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 $d{s}^2\mathrm{}(R,T)=\frac{d{R}^2}{A(R,T)\mathrm{}}+R^{2}d{\Omega }^2-B(R,T)d{T}^2$. The Einstein field equations, including cosmological constant, are given for arbitrary $T_{\nu }^{\mu }$, and the timelike geodesic equations are solved for radial motion. We then show how to replace $T$ with a new time coordinate $\tau$ that is equal to the time measured by radially moving geodesic clocks. Cosmology is brought into the picture by setting $T_{\nu }^{\mu }$ equal to the stress-energy tensor for a perfect fluid composed of geodesic particles, and letting $\tau$ be the time measured by clocks coincident with the fluid particles. We solve the field equations in terms of ($R$,$\tau$) coordinates to get the metric coefficients in terms of the pressure and density of the fluid. The metric on the subspace $\tau =\mathrm{const}$ is equal to $d{R}^2+R^{2}d{\Omega }^2$, 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 ($R$,$\tau$) spacetime diagram, all trajectories in an Einstein—de Sitter universe are emitted from $R=0\mathrm{}$ at the "big bang" at $\tau =0\mathrm{}$. Further, a light signal coming toward $R=0\mathrm{}$ at some time $\tau >0\mathrm{}$ will, in its past history, have started from $R=0\mathrm{}$ at $\tau =0\mathrm{}$, and have turned around on the line $2R=3\tau$. A consequence of this is a "tilting" of the null cones along the trajectory of a cosmological particle. The turnaround line $2R=3\tau$ marks the transition where an $R=\mathrm{const}$ 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.