Mathematics – Logic
Scientific paper
Jan 1984
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1984phrvd..29..186g&link_type=abstract
Physical Review D (Particles and Fields), Volume 29, Issue 2, 15 January 1984, pp.186-197
Mathematics
Logic
23
Scientific paper
We develop cosmological theory from first principles starting with curvature coordinates (R,T) in terms of which the metric has the form ds2(R,T)=dR2A(R,T)+R2dΩ2-B(R,T)dT2. The Einstein field equations, including cosmological constant, are given for arbitrary Tμν, and the timelike geodesic equations are solved for radial motion. We then show how to replace T with a new time coordinate τ that is equal to the time measured by radially moving geodesic clocks. Cosmology is brought into the picture by setting Tμν equal to the stress-energy tensor for a perfect fluid composed of geodesic particles, and letting τ be the time measured by clocks coincident with the fluid particles. We solve the field equations in terms of (R,τ) coordinates to get the metric coefficients in terms of the pressure and density of the fluid. The metric on the subspace τ=const is equal to dR2+R2dΩ2, 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,τ) spacetime diagram, all trajectories in an Einstein-de Sitter universe are emitted from R=0 at the "big bang" at τ=0. Further, a light signal coming toward R=0 at some time τ>0 will, in its past history, have started from R=0 at τ=0, and have turned around on the line 2R=3τ. A consequence of this is a "tilting" of the null cones along the trajectory of a cosmological particle. The turnaround line 2R=3τ 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.
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