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Harmonic gauges

Abstract:

In fully non-linear GR, perturbative methods are non longer valid when gravity is too strong. As a consequence we have to rely on the fully non-linear Einstein’s system of equations. It constitutes a system of second order partial differential equations for the metric gαβ. In order to build gravitational geons, we focus on the 4-dimensional Einstein’s equation in vacuum with AAdS asymptotics. The unknowns are the ten metric components, and Einstein’s system provides the same number of equations. However, not all are independent: they display six degrees of freedom only. In order to get an invertible system of equations, we thus have to fix one way or another four degrees of freedom of the metric, and solve the remaining six others via Einstein’s equation. This is the so-called gauge freedom. Namely, the four degrees of freedom to be fixed a priori are equivalent to a choice of a coordinate system. So there are infinitely many possibilities to fix the gauge freedom. However, not all choices are well behaved mathematically, and not all choices make Einstein’s system of equations invertible. This is called ill-posedness and is due to the gauge invariance of the theory. Actually, there are very few gauges which are known to give robust results in the field of numerical relativity. We refer to standard textbooks [238–240] for reviews of all known and successful gauges in this area.