Radiant gravitational collapse with anisotropy in pressures and bulk viscosity

Abstract

We model a compact radiant star that undergoes gravitational collapse from a certain initial static configuration until it becomes a black hole. The star consists of a fluid with anisotropy in pressures, bulk viscosity, in addition to the radial heat flow. A solution of Einstein’s field equations with temporal dependence was presented to study the dynamic evolution of physical quantities, such as the mass-energy function, the luminosity seen by an observer at infinity and the heat flow. We checked the acceptability conditions of the initial static configuration to obtain a range of mass-to-radius ratio in which the presented star model is physically reasonable. The energy conditions were analyzed for the dynamic case, in order to guarantee that the model is composed of a physically acceptable fluid within the range of the mass-to-radius ratio obtained for the static configuration or if they will be modified during the collapse.

Appendix: Junction condition

The requirement of the matching between the inner (matter distribution) and outer (radiation zone around the distribution) solution across the $\Sigma$ hipersurface is called as the junction condition problem. This condition imposes continuity of the first and second fundamental forms, the first being the metric and the second the extrinsic curvature, such as the conditions established by Israel [42]. So, when we approach $\Sigma$ through interior or exterior spacetime we must demand that

$$\begin{aligned} (ds^2_-)_\Sigma =(ds^2_+)_\Sigma =ds^2_\Sigma \end{aligned}$$

(87)

where $()_\Sigma$ means the value of () over $\Sigma$. Thus, the equations when taken on the hypersurface $\Sigma$ obtained from the inner and outer spacetimes are $\chi _\pm ^\alpha =\chi _\pm ^\alpha (\xi ^i)$. The extrinsic curvature of $\Sigma$ is given by [43]

$$\begin{aligned} K_{ij}^\pm \equiv -\eta ^\pm _\alpha \frac{\partial ^2\chi _\pm ^\alpha }{\partial \xi ^i\partial \xi ^j}-\eta _\alpha ^\pm \Gamma _{\beta \gamma }^\alpha \frac{\partial \chi _\pm ^\beta }{\partial \xi ^i}\frac{\partial \chi _\pm ^\gamma }{\partial \xi ^j} \ , \end{aligned}$$

(88)

where $\eta ^\pm _\alpha$ are the components of the vector normal to $\Sigma$ at coordinates $\chi ^\alpha _\pm$. Consequently, the second continuity condition imposed on $\Sigma$ takes the form

$$\begin{aligned} (K_{ij}^+)_\Sigma =(K_{ij}^-)_\Sigma \ . \end{aligned}$$

(89)

Following Nogueira and Chan [44], and considereing the metric given by (2), the continuity of the first fundamental form provides

$$\begin{aligned}{} & {} \left( \frac{dt}{d\tau }\right) _\Sigma =\frac{1}{A(r_\Sigma ,t)} \ , \end{aligned}$$

(90)

$$\begin{aligned}{} & {} \left( \frac{dv}{d\tau }\right) _\Sigma ^{-2}=1-2\frac{d{\textbf {r}}_\Sigma }{dv}-\frac{2m(v)}{{\textbf {r}}_\Sigma } \ , \end{aligned}$$

(91)

$$\begin{aligned} C(r_\Sigma ,t)={\textbf {r}}_\Sigma (v) = R(\tau ) \ . \end{aligned}$$

(92)

On the other hand, the continuity of the second fundamental form leads us to obtain the mass-energy function in the form

$$\begin{aligned} m= \left\{ \frac{C}{2}\left[ 1+\left( \frac{\dot{C}}{A}\right) ^2-\left( \frac{C'}{B}\right) ^2\right] \right\} _\Sigma \ \end{aligned}$$

(93)

which represents the total energy stored inside the hypersurface. This expression is equivalent to the one given by Cahill & McVittie [45].

In addition, we can take the gravitational redshift in the form

$$\begin{aligned} \left( \frac{dv}{d\tau }\right) ^{-1}_\Sigma = \frac{1}{1+z_\Sigma } = \left( \frac{C'}{B}+\frac{\dot{C}}{A}\right) _\Sigma \ , \end{aligned}$$

(94)

and we can see that this expression diverges when we do

$$\begin{aligned} \left( \frac{C'}{B}+\frac{\dot{C}}{A}\right) _\Sigma = 0\ \end{aligned}$$

(95)

The continuity of the extrinsic curvature also leads us to write the identity

$$\begin{aligned} \left( \frac{C}{2B^2}G_{11}^-\right) _\Sigma = \left( -\frac{C}{2AB}G_{01}^-\right) _\Sigma \ . \end{aligned}$$

(96)

Applying the equations (10) and (12) to the equality above we find [44]

$$\begin{aligned} (P_r+4\eta \sigma -\zeta \Theta )_\Sigma = (qB)_\Sigma \ , \end{aligned}$$

(97)

which represents a generalization of the junction condition $(P_r)_\Sigma =0$ in the presence of heat dissipation and viscosities.

Finally, the total luminosity of the star received by an observer at rest at infinity, denoted by $L_\infty$, is given by [46]

$$\begin{aligned} L_\infty =-\left( \frac{dm}{dv}\right) _\Sigma =-\left[ \frac{dm}{dt}\frac{dt}{d\tau }\left( \frac{dv}{d\tau }\right) ^{-1}\right] _\Sigma \ . \end{aligned}$$

(98)

So, replacing (90), (93), (94) and (97) in (98) we can write

$$\begin{aligned} L_\infty =\frac{k}{2}\left[ C^2\ (P_r+4\eta \sigma -\zeta \Theta )\left( \frac{\dot{C}}{A}+\frac{C'}{B}\right) ^2\right] _\Sigma \ . \end{aligned}$$

(99)