In order to write the conformal curve equations (24a)-(24b) for the Reissner-Nordström metric, it is noticed that the metric l in the warped product line element (23) is given by
l = D(r)dt ⊗ dt − D −1 (r)dr ⊗ dr. Equations (24a)-(24b) implyt + ∂rD(r) D(r)r t = 1 D(r) βr ,(32a)r − ∂rD(r) 2D(r)r 2 + D(r)∂rD(r) 2t 2 = D(r)βt ,(32b)
where consistent with the notation of section 3.2 we have setr ≡ r(τ ),t ≡ t(τ ). Initial **data** for these equations is prescribed by observing the discussion of Section 4.1, and by requiringẋ to be given initially by the unit normal toS. It follows that
t * = 0, r * > r + ,t * = 1 √ D * ,r * = 0, (b t ) * = 0, (b r ) * = − 2 r * ,(33)whereb t ≡ b , ∂ t ,b r ≡ b , ∂ r .
Notice that r * =r * , t * =t * . As a consequence of the symmetry of the hypersurfaceS with respect to the bifurcation sphere at r * = r + , it is only necessary to consider the case r * > r + . The equations (32a)-(32b) can be decoupled by making use of thẽ g-normalisation condition
D(r)t 2 − 1 D(r)r 2 = 1.(34)
Solving the latter for t ≥ 0 and substituting into (32b), one obtains that
r + 1 2 ∂rD(r) − β D(r) +r 2 = 0.(35)
This equation can be integrated once to yield
D(r) +r 2 − βr = γ,
where γ is a constant given in terms of the initial **data** by
γ = − D * . It follows thatr = ± (γ + βr) 2 − D(r),(36)
with the sign depending on the value of r * .