Ingoing Eddington-Finkelstein coordinates

The advanced coordinate

v = t + r_{*}

puts the radial Schwarzschild metric into the form

d s^{2} = - (1 - \frac{2 M}{r}) d v^{2} + 2 d v d r + r^{2} d Ω^{2},

which is regular at the future horizon.

Finkelstein diagram

r

against

T = v - r

in Ingoing Eddington-Finkelstein coordinates. Its radial null directions show explicitly that both future light-ray families decrease

r

Radial null curves comprise the ingoing family

v = constant

and the outgoing family

\frac{d r}{d v} = \frac{1}{2} (1 - \frac{2 M}{r}) .

Outside the horizon the outgoing family moves to larger

r

; inside it both future-directed families move to smaller

r

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