Numerical Analysis of Metric Perturbations, Spacetime Curvature, and Gravitational Wave Polarizations in Linearized General Relativity Using the Finite Difference Method - Ariefka, Reza; Siagian, Ruben Cornelius; Sahroni, Taufik Roni; Nurahman, Arip (Fakultas Matematika dan Ilmu Pengetahuan Alam, Univetsitas Tanjungpura, 2026)
The general theory of relativity states that small perturbations in the spacetime metric in a weak gravitational field can be treated as perturbations propagating on the Minkowski background. Although the analytical formulation of Einstein's linear equations has been well developed, numerical studies that integrate local metric perturbations, the curvature structure of space-time through the Ricci tensor, the dynamics of gravitational wave polarization, and reduction to Newton's law of gravity in a two-dimensional domain are still limited. This research aims to numerically analyze the behavior of metric perturbations and space-time curvature in the framework of linear general relativity and verify its consistency with gravitational wave theory and Newtonian gravity at weak field and low velocity limits. The study is limited to a linearized approach, two-dimensional flat spacetime, harmonic and transverse–traceless gauge conditions, and ideal sources in the form of Gaussian perturbations and source-free waves. The methods used include the theoretical formulation of linearized general relativity and the Finite Difference Method (FDM) numerical approach to discretize the Laplace operator and wave equation with the FTCS scheme. The results show that Gaussian perturbations produce highly localized curvature with a global Ricci tensor contribution approaching zero. Gravitational wave simulations show the dominance of energy in polarization (h_+) over (h_×), as well as numerical verification of the relationship (h_00=-2φ), which confirms the reduction of general relativity to Newton's laws. This research fills a research gap by integrating metric analysis, curvature, and gravitational waves into a single consistent numerical framework. The novelty of this research lies in Gaussian modeling and integrated FDM-based numerical analysis as a conceptual bridge between general relativity and classical mechanics.