The additional invariant is the squared norm of the affine tangent. Metric compatibility and the geodesic equation make it constant, and a photon has the null condition

g_{μν} \overset{x}{˙}^{μ} \overset{x}{˙}^{ν} = 0.

On the equatorial plane this gives

0 = - f c^{2} \dot{t}^{2} + f^{- 1} \overset{r}{˙}^{2} + r^{2} \dot{ϕ}^{2},

and the conserved quantities of part (a) yield

\overset{r}{˙}^{2} = c^{2} ε^{2} - \frac{ℓ ^{2}}{r ^{2}} (1 - \frac{2 m}{r}) .

For a nonradial ray

ℓ \neq = 0

, put

u = 1/ r

. Since

\frac{d u}{d ϕ} = \frac{- r ˙ / r ^{2}}{ℓ / r ^{2}} = - \frac{r ˙}{ℓ},

(\frac{d u}{d ϕ})^{2} + u^{2} (1 - 2 m u) = C, C = \frac{c ^{2} ε ^{2}}{ℓ ^{2}} = \frac{1}{b ^{2}} .

Here

b = ∣ ℓ ∣/ (c ε)

is the positive impact parameter for a future-directed ray arriving from infinity. A strictly radial ray has

ℓ = 0

and must be described by the radial equation rather than by

ϕ

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