Asymptotically shearfree congruences in (2,2) spacetimes and Burgers' equation

TL;DR

The paper demonstrates that asymptotically shearfree congruences in (2,2) spacetimes are governed locally by solutions to a pair of forced Burgers' equations, linking geometric structures at infinity to integrable PDEs.

Contribution

It establishes a novel connection between shearfree congruences in (2,2) spacetimes and solutions to specific forced Burgers' equations, revealing a new geometric-PDE correspondence.

Findings

Shearfree congruences are determined by solutions to Burgers' equations.

The functions {\sigma} and {\sigma}' are derived from the projective structure on scri.

The approach links conformal infinity geometry to integrable PDEs.

Abstract

The paper proves that any asymptotically shearfree congruence at the conformal infinity (scri) in a (2,2)-signature spacetime is determined locally by a solution to the pair of forced inviscid Burgers' equations L_u+LL_x={\sigma}(u,x,y,L) and M_u+MM_y={\sigma}'(u,x,y,M) where u,x,y are Bondi coordinates of scri. The functions {\sigma} and {\sigma}' are determined naturally by the projective structure on the {\alpha} and {\beta} surfaces that foliate scri.