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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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Tensors in general relativity

Tensors in general relativity

Chapter:
(p.298) 13 Tensors in general relativity
Source:
Relativity, Gravitation and Cosmology
Author(s):

Ta-Pei Cheng

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199573639.003.0013

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

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