Self-similar scalar field collapse

A spherically symmetric collapsing scalar field model is discussed with a dissipative fluid which includes a heat flux. This vastly general matter distribution is analyzed at the expense of a high degree of symmetry in the space-time, that of conformal flatness and self-similarity. Indeed collapsing models terminating into a curvature singularity can be obtained. The formation of black holes or the occurrence of naked singularities depends on the initial collapsing profiles.

Figure 1

Evolution of the conformal factor $B(z)$ with respect to $z$ for $z>10$. The potentials are taken to be $V(\varphi )=\frac{5{\varphi }^{2/5}}{2}$ (graph on the left side) and $V(\varphi )=\frac{{\varphi }^4}{4}$ (graph on the right side).

Figure 2

Evolution of $B(z)$ with respect to $z$ for $V(\varphi )=\frac{5{\varphi }^{2/5}}{2}$, for different choices of the parameter $C_1$.

Figure 3

Evolution of $B(z)$ with respect to $z$ for $V(\varphi )=\frac{{\varphi }^4}{4}$ for different negative values of the parameter $C_1$.

Figure 4

Evolution of $B(z)$ with respect to $z$ for $V(\varphi )=-\frac{2}{{\varphi }^{1/2}}$ for different positive values of $C_1$. The qualitative behavior remains the same over different choices of the parameter.

Figure 5

Evolution of $\frac{1}{B(z)}$ with respect to $z$ for different choices of potential where $m>0$: $m=3.0$, 1.5, 0.5, i.e., $V(\varphi )=\frac{{\varphi }^2}{2}+2\frac{{\varphi }^{5/2}}{5}$, $V(\varphi )=\frac{{\varphi }^2}{2}+\frac{{\varphi }^4}{4}$ and $V(\varphi )=\frac{{\varphi }^2}{2}+2\frac{{\varphi }^{3/2}}{3}$, respectively; for different initial conditions.

Figure 6

Evolution of $\frac{1}{B(z)}$ with respect to $z$ for $m=\frac{1}{10}$ or $V(\varphi )=\frac{{\varphi }^2}{2}+10\frac{{\varphi }^{11/10}}{11}$. The case where $|D^{\circ }(z)|\ll D(z)$ is plotted on the lhs and the case where $|D^{\circ }(z)|\sim D(z)$ is on the rhs.

Figure 7

Evolution of $\frac{1}{B(z)}$ with respect to $z$ for $m=-\frac{1}{10}$ or $V(\varphi )=\frac{{\varphi }^2}{2}+10\frac{{\varphi }^{9/10}}{9}$ and for $m=-2.0$ or $V(\varphi )=\frac{{\varphi }^2}{2}-\frac{1}{\varphi }$ for different initial conditions defined by $|D^{\circ }(z)|\ll D(z)$ or $D^{\circ }(z)\sim D(z)$. The qualitative behavior seems to be independent of the initial choice of parameters.

Figure 8

Evolution of $B(z)$ with respect to $z$ for $m=-7.0$, or $V(\varphi )=\frac{{\varphi }^2}{2}-\frac{1}{6{\varphi }^6}$ for different initial conditions.