Chapter

Tensors in general relativity

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/9780199573639.003.0013

Series: Oxford Master Series in Physics

                      Tensors in general relativity

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Differentiation of tensor components in a curved space must be handled with extra care. By adding another term (related to Christoffel symbols) to the ordinary derivative operator, we can form a “covariant derivative”; such a differentiation operation does not spoil the tensor property. The relation between Christoffel symbols and first derivatives of metric functions is re-established. Using the concept of parallel transport, the geometric meaning of covariant differentiation is further clarified. The curvature tensor for an n-dimensional space is derived by the parallel transport of a vector around a closed path. Symmetry and contraction properties of the Riemann curvature tensor are considered. We find just the desired tensor needed for GR field equation.

Keywords: differentiation in a curved space; parallel transport; Christoffel symbols; metric tensor; curvature; Riemann tensor

Chapter.  11440 words.  Illustrated.

Subjects: Astronomy and Astrophysics

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