Transcription of Introduction to Tensor Calculus for General Relativity
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Introduction to Tensor Calculus for General Relativity
Massachusetts Institute of Technology Department of Physics Physics Spring 1999. Introduction to Tensor Calculus for General Relativity c 1999 Edmund Bertschinger. All rights reserved. 1 Introduction There 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 a pseudo-Riemannian manifold. In brief, time and space together comprise a curved four- dimensional non-Euclidean geometry. Consequently, the practitioner of GR must be familiar 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 any coordinate system used to label points in spacetime.
primitive concept, familiar from undergraduate physics and mathematics, applies equally in general relativity. An example of a vector is d~x, the difference vector between two infinitesimally close points of spacetime. Vectors form a linear algebra (i.e., a vector space). If A~is a vector and ais a real number (scalar) then aA~is a vector ...
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