Asymptotics to all orders of the Euler–Darboux equation in a triangle

Abstract

In Einstein's theory of relativity, the interaction of two collinearly polarized plane gravitational waves can be described by a Goursat problem for the Euler-Darboux equation in a triangular domain. In this paper, using a representation of the solution in terms of Abel integrals, we give a full asymptotic expansion of the solution near the diagonal of the triangle. The expansion is related to the formation of a curvature singularity of the spacetime. In particular, our framework allows for boundary data with derivatives which are singular at the corners. This level of generality is crucial for the application to gravitational waves.PDEs in the case of colliding electromagnetic plane waves [3,7]. In the linear limit, both of these coupled equations reduce to the Euler-Darboux equation (1.1).Of particular interest is the behavior of the solution of (1.1) near the triangle's diagonal edge x+y = 1. Indeed, this behavior is related to the formation of a curvature singularity of the spacetime by the mutual focusing of the colliding waves. In [26], it was observed that the solution must behave like ln(1 − x − y) as x + y → 1. In this paper, using an Abel integral representation of the solution, we derive an asymptotic expansion to all orders as x + y → 1.Our first result (Theorem 1) establishes a mathematically precise version of the classical Abel integral representation of the solution of the Goursat problem for (1.1) with given boundary data. This type of representation is well known (cf. e.g. [11,14,15]) and the main purpose of Theorem 1 is to provide a formulation that is suitable for our needs. We discuss regularity, the behavior at the boundary of D, and uniqueness of the solution under reasonable assumptions on the boundary data. In this context, reasonable means that our results can be applied to the common examples of collinear solutions (cf. e.g. [9,19,26]). In particular, we allow for boundary data with singular derivatives at the corners of D. As a consequence, our representation of the solution of (1.1) contains singular integrands, which is the main challenge in the analysis of the Goursat problem.Our second result (Theorem 2) describes the asymptotic behavior of the solution near the diagonal of D. We show that V (x, 1 − x − ) admits an asymptotic expansion to all orders of the formwhere the coefficients f j , g j are given explicitly in terms of the boundary data V 0 (x) = V (x, 0) and V 1 (y) = V (0, y), and where the error term is uniform on compact subsets

The hyperbolic Ernst equation in a triangular domain

“…Solving these two equations for E 0 andĒ 0 , we find E 0 (x) = 1 + (m 0 (x, 0)) 11 − (m 0 (x, 0)) 21 1 + (m 0 (x, 0)) 11 + (m 0 (x, 0)) 21 . for (x, y) ∈ int D δ and P ∈ F −1 (x,y) (Ω ∞ ).…”

“…However, the representation (4.10) allows for applying more classical techniques. This approach is used in [21] to compute an asymptotic expansion of the solution near the diagonal of D.…”