Numerical Analysis of Metric Perturbations, Spacetime Curvature, and Gravitational Wave Polarizations in Linearized General Relativity Using the Finite Difference Method

Authors

DOI:

https://doi.org/10.26418/positron.v16i1.92372

Keywords:

Linearized General Relativity, Metric Perturbations, Gravitational Waves, Ricci Curvature, Finite Difference Method, Numerical Relativity, Spacetime Curvature

Abstract

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_\times), as well as numerical verification of the relationship (h_{00}=-2\phi), 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.

Author Biographies

Reza Ariefka, Physics Education, Faculty of Teacher Training and Education, Muhammadiyah University of OKU Timur, Indonesia

Reza Ariefka, M.Ed., is an academic and researcher in the field of Physics Education who is recognized for his strong commitment to discipline, hard work, and the ability to work both independently and collaboratively. He currently serves as a lecturer in the Physics Education Study Program, Faculty of Teacher Training and Education, Muhammadiyah University of OKU Timur, South Sumatra, Indonesia. In his role as an educator, Reza not only teaches and evaluates hundreds of undergraduate students but is also actively involved in developing physics experimental learning and strengthening students’ understanding of fundamental physics concepts. His academic journey reflects a consistent dedication to the advancement of physics, particularly in fluid dynamics, computational physics, and material and solid-state physics. He completed his Bachelor’s degree in Physics Education at Ahmad Dahlan University with strong academic performance, before continuing his Master’s degree in Physics Education at the same institution with excellent academic achievement. His research focus, from his academic years until the present, has largely emphasized numerical and experimental studies, including research on the motion of cylinders on inclined planes in fluid environments using computational approaches with scientific software. In addition to his responsibilities as a lecturer, Reza also serves as a Master Advisor Assistant in the Master Program of Physics Education at Ahmad Dahlan University. In this role, he contributes to mentoring graduate students, particularly in developing physics equations, numerical solutions, and experimental tool development based on fluid dynamics and Computational Fluid Dynamics (CFD). His academic experience has been further strengthened through his involvement as a laboratory assistant and lecturer’s assistant in various courses, including modern physics, fundamental physics, and physics experiments, providing him with extensive experience in practicum-based learning and research. Reza’s technical competencies are supported by his proficiency in various scientific analysis and simulation software, including Scilab, Matlab, SPSS, Tracker, and Stellarium, which he utilizes to support research and technology-based learning development. He has also actively participated in various international training programs, including courses organized by the International Centre for Theoretical Physics (ICTP) in solid-state physics, integrated energy systems, and statistical physics of complex systems, which have further enriched his academic insight and professional network. In his research activities, Reza has produced numerous scientific publications in internationally reputable Scopus-indexed journals, focusing on numerical modeling, fluid dynamics, and the application of computational and machine learning methods in physics and environmental studies. Furthermore, he actively contributes as a reviewer for several reputable international journals, demonstrating recognition of his academic competence and scientific integrity. Organizational involvement has also been an essential part of his professional journey. He is actively engaged in various academic organizations, including as a member of the Indonesian Physical Society and the Indonesian Young Lecturer Association, and previously held leadership roles in student organizations during his academic years. Through these activities, Reza continues to demonstrate his commitment to advancing physics education, strengthening academic collaboration, and encouraging research innovation relevant to the development of science and technology. With a combination of teaching experience, research contributions, academic service, and national and international collaborative networks, Reza Ariefka continuously strives to contribute to the development of applied, innovative, and impactful physics education and research for the advancement of science and educational development in Indonesia.

Ruben Cornelius Siagian, Departemen of Physics, Faculty of Mathematics and Natural Science, Universitas Negeri Medan, Medan, North Sumatera, Indonesia

Ruben Cornelius Siagian is a multidisciplinary researcher with significant contributions in astrophysics, machine learning, computational physics, and global social and political analysis. With a strong academic background, Ruben has been active in various researches covering topics ranging from black hole dynamics to geopolitical analysis and nuclear policy.

In astrophysics and theoretical physics, Ruben has researched various aspects of space-time structure, black hole thermodynamics, and particle dynamics around extreme astrophysical objects. One of his prominent publications is on "Non-Linear Force Dynamics in Black Hole Accretion Disks Using Logarithmic and Trigonometric Approximations," published in the Journal of Science Education Research (JPPIPA). His research also includes the development of topological concepts in black hole state modeling and numerical approaches in understanding the energy-cosmology relationship.

