Separation of Variables Method in Solving Partial Differential Equations and Investigating the Relationship between Gravitational Field Tensor and Energy-Momentum Tensor in Einstein's Theory of Gravity

Abstract

This research delves into the study of partial differential equations (PDEs) and gravitational fields in spacetime. It focuses on solving PDEs using the Separation of Variables method and explores the relationship between the gravitational field tensor and the energy-momentum tensor, leading to the final equation for the gravitational field tensor. The research also investigates Einstein's theory of gravity and the energy-momentum tensor integral, providing the general solution for the gravitational potential and its implications. Additionally, the mean integration of the gravitational wave tensor is analyzed, yielding an expression for the tensor strain of gravitational waves over an infinitely long period. The components of the gravitational wave tensor and their effect on gravitational sources are examined. Furthermore, the propagation of electromagnetic fields in spacetime is studied using the Retarded Green's Function. The primary objectives of this research are to understand and explore mathematical techniques for solving PDEs and analyzing gravitational fields and their interactions in spacetime. The integration of multiple theoretical concepts related to PDEs, gravitational fields, and electromagnetic fields enhances our understanding of fundamental physics principles. This contributes to the advancement of theoretical physics and opens avenues for potential practical applications, such as gravitational wave detection and electromagnetic field propagation in complex media. In conclusion, this research provides valuable insights into fundamental physics principles and fosters a deeper understanding of their interconnections and implications