Fishing in Black Holes

Abstract

The coordinate system $(\bar{x},\bar{t})$ defined by $r=2m+K\bar{x}-cK\bar{t}$ 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: \[ds^2=c^2 d\bar{t}^2 + (W^2/K^2 - 2W/K) d\bar{x}^2 + 2c (1 + W/K) d\bar{x}d\bar{t} - r^2 (dθ^2 + cos^2θdϕ^2)\] 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).

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