$$F^{\alpha\beta}_{~{}~{}~{};\beta}=4\pi j^{\alpha},~{}~{}{\rm and~{}~{}}F_{[\alpha\beta;\tau}+F_{[\tau\alpha;\beta]}+F_{[\beta\tau;\alpha]}=0.$$

(2.11)

Given the form of $F_{\alpha\beta}$ (2.10), the second set of equations is identically zero, also since $F_{\alpha\beta}$ is anti-symmetric the first set of equations can be written as

$$\frac{1}{\sqrt{-g}}(\sqrt{-g}F^{\alpha\beta})_{,\beta}=4\pi j^{\alpha},$$

(2.12)

where $j^{\alpha}$ is the current 4-vector. If the charge density $\sigma$ is given then

$$j^{\alpha}=\sigma u^{\alpha}.$$

(2.13)

Using the explicit forms of $F_{\alpha\beta}$ and $u_{\alpha}$ in equation (2.12) we find that

$$(r^{2}e^{-(\nu+\lambda)}E(r))_{,r}=4\pi r^{2}\sigma e^{\lambda}.$$

(2.14)

Integrating this equation gives an expression for $E(r)$:

$$E(r)=\frac{e^{\nu+\lambda}q(r)}{r^{2}},$$

(2.15)

with

$$q(r)=\int_{0}^{r}4\pi r^{2}\sigma e^{\lambda}dr.$$

(2.16)

We note that since $\sigma=j^{t}e^{\nu}$, $q(r)$ can be written as

$$q(r)=\int_{0}^{r}\int_{0}^{\pi}\int_{0}^{2\pi}j^{t}e^{\nu+\lambda}r^{2}\sin\theta d\theta d\phi dr=\int\limits_{V}j^{t}\sqrt{-g}\,dV.$$

(2.17)

Thus $q(r)$ represents the total charge contained in a sphere of radius $r$. Using (2.15) we can write

$$F_{\alpha\beta}=\delta_{\alpha}^{r}\delta_{\beta}^{t}\frac{e^{\nu+\lambda}q(r)}{r^{2}},$$

(2.18)

and the electromagnetic energy-momentum tensor takes the from

$$(T_{\alpha\beta})_{em}=\frac{q(r)^{2}}{8\pi r^{4}}{\rm diag}(e^{-2\nu},-e^{2\lambda},r^{2},r^{2}\sin^{2}\theta).$$

(2.19)

We can now write out complete set of equations that describe the interior gravitational field of static charged spheres. They are:

$$G^{t}_{~{}t}=e^{-2\lambda}\left(\frac{2\lambda^{\prime}}{r}-\frac{1}{r^{2}}\right)+\frac{1}{r^{2}}=8\pi\rho+\frac{q^{2}}{r^{4}},$$

(2.20)

$$G^{r}_{~{}r}=e^{-2\lambda}\left(\frac{2\nu^{\prime}}{r}+\frac{1}{r^{2}}\right)-\frac{1}{r^{2}}=8\pi p-\frac{q^{2}}{r^{4}},$$

(2.21)

and

$$G^{\theta}_{~{}\theta}=G^{\phi}_{~{}\phi}=e^{-2\lambda}\left(\nu^{\prime\prime}+{\nu^{\prime}}^{2}-\nu^{\prime}\lambda^{\prime}+\frac{\nu^{\prime}}{r}-\frac{\lambda^{\prime}}{r}\right)=8\pi p+\frac{q^{2}}{r^{4}}.$$

(2.22)

We will now develop solutions for the field equations. We start by noting that (2.20), can be written as

$$\frac{d}{dr}(re^{-2\lambda})=1-8\pi\rho r^{2}-\frac{q^{2}}{r^{2}}.$$

(3.1)

This equation can be immediately integrated to give

$$e^{-2\lambda(r)}=1-\frac{2m_{i}(r)}{r}-\frac{f(r)}{r},$$

(3.2)

with

$$m_{i}(r)=4\pi\int_{0}^{r}\rho(r^{\prime}){r^{\prime}}^{2}dr^{\prime}~{}~{}~{}{\rm and}~{}~{}~{}f(r)=\int_{0}^{r}\frac{q(r^{\prime})^{2}}{{r^{\prime}}^{2}}dr^{\prime}.$$

