We introduce and discuss the Schwarzschild metric, at a level suitable for beginners with basic calculus knowledge. First, the exterior vacuum metric is examined, and several techniques about how to work with metrics are demonstrated, and their physical significance explained. A geometric understanding of the various terms in the metric is developed, and some fully worked examples are given. We then progress on to the interior metric, and present its peculiarities and physical consequences. This leads on to a discussion of Schwarzschild black holes, the meaning of event horizons, and what physically happens if we allow a test particle to freely fall into such black holes, from the points of view of different observers. Animations and a video are presented that help visualise these principles.
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.
Quick and dirty overview of the tools and concepts of tensor calculus. Discusses indices, the metric, the line element, as well as some simple index gymnastics.