d v = d t + \frac{d r}{f ( r )}, d χ = d ϕ + \frac{Ω ( r )}{f ( r )} d r .

Thus

d t = d v - d r / f

and

d ϕ = d χ - Ω d r / f

, so the potentially singular terms cancel:

d ϕ - Ω d t = d χ - Ω d v,

and

- f (d v - \frac{d r}{f})^{2} + \frac{d r ^{2}}{f} = - f d v^{2} + 2 d v d r .

The metric becomes

d s^{2} = - f (r) d v^{2} + 2 d v d r + r^{2} [d χ - Ω (r) d v]^{2} .

For

r_{+} > r_{-}

v = t + \frac{L ^{2}}{2 ( r _{+}^{2} - r _{-}^{2} )} [r_{+} lo g \frac{r - r _{+}}{r + r _{+}} - r_{-} lo g \frac{r - r _{-}}{r + r _{-}}]

and

χ = ϕ + \frac{L}{2 ( r _{+}^{2} - r _{-}^{2} )} [r_{-} lo g \frac{r - r _{+}}{r + r _{+}} - r_{+} lo g \frac{r - r _{-}}{r + r _{-}}],

with the extremal case obtained by taking the limit. The transformed metric contains neither

1/ f

nor any other singular coefficient at

r = r_{+}

; since

f

Ω

, and

r^{2}

are analytic there, it gives an analytic extension across the outer horizon.

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