# Generating solutions for charged geodesic anisotropic spherical collapse with shear and heat radiation

## Abstract

In this paper we give a generating function for solutions of the type of collapse mentioned in the title. It satisfies a simple Riccati equation, derived from a formula, which holds in the interior of the star. The generating function is unique, unlike in the neutral case. Every neutral solution has a charged companion with the same generating function. The charged solution has bigger radius, surface redshift, mass and compactness than the neutral one. Its rate of collapse is slower. A class of known exact neutral solutions, containing generalised travelling waves, is charged.

## 1 Introduction

Gravitational collapse is an important issue in relativistic astrophysics. The collapse of a dust cloud was studied first [1]. This was followed by studies of fluid collapse. There are many indications that the collapsing fluid in the star models is anisotropic [2]. In addition, this process is highly dissipative, required to account for the enormous binding energy of the resulting object [3]. Thus a realistic scenario is anisotropic collapse with heat flow [4]. For simplicity, shearless fluid is used quite often. Even in the isotropic case the amount of interior solutions is enormous [5]. The stability of the shear-free condition during collapse, however, requires fine tuning of one of the structure scalars, which is also the complexity factor [6, 7, 8]. Thus collapse with shear seems to be the general case. It is described by a diagonal metric with three independent components. The exterior solution is the Vaidya shining star [9]. The two solutions should be matched on the stellar surface. The main junction condition states that the radial pressure should equal the heat flux. This gives a non-linear differential equation in partial derivatives (along radius and time). One can reduce the metric components to two by studying the geodesic case, \( g_{00}=1\). Shearless solutions were discussed in [10, 11, 12, 13].

Interior anisotropic geodesic solutions with shear and without radiation have been discussed in [14]. No matching to the exterior Schwarzschild solution was done. The same problem in non-comoving coordinates, but with matching, was solved in [15].

The first exact solution with radiation was obtained in [16]. After that in [17] it was noticed that the junction condition is a Riccati equation for \(g_{rr}\). Two simple regular solutions in separated variables were found. The solution of [16] is regained when certain parameters are set to zero. Later, the authors of [18] found even more general exact solutions depending on arbitrary functions of the coordinate radius. They encompass the previous solutions. The authors of [19] further expanded the number of analytic solutions by studying the Lie point symmetries of the boundary condition. Generalized travelling waves and self-similar solutions were derived. A class of these solutions was studied in detail [20]. One should mention also the dissipative LTB solutions [21].

Recently, it was shown that the junction equation simplifies for the so-called horizon function [22, 23] both in the general and the geodesic case. It is directly related to the redshift and the formation of a horizon, which means the appearance of a black hole at the end of collapse. It enters the expressions for the mass of the star, the heat flow and the luminosity at infinity. In the geodesic case a generating function was found for the solutions.

The collapse of charged anisotropic and dissipative fluids has also been studied in the shearless case as well as in the case with shear [24, 25, 26, 27, 28, 29, 30, 31, 32, 33].

The main purpose of the present paper is to generalize the junction equation for the horizon function to the case of charged anisotropic dissipative geodesic fluid. Like in the neutral case, the equation is easily integrated with the help of a generating function.

In Sect. 2 we present the Einstein–Maxwell equations, which in the anisotropic case are expressions for the energy density, the radial and the tangential pressure, the heat flow and the four-potential. The definitions of the different stellar characteristics are given. In Sect. 3 the results of the matching to the exterior charged Vaidya solution are given. The most important of them is a Riccati equation with simple coefficients. It is algebraic for the radius of the star. This allows to derive charged geodesic solutions from a generating function. Simple expressions are given for the mass and its time derivative. In Sect. 4 some stellar characteristics are written in terms of the generating function and its time derivatives. In Sect. 5 we show that one and the same generating function gives rise to a pair of solutions, one neutral and one charged. Certain inequalities between the stellar characteristics of the pair are derived. In Sect. 6 some recently found neutral solutions are charged. Sect. 7 contains conclusions.

## 2 Stellar characteristics and field equations

*B*and

*R*are independent functions of the time

*t*and the radius

*r*. The spherical coordinates are numbered as \(x^0=t\), \(x^1=r\), \(x^2=\theta \) and \(x^3=\varphi \). The energy-momentum tensor reads

*s*is the charge density and \(l\left( r\right) \) is the total charge up to radius

*r*. It is time-independent. We use relativistic units with \( G=1,c=1,k=8\pi \).

