Abstract
As we saw in the previous chapter, multi-index quantities exist in Differential Geometry. We highlighted that we did not see yet the mathematical nature of the first fundamental form; or the nature of Riemann symbols, Ricci symbols, or some important properties for geodesics. In this chapter, we take into account this nature and how general are the concepts introduced when we studied surfaces and curves on surfaces. We are interested in proving that the coefficients of the metric, the Riemann symbols, the Ricci symbols or the geodesics remain invariant when we deal with a change of coordinates. The substance of the General Relativity is related to the invariance under changes of coordinates and to the tensor structure of objects that have to present the same form in any reference frame. These two main properties can be related to the deep meaning of General Relativity which has the Equivalence Principle as the physical starting point.
Geometria substantia rerum.
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Notes
- 1.
It is worth noticing that spacetime coordinates can be indicated with Latin indexes \(i, j,\ldots =0,1,2,3,\ldots \) while space coordinates can be indicated with Greek indexes \(\alpha , \beta ,\ldots =1,2,3\ldots \). However, in literature, there is also the opposite choice, that is \(\alpha ,\beta ,\ldots = 0,1,2,3\ldots \) for spacetime coordinates and \( i, j,\ldots =1,2,3,\ldots \) for purely spatial coordinates. Here, we will adopt the first notation.
- 2.
It is worth noticing that General Relativity, from a mathematical point of view, is the physical theory whose objects are invariant under the group of linear transformations in four dimension, i.e. GL(4).
- 3.
In several textbook, the signature \((-+++)\) is adopted. In this case, time-like and space-like vectors have opposite sign.
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Boskoff, WG., Capozziello, S. (2020). Basic Differential Geometry. In: A Mathematical Journey to Relativity. UNITEXT for Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-47894-0_5
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DOI: https://doi.org/10.1007/978-3-030-47894-0_5
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