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.
Comparing equations (2) and (3), we see that vectors and one-forms are linear operators on each other, producing scalars. It is often helpful to consider a vector as being a linear scalar function of a one-form. Thus, we may write hP,˜ V~i = P˜(~V) = V~(P˜). The set of
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