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For $0<a<M$, the two exterior equatorial circular null geodesics of the Kerr metric have radii solving the signed equation above. The plus sign gives the prograde branch and the minus sign gives the retrograde branch. To derive it put $b=L/E$ in the double-root conditions of the equatorial Kerr null geodesic radial equation: $r^2=(b^2-a^2)/3$ and $r^3=M(b-a{)}^2$. Both exterior radii exceed $r_{+}=M+\sqrt{M^2-a^2}$ and approach $3M$ when $a\to 0$. The horizon generators are a separate case with $E=L=0$; dividing by $E$ excludes them. Constant-radius null directions at other exterior radii generally have nonzero acceleration and are not geodesics.

Reflection across $\theta =\pi /2$ is an isometry; a geodesic initially tangent to its fixed equatorial plane remains in that plane by uniqueness of the geodesic equation. Thus take $\theta =\pi /2$, $\dot{\theta }=0$, and $\Sigma =r^2$. Define the algebraic abbreviationsThe conserved quantities of part (b) becomeFor $\Delta \ne 0$, inversion givesThe null condition can be written without expanding a quadratic square:Substituting the preceding expressions yields the equatorial Kerr null geodesic radial equationAn equivalent useful polynomial form isThese expressions use an affine parameter $s$; a common rescaling of $E$ and $L$ accompanies its rescaling. The inversion was performed in the exterior $r>r_{+}$, where Boyer-Lindquist coordinates are regular. The polynomial radial identity has a smooth continuation to a horizon, but the formulas for $\dot{t}$ and $\dot{\varphi }$ there must be replaced by regular-coordinate expressions. In the limit $a=0$, the result reduces to the Schwarzschild spacetime equation ${\dot{r}}^2=E^2-(1-2M/r)L^2/r^2$.