Collision of localized shocks in AdS$_5$ as a series expansion in transverse gradients

TL;DR

This paper presents a new computational method using transverse gradient expansion to efficiently simulate localized shock collisions in AdS$_5$, achieving accurate results with significantly reduced computational effort.

Contribution

The authors develop a transverse gradient expansion framework that simplifies Einstein equations for shock collisions, enabling faster and accurate numerical simulations in AdS$_5$.

Findings

First-order expansion agrees well with exact solutions.

Computational speed increases by roughly tenfold.

Errors are within 10% at hydrodynamization time.

Abstract

We introduce a computational framework to more efficiently calculate the collision of localized shocks in five dimensional asymptotically Anti-de Sitter space. We expand the Einstein equations in transverse gradients and find that our numerical results agree well with exact solutions already at first order in the expansion. Moreover, the Einstein equations at first order in transverse gradients can be decoupled into two sets of differential equations. The bulk fields of one of these sets has only a negligible contribution to boundary observables, such that the computation on each time slice can be simplified to the solution of several planar shockwave equations plus four further differential equations for each transverse plane `pixel'. At the cost of errors of $\lesssim 10 \%$ at the hydrodynamization time and for low to mid rapidities, useful numerical solutions can be sped up by roughly one order of magnitude.