Abstract
Assuming that the reader is familiar with the notion of vectors, within a few pages, with a few examples, the reader will get to be familiar with the generic picture of tensors. With the specific notions given in this chapter, the reader will be able to understand more advanced tensor courses with no further effort. The transition between tensor algebra and tensor calculus is done naturally with a very familiar example. The notion of manifold and a few basic key aspects on Special Relativity are also presented.
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Notes
- 1.
In general it could be defined over \(\mathbb {C}\), but here, we are not interested in this case. Once the reader is familiarized with the notions presented here, it is easy to further study the generalization to complex spaces.
- 2.
We shall only be concerned with identical copies of vector spaces and their duals, therefore all spaces considered from now on will be of dimension n.
- 3.
Here we will use the short-hand notation \(f(x^i) \equiv f(x)\).
- 4.
M.P. Hobson, G.P. Efstathiou and A.N. Lanseby, General Relativity, An Introduction for Physicists..
- 5.
However, there are extensions of the theory which also include torsion.
- 6.
See Further Reading.
- 7.
The temporal coordinate should really be ct where c is the speed of light in the vacuum, but, as it is usual in Quantum Field Theory, we shall work using natural coordinates \(c=1=\hbar \).
- 8.
More on boosts and relativistic kinematics will be seen in Chap. 3.
- 9.
More on the Levi-Civita tensor density in four dimensions will be discussed in Chap. 5.
Further Reading
C.W. Misner, K.S. Thorne y John Archibald Wheeler, Gravitation, W. H. Freeman, New York
M.P. Hobson, G.P. Efstathiou, A.N. Lanseby, General Relativity, An Introduction for Physicists, Cambridge University Press, Cambridge
J.N. Salas, J.A de Azcarraga, Class notes
S. Weinberg, Gravitation and Cosmology , Wiley, New York (1972)
A.T. Cantero, M.B. Gambra, Variedades, tensores y fsica, http://alqua.tiddlyspace.com/
Y. Choquet-Bruhat, C. De Witt-Morette, M. Dillard-Bleick, Analysis Manifolds and Physics (North Holland, 1977)
R. Abraham, J.E. Marsden, T.S. Ratiu, Manifolds, Tensor Analysis, and Applications
J.B. Hartle, Gravity: An Introduction to Einstein’s General Relativity
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Ilisie, V. (2016). Vectors, Tensors, Manifolds and Special Relativity. In: Concepts in Quantum Field Theory. UNITEXT for Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-22966-9_1
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DOI: https://doi.org/10.1007/978-3-319-22966-9_1
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