Penrose Limits vs String Expansions

Understand

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.