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The Physical WorldAn Inspirational Tour of Fundamental Physics$
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Nicholas Manton and Nicholas Mee

Print publication date: 2017

Print ISBN-13: 9780198795933

Published to Oxford Scholarship Online: July 2017

DOI: 10.1093/oso/9780198795933.001.0001

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Curved Space

Curved Space

Chapter:
(p.126) 5 Curved Space
Source:
The Physical World
Author(s):

Nicholas Manton

Nicholas Mee

Publisher:
Oxford University Press
DOI:10.1093/oso/9780198795933.003.0006

This chapter develops the mathematical technology required to understand general relativity by taking the reader from the traditional flat space geometry of Euclid to the geometry of Riemann that describes general curved spaces of arbitrary dimension. The chapter begins with a comparison of Euclidean geometry and spherical geometry. The concept of the geodesic is introduced. The discovery of hyperbolic geometry is discussed. Gaussian curvature is defined. Tensors are introduced. The metric tensor is defined and simple examples are given. This leads to the use of covariant derivatives, expressed in terms of Christoffel symbols, the Riemann curvature tensor and all machinery of Riemannian geometry, with each step illustrated by simple examples. The geodesic equation and the equation of geodesic deviation are derived. The final section considers some applications of curved geometry: configuration space, mirages and fisheye lenses.

Keywords:   spherical geometry, hyperbolic geometry, geodesic, Riemann curvature, metric tensor, covariant derivative, Christoffel symbol, fisheye lens, tensor, curved space

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