The many ways of the characteristic Cauchy problem
This paper first reviews and then presents new approaches to the characteristic initial value problem for the Einstein equations formulated with initial data given on either two transverse hypersurfaces or on the light cone. It will be of interest to any researcher interested in the formulation and solution of such characteristic initial value problems.NEWLINENEWLINE\textit{A. D. Rendall}'s [Proc. R. Soc. Lond., Ser. A 427, No. 1872, 221--239 (1990; Zbl 0701.35149)] pioneering mathematically sophisticated approach for vacuum Einstein is reviewed together with its utility in analyzing certain non-vacuum situations, e.g., Einstein-scalar, Einstein-Maxwell, and Einstein-Yang-Mills. It is noted, however, that his approach will not work for Einstein-Vlasov. A new approach is needed and is given. Whereas Rendall's approach uses a family of two-dimensional conformal Riemannian metrics as initial data, with the Raychaudhuri equation determining the conformal factor, here a simpler more physical interpretation is posited. The entire metric tensor is viewed as the initial datum on the characteristic surface and it must satisfy one single constraint equation, the Raychaudhuri equation. With this in mind the authors illustrate the construction of solutions thereof. The Hayward gauge condition is employed, as is the affine-parameterization gauge. (In an appendix complications of the Hayward gauge, in the light cone case, are addressed).NEWLINENEWLINEFinally, other proposals for initial data are considered. A past proposal to use the shear tensor as the free data for the gravitational field is considered and technical difficulties overcome by using a frame formulation to account for its tracelessness. Another past proposal, this one by \textit{H. Friedrich} [Proc. R. Soc. Lond., Ser. A 375, 169--184 (1981; Zbl 0454.58017)], to use certain components of the Weyl tensor as initial data is extended to any dimension and technical difficulties again overcome using a frame formulation to account for the tracelessness.
scientific article; zbMATH DE number 6067935