Perturbation of the junction condition and doubly gauge-invariant variables

The junction condition across a singular surface in general relativity, for mulated by Israel, has double covariance. In this paper, a general perturba tion scheme of the junction condition around an arbitrary background is giv en in a doubly covariant way. After that, as an application of the general scheme, we consider perturbation of the junction condition around a backgro und with the symmetry of a (D - 2)-dimensional constant-curvature space, wh ere D is the dimensionality of the spacetime. The perturbed junction condit ion is written in terms of doubly gauge-invariant variables only. Since the symmetric background includes cosmological solutions in the brane-world as a special case, the doubly gauge-invariant junction condition can be used as basic equations for perturbations in the brane-world cosmology.