Scalar potential model of redshift and discrete redshift

On the galactic scale the universe is inhomogeneous and redshift $z$ is occasionally less than zero. A scalar potential model (SPM) that links the galaxy scale $z$ to the cosmological scale $z$ of the Hubble Law is postulated. Several differences among galaxy types suggest that spiral galaxies are Sources and that early type, lenticular, and irregular galaxies are Sinks of a scalar potential field. The morphology-radius and the intragalactic medium cluster observations support the movement of matter from Source galaxies to Sink galaxies. A cell structure of galaxy groups and clusters is proposed to resolve a paradox concerning the scalar potential like the Olber's paradox concerning light. For the sample galaxies, the ratio of the luminosity of Source galaxies to the luminosity of Sink galaxies approaches $2.7 \pm 0.1$. An equation is derived from sample data, which is anisotropic and inhomogeneous, relating $z$ of and the distance $D$ to galaxies. The calculated $z$ has a correlation coefficient of 0.88 with the measured $z$ for a sample of 32 spiral galaxies with $D$ calculated using Cepheid variable stars. The equation is consistent with $z<0$ observations of close galaxies. At low cosmological distances, the equation reduces to $z \approx \exp(KD) \, -1 \approx KD$, where $K$ is a constant, positive value. The equation predicts $z$ from galaxies over 18 Gpc distant approaches a constant value on the order of 500. The SPM of $z$ provides a physical basis for the $z$ of particle photons. Further, the SPM qualitatively suggests the discrete variations in $z$, which was reported by W. G. Tifft, 1997, Astrophy. J. 485, 465 and confirmed by others, are consistent with the SPM.