Relativity Lite

An exploration of the special and general theories of relativity in five parts: Section ?? covering notation and conventions, Section ?? being a collection of elegant proofs of the conclusions of the special theory of relativity, Section ?? being notes from Ronald Mallett’s astrophysics lectures on the general theory of relativity, Section ?? supplementing the aforementioned lectures, and Section ?? being a special bonus section discussing gravimagnetics. 1. Conventions and Notations While Kandra’s notes allow us to make a faithful reproduction of the content of Mallett’s lectures, we have made some changes in the form of lectures. [Note James: we can arrive at a formating agreement.] Only the equations that need to be referred to later are numbered. In order to distinguish the important equations or concepts from equations that are used to derive them, important equations and concepts are summarized in numbered propositions. Propositions that are used only to derive important points are unnumbered. The use of propositions also allows us to present our conclusions first, so that one has a sense of where the maze of equations that follows is leading. Mallett’s lectures retain his use of component notation. Other sections rely heavily on differential forms. ?? provides a quick and hopefully useful summary of this way of representing vectors and tensors, without the derivations. We use bold upright sans-serif font for tangent vector operators and tensors in four-dimensional spacetime: dx, u, p, g, T, Riemann, Ricci, and G. We use bold italicized serif fonts for tangent vector operators and tensors in spaces with less than four dimensions: x, v, p, g, T , Riemann, Ricci, and G. 1-forms are represented by bold greek letters regardless of the space’s dimension: dξ, and π. Geometric vectors and their operators use arrows: ~x, ~v, and ~ p. We use η for the Minkowski metric η =  1 −1 −1 −1 dxα ⊗ dx . This is the sign convention that Mallett used in his lectures. We use geometric units everywhere except Section ??. Everything is measured in meters with the fundamental constants such as c, G, and k set to dimensionless unity. That’s meters, not centimeters. How astrophysicists, who deal with the radius, age, and mass of the universe can still be using tiny itsy-bitsy little units like centimeters and grams is beyond the imagination of one of the authors, especially given science’s and the world’s predilection for SI units. Date: September 22, 2004.