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Gravity in general relativity, attractive and repulsive contributions

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Abstract

The gravitational equations of Einstein are solved for a sphere filled with a dust gas and floating in infinite empty space. It turns out that the radial acceleration of the gas has negative and positive contributions which may be interpreted as attractive and repulsive gravitational forces respectively. Two cases are considered: the collapse of a gas initially at rest and with uniform density, and the expansion of the gas with initial conditions appropriate to the observed Hubble diagram of Type Ia supernovae. The latter case may be seen as a proposal for a new type of cosmology.

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Notes

  1. Greek indices run from 0 to 3, Latin ones from 1 to 3, such that \(x^{0}=ct\), \(x^{b}=\left( r,\vartheta ,\varphi \right)\). \(g_{\alpha \beta }\) is the metric tensor in space–time. \(g_{\alpha \beta }\) and its inverse \(g^{\alpha \beta }\) may be used to lower and raise indices in the usual manner.

  2. Oppenheimer and Volkoff [4] and Oppenheimer and Snyder [5] have used Schwarzschild coordinates for the description of a neutron star. Their notation is different, however: they set \(Z=-\exp (-2\Phi )\) and \(S=\exp (-2\Lambda )\).

  3. We drop the angular part of \(ds^{2}\) for brevity; it is unimportant for radial motion.

  4. While (6) is an obvious consequence of (5), the relation (7) requires a few intermediate lines of calculation with partial derivatives in order to determine that \(\left( \frac{\partial t}{\partial r}\right) _{\theta }=\frac{V}{c^{2}}\frac{S}{|Z|}\) holds.

  5. For simplicity we shall often drop the parameters \(t_{f}\), \(r_{f}\), because it will be clear in each case from the text or the diagrams which trajectory we consider, usually the one that passes through the event of the observer.

  6. For stellar masses and radii the value of Q is so small as to be negligible except possibly for neutron stars. However, for the mass and radius of the universe—often estimated as \(\mathcal {O}(10^{52}\,{\text {kg}})\) and \(\mathcal {O}(10^{26}\,{\text {m}})\)—the value of Q may be close to 1.

  7. In (26) we have dropped the hats indicating non-dimensional quantities. Instead we have introduced tildas for functions of \(\hat{\tau }\) and \(\hat{r}\) instead of \(\hat{t}\) and \(\hat{r}\).

  8. For the fields shown in Fig. 3 this equation is an identity, of course, since those fields were calculated from the set (16).

  9. In elementary non-relativistic mechanics this term represents the gravitational attraction which grows linearly with r if the density is homogeneous. The other terms are absent in the non-relativistic case.

  10. Hubble [8] observed rather few nearby galaxies and found them to move outwards, away from us, with speeds proportional to their distance. Kirshner [9] gives an instructive review of Hubble’s work and the Hubble diagram.

  11. Figure 7 compiles the astronomical observations reported by Riess et al. [10], Amanullah et al. [11], Hicken et al. [12]. They are collected in the Union 2.0 and 2.1 catalogues [13].

  12. Our calculations were first presented in Müller and Weiss [3]; here they are summarized. For more detail we refer the reader to that preview.

  13. We trust that there be no confusion with the trajectories of Chap. 4—also denoted by \(r_{T}(t)\)—which passed through different events.

  14. Earlier in Sect. 4.1 a calculation like that was done by hand; now it must be done numerically, because of the complexity of (28), in particular the complex form of B(r).

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  1. TU Berlin, Berlin, Germany

    Ingo Müller & Wolf Weiss

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  1. Ingo Müller
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  2. Wolf Weiss
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Correspondence to Ingo Müller.

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Dedicated to the memory of Piero Villaggio (1932–2014), noted mechanician and mathematician.

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Müller, I., Weiss, W. Gravity in general relativity, attractive and repulsive contributions. Meccanica 51, 2933–2948 (2016). https://doi.org/10.1007/s11012-016-0508-x

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