Radial distance on a stationary frame in a homogeneous and isotropic universe

Abstract

This paper presents a physical distance to all radial events in a homogeneous and isotropic universe as a transform from Friedman-Lemaitre-Robertson-Walker (FLRW) coordinates, the model that solves the Einstein Field equation for an ideal fluid. Any well behaved transform is also a solution. The problem is relating the coordinates of the transform to observables. In the present case the objective is to find T,R on a stationary frame that has the R be a physical observable for all distances. We do this by working backwards, assuming the form of the metric that we desire, with some undetermined coefficients. These coefficients are then related to the partial derivatives of the transform. The transformed coordinates T,R are found by the integration of partial differential equations in the FLRW variables. We show that dR has the same units as the radial differential of the FLRW metric, which makes it observable. We develop a criterion for how close the transformed T comes to an observable time. Close to the space origin at the present time, T also becomes physical, so that the stationary acceleration becomes Newtonian. We show that a galactic point on a R,T plot starts close to the space origin at the beginning, moves out to a physical distance and finite time where it can release light that will be seen at the origin at the present time. Lastly, because the observable R has a finite limit at a finite T for t = 0 where the galactic velocity approaches the light speed, we see that the universe filled with an ideal fluid as seen on clocks and rulers on the stationary frame has a finite extent like that of an expanding empty universe, beyond which are no galaxies and no space.