Gravitational Curvature

An Introduction to Einstein's Theory

Theodore Frankel

• Publisher: Dover Publications
• Year: 2013
• eBook ISBN: 9780486289151
• Topic: Physical Sciences
• Subtopic: Physics
• Index: Physics

This classic text and reference monograph applies modern differential geometry to general relativity. A brief mathematical introduction to gravitational curvature, it emphasizes the subject's geometric essence, replacing the often-tedious analytical computations with geometric arguments. Clearly presented and physically motivated derivations express the deflection of light, Schwarzchild's exterior and interior solutions, and the Oppenheimer-Volkoff equations.
A perfect choice for advanced students of mathematics, this volume will also appeal to mathematicians interested in physics. It stresses the global aspects of cosmology and is suitable for independent study as well as for courses in differential geometry, relativity, and cosmology. Prerequisites include a background in Riemannian geometry and basic physics or a familiarity with relativity theory. Background chapters, with derivations, cover special relativity, continuum mechanics, and electromagnetism.

1

Special Relativity

The Lorentz Transformations as Viewed by Einstein

Classical mechanics rests on the notions of absolute space and absolute time. Absolute space is assumed to be an affine 3-space with a Euclidean metric that is unique up to constant multiples. One can introduce Cartesian coordinates x, y, z. Absolute time is measured by a time coordinate t, unique up to transformations t → at + b, where a and b are constants. Newton’s laws of motion are assumed valid when x, y, z, t are used as coordinates for space and time. Newton thought of absolute space as existing independently of matter in the universe, whereas Bishop Berkeley interpreted absolute space as being fixed with respect to the bulk of matter in the universe (the so-called “fixed stars”).

An inertial (coordinate) system is a Cartesian coordinate system that moves uniformly, i.e., without acceleration, and, in particular, without rotation, with respect to absolute space. It was realized that Newton’s laws of motion are also valid in any inertial system, since accelerations are unaffected by uniform motion. Any two inertial systems are in uniform translational motion with respect to each other.

The existence of an absolute space or absolute frame of reference became dubious much later when highly accurate optical experiments were performed. It was found, toward the end of the last century, that light is propagated isotropically, i.e., with the same speed in all directions, in each supposed inertial system. Consider two inertial systems S and S′ passing one another. A light pulse is emitted at their common origin at time t = 0. It is observed (essentially in the Michelson-Morley experiment) that both systems see their respective origins as the centers of the resulting spherical light pulse for all time (the “light pulse paradox”)!

This, together with other electromagnetic considerations, led Einstein (among others) to reject the notion of an absolute space. He still retained, however, the notion of a distinguished (but undefined) class of inertial systems. Einstein then showed that this rejection of an absolute space and the resulting notion of absolute motion of an inertial system forces us to abandon also the idea of an absolute time! Einstein (1905) reasoned as follows.

Consider two space-time events E₁ and E₂. When viewed from an inertial system S these events have coordinates (x₁, y₁, z₁, t₁) and (x₂, y₂, z₂, t₂). Since light is propagated isotropically in the system S, the two events will occur simultaneously (at the same time t) in S if and only if light pulses emitted E₁ and E₂ reach the spatial midpoint

at the same instant. Consider, for example, two lightning bolts that strike a railway embankment in S at t = 0 at spatial coordinates (x, 0, 0) and (−x, 0, 0). These strikes are seen simultaneously at the origin of S at time x/c, where c is the speed of light. Consider an inertial system S′ (a train) moving along the tracks (the x direction in S) and suppose that the two ends of the train are struck by the same bolts. If the origin of S′ is at the midpoint of the train (equal number of cars forward and behind), it is clear that the forward bolt will be seen at this midpoint before the backward bolt. (It is important here that the speed of light is not infinite.) Two inertial systems in relative motion will disagree as to whether or not certain spatially separated events are simultaneous. S and S′ must be keeping different times. This simple observation by Einstein distingui...

Table of contents

• Cover Page
• Title Page
• Copyright Page
• Dedication
• Contents
• Preface to the Dover Edition
• Preface
• Notation
• 1 Special Relativity
• 2 Clocks and Gravitational Potential
• 3 A Heuristic Derivation of Einstein’s Equations
• 4 The Geometry of Einstein’s Equations
• 5 The Schwarzschild Solution
• 6 The Classical Motion of a Continuum
• 7 The Relativistic Equations of Motion
• 8 Light Rays and Fermat’s Principle
• 9 Electromagnetism in Three-Space and Minkowski Space
• 10 Electromagnetism in General Relativity
• 11 The Interior Solution
• 12 Cosmology
• References
• Index
• Back Cover