Acceleration of particles in spacetimes of black string

The particle acceleration mechanism in $S^2\times {\mathbb{R}}^1$ topology, namely, in the spacetime of the five-dimensional compact black string, has been studied. The expression of center-of-mass energy of the colliding neutral particles near static black string has been found. The collision of a charged particle moving at the innermost stable circular orbit with a neutral particle coming from infinity has been considered when black string is immersed in external uniform magnetic field. It has been shown that the unlimited center-of-mass energy can be approached in the case of the extremal rotation of the black string which is similar to the analogous effect in Kerr spacetime. We have also obtained that the scattering energy of particles in the center-of-mass system can take arbitrarily large values not only for extremal black string but also for the nonextremal one. It has been derived that the presence of the extra dimension can, in principle, increase the upper limit of efficiency of energy extraction from the extremely rotating black string up to 203% versus 143% which can be extracted from the extreme Kerr black hole.

Figure 1

Schematic picture of two particles colliding near the event horizon of a black string.

Figure 2

Effective potential of the motion for the different values of parameters $\mathcal{L}$ and $\mathcal{J}$. Thick solid curves correspond to the particles with the critical angular momentum ${\mathcal{L}}_{\mathrm{cr}}$ while dashed lines are the levels of the energy of a free particle at infinity. Thin solid curves are the potentials of the particles with the representative values of the angular momentum $\mathcal{L}=1$ and $\mathcal{L}=6$.

Figure 3

Examples of particle trajectories moving at the equatorial plane around black string. We compare motion with $\mathcal{J}=0$ (solid curve) with $\mathcal{J}=0.5$ (dashed) for all TO, BO, and EO possible orbits. Particles start from the position $r_0=10$ but with the different energies. (a) TO $\mathcal{L}=3$. (b) BO $\mathcal{L}=4$. (c) EO $\mathcal{L}=6$.

Figure 4

Acceleration of a freely falling particle along the axis $r$ by the static black string for three different values of parameter $\mathcal{J}$, $\mathcal{J}=0$ (solid), $\mathcal{J}=0.5$ (dashed), and $\mathcal{J}=1.5$ (dotted), and three representative values of the angular momentum. The left panel corresponds to the case when $\mathcal{L}=1$, the middle panel corresponds to the case when $\mathcal{L}=4.47$ and the right panel corresponds to the case when $\mathcal{L}=8$. The critical case corresponds to the dashed curve in the middle panel.

Figure 5

The variation of the energy of the collision of two particles with the opposite momenta near the static black string in the center-of-mass frame for three different values of parameter $\mathcal{J}$: ${\mathcal{J}}_{1,2}=0$ (solid), ${\mathcal{J}}_{1,2}=\pm 1$ (dashed), ${\mathcal{J}}_{1,2}=\pm 1.5$ (dotted). The event horizon is indicated by the vertical thin line.

Figure 6

Maximal energy of the collision as function of the parameter $\mathcal{J}$.

Figure 7

Velocity of the particle and its Lorenz gamma factor at the ISCO as the function of its radius for the positive angular momentum and different values of parameter $\mathcal{J}$: $\mathcal{J}=0$ (solid), $\mathcal{J}=0.7$ (dashed), and $\mathcal{J}=1.1$ (dotted).

Figure 8

Velocity of the particle at the ISCO as the function of parameter $\mathcal{J}$.

Figure 9

Dependence of the innermost stable circular orbits $r_{\mathrm{ISCO}}$ on the magnetic field parameter $\mathcal{B}$. The left panel corresponds to the motion of the particle with positive angular momentum, while the right panel corresponds to the particle with negative angular momentum $\mathcal{L}$.

Figure 10

The limiting value of the angular momentum of the particle near horizon $r=r_{+}$ of the rotating black string in dependence on the spin parameter $a$ for different values of the parameter $\mathcal{J}:\mathcal{J}=0$ (solid), $\mathcal{J}=1$ (dashed), $\mathcal{J}=1.7$ (dotted).

