implies their motion along null geodesics obeying the equations ($\lambda$ is an affine parameter) $$\frac{\mathrm{D} p^{\mu}}{\mathrm{d} \lambda} = 0,\qquad
      p^{\mu}p_{\mu} = 0.                                          \label{EQgeod}$$

Due to the existence of two Killing vector fields: the temporal $\partial/\partial t$ one, and the azimuthal $\partial/\partial \phi$ one, two conserved components of the 4-momentum must exist: $$E=-p_{t}\quad\mbox{(energy),}\qquad
      L=p_{\phi}\quad\mbox{(axial angular momentum)}.              \label{EQconserv}$$ Moreover, the motion plane is central. For a single-particle motion, one can set $\theta=\pi/2=\mathrm{const}$, choosing the equatorial plane.

The motion along null-geodesics is independent of energy (frequency) and can conveniently be described in terms of the impact parameter $$\ell = \frac{L}{E}.                                           \label{EQimpar}$$ Then (\[EQgeod\]) yields the relevant equation governing the radial motion in the form $$(p^{r})^{2} = A^{-2}(r)B^{-2}(r)E^{2}
        \left(1-A^{2}(r)\frac{\ell^{2}}{r^{2}}\right).              \label{EQgovmot}$$ Clearly, the energy $E$ is irrelevant and can be used for rescalling of the affine parameter $\lambda$. The radial motion is restricted by an effective potential related to the impact parameter $\ell$, and defined by the relations $$\ell^{2} \leq V{}_{\mathrm{eff}} =\left\{
      \begin{array}{lll}
        V{}_{\mathrm{eff}}^{\mathrm{int}} = \displaystyle\frac{r^2}{(A^{-}(r))^2}  & \quad\mbox{for} & r\leq R\\
        V{}_{\mathrm{eff}}^{\mathrm{ext}} = \displaystyle\frac{r^2}{(A^{+}(r))^2}  & \quad\mbox{for} & r > R.
      \end{array}\right.$$ $V{}_{\mathrm{eff}}^{\mathrm{int}}$ is the effective potential of the null-geodetical motion in the internal braneworld spacetime and $V{}_{\mathrm{eff}}^{\mathrm{ext}}$ is the effective potential of the null-geodetical motion 