List of Footnotes

1 In this work, the linearized Einstein equations in terms of the conformal metric in a neighborhood of 𝒥 were regularized and made hyperbolic. Thus local existence results applied to these equations showed that perturbations with smooth initial data at a hypersurface reaching 𝒥 stayed bounded during evolution, and so the asymptotic structure was preserved to the future of that hypersurface.
2 Actually, for the machinery of proving well posedness, finite differentiability to some higher order is needed.
3 Extra conditions have to be imposed, like entropy growth across shocks, to obtain uniqueness.
4 That is, sufficiently smooth for the Sobolev energy, with minimum differentiability possible to close the bounds, to exist.
5 One could of course put a discontinuity on the derivatives of the metric on initial data. The discontinuity would propagate along the characteristics. In general this is not considered a shock, for it is not generated by the dynamics and does not propagate along a different characteristic than the neighboring continuous regions. These are called contact discontinuities.
6 If matter sources are included, like fluids, then one might need to consider flux conservative schema for the whole system of equations. But probably it would be much better to use relativistic dissipative fluids – whose global existence for small data has been proven recently [49] – to dispense of shocks altogether when considering weak data.
7 If 𝜀abcd is the Levi-Civita tensor corresponding to the physical metric, ab g and &tidle;𝜀abcd the one corresponding to background metric, ab &tidle;g, then √ - 𝜀abcd = g&tidle;𝜀abcd.
8 Note that the lapse used in this work already had the generalized harmonic time gauge built into it.
9 Basically one is able to bound norms on the solution by different norms on the initial data, with the last one involving more derivatives, but with smaller and smaller derivatives as the order of the derivatives in the norms increases.