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High-energy particle collisions at the speed of light

The gravitational solution for the each incoming particles can be found by boosting the rest-frame $ D$-dimensional Schwarzschild solution,

  $\displaystyle ds^2=-\left(1-\frac{16\pi G_DM}{(D-2)\Omega_{D-2}}\frac{1}{r^{D-3}}\right)dt^2$    
  $\displaystyle +\left(1-\frac{16\pi G_DM}{(D-2)\Omega_{D-2}}\frac{1}{r^{D-3}}\right)^{-1}dr^2 +r^2d\Omega_{D-2}^2,$ (1)

where $ d\Omega_{D-2}^2$ and $ \Omega_{D-2}$ are the line element and volume of the unit $ D-2$-sphere and $ G_D$ is the $ D$-dimensional gravitational constant. Aichelburg-Sexl solution [6] is found by taking the limit of large boost and small mass with the fixed total energy $ \mu$. The resulting metric represents a massless particle moving in the $ +z$ direction with the speed of light:

$\displaystyle ds^2=-d\bar{u} d\bar{v}+\sum_{i=1}^{D-2}d\bar{x}_i^2+\Phi(\bar{x}_i) \delta(\bar{u})d\bar{u}^2,$ (2)

where $ \bar{u}=\bar{t}-\bar{z}$ and $ \bar{v}=\bar{t}+\bar{z}$. $ \Phi$ depends only on the transverse radius $ \bar{\rho}=\sqrt{\bar{x}_i\bar{x}^i}$ and takes the form

  $\displaystyle \Phi=-8G_4\mu\log\bar{\rho},$   for$\displaystyle ~D=4,$ (3)
  $\displaystyle \Phi=\frac{16\pi\mu G_D}{\Omega_{D-3}(D-4)}\frac{1}{\bar{\rho}^{D-4}},$   for$\displaystyle ~D>4.$ (4)

A delta function appeared in (2) shows that two coordinate systems are discontinuously connected on $ \bar{u}=0$. The continuous coordinate system can be introduced by

  $\displaystyle \bar{u}=u,$    
  $\displaystyle \bar{v}=v+\Phi\theta(u) +\frac{u}{4}\theta(u)\left(\nabla_i\Phi\nabla^i\Phi\right),$ (5)
  $\displaystyle \bar{x}_i=x_i+\frac{u}{2}\nabla_i\Phi(x_i)\theta(u),$    

where $ \theta$ is the Heaviside step function and $ \nabla_i$ is the $ (D-2)$-dimensional flat-space derivative. We can superpose the two solutions to obtain the exact geometry outside the future light cone of the collision of the shocks:

  $\displaystyle ds^2=-dudv$    
  $\displaystyle +\left(H^{(1)}_{ik}H^{(1)}_{jk}+H^{(2)}_{ik}H^{(2)}_{jk} -\delta_{ij}\right)dx^idx^j,$ (6)

where

  $\displaystyle H^{(1)}_{ij}=\delta_{ij}+\frac{u}{2}\theta(u)\nabla_i\nabla_j\Phi^{(1)}(\boldsymbol{x}),$    
  $\displaystyle H^{(2)}_{ij}=\delta_{ij}+\frac{v}{2}\theta(v)\nabla_i\nabla_j\Phi^{(2)}(\boldsymbol{x}).$ (7)

Here $ \boldsymbol{x}\equiv(x^i)$ is the point in flat $ D-2$-space that is transverse to the direction of particle motion.

The apparent horizon is defined as a closed spacelike $ D-2$-surface on which the outer null geodesic congruence have zero convergence. It was shown that the apparent horizon exists in the union of the two shock waves, $ u=0>v$ and $ v=0>u$ [3]. This apparent horizon consists of two flat discs with radii

$\displaystyle \rho_0\equiv \left(\frac{8\pi\mu G_D}{\Omega_{D-3}}\right)^{1/(D-3)},$ (8)

and $ \rho _0$ gives a characteristic scale for each dimension $ D$.


next up previous
Next: Time slicing and apparent Up: High-energy head-on collisions of Previous: Introduction
Yasusada Nambu
2002-08-23