(3.3)

The quantity $m_{i}(r)$ is the inertial mass of the fluid in a sphere of radius $r$. A related quantity is the total gravitational mass $m_{g}(r)$ of the fluid in a sphere of radius $r$. Bekenstein [16] in his study of charged spheres introduced $m_{g}(r)$ in an $ad\,hoc$ manner. Since the exterior metric of a charged sphere, the Reissner-Nordström solution has

$$e^{-2\lambda(r)}=1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}$$

(3.4)

where $M$ is the total gravitational mass of the sphere and $Q$ is the total charge of the sphere, Bekenstein proposed that in the interior of a charged sphere, $e^{-2\lambda(r)}$ should have following form:

$$e^{-2\lambda(r)}=1-\frac{2m_{g}(r)}{r}+\frac{q^{2}(r)}{r^{2}}.$$

(3.5)

The function $q(r)$ is the total charge in a sphere of radius $r$. It is the charge function defined in (2.16) that we have been studying. In order for (3.2) to be equal to (3.5), $m_{g}(r)$ must be defined in the following manner:

$$m_{g}(r)\equiv\frac{1}{2}\int_{0}^{r}\left(8\pi\rho{r^{\prime}}^{2}+\frac{q^{2}(r^{\prime})}{{r^{\prime}}^{2}}\right)dr^{\prime}+\frac{q(r)^{2}}{2r}.$$

(3.6)

We note that in the absence of the electric field $m_{g}=m_{i}$.

The requirement that $e^{-2\lambda(r)}$ matches the Reissner-Nordström metric at surface of the sphere, $r=R$ gives

$$1-\frac{M}{R}+\frac{Q^{2}}{R^{2}}=1-\frac{1}{R}\int_{0}^{R}(8\pi\rho r^{2}+\frac{q^{2}}{r^{2}})dr.$$

(3.7)

An expression for the total mass can be found from this equation:

$$M=\frac{1}{2}\int_{0}^{R}(8\pi\rho r^{2}+\frac{q^{2}}{r^{2}})dr+\frac{Q^{2}}{2R}.$$

(3.8)

In this paper we will study charged spheres with the following inertial mass density $\rho(r)$ and charge distribution $q(r)$ profiles:

$$\rho(r)=\rho_{o}-\frac{b}{8\pi}\frac{Q^{2}}{R^{6}}r^{2}{\rm~{}~{}~{}~{}and~{}~{}~{}~{}}q(r)=\frac{Q}{R^{3}}r^{3},$$

(3.9)

in the expression for $\rho(r)$, $b$ is a number. In our models

$$e^{-2\lambda(r)}=1-\frac{8\pi\rho_{o}}{3}r^{2}+\frac{(b-1)}{5}\frac{Q^{2}}{R^{6}}{r^{4}}$$

(3.10)

$$m_{g}(r)=\frac{4\pi\rho_{o}}{3}r^{3}+\frac{(6-b)}{10}\frac{Q^{2}}{2R}\frac{r^{5}}{R^{5}}.$$

(3.11)

and

$$M=\frac{4\pi\rho_{o}}{3}R^{3}+\frac{(6-b)}{10}\frac{Q^{2}}{2R}.$$

(3.12)

With the assumed mass and charge distributions here, when $b=6$

$$\frac{m_{g}(r)}{r^{3}}=\frac{M}{R^{3}}=const,$$

(3.13)

thus the $b=6$ model is a sphere with a constant gravitational mass density. Also all models have

$$\frac{q(r)}{r^{3}}=\left(\frac{Q}{R^{3}}\right)=const.$$

(3.14)

thus the charge density is constant for all our models. The models that we will study in detail here include following three charged configurations:

1. 1.

   $b=0$ - a sphere with a constant inertial mass density and constant charge density.

2. 2.

   $b=1$ - a sphere with constant total energy density and constant charge density.

3. 3.

   $b=6$ - a sphere with constant gravitational mass density and constant charge density.

We can solve for $\rho_{o}$ from (3.12) and rewrite (3.10) as

$$e^{-2\lambda(r)}=1-\left(\frac{2M}{R}+\frac{(b-6)}{5}\frac{Q^{2}}{R^{2}}\right)\frac{r^{2}}{R^{2}}+\frac{(b-1)}{5}\frac{Q^{2}}{R^{2}}\frac{r^{4}}{R^{4}}.$$

(3.15)

Thus, if we are given $\rho(r)$ and $q(r)$ we have found $\lambda(r)$.

