The mathematics of Einstein's theories of relativity

In the early 20th century Albert Einstein proposed two startling theories that have since been confirmed at ever-increasing levels of accuracy: special relativity and general relativity. The "special" theory, for observers in uniform motion relative to each other, holds that time is a fourth dimension, that the four components of space-time for an event are measured differently by the different observers (with time running more slowly for a traveler than for an observer, and momentum increasing exponentially as objects approach the speed of light), and that the energy contained in an object at rest is equal to its mass multiplied by the square of the speed of light. The general theory incorporates both uniform and accelerated motion, and holds that gravity is not a force, as Newton held, but rather a matter of geometry: mass curves space-time, causing objects, and light, to travel on geodesics, such as planetary orbits. The author develops the concepts of relativity, and then derives the equations of special relativity with the tools of elementary algebra, Euclidean geometry, and univariate calculus, and the equations of general relativity with the tools of multivariate tensor calculus and Riemannian geometry.