Partial differential equations and the flow of information

A fundamental but generally unstated principle in physics is that all things happen locally, action at a distance is nonsense, and there is no such thing as magic. What all this means is that whenever some event A is deemed to have caused event B, there has to be some identifiable agent or agency that transmitted the signal, force, interaction, or influence from A to B in a well-defined process with a finite speed, relative to any frame of observers.

To illustrate the point, consider Isaac Newton, who stated his law of universal gravitation in The Principia [Newton, 1687]. This law asserts that the gravitational force on one mass due to another mass depends on the instantaneous distance between them. Therefore it seems at first sight that Newton believed that gravitational forces act instantaneously over any distance. To most physicists, this is anathema. In fact, Newton also rejected action-at-a-distance, writing in a letter to Bentley that [Newton, 2006]

Tis unconceivable that inanimate brute matter should (without the mediation of something else which is not material) operate upon & affect other matter without mutual contact; as it must if gravitation in the sense of Epicurus be essential & inherent in it. And this is one reason why I desired you would not ascribe {innate} gravity to me. That gravity should be innate inherent & {essential} to matter so that one body may act upon another at a distance through a vacuum without the mediation of any thing else by & through which their action or force {may} be conveyed from one to another is to me so great an absurdity that I believe no man who has in philosophical matters any competent faculty of thinking can ever fall into it. Gravity must be caused by an agent {acting} consta{ntl}y according to certain laws, but whether this agent be material or immaterial is a question I have left to the consideration of my readers.

When he formulated the law of gravitation, Newton had no way of measuring the speed of propagation of the forces of gravity, which to all intents and purposes appears to be infinite. It is currently commonly believed that gravitational disturbances are metrical (distance) perturbations in spacetime that propagate at the speed of light.

The way that physicists encode this locality principle into the laws of physics is by formulating them in differential equations. Differential equations represent the principle of locality in its purest form. Most of the great equations of physics are partial differential equations, such as Newton’s laws of motion, Maxwell’s equations of electromagnetism, Einstein’s field equation in GR, Schrodinger’s wave equation, Dirac’s electron equation, and many more.

Differential equations come in many varieties: ordinary, partial, homogeneous, inhomogeneous, linear, nonlinear, coupled, real, complex, and so on. Physicists are fortunate that the structure of spacetime seems to restrict the class of differential equations that they need to solve. Almost all the fundamental equations of physics are second-order or first-order in time PDEs. Some are linear, others are nonlinear: some are classical, others need quantum mechanical interpretation. We discuss now a class of PDE fundamental to all branches of science.