Penrose Limits vs String Expansions

TL;DR

This paper compares two different expansions of string equations in curved backgrounds, showing their equivalence under specific coordinate choices and gauges, and clarifying their relation to Penrose limits and Riemann coordinate expansions.

Contribution

It establishes an exact correspondence between Penrose-Fermi and Riemann coordinate expansions of string equations in curved backgrounds under certain gauges and coordinate systems.

Findings

Exact analogy between string equations in plane waves and general backgrounds.

Higher-order expansions agree in Fermi coordinates and lightcone gauge.

Conformal gauge restricts backgrounds to Brinkmann class with specific gauge conditions.

Abstract

We analyse the relation between two a priori quite different expansions of the string equations of motion and constraints in a general curved background, namely one based on the covariant Penrose-Fermi expansion of the metric G_{\mu\nu} around a Penrose limit plane wave associated to a null geodesic \gamma, and the other on the Riemann coordinate expansion in the exact metric G_{\mu\nu} of the string embedding variables around the null geodesic \gamma. Starting with the observation that there is a formal analogy between the exact string equations in a plane wave and the first order string equations in a general background, we show that this analogy becomes exact provided that one chooses the background string configuration to be the null geodesic \gamma itself. We then explore the higher-order correspondence between these two expansions and find that for a general curved background they agree to all orders provided that one works in Fermi coordinates and in the lightcone gauge. Requiring moreover the conformal gauge restricts one to the usual class of (Brinkmann) backgrounds admitting simultaneously the lightcone and the conformal gauge, without further restrictions.