Kinematic interpretation of string instability in a background gravitational field

The unstable regime in which the string oscillating modes develop imaginary frequencies is shown to be characterized, kinematically, by a positive relative acceleration among the different points of the string. Instability occurs when this acceleration, induced by the background curvature, is large enough to make the extension of the corresponding causally connected region smaller than the string maximal size. This kinematic characterization is applied, in particular, to discuss string instability in a static and spherically symmetric gravitational background.