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Ricci time in the Lemaître–Tolman model and the block universe

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Abstract

It is common to think of our universe according to the “block universe” concept, which says that spacetime consists of many “stacked” three-surfaces, labelled by some kind of proper time, \(\tau \). Standard ideas do not distinguish past and future, but Ellis’ “evolving block universe” tries to make a fundamental distinction. One proposal for this proper time is the proper time measured along the timelike Ricci eigenlines, starting from the big bang. This work investigates the shape of the “Ricci time” surfaces relative to the the null surfaces. We use the Lemaître–Tolman metric as our inhomogeneous spacetime model, and we find the necessary and sufficient conditions for these \(\{\tau = \) constant\(\}\) surfaces, \(S(\tau )\), to be spacelike or timelike. Furthermore, we look at the effect of strong gravity domains by determining the location of timelike S regions relative to apparent horizons. We find that constant Ricci time surfaces are always spacelike near the big bang, while at late times (near the crunch or the extreme far future), they are only timelike under special circumstances. At intermediate times, timelike S regions are common unless the variation of the bang time is restricted. The regions where these surfaces become timelike are often adjacent to apparent horizons, but always outside them, and in particular timelike S regions do not occur inside the horizons of black-hole-like models.

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Notes

  1. Equation (26) follows from \(R_{ab} = (T_{ab} - (1/2) T g_{ab}) \rightarrow \lambda _2 = \lambda _1 - (1/2) T\).

  2. We only need the ratio of absolute values, since the slopes of the incoming and outgoing light rays (31) have the same magnitude. It is convenient to put the simpler expression in the denominator.

  3. Several other plots, such as \(\mathscr {R}\) against t and R, are plotted but not used here.

  4. In fact, all the “black holes” we know about, such as at the centres of galaxies, are immersed in a CMB background and usually have accretion discs too, which means they are non-vacuum and thus do not have Shwarzschild or Kerr metrics. Nevertheless, non-vacuum black holes in LT have the same topology as the Shwarzschild-Kruskal-Szekeres manifold, i.e. two asymptotic regions joined by a neck, and they have a similar causal structure, though the horizons are dynamic rather than static (“isolated”). Other differences are more quantitative than qualitative.

  5. It is suggested that inflation will enormously reduce any bang time variations. However, after recombination, when LT models become reasonable models of near-spherical structures, the LT description will have an effective bang time variation, needed to best describe the matter evolution. This is because the evolution before recombination was controlled by a different equation of state, so the matching into a subsequent LT model will in general use all of the LT arbitrary functions. We don’t need to claim that the LT model is valid before recombination, but this effective bang time function still affects the Ricci-time calculations.

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Acknowledgments

YE thanks South Africa’s National Astrophysics and Space Science Programme (NASSP) for a bursary. CH and GFRE wish to thank South Africa’s National Research Foundation (NRF) and the University of Cape Town Research Committee (URC) for funding awards.

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Authors and Affiliations

  1. Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch, 7701, South Africa

    Yasser Elmahalawy, Charles Hellaby & George F. R. Ellis

  2. Department of Physics, Benha University, Banha, Egypt

    Yasser Elmahalawy

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  1. Yasser Elmahalawy
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  2. Charles Hellaby
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  3. George F. R. Ellis
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Correspondence to Yasser Elmahalawy.

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Elmahalawy, Y., Hellaby, C. & Ellis, G.F.R. Ricci time in the Lemaître–Tolman model and the block universe. Gen Relativ Gravit 47, 113 (2015). https://doi.org/10.1007/s10714-015-1950-0

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