  Approximations are commonly employed to find approximate solutions to the
Einstein equations. These solutions, however, are usually only valid in some
specific spacetime region. A global solution can be constructed by gluing
approximate solutions together, but this procedure is difficult because
discontinuities can arise, leading to large violations of the Einstein
equations. In this paper, we provide an attempt to formalize this gluing scheme
by studying transition functions that join approximate solutions together. In
particular, we propose certain sufficient conditions on these functions and
proof that these conditions guarantee that the joined solution still satisfies
the Einstein equations to the same order as the approximate ones. An example is
also provided for a binary system of non-spinning black holes, where the
approximate solutions are taken to be given by a post-Newtonian expansion and a
perturbed Schwarzschild solution. For this specific case, we show that if the
transition functions satisfy the proposed conditions, the joined solution does
not contain any violations to the Einstein equations larger than those already
inherent in the approximations. We further show that if these functions violate
the proposed conditions, then the matter content of the spacetime is modified
by the introduction of a matter-shell, whose stress-energy tensor depends on
derivatives of these functions.
