Abstract
General Relativity is not the end of the story. Several issues, ranging from Quantum Gravity to the Dark Side of the Universe need to be addressed in a self-consistent theory. Here we want to summarize some of these approaches. In particular, we want to show that the same theory, General Relativity, can be represented in different ways. This fact is questioning the basic foundations of the theory like the Equivalence Principle, the Lorentz Invariance, and so on. According to this picture, new possibilities can be taken into account in view of further theoretical and experimental developments. Topics of this chapter are more advanced with respect to the rest of the book and could be considered for some short advanced lectures.
The development of Physics, like the development of any science, is a continuous one.
Owen Chamberlain
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Notes
- 1.
Sometimes, capital Latin indexes, referring to local coordinate indexes, are also indicated by an over hat on the Greek indexes, i.e. \(e^A_{\ \mu }=e^{\hat{\nu }}_{\ \mu }\).
- 2.
We define the torsion tensor as minus of that defined in Eq. (14.23), for having the signs in agreement when compared to those of GR.
- 3.
Equation (14.111) is very important, but its derivation is also not trivial at all. Here, we provide an intuitive proof, although a more rigorous demonstration can be found in Sect. II.6 of Ref. [125]. Let us suppose to have the tetrads \(\mathring{e}^A_{\ \mu }\) in GR and \(\hat{e}^A_{\ \mu }\) in TG such that they have the same coefficients of anholonomy \(\mathring{f}^A_{\ BC}=\hat{f}^A_{\ BC}\), guaranteed by the fact that there exists an isomorphism assuring this property. This implies \(\mathring{\mathcal {D}}_\mu =\hat{\mathcal {D}}_\mu \), which then gives Eq. (14.111).
- 4.
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Boskoff, WG., Capozziello, S. (2024). Metric-Affine Theories of Gravity. In: A Mathematical Journey to Relativity. UNITEXT for Physics. Springer, Cham. https://doi.org/10.1007/978-3-031-54823-9_14
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DOI: https://doi.org/10.1007/978-3-031-54823-9_14
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