IX - Gravitation

Summary

Differential geometry originally sneaked into theoretical physics through Einstein's theory of general relativity. Never before had tensors played such a fundamental role in a physical theory, nor had so much emphasis been placed before on the transformation properties of physical quantities under coordinate change. Indeed, the class from which this book sprang was originally a general-relativity class. By now you have grasped the message of this book: differential geometry is useful everywhere in physics. That feeling, plus the number of good books specializing in general relativity, led me to include very little about gravitation here, just enough to fill out the geometric message of the book. Also, since general relativity is a metric theory, it could lead you into the bad habit of failing to distinguish between tangent vectors and 1-forms. By now you realize that that is a poor style of thinking.

In Chapter VI I argued that electrodynamics was a special classical field theory, arguing from the need of the force law to preserve rest masses. Since in electrodynamics like charges repel, no theory like electrodynamics can explain gravity. How, then, is gravity to be described? Give up the constancy of rest mass. Settle instead for the constancy only of mass ratios. This allows us to include one additional classical field theory, one not satisfying u ⌟ f = 0. This field theory must be universal, with the same coupling to all matter. We can finish by rescaling lengths and times so that masses appear constant. The price paid is the introduction of a curved spacetime.