Curvature coordinates in cosmology

We develop cosmological theory from first principles starting with curvature coordinates (R, T) in terms of which the metric has the form ds^2 (R, T) =dR^{2}A (R, T) +R^2d^2-B (R, T) dT^2. The Einstein field equations, including cosmological constant, are given for arbitrary T_^, and the timelike geodesic equations are solved for radial motion. We then show how to replace T with a new time coordinate that is equal to the time measured by radially moving geodesic clocks. Cosmology is brought into the picture by setting T_^ equal to the stress-energy tensor for a perfect fluid composed of geodesic particles, and letting be the time measured by clocks coincident with the fluid particles. We solve the field equations in terms of (R, ) coordinates to get the metric coefficients in terms of the pressure and density of the fluid. The metric on the subspace =const is equal to dR^2+R^2d^2, and so is flat, with R having the physical significance that it is a measure of proper distance in this subspace. As specific examples, we consider the de Sitter and Einstein---de Sitter universes. On an (R, ) spacetime diagram, all trajectories in an Einstein---de Sitter universe are emitted from R=0 at the "big bang" at =0. Further, a light signal coming toward R=0 at some time >0 will, in its past history, have started from R=0 at =0, and have turned around on the line 2R=3. A consequence of this is a "tilting" of the null cones along the trajectory of a cosmological particle. The turnaround line 2R=3 marks the transition where an R=const line changes from spacelike to timelike in character. We show how to apply the techniques developed here to the inhomogeneous problem of a Schwarzschild mass imbedded in a given universe in the paper immediately following this one.