Theorems on Null-Paths and Red-Shift
  In the present work, we prove the validity of two theorems on null-paths in a version of absolute parallelism geometry. A version of these theorems has been originally established and proved by Kermak, McCrea and Whittaker (KMW) in the context of Riemannian geometry. The importance of such theorems is their use, in applications, to derive a general formula for the red-shift of spectral lines coming from distant objects. The formula derived in the present work, can be applied for both cosmological and astrophysical red-shifts. It takes into account the shifts resulting from gravitation, different motions of the source of photons, spin of the moving particle (photons) and the direction of the line of sight. It is shown that this formula cannot be derived in the context of Riemannian geometry, but it can be reduced to a formula given by KMW under certain conditions. 
M.I. Wanas and A.B. Morcos