Curved-spacetime metric generated by Planckian energy string collisions

Abstract

We study Planckian energy string collisions in flat spacetime as the scattering of a string in the effective curved background produced by the others as the impact parameter $b$ decreases. We find the effective energy-density distribution $\ensuremath{\sigma}(\ensuremath{\rho})\ensuremath{\sim}\mathrm{exp}{\frac{{\ensuremath{-}\ensuremath{\rho}}^{2}}{{\ensuremath{\Delta}}^{2}}}$, generated by these collisions. Two different regimes can be studied: intermediate impact parameters ${x}^{d}<<b<<\sqrt{{\ensuremath{\alpha}}^{\ensuremath{'}}\mathrm{ln}s}\ensuremath{\simeq}\frac{\ensuremath{\Delta}}{2}$ (${x}^{d}$ characterizing the string fluctuations) and large impact parameters $b>>\sqrt{{\ensuremath{\alpha}}^{\ensuremath{'}}\mathrm{ln}s}\ensuremath{\simeq}\frac{\ensuremath{\Delta}}{2}>>{x}^{d}$. The effective metric generated by these collisions for large $b$ is a gravitational shock wave of profile $f(\ensuremath{\rho})\ensuremath{\sim}p{\ensuremath{\rho}}^{4\ensuremath{-}D}$, i.e., the Aichelburg-Sexl (AS) geometry for a pointlike particle of momentum $p$. For intermediate $b$, $f(\ensuremath{\rho})\ensuremath{\sim}q{\ensuremath{\rho}}^{2}$, corresponding to an extended source of momentum $q$. The scattering matrix in this geometry and its implications for the string collision process are analyzed. We show that the poles $iGs=n$, $n=0,1,2,\dots{}$, characteristic of the scattering by the AS geometry are absent here, due to the extended nature of the effective source.