Observable form of pulses emitted from relativistic collapsing objects

In this work, we discuss observable characteristics of the radiation emitted from a surface of a collapsing object. We study a simplified model in which a radiation of massless particles has a sharp in time profile, and it happens at the surface at the same moment of comoving time. Since the radiating surface has finite size the observed radiation will occur during some finite time. Its redshift and bending angle are affected by the strong gravitational field. We obtain a simple expression for the observed flux of the radiation as a function of time. To find an explicit expression for the flux we develop an analytical approximation for the bending angle and time delay for null rays emitted by a collapsing surface. In the case of the bending angle this approximation is an improved version of the earlier proposed Beloborodov-Leahy-approximation. For rays emitted at $R>2R_g$ the accuracy of the proposed improved approximations for the bending angle and time delay is of order (or less) than 2%–3%. By using this approximation we obtain an approximate analytical expression for the observed flux and study its properties.

Figure 1

The figure shows $D={10}^3{\Delta }_{\theta }$ (${\Delta }_{\theta }$ is a relative error for $\Theta$) as a function of $s=l/l_{\max }$. The plots are shown for the following four values of $R/M=4.5,5,6,7$ (the smaller value $R/M$ the higher is a plot in the vicinity of $s=1$). The curves are practically indistinguishable from $D=0$ line for $s<0.5$. For this reason we plotted only $s\ge 0.5$ parts of the graphs.

Figure 2

The figure shows $D={10}^3{\Delta }_T$ (${\Delta }_T$ is a relative error for $T$) as a function of $s=l/l_{\max }$. The plots are shown for the following four values of $R/M=4.5,5,6,7$ (the smaller value $R/M$ the lower is a plot in the vicinity of $s=1$). The curves are practically indistinguishable from $D=0$ line for $s<0.5$. For this reason we plotted only $s\ge 0.5$ parts of the graphs.

Figure 3

The figure shows $D={10}^2{\Delta }_{T_{,l}}$ (${\Delta }_{T_{,l}}$ is a relative error for $T_{,l}$) as a function of $s=l/l_{\max }$. The plots are shown for the following four values of $R/M=4.5,5,6,7$. In the region where $D<0$, the smaller value $R/M$ the lower is a plot in the vicinity of $s=1$. The curves are practically indistinguishable from $D=0$ line for $s<0.5$. For this reason we plotted only $s\ge 0.5$ parts of the graphs.

Figure 4

Bending angle as a function of the time parameter $\delta$. The upper curve shows the bending angle for a freely-collapsing surface for $R_0=5.4M$ and $R_e=4.8M$. The lower curve show the bending angle for a static surface with $R_e=4.8M$.

Figure 5

Bending angle as a function of the time parameter $\delta$. The three graphs show the bending angle for the same initial radius $R_0=9M$ for three different values of $R_e$: (a) $5.5M$, (b) $6.5M$, and (c) $7.5M$.

Figure 6

Redshift factor for a freely-collapsing surface as a function of the time parameter $\delta$ for the cases I, and IIa, b, c.

Figure 7

Light curves for $\mathbf{I}=1$ (IA) and $\mathbf{I}=\cos \beta (l)$ (IB) for a freely-collapsing surface with $R_e/M=4.8$ and $R_0/M=5.4M$.

Figure 8

Light curves for $\mathbf{I}=1$ (IA) and $\mathbf{I}=\cos \beta (l)$ (IB) for a static surface with $R_e/M=4.8$.

Figure 9

Light curves for $\mathbf{I}=1$ for the cases IIa, b, c.

Figure 10

Light curves for $\mathbf{I}=\cos \beta (l)$ for the cases IIa, b, c.