Curved-spacetime metric generated by Planckian energy string collisions

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 $\sigma \left(\rho \right)\sim \exp \left\{\frac{{-\rho }^2}{{\Delta }^2}\right\}$, generated by these collisions. Two different regimes can be studied: intermediate impact parameters $x^d<<b<<\sqrt{{\alpha }^{\prime }\ln s}\simeq \frac{\Delta }{2}$ ($x^d$ characterizing the string fluctuations) and large impact parameters $b>>\sqrt{{\alpha }^{\prime }\ln s}\simeq \frac{\Delta }{2}>>x^d$. The effective metric generated by these collisions for large $b$ is a gravitational shock wave of profile $f\left(\rho \right)\sim p{\rho }^{4-D}$, i.e., the Aichelburg-Sexl (AS) geometry for a pointlike particle of momentum $p$. For intermediate $b$, $f\left(\rho \right)\sim q{\rho }^2$, 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.