Relativistic gravitation is so rich a problem which is beyond any addition. Even then this book will help students of physics to get the essential ingredients of general relativity and find some solved problems. Most of the problems are published matter. The book is short but no doubt very rich in context. Graduate students will be very benefited from the book.
Table of Contents
Chapter 1: Introduction
Chapter 2: Principle of Equivalence
Chapter 3: Gravitational Forces
Chapter 4: Newtonian Limit
Chapter 5: Curvature Tensor
Chapter 6: Einstein's Field Equation
Chapter 7: Derivation of the Orbit Equation in Schwarzschild Field
Chapter 8: Motion of an Effective Particle in Kerr Field
Chapter 9: Orbital Shape and Periastron Shift in a Compact Binary
Chapter 10: Gravitomagnetic Clock Effect
Chapter 11: Gclock Effect in Spinning Particle
Chapter 12: Phase Quantization in Relativistic Gravitation
Chapter 13: Correspondence Between Test- and Effective-Particle Dynamics of Relativistic Gravitational Two-Body Problem
Objectives & Core Themes
This work explores the foundational concepts and specific applications of relativistic gravitation, aiming to bridge the gap between basic theory and advanced astrophysical problems. It focuses on reformulating gravitational dynamics using the effective one-body approach to address two-body systems in Schwarzschild and Kerr fields.
- Theoretical derivation of geodesic equations and orbit dynamics in curved space-time.
- Application of the effective one-body approach to relativistic two-body problems.
- Calculation of periastron shifts and the gravitomagnetic clock effect in compact binary systems.
- Investigation into quantum aspects of gravitation, including phase quantization.
- Comparative analysis of test-particle and effective-particle dynamics in spinning space-times.
Excerpt from the Book
7. Derivation of the Orbit Equation in Schwarzschild Field
A test mass is defined as a body of infinitesimal mass which is moving round another body of larger mass so that the test body does not influence the gravitational field of the central mass. Schwarzschild metric represents the exterior field of a massive non-rotating body. The motion of a test body in the Schwarzschild field of the massive body is described by geodesic equations in Schwarzschild metric. For example, the motion of the planet Mercury in the Sun’s gravitational field is described by geodesic equations in Schwarzschild metric.
The Schwarzschild metric is given by ds^2 = e^v c^2 dt^2 - r^2 (dθ^2 + sin^2 θ dφ^2) - e^λ dr^2 (1)
Explicitly, it is ds^2 = (1 - 2GM/c^2 r) c^2 dt^2 - r^2 (dθ^2 + sin^2 θ dφ^2) - dr^2 / (1 - 2GM/c^2 r) (2)
Summary of Chapters
Chapter 1: Introduction: Provides an overview of the transition from Newtonian gravity to Einstein's geometric formulation of space-time.
Chapter 2: Principle of Equivalence: Discusses the fundamental equality of gravitational and inertial masses and the role of non-inertial reference systems.
Chapter 3: Gravitational Forces: Derives the equations of motion for freely falling particles in curved space-time.
Chapter 4: Newtonian Limit: Examines how general relativity reduces to Newtonian gravity in the case of weak, stationary fields.
Chapter 5: Curvature Tensor: Introduces the Riemann-Christoffel curvature tensor and the Ricci tensor as metrics of space-time curvature.
Chapter 6: Einstein's Field Equation: Presents the derivation and conceptual basis of the field equations for gravitation.
Chapter 7: Derivation of the Orbit Equation in Schwarzschild Field: Details the calculation of test-mass orbits within a Schwarzschild space-time.
Chapter 8: Motion of an Effective Particle in Kerr Field: Extends orbit analysis to spinning central bodies using the effective one-body method.
Chapter 9: Orbital Shape and Periastron Shift in a Compact Binary: Analyzes the dynamics and periastron advancement in binary systems.
Chapter 10: Gravitomagnetic Clock Effect: Explains the difference in orbital periods for prograde and retrograde motion near a rotating mass.
Chapter 11: Gclock Effect in Spinning Particle: Investigates how intrinsic spin affects the clock effect for an orbiting particle.
Chapter 12: Phase Quantization in Relativistic Gravitation: Explores potential quantum phenomena within gravitomagnetic environments.
Chapter 13: Correspondence Between Test- and Effective-Particle Dynamics of Relativistic Gravitational Two-Body Problem: Summarizes the consistency between test-body approximations and effective one-body formulations.
Keywords
Relativistic Gravitation, General Relativity, Schwarzschild Metric, Kerr Field, Effective One-Body Approach, Geodesic Equations, Periastron Shift, Gravitomagnetic Clock Effect, Two-Body Problem, Space-Time Curvature, Orbital Mechanics, Spin-Orbit Interaction, Phase Quantization, Compact Binary Systems.
Frequently Asked Questions
What is the primary focus of this book?
The book provides a concise exploration of the fundamental concepts of general relativity and its application to specific astrophysical problems, such as orbital mechanics in Schwarzschild and Kerr fields.
What are the central research themes?
Key themes include the geometric nature of gravity, the effective one-body approach to two-body problems, and the investigation of relativistic effects like the gravitomagnetic clock effect.
What is the primary goal of the author?
The author aims to equip advanced undergraduate and graduate students with the essential equations and mathematical techniques needed to understand current research in post-Newtonian celestial mechanics.
Which scientific methods are employed?
The book utilizes differential geometry, Lagrangian and Hamiltonian mechanics, and the effective one-body formalism to solve gravitational equations of motion.
What topics are covered in the main body of the work?
The main chapters cover the Principle of Equivalence, Einstein's Field Equations, the derivation of orbit equations for test and effective particles, and the analysis of spinning bodies.
Which keywords best characterize the work?
The work is defined by terms like General Relativity, Schwarzschild Metric, Kerr Field, Gravitomagnetic Clock Effect, and Effective One-Body Approach.
How does the author treat the Kerr metric?
The author applies the Kerr metric to study particles orbiting a rotating massive body, specifically focusing on how the spin of the central object influences the orbital path and time duration.
What is the significance of the "Gclock Effect"?
The Gclock effect refers to the disparity in orbital periods for prograde versus retrograde motion, providing a measurable relativistic phenomenon that serves as a test for theories of gravity in curved space-time.
- Quote paper
- Syed B. Faruque (Author), 2019, Relativistic Gravitaion Basics and Special Topics, Munich, GRIN Verlag, https://www.grin.com/document/505903
