"Top Down" versus "Bottom Up"

Physics Pedagogy versus Physics Research in General Relativity

Most courses (this one included) are taught in a “top down” fashion: Starting with the principles of relativity then exploring differential geometry as a beautiful mathematical structure that progresses to the field equations and then to the explanations of effects like the Anomalous precession of the perihelion of Mercury.

In reality, Einstein and his collaborators started with the experimental anomalies like the Anomalous precession of the perihelion of Mercury and the incompatibilities between special relativity and Newtonian mechanics and found a way (from the “bottom up”) to the laws, equations and principles, via a large number of dead ends.

Just two years before the “final form” of General Relativity was published, Einstein and Hilbert published a paper where they “proved” that there was no way the correct theory of gravitation could be “generally covariant” (more on this later). They even included, as one of the possible equations that could be discounted due to it possessing the property of being generally covariant, an equation that turned out, in the end, to be Einstein’s field equation (the central equation of general relativity, describing the curvature of space under the influence of mass)!