Gauge structure of the Einstein field equations in Bondi-like coordinates

Abstract

A classic strategy [55,56] for well-posedness analysis of the CIVP is to reduce to an initial value problem (IVP). Well-posedness in L 2 for the IVP is characterized by strong hyperbolicity [57,58]. The IVP of a WH PDE system is ill-posed in L 2 , but it may be wellposed in a different norm [59]. This well-posedness is however delicate and, unlike the well-posedness of a SH PDE, can be broken by source terms. Well-posedness is a necessary condition for a numerical approximation of a PDE problem to converge to the continuum solution with increasing resolution. Convergence here is to be understood in terms of a discretized version of the norm in which the continuum problem is well-posed. An error estimate obtained from the numerical solutions of an ill-posed PDE problem should, a priori, be treated with great care. Therefore, well-posedness of the Bondi-like CIVP and CIBVP in GR is a particularly pressing open question for studies that focus on accuracy.The result of [54] was that two commonly used Bondilike gauges give rise to second order PDE systems that are only WH outside of the spherical context. Toy models that mimic this structure were used to demonstrate the effect of weak hyperbolicity in numerical experiments. In this paper we examine the cause of this result and, following [60,61], identify this weak hyperbolicity as a pure gauge effect. We argue that the construction upon radial null geodesics renders the vacuum EFE only WH in all Bondi-like gauges. We explicitly show the effect of weak hyperbolicity in numerical experiments in full GR formulated in the Bondi-Sachs proper gauge. This result implies that the CIVP and CIBVP of GR in vacuum are ill-posed in the simplest norms one might like to employ when formulated in these gauges. This issue can potentially be sidestepped by working with alternative norms, or higher derivatives of the metric, which might be taken explicitly as evolved variables, or simply placed within the norm under consideration. The latter tack has been successfully followed in for example [62][63][64][65].In Sec. II we map the Bondi-like equations and variables to the ADM ones, so that we can straightforwardly apply the aforementioned tools. In Sec. III we summarize the relevant theory and the structure of the principal part resulting from gauge freedom. In Sec. IV we analyze the affine null gauge [16], showing it to be only WH, both in the characteristic and in the equivalent ADM setups. In Sec. V A this analysis is repeated for the Bondi-Sachs gauge proper [66,67] in the ADM setup. In Sec. V B the calculation is repeated for the double null gauge [68]. We argue that all Bondi-like gauges possess this pure gauge structure. In Sec. VI we examine the numerical consqeuences of WH by performing robust-stabilitylike [69][70][71] tests. The results are compared against those of an artificial SH system. The tests are performed using the publicly available PITTNull thorn of the Einstein Toolkit [72]. We conclude in Sec. VII. Geometric units are used throughout. Our scripts are available in...