Natural flat observer fields in spherically symmetric space-times

An observer field in a space-time is a time-like unit vector field. It is natural if the integral curves (field lines) are geodesic and the perpendicular three-plane field is integrable (giving normal space slices). We prove that a natural observer field determines a coherent notion of time: a coordinate that is constant on the perpendicular space slices and whose difference between two space-slices is the proper time along any field line. A natural observer field is flat if the normal space slices are metrically flat. For static spherically symmetric space-times we find a necessary and sufficient condition for possession of a spherically symmetric natural flat observer field. In this case, which includes the Schwarzschild and the Kottler space-times, there is in fact a dual pair of spherically symmetric natural flat observer fields. One of these observer fields is expanding and the other contracting and it is natural to describe the expanding field as the 'escape' field and the dual contracting field as the 'capture' field. Observer fields are useful for understanding redshift and the fields described here are used in a possible explanation of redshift explored in [8].