Abstract
Einstein developed his ideas about the relativistic approach to gravity over many years, culminating with his 1915 papers presented to the Prussian Academy of Science.
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Notes
- 1.
All these formulas are written in SI.
- 2.
Actually, the notion of Levi-Civita parallel transport includes two components: the first is the one mentioned above, namely the fact that under parallel transport the inner product of vectors is preserved (in our case, the inner product between the vector \(A_\mu \) and the tangent to the geodesic); the second component is more subtle, as it is the requirement that the vectors do not “twist”. Roughly speaking, this means that the result of parallel transporting a vector field X along a vector field Y is the ‘same’ as parallel transporting Y along X.
- 3.
However, we have seen that at any point of the Riemannian manifold we can set the metric to the Minkowski one, which means that the derivatives of the metric are set to zero. So we have to involve necessarily in the Lagrangian also second derivatives of the metric.
- 4.
This is a formal writing, because \(G^{\mu \nu }\) is not a contravariant tensor, but the product between the minor corresponding to \(g_{\mu \nu }\) and the sign \((-1)^{\mu +\nu }\).
- 5.
Incidentally, if we allow the mass parameter M to turn into a function of the corresponding null coordinate , M (u) or M(v), we obtain the simplest non-static generalization of the non-radiative Schwarzschild solution, known as the Vaidya metric (ingoing and outgoing):
$$\begin{aligned}ds^2 = \left( 1-{r_S(v) \over r}\right) c^2\,dv^2-2\,dv\,dr - r^2\left( d\theta ^2 + \sin ^2\theta \,d\varphi ^2\right) ,\nonumber \end{aligned}$$$$\begin{aligned}ds^2 = \left( 1-{r_S(u) \over r}\right) c^2\,du^2+2\,du\,dr - r^2\left( d\theta ^2 + \sin ^2\theta \,d\varphi ^2\right) .\nonumber \end{aligned}$$The Vaidya metric describes the non-empty external space-time of a spherically symmetric and non-rotating star which is either emitting or absorbing null dust (sometimes called null fluid , i.e. a fluid for which the Einstein tensor is null). It is named after the Indian physicist and mathematician Prahalad Chunnilal Vaidya (1918–2010) and it is also called the radiating/shining Schwarzschild metric.
- 6.
To choose spatial coordinates to comove with the matter means to take \({dx^i \over dt}=0\).
- 7.
The method used here was suggested by D. K. Ross and L. I. Schiff, Phys. Rev., 1215, 141, 1960.
- 8.
The observation was reported in B.P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration ), Phys. Rev. Lett. 116, 061102 (2016).
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Chaichian, M., Merches, I., Radu, D., Tureanu, A. (2016). General Theory of Relativity. In: Electrodynamics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17381-3_9
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