Global existence of smooth solutions to relativistic string equations in Schwarzschild space-time for small initial data

Abstract

This paper is concerned with the motion of relativistic strings in Schwarzschild space-time. In a general framework, we first analyze the basic equations for the motion of a [Formula: see text]-dimensional extended object in a general enveloping space-time [Formula: see text], which is a given Lorentzian manifold, and then we investigate some important properties enjoyed by the equations for the motion of relativistic strings in Schwarzschild space-time. In particular, the equations are shown to form a totally linearly degenerate system of first-order hyperbolic equations in [Formula: see text] dimensions. Based on this observation and under suitable assumptions, we are able to prove the global existence of smooth solutions to the Cauchy problem for the equations of motion of relativistic strings (in Schwarzschild space-time) with sufficiently small arc length.