The Cauchy problem on a lightcone differs from the one on a smooth spacelike surface in several ways; the initial surface cannot be smooth everywhere, and the initial surface being characteristic also has implications for the values the initial data may take. We show well-posedness of the vacuum Cauchy problem for a smooth spacelike initial surface in arbitrary dimension. The initial data for this problem is related to initial data for the characteristic Cauchy problem. We derive the Raychaudhuri equation for lightlike geodesics and use it to analyze initial data for the Cauchy problem on a lightcone.
We show how to derive the extrinsic curvature of a lightcone using a field of connection coefficients and the metric. We discuss the Raychaudhuri equation as a constraint equation if one uses the metric as initial data. Alternatively, we show how to use the Raychaudhuri equation to complete initial data when a conformal class of metrics is used as initial data instead. In this case, we give an exact expression for the conformal factor associated with a representative of the conformal class. ... Read more