Abstract
In this initial chapter we give a very short introduction to special and general relativity for mathematicians. In particular, we relate the index-free differential geometry notation used in mathematics to the index notation used in physics. As an exercise in index gymnastics, we derive the contracted Bianchi identities.
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Notes
- 1.
We use Einstein’s summation convention, with Greek indices running from 0 to 3 and Latin indices running from 1 to 3.
- 2.
Thus \(T_{[\alpha \beta ]}=\frac {1}{2}\left (T_{\alpha \beta }-T_{\beta \alpha }\right )\), \(T_{[\alpha \beta \gamma ]}=\frac 16\left (T_{\alpha \beta \gamma } + T_{\beta \gamma \alpha } + T_{\gamma \alpha \beta } - T_{\beta \alpha \gamma } - T_{\alpha \gamma \beta } - T_{\gamma \beta \alpha }\right )\), etc.
- 3.
In the formula below the indices between vertical bars are not anti-symmetrized.
References
W. Boothby, An introduction to differentiable manifolds and Riemannian geometry, Academic Press, 2003.
M. do Carmo, Riemannian geometry, Birkhäuser, 1993.
L. Godinho and J. Natário, An introduction to Riemannian geometry: With applications to mechanics and relativity, Springer, 2014.
S. Hawking and G. Ellis, The large scale structure of space-time, Cambridge University Press, 1995.
C. Misner, K. Thorne, and J. A. Wheeler, Gravitation, Freeman, 1973.
B. O’Neill, Semi-Riemannian geometry, Academic Press, 1983.
R. Wald, General relativity, University of Chicago Press, 1984.
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Natário, J. (2021). Preliminaries. In: An Introduction to Mathematical Relativity. Latin American Mathematics Series(). Springer, Cham. https://doi.org/10.1007/978-3-030-65683-6_1
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DOI: https://doi.org/10.1007/978-3-030-65683-6_1
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