Figure 11

Examples of particle trajectories moving in the equatorial plane around rotating black string when the spin parameter $a=0.99$. We compare motion with $\mathcal{J}=0$ (solid curve) and with $\mathcal{J}=0.5$ (dashed) for all TO, BO, and EO possible orbits. Particles start from the initial position $r_0=10$ but with the different energies. (a) TO $\mathcal{L}=2$. (b) BO $\mathcal{L}=3$. (c) EO $\mathcal{L}=6$.

Figure 12

We compare TO for particle motion with $\mathcal{J}=0$ (solid curve) and with $\mathcal{J}=1$ (dashed one), depicting dependence of “ending angle” ${\phi }_{\mathrm{end}}$ on the specific angular momenta $\mathcal{L}$ for the cases when the spin parameter is $a=0$ and $a=0.99$. Particles start from the “infinity” (at $r_0=1000$) with energies $E\doteq 1$ (for $\mathcal{J}=0$) and $\mathcal{E}\doteq 1.4$ (for $\mathcal{J}=1$). For some critical values of angular momentum $\mathcal{L}$, the TO orbits become BO and hence are not visible on the graphs. Such critical value ${\mathcal{L}}_{\mathrm{c}}$ is growing with parameter $\mathcal{J}$. (For the case when $a=0$, we have ${\mathcal{L}}_{\mathrm{c}}=4\sqrt{1+{\mathcal{J}}^2}$.) We also see strong “winding” of particle trajectories onto the event horizon of black string when $a=0.99$. Even orbits with $\mathcal{L}=0$ have nonzero ending angle ${\phi }_{\mathrm{end}}$.

Figure 13

The radial dependence of the effective potential of the particle moving around an extremely rotating black string with $a=1$ for the different values of the angular momentum $\mathcal{L}$ in two different cases of the absence (left panel) and presence (right panel) of the parameter $J$. Dashed lines indicate the energy level of a free particle at infinity.

Figure 14

Acceleration of particle along $r$ by extremely rotating black string for three different values of the parameter $\mathcal{J}$, $\mathcal{J}=0$ (solid), $\mathcal{J}=0.5$ (dashed), and $\mathcal{J}=1$ (dotted), and three different values of the angular momentum $\mathcal{L}=0.5$ (left panel), $\mathcal{L}\approx 2.24$ (middle panel), and $\mathcal{L}=4$ (right panel). The critical case corresponds to the dashed curve in the middle panel.

Figure 15

Dependence of the critical angular momentum ${\mathcal{L}}_{\pm }$ of the particle on the parameter $\mathcal{J}$ for the different values of the spin parameter $a$ of the black string.

Figure 16

The variation of the energy of the collision of two particles with opposite momenta near the static black string in the center-of-mass frame for three different values of parameter $\mathcal{J}$. ${\mathcal{J}}_{1,2}=0$ (solid), ${\mathcal{J}}_{1,2}=\pm 1$ (dashed), ${\mathcal{J}}_{1,2}=\pm 1.5$ (dotted). The event horizon is indicated by vertical thin line.

Figure 17

The center-of-mass energy of the collision of two freely falling particles close to the horizon of the rotating black string in dependence on the spin parameter $a$ for three representative values of the parameter ${\mathcal{J}}_{1,2}$ of each particle: ${\mathcal{J}}_{1,2}=0$ (solid), ${\mathcal{J}}_{1,2}=\pm 0.5$ (dashed), and ${\mathcal{J}}_{1,2}=\pm 1$ (dotted).

Figure 18

The effective potential of the particle moving around nonextremal black string with $a=0.9$ in the absence (left panel) and the presence (right panel) of the parameter $\mathcal{J}$. The allowed zone for $\mathcal{L}=2.5$ is shaded. Dashed lines represent the energy level of a free particle at infinity.

Figure 19

The radial dependence of impact parameters for the different values of $\kappa$. The solid, dashed, and dotted lines correspond to values of $j=0$, $j=0.5$, and $j=1$, respectively.

Figure 20

Upper limits of the mass and energy of the emitted particle as functions of the energy of the incident critical particle. The left and right panels show the ratio to the mass and to the energy of the incident critical particle, respectively. The mass limits lie below the energy limits. The solid, dashed, and dotted lines correspond to the values of $\mathcal{J}=0$, $\mathcal{J}=0.5$, and $\mathcal{J}=1$, respectively.