Apart from astrophysics, Ruben is also active in data analytics and machine learning. He has developed predictive models to understand urbanization growth patterns and UV exposure using clustering and machine learning techniques. One of his works in this area is the research on "Impact of Urban Development on UV Exposure: A Clustering and Machine Learning Assessment," which shows its application in the field of environment and sustainability.

His contributions to geopolitics and global policy studies are also noteworthy. Ruben has written numerous strategic analyses on nuclear policy, international conflict, and global diplomatic relations. One of his significant works is "Analysis of the Evolution of the Conference on the Treaty on the Prohibition of Nuclear Weapons (TPNW) 2012-2024: A SWOT Method Approach," an in-depth look at the development of world nuclear policy.

In recent years, he has collaborated with various academics from renowned institutions, including experts from physics, engineering, and social sciences. His widespread publications show substantial contributions to the development of science.

With an ever-increasing number of academic citations and profound multidisciplinary contributions, Ruben Cornelius Siagian continues to be one of the figures instrumental in the development of science in various fields. His dedication to research and academic innovation has had a far-reaching impact, both in the academic community and in global policy.

Taufik Roni Sahroni, Industrial Engineering Department, BINUS Graduate Program - Master of Industrial Engineering, Bina Nusantara University, Jakarta 11480, Indonesia

Taufik Roni Sahroni is a researcher and academic affiliated with Bina Nusantara University, Jakarta, Indonesia. He is active in multidisciplinary research covering systems engineering, data security, ergonomics, and sustainable systems analysis and development. According to the Scopus database, he has a Scopus ID of 57223289031 and an ORCID of 0000-0002-8497-3947, with a total of 84 scientific publications that have received more than 200 citations and an h-index of 8. His works are spread across journal articles and international conference proceedings, with topics ranging from the development of pseudo random number generators for embedded systems and smart factories, data exchange security analysis, ergonomic studies on public transportation systems, to the evaluation of comfort, safety, and workload in the sustainable transportation sector. In addition, he is also involved in applied geophysical research, specifically the identification of geological structures using satellite data. His research contributions reflect a strong and relevant applied approach to technological, industrial, and sustainability challenges in the context of urban and smart systems.

Arip Nurahman, Department of Physics Education, Indonesian Institute of Education, Jl. Terusan Pahlawan N

Arip Nurahman, S.Pd., M.Pd., CPSP., is an academic and researcher in the field of physics education who currently serves as a Physics lecturer at the Institut Pendidikan Indonesia. He is recognized for his scholarly contributions in integrating Science, Technology, Engineering, and Mathematics (STEM) with Education for Sustainable Development (ESD), as well as incorporating generative artificial intelligence into innovative physics learning models. His academic work reflects a multidisciplinary approach that encompasses renewable energy research, particularly Dye-Sensitized Solar Cells (DSSC), curriculum innovation in physics education, technology-assisted learning media development, and studies in astronomy and theoretical physics. Throughout his academic career, Arip Nurahman has actively produced numerous scientific publications that contribute to the advancement of modern science education. Many of his studies focus on strengthening renewable energy literacy through experimental DSSC development using natural dyes such as spinach, dragon fruit peel, and black rice, which are integrated into project-based learning frameworks supported by artificial intelligence. In addition, he has contributed to the development of instructional modules, academic books, and intellectual property rights related to physics learning design, STEM-ESD laboratory practices, and innovative science education media. Arip Nurahman maintains strong collaborative research networks with national and international scholars from various institutions, including Universitas Pendidikan Indonesia, Universitas Ahmad Dahlan, Institut Teknologi Bandung, and global research communities in astronomy and solar energy materials. These collaborations reinforce his position as an interdisciplinary researcher working at the intersection of computational science, science education, and sustainable technology. Several of his research works also explore cosmological variables, solar activity relationships, and dark matter distribution models using computational and statistical approaches. His academic track record demonstrates consistent scientific productivity, reflected in numerous scholarly publications, educational modules, and academic contributions that have received increasing citations since 2021. Beyond research and publication, he is actively involved in community engagement through digital literacy programs, community-based astronomy education initiatives, and human resource capacity development for sustainable tourism village management. As an educator, Arip Nurahman is deeply committed to developing innovative, contextual, and future-oriented physics education. He views science education as a strategic platform for cultivating technologically literate generations with strong environmental awareness, critical thinking skills, and adaptive problem-solving abilities to address global scientific and societal challenges.

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Published

2026-03-31

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Vol. 16 No. 1 (2026): Vol. 16 No. 1 Edition

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