Planckian Scatterings of Massive Particles and Gravitational Shock Waves

The scattering process of two particles at Planck energies or beyond is calculated using the gravitational shock wave metric for a massive black hole. Then, the scattering between a heavy mass particle and a small mass one is deal with. The cross section contains an extra new term of σ ∝ G3 as the correction term of the leading term derived by ’t Hooft. The ultrarelativistic limit of the Lorentz boosted Reissner– Norstrom black hole is also calculated. e-mail address: [email protected] 1 After the paper of ’t Hooft [1], many articles have been devoted to the study of the particle scatterings at Planck scale. When particle scatterings take place at energies of the order or larger than Planck mass, MP lanck, the gravitational interaction dominates their collision processes. Therefore this would be useful to the understanding of quantum gravity. Although a conventional quantum field theory of the gravitation is unrenormalizable, physical insights at Planck scale can be obtained from the study of quantum fields in classical geometries. For example, the classical background geometry of the ultrarelativistic particle is chosen as the gravitational shock wave ( G. S. W.) metric of a black hole. The G. S. W. metric of a black hole is the one which makes a black hole move with an ultrarelativistic speed ( Lorentz γ factor → ∞). As the black hole is Lorentz contracted, it looks like a plane black hole at this limit. In usual cases, G. S. W. metrics for the massless Schwarzschild black hole is obtained by Aichelburg and Sexl (AS) [2]. On the other hand, Loustó and Sánchez derived the G. S. W. metric for the massless Kerr– Newman black hole [3] and the massless Reissner–Norstrom (RN) black hole[4]. Recently, the authors have derived the G. S. W. metric for a massive black hole ( the Schwarzschild and Kerr black hole) [5] (HS below). We have assumed that γ factor becomes very large but finite. This is nothing but leading contribution for large γ ( not mathematically infinite) in 1/γ expansion is sought. For a particle scattering at Planckian energy or beyond, the γ factor is very large but finite. Therefore our derived metrics can be applied for the scattering of the massive particles. The mass of the black hole takes an arbitrary and finite value. Using the method of Loustó and Sánchez, the G. S. W. metric for a massive Schwarzschild black hole becomes [5]: lim γ→∞ dsS −→ −du dv + dx + dy −4p [ log ( ρ 2GM )2 − ρ 4M (