In this article we will be presenting an overview of the mathematical formulation and foundations of General Relativity. Basic objects such as the metric tensor and the connection are introduced, and given a geometric interpretation. The structure and meaning of the Einstein Field Equations will be discussed in more detail, a recipe for solving them will be presented, and the calculation of geodesics will be explained.
We introduce the basic concepts of differential geometry on manifolds. We define manifold, and explain the reasoning behind connections. This leads us on to the covariant derivative, and eventually to the Riemann curvature tensor, as well as the Ricci tensor. The geometric meaning of these objects is explained.
Consider Einstein’s gravitational field equations (1) We can perform a trace reversal on this, to obtain the Ricci tensor in terms of the source term : (2) In vacuum, the energy-momentum tensor field vanishes, so the vacuum equations reduce to the very simple form (3) which means that the vacuum, in the […]