Abstract
We know that, contrary to electromagnetism, the Newtonian theory of gravity is compatible with Newtonian dynamics, and because we have just realized that the latter has to be replaced by special relativity, it is easy to understand that the former has to be superseded by another theory of gravity which is compatible with the new dynamics.
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Notes
- 1.
For a comprehensive historical review of this subject in English the interested reader can refer to Norton (2007).
- 2.
The “\(\eta _{\alpha \beta }\) to \(g_{\alpha \beta }\)” operation is one of the characteristics of the transition from special to general relativity. As we have seen, this derives directly from the equivalence principle which on its turn is connected to the way matter couples with gravity. This explains why it also enters into action in the so-called minimal coupling mentioned in the next chapters.
- 3.
We are using \(\varphi \) as the longitude of the polar coordinates to avoid misunderstandings with the scalar field \(\phi \).
- 4.
Actually in the literature there are several definitions of the different versions of this principle, each with its own peculiar interpretation. This is especially valid for the Einstein equivalence principle, which is also called the semi-strong equivalence principle.
- 5.
Actually the same kind of reasoning could be followed in the classical case by noting that the Lagrangian \(L=T-V\) in the case of the gravitational potential does not depend on the mass of the moving body.
- 6.
This and the next apparently strange initial conditions are taken just for a matter of convenience. In this case in fact we obtain the simplest expressions as a function of the proper time, and it also helps us also in the exercises.
- 7.
It is worth stressing that here we are not changing the reference system, in the sense that we are still in \(\bar{S}\) defined as the reference system comoving with the particle. Rather, this is just a change of one coordinate labeling the axis of \(\bar{S}\), and precisely one where \(\bar{x}\) is shifted by the quantity \(c^{2}/g\) at \(x'=0\).
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Vecchiato, A. (2017). Gravity and Special Relativity. In: Variational Approach to Gravity Field Theories. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-51211-2_7
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DOI: https://doi.org/10.1007/978-3-319-51211-2_7
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