Collision of two general geodesic particles around a Kerr black hole

We obtain an explicit expression for the center-of-mass (CM) energy of two colliding general geodesic massive and massless particles at any spacetime point around a Kerr black hole. Applying this, we show that the CM energy can be arbitrarily high only in the limit to the horizon and then derive a formula for the CM energy of two general geodesic particles colliding near the horizon in terms of the conserved quantities of each particle and the polar angle. We present the necessary and sufficient condition for the CM energy to be arbitrarily high in terms of the conserved quantities of each particle. To have an arbitrarily high CM energy, the angular momentum of either of the two particles must be fine-tuned to the critical value $L_{i}=Ω_{H}^{-1}E_{i}$, where $Ω_{H}$ is the angular velocity of the horizon and $E_{i}$ and $L_{i}$ are the energy and angular momentum of particle $i$ ($=1,2$), respectively. We show that, in the direct collision scenario, the collision with an arbitrarily high CM energy can occur near the horizon of maximally rotating black holes not only at the equator but also on a belt centered at the equator. This belt lies between latitudes $\pm acos(\sqrt{3}-1)\simeq \pm 42.94^{\circ}$. This is also true in the scenario through the collision of a last stable orbit particle.

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