Chapter

Metric description of a curved space

Ta-Pei Cheng

in Relativity, Gravitation and Cosmology

Second edition

Published in print November 2009 | ISBN: 9780199573639
Published online February 2010 | e-ISBN: 9780191722448 | DOI: http://dx.doi.org/10.1093/acprof:oso/9780199573639.003.0005

Series: Oxford Master Series in Physics

                      Metric description of a curved space

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Einstein's new theory of gravitation is formulated in a geometric framework of curved spacetime. Here the subject of non-Euclidean geometry is introduced by way of Gauss's theory of curved surfaces. Generalized (Gaussian) coordinates: A systematic way to label points in space without reference to any objects outside this space. Metric function: For a given coordinate choice, the metric determines the intrinsic geometric properties of a curved space. Geodesic equation: It describes the shortest and the straightest possible curve in a warped space and is expressed in terms of the metric function. Curvature: It is a nonlinear second derivative of the metric. As the deviation from Euclidean relations is proportional to the curvature, it measures how much the space is warped.

Keywords: curved spacetime; non-Euclidean geometry; Gaussian coordinates; metric; geodesic equation; Gaussian curvature

Chapter.  9972 words.  Illustrated.

Subjects: Astronomy and Astrophysics

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