Schwarzschild tortoise coordinate

The tortoise coordinate satisfies

\frac{d r _{*}}{d r} = (1 - \frac{2 M}{r})^{- 1}, r_{*} = r + 2 M lo g \frac{r}{2 M} - 1 .

Outgoing Eddington-Finkelstein coordinates ( $u = t - r_{*}$ )

The retarded coordinate

u = t - r_{*}

puts the radial Schwarzschild metric into

d s^{2} = - f d u^{2} - 2 d u d r + r^{2} d Ω^{2}

, with

f = 1 - 2 M / r

. Outgoing radial null geodesics have constant

u

. These coordinates are regular at the past horizon; an infaller crossing the future horizon has

u \to + \infty

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.

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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