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Relativity, Gravitation and CosmologyA Basic Introduction$
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Ta-Pei Cheng

Print publication date: 2009

Print ISBN-13: 9780199573639

Published to Oxford Scholarship Online: February 2010

DOI: 10.1093/acprof:oso/9780199573639.001.0001

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Metric description of a curved space

Metric description of a curved space

Chapter:
(p.81) 5 Metric description of a curved space
Source:
Relativity, Gravitation and Cosmology
Author(s):

Ta-Pei Cheng

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199573639.003.0005

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

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