Steven Tieu

Introducing Einstein’s General Relativity

In the preceding chapters, we witnessed the new aspects of physics brought about by Einstein’s Relativity. All of this arose because: a) there is a speed limit in nature, \(c\), the maximum speed at which influences can be propagated and b) because all inertial observers are physically equivalent, they must all agree on this speed limit. However, not a word was mentioned about gravity and it is with the inclusion of gravity that Einstein’s Relativity takes on a whole new and exciting complexion. Einstein’s Relativity without gravity is called “Special Relativity” to distinguish it from Relativity with gravity which is called “General Relativity”. In brief, General Relativity is Einstein’s theory of gravity. In what follows, we will delve into General Relativity, showing how it is the curving of spacetime in the general theory that replaces the old Newtonian idea of gravity being just another force.

The Motion of Galaxies in Galaxy Clusters

Zwicky’s observation of the unexpectedly high velocities of the individual galaxies in the cluster of galaxies known as the ComaCluster led to the first appearance of the dark matter hypothesis. Unlike the organized motions in a nearly steady-state of the individual stars in a single spiral galaxy, the galaxies within a cluster are generally observed to move chaotically. Since we are not yet in a position to model such intrinsically time-dependent chaotic systems in General Relativity, we took the first step in the direction of dealing with time-dependence, again with an idealization. In what follows, we will describe our idealized model that brings into play intrinsic time-dependence. As a bonus, we are able to counter some of the criticism that had been levied against our earlier work.

A Simple Approach to Einstein’s Special Relativity

Your everyday experience is governed by the “where” and the “when”. If you arrange to have lunch with your friend, you might tell him, “Meet me at Joe’s Diner at noon tomorrow.” In your mind, you know the location of Joe’s Diner in the two-dimensional grid of your street map, let us say the corner of 53rd Street and 6th Avenue and if the diner is on the third floor, you know the height that you have to climb in the vertical direction. He knows it and you know it. And your watch tells you when to be there—the same for your friend. We are back to those four numbers again; we have the event of meeting the friend, where and when to do so. A presupposition is that there is no ambiguity about clocks, that there is a universal time, an absolute time (assuming that we all use good watches). We will see that Relativity changes all that.

Time Machines, the Multiverse and Other Fantasies

It is our everyday life experience that we re-visit previous places. We take it for granted that we always do so at later times. Even with the wonders of Special Relativity, when we considered the experiences of Alicia and Beatrice in Chap. 3, we saw that Beatrice re-visited her old home at a later time for her and the re-visit was at a much later time for Alicia, so much later in fact, that she was no longer a member of the living.

Testing Einstein’s General Relativity

It has been said that Newton was the first to state, with exemplary modesty, “If I have seen further it is only by standing on the shoulders of giants.” While we would place Newton and Einstein, as the ultimate giants at the very highest level of achievement in the history of physics, there is much wisdom and justification in Newton’s homage to earlier researchers. After all, the greatest advances were made with the help of important advances by those who preceded Newton and Einstein. Arguably the greatest of these was made by Copernicus who replaced the Earth with the Sun as the central body in what we now recognize as the Solar System.

The Universe According to Relativity

In the previous chapters, we have already witnessed some of the greatest marvels that Einstein’s Relativity has provided for us. In this chapter, we will describe one marvel that many consider the greatest of all: the power of Relativity to describe the overall structure, evolution and ultimate fate of the entire universe. Can one imagine a more profound, weighty, intellectual achievement? To get us started, we will try to give the reader a real feeling for the basic elements and incredible distances that we deal with in cosmology.

Spacetime Diagrams for General Relativity

In the earlier chapters, we saw that much can be appreciated pictorially by suppressing two of the three spatial dimensions in order to bring time into the illustration for Special Relativity. We now do this with the universe itself where General Relativity comes into play. Gravity is the primary mover in cosmology, and this justifies our dependence upon General Relativity, our best theory of gravity, to describe the overall dynamics of the universe.

The Motion of the Stars in the Galaxies

In the 1930s, F. Zwicky discussed the motions of the galaxies that comprised the elements of the Coma Cluster of galaxies. He noted that their velocities were not falling off with distance from the center of the cluster as would be expected on the basis of Newton’s theory of gravity.

Gravitational Waves and Energy-Momentum

In earlier chapters, we discussed sound waves, the actions of vibrating atoms and molecules passing along their motions in a linear chain which finally elicit vibrations in our ear drums. Our brains interpret these as sound, the information by which we communicate with each other and the pleasure that we gather from great vocalists and musical instruments that generate these waves in a skillful, blended manner.

Preliminary Aspects of Classical Physics

Before we get to Einstein’s Relativity, we should start with Isaac Newton, who was another genius and also, in his own way, a relativist. Many, including ourselves, feel that Newton and Einstein were the greatest physics geniuses of all time.