Chapter 1 Physics in Euclidean Space and Flat Spacetime : Geometric Viewpoint

In this book, we shall adopt a different viewpoint on the laws of physics than that found in most elementary texts. In elementary textbooks, the laws are expressed in terms of quantities (locations in space or spacetime, momenta of particles, etc.) that are measured in some coordinate system or reference frame. For example, Newtonian vectorial quantities (momenta, electric fields, etc.) are triplets of numbers [e.g., (1.7, 3.9, −4.2)] representing the vectors' components on the axes of a spatial coordinate system, and relativistic 4-vectors are quadruplets of numbers representing components on the spacetime axes of some reference frame. By contrast, in this book, we shall express all physical quantities and laws in a geometric form that is independent of any coordinate system. For example, in Newtonian physics, momenta and electric fields will be vectors described as arrows that live in the 3-dimensional, flat Euclidean space of everyday experience. They require no coordinate system at all for their existence or description—though sometimes coordinates will be useful. We shall state physical laws, e.g. the Lorentz force law, as geometric, coordinate-free relationships between these geometric, coordinate free quantities. By adopting this geometric viewpoint, we shall gain great conceptual power and often also computational power. For example, when we ignore experiment and simply ask what forms the laws of physics can possibly take (what forms are allowed by the requirement that the laws be geometric), we shall find remarkably little freedom. Coordinate independence strongly constrains the laws (see, e.g., Sec. 1.4 below). This power, together with the elegance of the geometric formulation, suggests that in some deep (ill-understood) sense, Nature's physical laws are geometric and have nothing whatsoever to do with coordinates or reference frames. The mathematical foundation for our geometric viewpoint is differential geometry (also 1