In the present work, we prove the validity of two theorems on null-paths in a version of absolute parallelismgeometry. 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 lies in their applications to derive a general formula for the redshift of spectral lines coming from distant objects. The formula derived in the present work can be applied to both cosmological and astrophysical redshifts. 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.

ABSTRACT
1. INTRODUCTION
2. KMW THEOREMS
    Theorem (I)
    Theorem (II)
3. SPIN-DEPENDENT PATH EQUATION
4. THEOREMS ON NULL PATHS IN PAP SPACES
    4.1 Theorem (I)
    4.2 Theorem (II)
5. GENERAL EXPRESSION OF REDSHIFT IN PAP SPACE
6. CONCLUDING REMARKS
REFERENCES

• M. I. WANAS(Astronomy Department, Faculty of Science, Cairo University, Center of Theoretical Physics(CTP), British University(BUE), Egyptian Relativity Group)
• A. B. MORCOS(Astronomy Department, National Research Institute of Astronomy and Geophysics, Center of Theoretical Physics(CTP), British University(BUE), Egyptian Relativity Group) Corresponding Author