This paper presents a compelling argument for the physical light speed in the Friedman-Lemaitre-Robertson-Walker (FLRW) universe to vary with the cosmic time coordinate t of FLRW. It must be variable when the radial comoving differential coordinates of FLRW is interpreted as physical and therefore transformable by a Lorentz transform locally to differentials of stationary physical coordinates. Because the FLRW differential radial distance has a time varying coefficient a(t), integration of the transformed differentials to obtain stationary coordinates for a short radial distance requires the light speed c(t) to be proportional to the square root of da/dt. Since we assume homogeneity of space, this derived c(t) is the physical light speed on all points of the FLRW universe. This impacts the interpretation of all astronomical observations of distant phenomena that are sensitive to light speed. A world transform from FLRW that has a Minkowski metric close to the origin is shown to have a physical radius out to all points of the visible universe. In order to obtain numerical values for c(t), the general relativity (GR) field equation is extended by using a variable gravitational constant and rest mass that keeps constant the gravitational and particle rest energies. This also keeps constant the proportionality constant between the GR tensors of the field equation and conserves the rest stress-energy tensor of the ideal fluid used in the FLRW GR field equation. In the same way all of special and general relativity is extended to include a variable light speed.
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