We now need to solve for $\nu(r)$. We start by transforming (2.22). First we subtract (2.21) from (2.22 ) to get

$$e^{-2\lambda}\left(\nu^{\prime\prime}+{\nu^{\prime}}^{2}-\nu^{\prime}\lambda^{\prime}-\frac{\nu^{\prime}}{r}\right)-\frac{\lambda^{\prime}e^{-2\lambda}}{r}-\frac{e^{-2\lambda}}{r^{2}}+\frac{1}{r^{2}}=2\frac{q^{2}}{r^{4}}.$$

(3.16)

Then we substitute for $1/r^{2}$ from (2.20) to get the following equation

$$\nu^{\prime\prime}+{\nu^{\prime}}^{2}-\nu^{\prime}\lambda^{\prime}-\frac{\nu^{\prime}}{r}=3\frac{\lambda^{\prime}}{r}-\left(8\pi\rho-\frac{q^{2}}{r^{4}}\right)e^{2\lambda}.$$

(3.17)

We now multiply both sides of this equation by $e^{-\lambda+\nu}/r$, and we find that the left hand-side becomes an exact differential: $((\nu^{\prime}e^{-\lambda+\nu})/r)^{\prime}$. Introducing $\zeta(r)\equiv e^{\nu(r)}$, we can write (3.17) in the following form

$$\left(\frac{1}{r}e^{-\lambda}\zeta^{\prime}\right)^{\prime}=\left[\frac{3\lambda^{\prime}e^{-2\lambda}}{r^{2}}-\frac{8\pi\rho}{r}+\frac{q^{2}}{r^{5}}\right]e^{\lambda}\zeta.$$

(3.18)

This equation was derived by Giuliani and Rothman [15] in their study of charged spheres. The transformation of the left hand-side (3.17) into that of (3.18) was introduced by Weinberg [14] in deriving the equation he used to prove the Buchdahl limit of $M/R$ for neutral perfect fluid spheres.

Substituting for $\rho$, $q$ (from (3.9)) and $\lambda^{\prime}e^{-2\lambda}$ (from (3.71)) we find that (3.18) becomes

$$\left(\frac{1}{r}e^{-\lambda}\zeta^{\prime}\right)^{\prime}=\left(\frac{11}{5}-\frac{b}{5}\right)\frac{Q^{2}}{R^{6}}re^{\lambda}\zeta.$$

(3.19)

We now define a new variable

$$\tilde{\zeta}(u(r))\equiv\zeta(r){\rm~{}~{}~{}~{}~{}with~{}~{}~{}~{}}u(r)=\frac{1}{R^{2}}\int_{0}^{r}se^{\lambda(s)}ds.$$

(3.20)

Then (3.19) becomes

$$\frac{d^{2}\tilde{\zeta}}{du^{2}}=\left(\frac{11}{5}-\frac{b}{5}\right)\frac{Q^{2}}{R^{2}}\tilde{\zeta}(u).$$

(3.21)

This is our master equation. Our task now is to find solutions for it given various values of $b$. There are three types of solutions of the master equation depending on the value of $b$:

$$(i)~{}~{}b<11:~{}~{}\tilde{\zeta}(u(r))=Ae^{a\frac{Q}{R}u(r)}+Be^{-a\frac{Q}{R}u(r)}~{}~{}~{}{\rm with}~{}a={\left(\frac{11}{5}-\frac{b}{5}\right)}^{\frac{1}{2}},$$

(3.22)

$$(ii)~{}~{}b=11:~{}~{}\tilde{\zeta}(u(r))=A+Bu(r),$$

(3.23)

$$(iii)~{}~{}b>11:~{}~{}\tilde{\zeta}(u(r))=A\sin\left(d\frac{Q}{R}u(r)\right)+B\cos\left(-d\frac{Q}{R}u(r)\right)~{}~{}~{}{\rm with}~{}d={\left|\frac{11}{5}-\frac{b}{5}\right|}^{\frac{1}{2}}.$$

(3.24)