Revisiting the characteristic initial value problem for the vacuum Einstein field equations

Abstract

Using the Newman–Penrose formalism we study the characteristic initial value problem in vacuum General Relativity. We work in a gauge suggested by Stewart, and following the strategy taken in the work of Luk, demonstrate local existence of solutions in a neighbourhood of the set on which data are given. These data are given on intersecting null hypersurfaces. Existence near their intersection is achieved by combining the observation that the field equations are symmetric hyperbolic in this gauge with the results of Rendall. To obtain existence all the way along the null-hypersurfaces themselves, a bootstrap argument involving the Newman–Penrose variables is performed.

A symmetric hyperbolic formulation of the vacuum Einstein equations in affine-null coordinates

“…Studying the domain of existence of solutions could be studied using the techniques used by the authors of, e.g. [12].…”

“…We conclude that the evolution system defined by defined by Eqs, ( 8), (7), and ( 5) is symmetric hyperbolic with respect to the timelike direction t µ (c.f. [12] for a similar recent construction, but which instead works in double null coordinates).…”

“…Surprisingly, those authors found that the Einstein Equations, when written as an evolution system for the metric in those Bondi and affine-null coordinates, formed only a weakly-but not strongly-hyperbolic system of partial differential equations for the metric components. Following the general approach of [10][11][12], we show that it is relatively straightforward to write down a symmetric (and hence strongly) hyperbolic formulation of the Einstein equations when one works in the tetrad-based based Newman-Penrose [13] formalism [14]. Unfortunately, our symmetric hyperbolic evolution system does not have the same convenient nested structure as the Einstein equations written as an evolution system for the metric components in Bondi-like coordinates.…”

The conformal Einstein field equations and the local extension of future null infinity

“…This is very similar to the one used in Papers I and II and makes use of a gauge which we will call Stewart's gauge. The reader is referred to [3,5] for further details and discussion -see also [18,4].…”

“…This research programme is motivated by the techniques introduced by Luk in [1] to obtain local existence results for the Einstein field equations which are optimal in the sense that one obtains a solution in a neighbourhood of both the initial null hypersurfaces and not only in a neighbourhood of their intersection as in Rendall's original approach [2]. In Paper I of this series, see [3], we obtained an improved local existence result for the CIVP for the Einstein field equations expressed in terms of the Newman-Penrose formalism and a gauge due to Stewart -see [4]. This result demonstrates the robustness of Luk's approach, showing that the specific choice of gauge employed in [1] is not crucial.…”