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arXiv · 2111.14794

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

Thanasis Giannakopoulos·Nigel T. Bishop·David Hilditch·Denis Pollney·Miguel Zilhao
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Abstract

The characteristic initial (boundary) value problem has numerous applications in general relativity (GR) involving numerical studies, and is often formulated using Bondi-like coordinates. Recently it was shown that several prototype formulations of this type are only weakly hyperbolic. Presently we examine the root cause of this result. In a linear analysis we identify the gauge, constraint and physical blocks in the principal part of the Einstein field equations in such a gauge, and show that the subsystem related to the gauge variables is only weakly hyperbolic. Weak hyperbolicity of the full system follows as a consequence in many cases. We demonstrate this explicitly in specific examples, and thus argue that Bondi-like gauges result in weakly hyperbolic free evolution systems under quite general conditions. Consequently the characteristic initial (boundary) value problem of GR in these gauges is rendered ill-posed in the simplest norms one would like to employ. The possibility of finding good alternative norms, in which well-posedness is achieved, is discussed. So motivated, we present numerical convergence tests with an implementation of full GR which demonstrate the effect of weak hyperbolicity in practice.

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Thanasis Giannakopoulos, Nigel T. Bishop, David Hilditch, Denis Pollney, Miguel Zilhao. 2021-11-29. Gauge structure of the Einstein field equations in Bondi-like coordinates. https://doi.org/10.1103/physrevd.105.084055

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