Fermi Coordinates and Penrose Limits

TL;DR

The paper presents a covariant framework to formulate the Penrose plane-wave limit using null Fermi coordinates, yielding a direct, curvature-based expansion of the original metric around its Penrose limit. By systematically constructing null Fermi coordinates and deriving the metric expansion in terms of Riemann tensor components, the authors obtain the Penrose limit at leading order and explicit higher-order corrections, including a first-order term compatible with light-cone gauge in string theory. They demonstrate a Weyl-tensor peeling analogue in any dimension and apply the method to AdS$_5\times$S$^5$, recovering known quadratic corrections with a transparent curvature interpretation. The results offer a geometrically natural, covariant alternative to Rosen-Brinkmann constructions, with practical relevance to string theory in plane-wave backgrounds and perturbations thereof.

Abstract

We propose a formulation of the Penrose plane wave limit in terms of null Fermi coordinates. This provides a physically intuitive (Fermi coordinates are direct measures of geodesic distance in space-time) and manifestly covariant description of the expansion around the plane wave metric in terms of components of the curvature tensor of the original metric, and generalises the covariant description of the lowest order Penrose limit metric itself, obtained in hep-th/0312029. We describe in some detail the construction of null Fermi coordinates and the corresponding expansion of the metric, and then study various aspects of the higher order corrections to the Penrose limit. In particular, we observe that in general the first-order corrected metric is such that it admits a light-cone gauge description in string theory. We also establish a formal analogue of the Weyl tensor peeling theorem for the Penrose limit expansion in any dimension, and we give a simple derivation of the leading (quadratic) corrections to the Penrose limit of AdS_5 x S^5.