Fishing in Black Holes

The coordinate system $(\bar{x},\bar{t})$ defined by $r = 2m + K\bar{x}- c K \bar{t}$ and $t=\bar{x}/cK - 1 /cK \int_{r_a}^r (1- 2m/r + K^2)^{1/2} (1 - 2m/r)^{-1}dr$ allow us to write the Schwarzschild metric in the form: $$ds^2=c^2 d\bar{t}^2 + (W^2/K^2 - 2W/K) d\bar{x}^2 + 2c (1 + W/K) d\bar{x}d\bar{t} - r^2 (d\theta^2 + cos^2\theta d\phi^2)$$ with $W=(1 - 2m/r + K^2)^{1/2}$, in which the coefficients' pathologies are moved to $r_K = 2m/(1+K^2)$. This new coordinate system is used to study the entrance into a black hole of a rigid line (a line in which the shock waves propagate with velocity c).