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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.

We begin with vectors. A vector is a quantity with a magnitude and a direction. This 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 ...

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