Are time dilation and frequency shift really infinite at the event horizon of a Schwarzschild black hole ?
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.
Not an easy question to answer, because there really isn’t any way to be completely sure, unless we develop the ability to travel to very distant regions of the universe. And even then, testing whether the laws of physics have remained the same may not be a trivial task. However, we can ask ourselves whether […]
Prerequisits : Special Relativity for Laypeople, Special Relativity : The Rest Of The Story Recommended : Manifolds and Curvature The purpose of this article is to introduce the basic concepts of what GR is about, without getting lost in mathematics, but also without succumbing to the “easy way out”, as given by seemingly simple yet highly misleading […]
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.
Further concepts of Special Relativity are introduced. We present Lorentz transformations in more detail, talk about Lorentz invariance as a fundamental symmetry of nature, and then introduce Noether’s theorem. This leads to a discussion of the energy-momentum tensor, its properties and conservation.