Title: Extremal limits and Bañados-Silk-West effect Show Chinese title

Title: 极端极限和巴纳多斯-西尔克-韦斯特效应

Abstract: A fascinating property of extremal Kerr black hole (BH) is that it could be act as a particle accelerator with infinite high center-of-mass (CM) energy \cite{bsw}. In this note, we would like to discuss about such fascinating result and to point out that this infinite energy at the event horizon comes solely due \emph{to the singular nature of the extremal limit}. We also show that a non-extremal Kerr BH can \emph{not} transform into extremal Kerr BH by the Ba\~{n}ados-Silk-West mechanism. Moreover, we discuss about three possible geometries (near extremal, purely extremal and near horizon of extremal Kerr) of this mechanism. We further prove that near extremal geometry and near horizon geometry, precisely extremal geometry of extremal Kerr BHs are qualitatively different. Near extremal geometry and near horizon geometry gives the CM energy is finite, whereas precisely extremal geometry gives the diverging energy. Thus, we can argue that extremal Kerr BH and non-extremal Kerr BH are quite distinct objects. Finally, we show that the CM energy of collisions of particles not only diverges at infinite red-shift surface ($r_{+}$) but it could also diverges at the ISCO ($r_{isco}$) or at the circular photon orbit ($r_{cpo}$) or at the marginally bound circular orbit ($r_{mbco}$) or at the Cauchy horizon i.e. at $r \equiv r_{isco}=r_{cpo}=r_{mbco}=r_{+}=r_{-}=M$. Show Chinese abstract

Abstract: 极端Kerr黑洞（BH）的一个迷人特性是，它可能作为粒子加速器，具有无限高的质心（CM）能量\cite{bsw}。在本注释中，我们想讨论这种迷人的结果，并指出事件视界处的无限能量仅仅是由于\emph{对极值极限的奇异性质}。我们还表明，非极端Kerr BH可以通过Bañados-Silk-West机制\emph{不}转变为极端Kerr BH。此外，我们讨论了该机制的三种可能几何结构（接近极端、纯极端和极端Kerr的视界附近）。我们进一步证明，接近极端几何和视界附近几何，即极端Kerr BH的精确极端几何在定性上是不同的。接近极端几何和视界附近几何给出的CM能量是有限的，而精确极端几何给出的是发散的能量。因此，我们可以认为极端Kerr BH和非极端Kerr BH是截然不同的物体。最后，我们展示了粒子碰撞的CM能量不仅在无限红移表面（$r_{+}$）发散，也可能在ISCO（$r_{isco}$）或圆形光子轨道（$r_{cpo}$）或边缘束缚圆轨道（$r_{mbco}$）或柯西视界处发散。 在$r \equiv r_{isco}=r_{cpo}=r_{mbco}=r_{+}=r_{-}=M$.