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Unformatted text preview: Semi-Riemann Geometry and General Relativity Shlomo Sternberg September 24, 2003 2 0.1 Introduction This book represents course notes for a one semester course at the undergraduate level giving an introduction to Riemannian geometry and its principal physical application, Einsteins theory of general relativity. The background assumed is a good grounding in linear algebra and in advanced calculus, preferably in the language of differential forms. Chapter I introduces the various curvatures associated to a hypersurface embedded in Euclidean space, motivated by the formula for the volume for the region obtained by thickening the hypersurface on one side. If we thicken the hypersurface by an amount h in the normal direction, this formula is a polynomial in h whose coefficients are integrals over the hypersurface of local expressions. These local expressions are elementary symmetric polynomials in what are known as the principal curvatures. The precise definitions are given in the text.The chapter culminates with Gauss Theorema egregium which asserts that if we thicken a two dimensional surface evenly on both sides, then the these integrands depend only on the intrinsic geometry of the surface, and not on how the surface is embedded. We give two proofs of this important theorem. (We give several more later in the book.) The first proof makes use of normal coor- dinates which become so important in Riemannian geometry and, as inertial frames, in general relativity. It was this theorem of Gauss, and particularly the very notion of intrinsic geometry, which inspired Riemann to develop his geometry. Chapter II is a rapid review of the differential and integral calculus on man- ifolds, including differential forms,the d operator, and Stokes theorem. Also vector fields and Lie derivatives. At the end of the chapter are a series of sec- tions in exercise form which lead to the notion of parallel transport of a vector along a curve on a embedded surface as being associated with the rolling of the surface on a plane along the curve. Chapter III discusses the fundamental notions of linear connections and their curvatures, and also Cartans method of calculating curvature using frame fields and differential forms. We show that the geodesics on a Lie group equipped with a bi-invariant metric are the translates of the one parameter subgroups. A short exercise set at the end of the chapter uses the Cartan calculus to compute the curvature of the Schwartzschild metric. A second exercise set computes some geodesics in the Schwartzschild metric leading to two of the famous predictions of general relativity: the advance of the perihelion of Mercury and the bending of light by matter. Of course the theoretical basis of these computations, i.e....
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