Abstract
Isaac Newton: Dear Prof. Einstein, my Universe is very simple. I can describe it using vectors and calculus. Between any two objects, a gravitational force is acting and, according to the masses of objects and the distance between them, the gravitational force law is \(F=G\dfrac{mM}{r^2}\). The gravitational field, in this case, is \(A=\dfrac{GM}{r^2}\).
Quod erat demostrandum.
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Notes
- 1.
This request is an important property that any physical solution has to posses. In fact, very far from the source, a gravitational field has to go to zero. This means that Minkowski spacetime has to be recovered. This property is called “asymptotic flatness” and characterizes any physical gravitational field. It is worth noticing that this feature is fundamental for black hole solutions having physical meaning.
- 2.
From observational surveys, the Universe can be considered homogeneous and isotropic beyond scales of the order 100–120 Megaparsecs. See [62] for details. This means that, over these scales, no large scale structure, like clusters or super-clusters of galaxies are detected. According to these data, matter density can be considered homogeneously distributed in all directions.
- 3.
Actually the recombination of hydrogen happened at a redshift \(z=1089\) corresponding to a period of \(3.79\times 10^5\) years after Big Bang. Here the redshift correspond to the above \(a_{today}/a_{ionized}\). See [63].
- 4.
\(H_0\) is assumed constant because \(\rho _0\) is constant.
- 5.
It is important to note that any form of standard matter, in the interval \(0\le w\le 1\), gives rise to decelerated expansion.
- 6.
It is interesting saying that the paper reporting these results was the first one written by Enrico Fermi when he was student at Scuola Normale Superiore di Pisa [67].
- 7.
The story of this solution is very nice. Kurt Gödel gave it to Albert Einstein as a gift for his 70th birthday when they both lived in Princeton.
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Boskoff, WG., Capozziello, S. (2020). General Relativity and Relativistic Cosmology. In: A Mathematical Journey to Relativity. UNITEXT for Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-47894-0_9
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