The redshift-distance and velocity-distance laws

Abstract

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) = cH_0_ {integral} dz/H(z) where H_0_ 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