*H*[22, 23]:

*B*, which becomes

*r*is given by the expression [26, 29]

*H*

*R*and \(D\equiv HR\)

## 3 Junction conditions

*v*by an observer at infinity,

*Q*is the total charge, while \(\rho \) is the exterior coordinate radius. Both solutions should be joined smoothly at \( \varSigma \), which leads to the following junction conditions:

*R*and

*H*while the other equations are definitions of different stellar characteristics. Replacing Eq. (21) in Eq. (30) we obtain on the surface of the star

*D*. Here we have derived it from Eq (21), which also holds in the bulk of the star. The charge alters its free term.

*H*. The redshift is positive during collapse. Then Eq. (22) shows that \(0\le H_\varSigma \le 1\). When \(H_\varSigma =0\) we obtain from Eq. (16) and the junction conditions

*D*, but

*R*enters in an algebraic way, like in the neutral case. Thus we get an expression for

*R*in terms of

*D*and

*Q*

*H*

*Q*by \( l\left( r\right) \), then the junction equation \(p_r=qB\) will hold everywhere in the interior, due to Eq. (21). A simpler possibility is to keep

*Q*constant, which we choose. All stellar characteristics become functions of

*D*and

*Q*. Obviously, \(D\ge 0\), \(\dot{D}<0\), since both

*H*and

*R*are non-negative and decreasing. Furthermore, Eqs. (33, 34) show that \(\dot{D}>-1/2 \). Thus

*H*. With its help, Eq. (16) acquires a simple form on \(\varSigma \)

*H*as a generating function. In the charged case this is not possible.

All formulas in this section reduce to the formulas in [23] when \( Q=0\).

## 4 Going to the D-level

*Q*,

*D*and its time derivatives. The derivative of Eq. (33) gives

## 5 Pairs of solutions

*D*gives rise to two solutions – a neutral one with \(R_0\), \(H_0\) and \(Q=0\), and a charged one with \(R_1\), \(H_1\) and \(Q\ne 0\). We have

*Q*, hence, \(R_1\dot{R}_1>R_0\dot{R}_0\) or \(\alpha \mid \dot{R}_1\mid <\mid \dot{R}_0\mid \). Therefore

## 6 Charging known solutions

*x*is

*a*is a constant, while \(f\left( r\right) \) is an arbitrary function. When \( f\left( r\right) =1\) we have a travelling wave with speed 1 /

*a*. One easily finds that

*x*. Equations (33, 34) show that \(R=R\left( x\right) \) and \( H=H\left( x\right) \) because

*Q*is a constant and from Eq. (14)

*h*satisfies a Riccati equation because

*B*does so. The latter can be seen by plugging Eqs. (8, 10) into the junction condition (30) and multiplying by \( -R^2B^2\). The result is

*B*no matter what

*Q*is. One can replace the time derivatives by

*x*derivatives as in Eq. (54). Another way to derive it is to plug the definition of

*H*(Eq. (13)) into Eq. (31), which is equivalent to Eq. (56). In the neutral case the second class of solutions in [19], which are self-similar, can be restored too. This doesn’t seem possible in the charged case.

*R*when one is working with Eq. (56) (with \(Q=0\) in this case) and then solve it for

*B*. One has to integrate the Riccati equation for every value of \( \varepsilon \).

*D*for a given

*R*by solving this equation. Let us set \(z=\beta x+\gamma \). Then

*z*it becomes the Special Riccati equation (Example 1.2.2.4)

*n*an integer, and for imaginary argument, they reduce to derivatives of exponents. This happens for

*B*it is useful to transform Eq. (14) into

*R*is given. Knowing

*D*, its charged companion follows from Eq. (45) for \(R_1\) and \(H_1=D/R_1\).

## 7 Conclusions

In this paper we give a generating function *D* for solutions of the main junction condition (31). It holds for charged, geodesic, anisotropic spherical collapse. For this purpose we have used the physically important object *H*, called the horizon function [22, 23]. It rules the appearance of a black hole (\(H=0\)). Fortunately, it and \(D=HR\) satisfy Riccati equations, simpler than the previous such equation for *B*. The junction equation has been derived from a formula, which holds in the interior of the star, unlike its previous derivations in [22, 23]. The total charge of the star *Q* enters the equation, but it remains of Riccati type for *D*, while the stellar radius *R* enters algebraically, just like in the neutral case.

The generating function is unique. One cannot choose *H* or *Z* as generating functions, as in the neutral case.

It is shown that every neutral solution has a charged companion with the same *D*. Certain inequalities between the members of such pairs have been derived. The charged member has bigger radius, surface redshift, mass and compactness than the neutral one. Its horizon function is smaller and the rate of collapse slower.

A class of the known exact neutral solution [19], containing generalised travelling waves, has been charged. We also show in a simple universal way how to find the generating function when *R* is given, like in [20], and then charge the solution.

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