Figure 1

Effective potential (left) and orbits (right) for radial timelike particles on the brane. These graphs correspond to $l=1$, $M=0,5$. Notice that geodesics have an oscillatory behavior outside the event horizon; however, there are no stable orbits since some of the particles could cross it and never return. Particles with $E<0$ are not allowed since $V_{\mathrm{eff}}=0$ at the event horizon.Reuse & Permissions

Figure 2

Effective potential for lightlike (left) and timelike (right) brane particles with different angular momenta ($L=1,2,5$). Remark that all the potentials cross themselves at the event horizon. The lightlike potential has an asymptote given by $L/l$, while the timelike potential grows with no limit.Reuse & Permissions

Figure 3

Orbits for lightlike (left) and timelike (right) brane geodesics with angular momentum ($L=2$). The energies of the particles are shown in the legend. Notice that in the lightlike case orbits with low energy fall into the event horizon whereas particles with high energy can escape from the black hole. For timelike geodesics the orbits have basically the same shape as those in the radial case [Fig. 1b]; the only effect of $L$ is to increase the amplitude of the oscillation.Reuse & Permissions

Figure 4

Effective potential for timelike geodesics on the brane for BTZ with charge (left) and with scalar field (right). These graphs correspond to $Q=1,2,3$ and $\zeta =1,5,10$. In the former case the potential admits oscillating orbits that can fall into the event horizon when $Q$ is small; however, as $Q$ grows, the minimum of the potential is shifted outside the horizon making possible the existence of stable oscillating or bounded geodesics. In the latter case only unstable oscillations are allowed.Reuse & Permissions

Figure 5

Effective potential and timelike orbits for radial particles in the bulk. From the oscillating behavior of the trajectories we can infer the existence of shortcuts since particles leaving the brane can return to it. Note that the trajectory with the energy that corresponds to the minimum of the effective potential ($E=1$) is entirely on the brane.Reuse & Permissions

Figure 6

Orbits for lightlike (left) and timelike (right) bulk particles with angular momentum ($K=2$, $L=0$). Lightlike geodesics show a scattering behavior. While timelike particles oscillate near the brane, and the higher their energies are, the closer to the brane they can reach, but never cross it.Reuse & Permissions

Figure 7

Bulk effective potential for $f=1$, with $L=0,2$ and $K=1$. We can see that close to the brane the potential has a scattering behavior, while far from the brane it becomes constant, so that any bulk geodesic behaves like a free particle.Reuse & Permissions

Figure 8

Lightlike (left) and timelike (right) brane geodesics for BTZ solution with angular momentum. Both positive and negative branches are plotted. The energies of each geodesic are displayed on the legend. Notice that in both cases geodesics can cross the horizon, some of them after a roundabout and some others directly. Only in the lightlike case particles with high energies can escape from the black hole allowing energy extraction.Reuse & Permissions

Figure 9

Lightlike (left) and timelike (right) geodesics in the bulk for BTZ with angular momentum case with $f=\cosh (\rho /2\sqrt{\alpha })$ (up) and $f=1$ (down). In both cases we observe that particles cannot reach the brane. We can infer that the brane acts as a repulsive barrier both confining particles already on the brane and preventing the entrance of bulk particles.Reuse & Permissions