Introduction to Tensor Calculus for General Relativity

Massachusetts Institute of TechnologyDepartment of PhysicsPhysics 1999 Introduction to Tensor Calculus for GeneralRelativityc 1999 Edmund Bertschinger. All rights IntroductionThere are three essential ideas underlying General Relativity (GR). The first is that space-time may be described as a curved, four-dimensional mathematical structure called apseudo- riemannian manifold. In brief, time and space together comprise a curved four-dimensional non-Euclidean geometry. Consequently, the practitioner of GR must befamiliar with the fundamental geometrical properties of curved spacetime. In particu-lar, the laws of physics must be expressed in a form that is valid independently of anycoordinate system used to label points in second essential idea underlying GR is that at every spacetime point there existlocally inertial reference frames, corresponding to locally flat coordinates carried by freelyfalling observers, in which the physics of GR is locally indistinguishable from that ofspecial Relativity .

pseudo-Riemannian manifold. In brief, time and space together comprise a curved four- ... Our metric has signature +2; the flat spacetime Minkowski metric ... also called a (m,n) tensor, is defined to be a scalar function of mone-forms and nvectors

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  Metrics, Tensor, Riemannian

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