2 - The theory of relativity: a mathematical overview

Summary

In the theory of relativity space and time loose their individuality and become indistinguishable in a continuous network termed space-time. The latter provides the unique environment where all phenomena occur and all observers and observables live undisclosed until they are forced to be distinguished according to their role. Unlike other interactions, gravity is not generated by a field of force but is just the manifestation of a varied background geometry. A variation of the background geometry may be induced by a choice of coordinates or by the presence of matter and energy distributions. In the former case the geometry variations give rise to inertial forces which act in a way similar but not fully equivalent to gravity; in the latter case they generate gravity, whose effects however are never completely disentangled from those generated by inertial forces.

The space-time

A space-time is described by a four-dimensional differentiable manifold M endowed with a pseudo-Riemannian metric g. Any open set U ∈ M is homeomorphic to ℜ⁴ meaning that it can be described in terms of local coordinates x^α, for example, with α = 0, 1, 2, 3. These coordinates induce a coordinate basis {∂/∂x^α ≡ ∂_α} for the tangent space TM over U with dual {dx^α}.