Bondi-Sachs Formalism

Abstract

The Bondi-Sachs formalism of General Relativity is a metric-based treatment of the Einstein equations in which the coordinates are adapted to the null geodesics of the spacetime. It provided the first convincing evidence that gravitational radiation is a nonlinear effect of general relativity and that the emission of gravitational waves from an isolated system is accompanied by a mass loss from the system. The asymptotic behaviour of the Bondi-Sachs metric revealed the existence of the symmetry group at null infinity, the Bondi-Metzner-Sachs group, which turned out to be larger than the Poincare group.Comment: Scholarpedia review article, an up-to-date version can be found under http://scholarpedia.org/article/Bondi-Sachs_Formalis

Sachs’ free data in real connection variables

“…Among their special features, they admit a gauge-fixing for which the Einstein's equations can be integrated hierarchically, and constraint-free initial data identified, as shown by Sachs [1]; and provide a framework for the description of gravitational radiation from isolated systems and of conserved charges, starting from the seminal work of Sachs, of Bondi, van der Burg and Metzner (henceforth BMS), Newman and Penrose (NP), Geroch and Ashtekar [2][3][4][5][6][7][8][9][10][11][12] (see also [13][14][15] and reference therein). These classic results are based on the Einstein-Hilbert action and the spacetime metric as fundamental variable, and provide a clear geometric picture of the physical degrees of freedom of general relativity at the non-linear level.…”

“…Its location can be fixed with a measure-zero gauge-fixing g 11 | r 0 = 0. Then, as shown originally in [1] (see also [13,35,36]), constraint-free initial data for general relativity can be identified with the conformal class of 2d space-like metrics γ AB , of which we take the uni-modular representativeγ AB := γ −1/2 γ AB ; supplemented by boundary data at the corner S 0 between the two initial slices. 6 Up to the measure-zero corner data, the two independent components ofγ AB are the two physical degrees of freedom of general relativity on a null hypersurface.…”

“…In particular, the projection (χ i +χ i )ϕ i gives the nullfoliation condition χ 2 = 1, namely g 00 = 0, and its stabilisation plays an important role in recovering all of Einstein's equations. 14 13 In [24] we rescaled the action by a factor 1/2, to avoid a number of factors of 2 when computing Poisson brackets. Here we restore the conventional units.…”

“…For an expression of this equation in metric language, see e.g. [13]. The point is that if the connection is initially independent from the metric, this is a PDE with a single time derivative in the term ∆σ; but this term can be eliminated using an algebraic Bianchi identity, or 'eliminant relation' in the terminology of [55].…”

“…The use of a tetrad adapted to a 2 + 2 foliation is common, e.g. [13,66], but not universal. In particular in [1] the partial Bondi gauge is completed with N 11 = 0 = N 00 = c 0 , so to have both 1-forms dφ α null.…”