Core Principles of Special and General Relativity

James Luscombe

• Publisher: CRC Press
• Year: 2018
• ISBN:
  9780429657092
• Edition: 1
• Topic: Physical Sciences
• Subtopic: Physics
• Index: Physics

This book provides an accessible, yet thorough, introduction to special and general relativity, crafted and class-tested over many years of teaching. Suitable for advanced undergraduate and graduate students, this book provides clear descriptions of how to approach the mathematics and physics involved. It is also contains the latest exciting developments in the field, including dark energy, gravitational waves, and frame dragging.

The table of contents has been carefully developed in consultation with a large number of instructors teaching courses worldwide, to ensure its wide applicability to modules on relativity and gravitation.

Features:

• A clear, accessible writing style, presenting a sophisticated approach to the subject, that remains suitable for advanced undergraduate students and above
• Class-tested over many years
• To be accompanied by a partner volume on 'Advanced Topics' for students to further extend their learning

Chapter 1

Relativity

A theory of space, time, and gravity

Relativity is a theory of space and time that provides the foundation for much of physics. It applies to any branch of physics that makes use of the four variables x, y, z, t, where x, y, z are independent spatial coordinates and t denotes time.1 While originating from a reasonable premise (see below), the theory of relativity2 implies conceptions of space, time, matter, and motion vastly different from what our everyday experience of the world leads us to formulate. To understand physics in full, as applied to phenomena beyond ordinary experience, one must study relativity (as well as quantum mechanics); our everyday experience is but a special case of all that’s possible in the universe. We’ll see that relativity consists of two theories: the special theory of relativity (SR) and the general theory of relativity (GR).

1.1The principle of relativity

to vanquish coordinates, transcend them

In broadest terms, relativity holds that the universe doesn’t care what systems of coordinates, or reference frames we use to describe physical phenomena.3 Such a statement hardly sounds revolutionary, yet its implications are far-reaching because in the theory of relativity time is taken as a coordinate in a four-dimensional geometry of space and time, rather than as a parameter in pre-relativistic physics.4 Coordinates are essential for making measurements and performing calculations, yet they’re not fundamental—they don’t exist in nature—they’re artifacts of our thinking, what we as humans impose on the world. Therein lies the rub. We need coordinates for practical purposes, yet the goal of physics is to formulate laws of nature as manifestations of an objective reality, that which occurs independently of human beings.5 The laws of physics should be expressed in a way that’s independent of coordinate system. Relativity is an outgrowth of a single idea, the principle of relativity, that physical laws be independent of the reference frame used to represent them. Relativity is therefore a law about laws.6 Albert Einstein said: “… time and space are modes by which we think, and not conditions in which we live.”[2, p81] The program of relativity is to express equations of physics in such a way that, if true in one system of space-time coordinates, are true in any coordinate system, and thereby transcend coordinates. We will travel far in the theory of relativity in pursuit of this goal, which, as we’ll see, is achieved by expressing equations as relations between tensors,7 tensors defined on a four-dimensional geometry where time is a dimension.

1.2The Law of inertia: Foundation of special relativity

Motion exists … relatively to things that lack it.—Galileo, 1632[3, p116]

Motion is ubiquitous, yet learning to describe it correctly took a long time to achieve. Galileo taught, for the purposes of formulating laws of motion, that states of uniform motion are the same as rest,8 when observed from reference frames in which the law of inertia holds, inertial reference frames (IRFs)....

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Contents
• Preface
• Chapter 1: Relativity: A theory of space, time, and gravity
• Chapter 2: Basic special relativity
• Chapter 3: Lorentz transformation, I
• Chapter 4: Geometry of Lorentz invariance
• Chapter 5: Tensors on flat spaces
• Chapter 6: Lorentz transformation, II
• Chapter 7: Particle dynamics
• Chapter 8: Covariant electrodynamics
• Chapter 9: Energy-momentum of fields
• Chapter 10: Relativistic hydrodynamics
• Chapter 11: Equivalence of local gravity and acceleration
• Chapter 12: Acceleration in special relativity
• Chapter 13: Tensors on manifolds
• Chapter 14: Differential geometry
• Chapter 15: General relativity
• Chapter 16: The Schwarzschild metric
• Chapter 17: Physical effects of Schwarzschild spacetime
• Chapter 18: Linearized gravity
• Chapter 19: Relativistic cosmology
• Appendix A: Invariance of the wave equation
• Appendix B: The Doppler effect
• Appendix C: Topics in linear algebra
• Appendix D: Topics in classical mechanics
• Appendix E: Photon and particle orbits
• Bibliography
• Index