The redshift-distance and velocity-distance laws

The distinction between Hubble's linear redshift-distance z (L) law and the linear velocity-distance V (L) law that emerged later is discussed, using first the expanding space paradigm and then the Robertson-Walker metric. The z (L) and V (L) laws are theoretically equivalent only in the limit of small redshifts, and failure to distinguish between the two laws obscures the basic elementary principles of modern cosmology. The linear V (L) law V = HL, where H (t) is the Hubble term applies quite generally in expanding homogeneous and isotropic cosmological models, and recession velocities can exceed the velocity of light. The z (L) relation in its linear form (cz = HL), however, has no theoretical basis and can be used only in the limit of small redshifts. In general, the z (L) relation is nonlinear (with the exception of exponentially expanding spaces) and must be derived separately for each particular model. The general distance- redshift L (z) relation is obtained from the fundamental velocity-redshift relation V (z) = cH₀_ integral dz/H (z) where H₀_ is the value of the Hubble term at the present epoch. Possible historical reasons for the confusion between the z (L) and V (L) laws, and why both are indiscriminately referred to as Hubble's law, are discussed. Subject headings cosmology: theory - galaxies: distances and redshifts