On the Fine Structure of Spacetime

General relativity describes spacetime as far as large scales are concerned and is based on the geometric paradigm discovered by Riemann. It replaces the flat (pseudo) metric of Poincaré, Einstein, and Minkowski, by a curved spacetime metric whose components form the gravitational potential. The basic equations are Einstein equations (Figure 1) which have a clear geometric meaning and are derived from a simple action principle. Many processes in physics can be understood in terms of an action principle, which says, roughly speaking, that the actual observed process minimizes some functional, the action, over the space of possible processes. A simple example is Fermat’s principle in optics which asserts that light of a given frequency traverses the path between two points which takes the least time. In the case of Einstein’s equation the action is a functional on the set of all possible spacetime configurations and is computed by integrating the spacetime curvature over all of spacetime.