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Inhomogeneities in Cosmological “Backgrounds” in Einstein Theory

  • Chapter
Cosmological and Black Hole Apparent Horizons

Part of the book series: Lecture Notes in Physics ((LNP,volume 907))

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Abstract

This chapter discusses spherical inhomogeneities embedded in FLRW cosmological “backgrounds” in Einstein theory. We introduce the topic with the Schwarzschild-de Sitter-Kottler solution and then study a generalization, the McVittie metric (including its charged version). This family of spacetimes is further generalized and a late-time attractor of this class of solutions is found. We continue by discussing the Sultana-Dyer, Husain-Martinez-Nuñez, Fonarev, and generalized Fonarev solutions.

One has no right to love or hate anything if one has not acquired a thorough knowledge of its nature. Great love springs from great knowledge of the beloved object, and if you know it but little you will be able to love it only a little or not at all.

—Leonardo da Vinci

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Notes

  1. 1.

    This is hypothesis e) of Ref. [119].

  2. 2.

    Use \(m/\bar{r} = ma/R = m_{0}/R\), where ma is constant.

  3. 3.

    Also the known analytic solutions describing wormholes embedded in cosmological “backgrounds”, which are few, show that these wormholes evolve in time [10, 50, 109, 160].

  4. 4.

    A typographical error is present in the numerator of \(g_{00}\) in Ref. [64], but the line element appears correctly in the later references [65, 116, 117].

  5. 5.

    This fact is well known in the special case \(a \equiv 1,Q = 0\) corresponding to the Schwarzschild spacetime [19, 170].

  6. 6.

    The Einstein tensor appearing in Refs. [117] and [116] misses a scale factor in the denominators.

  7. 7.

    The analysis of Ref. [53] makes use of the line element of [64] which contains an error but the qualitative behaviour of the apparent horizons for \(\vert Q\vert \leq m_{0}\) does not change and the argument of Ref. [53], which is qualitative, is still valid.

  8. 8.

    A mixture of two perfect fluids is the matter source for the Sultana-Dyer solution (Sect. 4.7), which does not belong to the McVittie class.

  9. 9.

    In principle, one could take this vector to be spacelike instead of purely spatial.

  10. 10.

    Contrary to the McVittie spacetime, now ρ depends also on the radial coordinate.

  11. 11.

    This study analyzes a test fluid in great detail and finds the same qualitative behaviour for the mass of a black hole accreting cosmic fluid.

  12. 12.

    In principle energy can still flow superluminally inward across the cosmological horizon. The magnitude of the flux density q c decreases with the radial distance from the black hole.

  13. 13.

    This expression appears also in Ref. [43] and it can be derived also from Eq. (4.116) by expressing it in terms of R.

  14. 14.

    R(t, r) is an increasing function of r for \(r > m/2\) since, in this range, \(\frac{\partial R} {\partial r} = a\left (1 + \frac{M} {2ar}\right )\left (1 - \frac{M} {2ar}\right )\) is positive.

  15. 15.

    See Ref. [13] for scalar field sources of Lemaître-Tolman-Bondi models and the rest of this chapter for other scalar field solutions.

  16. 16.

    Beware of an error at the beginning of Ref. [48] consisting of imposing a coordinate condition which cannot be satisfied. This error was corrected in [29] and, later, in [159].

  17. 17.

    In a FLRW universe there are no spatial scalar field gradients (which would identify a preferred spatial direction) and the energy density and pressure are simply \(\rho ^{(\phi )} = \frac{\dot{\phi }^{2}} {2} + V (\phi )\), \(P^{(\phi )} = \frac{\dot{\phi }^{2}} {2} - V (\phi )\). If \(V (\phi ) = 0\), then it is \(P^{(\phi )} =\rho ^{(\phi )}\).

  18. 18.

    Sometimes one encounters in the literature also dynamical black hole spacetimes which are constructed by hand and are not known to be solutions of the Einstein equations or of the field equations of other theories of gravity (e.g., [16, 60, 105–107]).

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    Valerio Faraoni

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Faraoni, V. (2015). Inhomogeneities in Cosmological “Backgrounds” in Einstein Theory. In: Cosmological and Black Hole Apparent Horizons. Lecture Notes in Physics, vol 907. Springer, Cham. https://doi.org/10.1007/978-3-319-19240-6_4

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