This book serves as a textbook for senior undergraduate students who are learning the subject of general relativity and gravitational waves for the first time. Both authors have been teaching the course in various forms for a few decades and have designed the book as a one stop book at basic level including derivations and exercises.
A spectacular prediction of general relativity is gravitational waves. Gravitational waves were first detected by the LIGO detectors in 2015, hundred years after their prediction. Both authors are part of the LIGO Science Collaboration and were authors on the discovery paper. Therefore, a strong motivation for this book is to provide the essential concepts of general relativity theory and gravitational waves with their modern applications to students and to researchers who are new to the multi-disciplinary field of gravitational wave astronomy.
One of the advanced topics covered in this book is the fundamentals of gravitational wave data analysis, filling a gap in textbooks on general relativity. The topic blends smoothly with other chapters in the book not only because of the common area of research, but it uses similar differential geometric and algebraic tools that are used in general relativity.
Author(s): Sanjeev Dhurandhar, Sanjit Mitra
Series: UNITEXT for Physics
Edition: 1
Publisher: Springer Nature Switzerland AG
Year: 2022
Language: English
Pages: 207
City: Cham, Switzerland
Tags: Equivalence Principle, Tensors, Einstein Field Equation, Black Holes, Gravitational Waves
Preface
Contents
1 Overview of Special Relativity
1.1 Introduction
1.2 Postulates of Special Relativity (SR)
1.3 Space-Time Diagrams: A Picture Is Worth a Thousand Words
1.3.1 Setting Up the Axes for a Moving Observer
1.3.2 Lorentz Contraction and Time Dilation
1.4 Lorentz Transformations
1.4.1 The Lorentz Transformation Equations
1.4.2 Velocity Addition
1.5 Four-Vector Notation and Covariant Formalism
1.6 Relativistic Mechanics
1.7 Covariant Formulation of Electrodynamics
2 The Equivalence Principle
2.1 Equivalence Principle: Weak Form
2.2 Equivalence Principle: Strong Form (SEP)
2.3 Gravity and Curvature of Space-Time
2.3.1 Gravity Bends Light Rays
2.3.2 Gravity Affects Clocks: Gravitational Redshift
3 Tensor Algebra
3.1 Introduction
3.2 Basis Vectors
3.3 Contravariant and Covariant Components of a Vector
3.3.1 The Jacobian matrices
3.4 The Metric
3.4.1 Scalar Product
3.4.2 Raising and Lowering of Indices
3.5 Tensors of Higher Rank
3.5.1 Transformation Laws
3.5.2 Symmetric and Anti-symmetric Tensors
3.6 The Volume Element
3.6.1 Determinant of the Metric Tensor
3.6.2 Perfectly Anti-Symmetric Tensor: Levi-Civita Symbol
3.6.3 Volume Element
3.7 Example: Oblique Coordinate System on a Plane
3.7.1 Contravariant and Covariant Bases
3.8 Transition to Curved Spaces
4 The Geometry of Curved Spaces and Tensor Calculus
4.1 Introduction
4.2 Parallel Transport and the Covariant Derivative
4.3 Geodesics
4.3.1 Parallelly Transporting the Tangent
4.3.2 Extremal Distance
4.4 Locally Flat Coordinate System
4.5 Curvature
4.5.1 Riemann-Christoffel Curvature Tensor
4.5.2 Geodesic Deviation
4.5.3 Useful Tensors and Identities
5 Einstein's Equations
5.1 Einstein's Field Equations
5.2 The Energy-Momentum Stress Tensor
5.3 The Newtonian Limit of Einstein's Equations
6 Schwarzschild Solution and Black Holes
6.1 Introduction
6.2 The Schwarzschild Metric
6.3 Event Horizons and Black Holes
6.3.1 One Way Membranes in Special Relativity
6.3.2 The Event Horizon in Schwarzschild Spacetime
6.3.3 Infinite Redshift Surface and the Central Singularity
6.4 Orbits in Schwarzschild Spacetime
6.4.1 The First Integrals
6.4.2 The Effective Potential
6.4.3 Photon Orbits
6.5 Gravitational Redshift and the Infinite Redshift Surface
6.6 Kerr Spacetime and the Rotating Black Hole
6.6.1 The Metric and the Horizon
6.6.2 The Static Limit, Frame Dragging and the Ergosphere
7 Classical Tests of General Relativity
7.1 Introduction
7.2 The Deflection of Light by a Central Mass
7.3 The Perihelion Shift in the Orbit of Mercury
7.4 Gravitational Redshift: Pound-Rebka Experiment
7.5 Shapiro Time Delay
8 Gravitational Waves
8.1 Introduction
8.2 Linearised Gravity
8.3 Choice of the Lorenz Gauge
8.4 Plane Wave Solutions
8.4.1 The Transverse Traceless (TT) Gauge
8.4.2 TT—Gauge as an Algebraic Projection
8.5 Quadrupole Formula
8.6 Energy Carried by Gravitational Waves
8.7 GWs from an Inspiraling Binary in Circular Orbit
8.7.1 GW Amplitudes for a Equal Mass Binary in Circular Orbit with Constant Angular Frequency
8.7.2 GWs from an Inspiraling Binary
8.8 Sources
8.9 Effect of GW on a Ring of Test Particles
8.10 The Response of a Laser Interferometric Detector
8.11 Detection of GW
8.11.1 Hulse-Taylor Binary Pulsar
8.11.2 Bar Detectors
8.11.3 Ground-Based Laser Interferometric Detectors
8.11.4 Space-Based Detectors
8.11.5 Pulsar Timing Array (PTA)
8.11.6 Cosmic Microwave Background (CMB)
8.12 Direct Detection of Gravitational Waves
9 Gravitational Wave Data Analysis
9.1 The Task of a GW Data Analyst
9.2 Time Series Data and Its DFT
9.3 Characterisation of Noise
9.3.1 Random Variables and their Distributions
9.3.2 Stationary, Coloured and Gaussian Noise
9.4 Matched Filtering
9.4.1 A Simple Example
9.4.2 The Matched Filter and Signal to Noise Ratio
9.4.3 The Optimality of the Matched Filter
9.5 Statistical Significance of Detection
9.5.1 Binary hypothesis testing
9.5.2 Neyman-Pearson Criterion
9.6 Composite Hypothesis and Maximum Likelihood
9.6.1 The Signal Manifold
9.6.2 Maximum Likelihood
9.7 Metric on the Signal Manifold
9.7.1 Placing Templates in the Parameter Space
9.7.2 Fisher Information Matrix and Rao-Cramer Bound
9.8 Burst and Stochastic Searches
9.9 Present Challenges
Appendix Bibliography
Index
