Energy momentum prescriptions in the framework of rainbow gravitation theory for the inhomogenous space time models

In this thesis, the long-standing problem of energy–momentum localization in general relativity is examined through classical energy-momentum complexes (Einstein, Bergmann–Thomson, Landau–Lifshitz, Papapetrou, Tolman, Weinberg, and Møller) and generalized within the framework of the Rainbow Gravity theory. As a non-homogeneous space-time model, the Van Stockum rotating dust solution is employed, transformed into Cartesian coordinates, and energy density expressions are computed accordingly. Rainbow Gravity modifies the space-time metric through two energy-dependent functions, implying that test particles with higher energies may "experience" space-time differently. Calculations show that all energy-momentum complexes yield mutually consistent and physically meaningful results in the classical limit (f1=f2=1), matching general relativity. However, when Rainbow corrections are introduced, significant deviations appear, particularly in the Einstein, Bergmann–Thomson, and Møller prescriptions. Energy densities are re-evaluated for widely used Rainbow functions such as the Modified Dispersion and Exponential Models. While some functions produce strong modifications, others—especially in the Landau–Lifshitz and Weinberg prescriptions—lead to results nearly identical to general relativity. The Møller prescription, owing to its coordinate-independent formulation, provides the most general physical interpretation and reveals a monotonic decrease in energy density with increasing test-particle energy.