09 The Wave Equation in 3 Dimensions

We now turn to the 3-dimensional version of the wave equation, which can be used to describe a variety of wavelike phenomena, e.g., sound waves and electromagnetic waves. One could derive this version of the wave equation much as we did the one-dimensional version by generalizing our line of coupled oscillators to a 3-dimensional array of oscillators. For many purposes, e.g., modeling propagation of sound, this provides a useful discrete model of a three dimensional solid.https://digitalcommons.usu.edu/foundation_wave/1009/thumbnail.jp

Problem Set 5

Problem Set 5https://digitalcommons.usu.edu/foundation_wave/1029/thumbnail.jp

All homogeneous pure radiation spacetimes satisfy the Einstein–Maxwell equations

It is shown that all homogeneous pure radiation solutions to the Einstein equations admit electromagnetic sources. This corrects an error in the literature

02 Coupled Oscillators

Our next step on the road to a bona fide wave is to consider a more interesting oscillating system: two coupled oscillators.https://digitalcommons.usu.edu/foundation_wave/1002/thumbnail.jp

14 Conservation of Energy

After all of these developments it is nice to keep in mind the idea that the wave equation describes (a continuum limit of) a network of coupled oscillators. This raises an interesting question. Certainly you have seen by now how important energy and momentum — and their conservation — are for understanding the behavior of dynamical systems such as an oscillator. If a wave is essentially the collective motion of many oscillators, might not there be a notion of conserved energy and momentum for waves? If you’ve ever been to the beach and swam in the ocean you know that waves do indeed carry energy and momentum which can be transferred to other systems. How to see energy and momentum and their conservation laws emerge from the wave equation? One way to answer this question would be to go back to the system of coupled oscillators and try to add up the energy and momentum of each oscillator at a given time and take the continuum limit to get the total energy and momentum of the wave. Of course, the energy and momentum of each individual oscillator is not conserved (exercise). Indeed, the propagation of a wave depends upon the fact that the oscillators are coupled, i.e., can exchange energy and momentum. What we want to do here, however, is to show how to keep track of this energy flow in a wave, directly from the continuum description we have been developing. This will allow us to define the energy (and momentum) densities of the wave as well as the total energy contained in a region.https://digitalcommons.usu.edu/foundation_wave/1014/thumbnail.jp

Spinors, Jets, and the Einstein Equations

Many important features of a field theory, e.g., conserved currents, symplectic structures, energy-momentum tensors, etc., arise as tensors locally constructed from the fields and their derivatives. Such tensors are naturally defined as geometric objects on the jet space of solutions to the field equations. Modern results from the calculus on jet bundles can be combined with a powerful spinor parametrization of the jet space of Einstein metrics to unravel basic features of the Einstein equations. These techniques have been applied to computation of generalized symmetries and “characteristic cohomology” of the Einstein equations, and lead to results such as a proof of non-existence of “local observables” for vacuum spacetimes and a uniqueness theorem for the gravitational symplectic structure

Problem Set 7

Problem Set 7https://digitalcommons.usu.edu/foundation_wave/1031/thumbnail.jp

Problem Set 10

Problem Set 10https://digitalcommons.usu.edu/foundation_wave/1034/thumbnail.jp

The Spacetime Geometry of an Electromagnetic Wave

Since the 1920\u27s it has been known how to characterize almost all solutions to the Einstein-Maxwell equations in terms of geometric conditions built solely from the spacetime metric. These conditions are known as the Rainich conditions ; they provide a generalization to electrovacuum spacetimes of the geometry of vacuum (Ricci-flat) spacetimes. With the aid of modern computer algebra systems, the Rainich conditions also provide a novel approach to solving the Einstein-Maxwell equations. The Rainich conditions fail to describe solutions of the Einstein-Maxwell equations which have a null electromagnetic field, e.g., electromagnetic plane waves. In this talk I will review Rainich geometry and then describe geometric conditions on a spacetime which are necessary and sufficient for the existence of a solution to the Einstein-Maxwell equations with a null electromagnetic field. These conditions can be viewed as the analog of the Rainich conditions for null electrovacua, and they are equally amenable to computer implementation

15 Schrodinger Equation

An important feature of the wave equation is that its solutions q(r, t) are uniquely specified once the initial values q(r, 0) and (del)q(r, 0)/@t are specified. As was mentioned before, if we view the wave equation as describing a continuum limit of a network of coupled oscillators, then this result is very reasonable since one must specify the initial position and velocity of an oscillator to uniquely determine its motion. It is possible to write down other “equations of motion” that exhibit wave phenomena but which only require the initial values of the dynamical variable — not its time derivative — to specify a solution. This is physically appropriate in a number of situations, the most significant of which is in quantum mechanics where the wave equation is called the Schrodinger equation. This equation describes the time development of the observable attributes of a particle via the wave function (or probability amplitude) . In quantum mechanics, the complete specification of the initial conditions of the particle’s motion is embodied in the initial value of . The price paid for this change in the allowed initial data while asking for a linear wave equation is the introduction of complex numbers into the equation for the wave. Indeed, the values taken by are complex numbers. In what follows we shall explore some of the elementary features of the wave phenomena associated with the Schrodinger equation.https://digitalcommons.usu.edu/foundation_wave/1015/thumbnail.jp