Particle collision with an arbitrarily high center-of-mass energy near a Bañados-Teitelboim-Zanelli black hole

We consider a particle collision with a high center-of-mass energy near a Bañados-Teitelboim-Zanelli (BTZ) black hole. We obtain the center-of-mass energy of two general colliding geodesic particles in the BTZ black hole spacetime. We show that the center-of-mass energy of two ingoing particles can be arbitrarily large on an event horizon if either of the two particles has a critical angular momentum and the other has a noncritical angular momentum. We also show that the motion of a particle with a subcritical angular momentum is allowed near an extremal rotating BTZ black hole and that a center-of-mass energy for a tail-on collision at a point can be arbitrarily large in a critical angular momentum limit.

Figure 1

The examples of the effective potential $V(r)=-R(r)/2$ of the radial motion of a particle with $m=E=1$ and $\delta ={10}^{-2}$ in the BTZ black hole spacetime with $r_{+}=l=1$. The solid (red), dashed (green), and dotted (blue) curves denote the effective potential in the cases I ($r_{-}=0.99$; $V^{\prime }(r_{+})>0$), II ($r_{-}=0.999$; $V^{\prime }(r_{+})<0$), and III ($r_{-}=1$; $V^{\prime }(r_{+})<0$), respectively. Notice that the spacetime is not extremal in the cases I and II and it is extremal in the case III.

Figure 2

The examples of the effective potential $V(r)=-R(r)/2$ in the extremal case. The solid (red), dashed (green), dotted (blue) and dash-dotted (purple) curves denote the effective potential for the particle with a subcritical angular momentum with $\delta ={10}^{-2}$ in the cases I ($r_{+}=r_{-}=E=m=l=1$), II ($r_{+}=r_{-}=m=l=1$ and $E=1.1$), III ($r_{+}=r_{-}=E=l=1$ and $m=1.1$), and IV ($r_{+}=r_{-}=E=m=1$ and $l=1.1$), respectively.