Covariant Physics: From Classical Mechanics to General Relativity and Beyond endeavours to provide undergraduate students as well as self-learners with training in the fundamentals of the modern theories of spacetime, most notably the general theory of relativity as well as physics in curved spacetime backgrounds in general. This text does so with the barest of mathematical preparation. In fact, very little beyond multivariable calculus and a bit of linear algebra is assumed.
Throughout this textbook, the main theme tying the various topics is the so-called principle of covariance - a fundamental symmetry of physics that one rarely encounters in undergraduate texts. The material is introduced very gradually, starting with the simplest of high school mathematics, and moving through the more intense notions of tensor calculus, geometry, and differential forms with ease. Familiar notions from classical mechanics and electrodynamics are used to increase familiarity with the advanced mathematical ideas, and to emphasize the unity of all of physics under the single principle of covariance.
The mathematical and physical techniques developed in this book should allow students to perform research in various fields of theoretical physics as early as their sophomore year in college. The language the reader will learn in this book is the foundational mathematical language of many modern branches of physics, and as such should allow them to read and generally understand many modern physics papers.
Author(s): Moataz H. Emam
Edition: 1
Publisher: Oxford University Press
Year: 2021
Language: English
Pages: 403
Coordinate Systems and Vectors
Coordinate Systems
Measurements and the Metric
Vectors in Cartesian Coordinates
Derivatives in the Index Notation
Cross Products
Vectors in Curvilinear Coordinates
Tensors
What's a Vector, Really?
Defining Tensors
Derivatives and Parallel Transport
Calculating the Christoffel Symbols
More on the Divergence and Laplacian Operators
Integrals and the Levi-Civita Tensor
Classical Covariance
Point Particle Mechanics
The Geodesic Equation: Take One
The Rest of Point Particle Mechanics
Rigid Body Mechanics
Motion in a Potential
Continuum Mechanics
Galilean Covariance
Special Covariance
Special Relativity, the Basics
Four-Vectors and Four-Tensors
Lorentz Boosts as Rotations
A Bit of Algebra
Addition of Velocities
Time Dilation
Length Contraction
Specially Covariant Mechanics
Conservation of Four-Momentum
A Note on Units
Specially Covariant Electrodynamics
General Covariance
What is Spacetime Curvature?
Gravity as Curvature
The Newtonian Limit: The Metric of Weak Gravity
The Metric Outside a Spherical Mass
The Metric Inside a Spherical Mass
Areas and Volumes in Curved Spaces
Non-Rotating Black Holes
Cosmological Spacetimes
Physics in Curved Spacetime
Generally Covariant Mechanics
Time Dilation
Four-Velocity
Four-Acceleration, Geodesics, and Killing Fields
Four-Momentum
The Schwarzschild Metric as a Case Study
Escape Velocity
The Event Horizon and Light Cones
Falling Past the Event Horizon
The Geodesic Equations
The Way of the Potential
Circular Orbits
Interlude
Bound Non-Circular Orbits
Open Orbits
Generally Covariant Electrodynamics
Riemann and Einstein
Manifolds
Calculus on Manifolds
Curvature
The Vacuum Einstein Equations
The Stress Energy Momentum Tensor
The Einstein Field Equations
Einstein's Greatest Blunder
Least Action and Classical Fields
The Principle of Least Action
The Actions of Non-Relativistic Particles
The Euler–Lagrange Equations
The Actions of Relativistic Particles
Classical Field Theory
The Principle of Least Action for Scalar Fields
The Principle of Least Action for Maxwell Fields
The Stress Tensor from the Action
General Relativistic Actions
The Geodesic Equation One More Time
Some More Spacetimes
Differential Forms
An Easy-Going Intro
More Generally . . .
Hodge Duality
Maxwell's Theory and Differential Forms
Stokes' Theorem
Cartan's Formalism
Generalizing General Relativity
Brans–Dicke Theory
f(R) Theory
Gauss–Bonnet Gravity
Kaluza–Klein Theory
Einstein–Cartan Theory
Afterthoughts
References
Index
