How does a homogeneous FLRW metric arise from a cosmological model with black holes as the only source of gravitational field?

In astrophysical applications of general relativity we often need to apply the Einstein’s field equations to situations where the matter distribution, and consequently also the metric tensor, has a complicated form with relatively smooth large scale behavior and a complicated structure on smaller scales. The problems of this kind are usually approached in the following way: instead of solving the equations directly we apply them to an idealized metric with the small-scale structure removed by a coarse-graining procedure. As the effective stress-energy tensor on the right hand side of the field equations we take the average of the local stress-energy tensor over a relatively small region. Since the Einstein equations are non–linear, this procedure can only be justified as an approximation. It is very important to draw precisely its limits of validity and provide a way to improve it if needed. In the context of cosmology the additional terms we need to take into account in the effective stress–energy tensor are known as the backreaction. The problem of their derivation or estimation has been investigated in the recent years by many researchers.

The problem is most difficult and interesting if the metric in question is vacuum and the only source of the gravitational field are Schwarzschild or Kerr–like black holes. In this case there is no continuous matter distribution we may average over and the only contributions to the effective stress-energy tensor may come from summing over the discrete objects. A number of questions arises immediately as we apply coarse-graining to a collection of isolated objects, such as: whether the physical metric tensor approaches the coarse-grained one in any mathematical sense as the objects become smaller and smaller comparing with the large scale or whether the backreaction vanishes in the limit when the ratio between those scales goes to infinity. This is the problem of the continuum limit of a spacetime with discrete sources of gravitational field.

In my paper I address these issues by considering a particular class of solutions, namely the initial data for a closed FLRW model with dust as the continuous matter source. I present the construction of the corresponding vacuum initial data in which the dust has been replaced by an arbitrary number of Schwarzschild–like black holes. I then consider

the properties of the initial data when the number of black holes goes to infinity.

The main result of the paper consists of 2 inequalities. The first one estimates the difference between the metric tensor at a given point and the 3-metric of a suitably fitted FLRW initial data. The estimate depends on the distance to the nearest black hole as well as on the way the black holes are positioned. In the second inequality the relative difference between the effective total mass of the model, inferred from the fitted FLRW model, and the sum of the masses of the black holes is bounded by a function of various parameters characterizing the distribution of black holes. Since the coarse-grained FLRW model is homogeneous, the difference between both masses contains all the information about backreaction. In both theorems the more evenly the black holes cover the sphere, the tighter bounds we obtain.

I finally consider an infinite sequence of configurations of black holes for which the inequalities proved above suffice to prove that the continuum limit is attained as the number of black holes goes to infinity. I prove that for very large number of black holes the spacetime can be divided into two regions: the region lying within a finite number of Schwarzschild radii from the nearest black hole and the faraway region. In the latter we do not feel the gravitational field of any single black hole, but rather the collective influence of all of them. Consequently the metric converges to the corresponding FLRW metric at a known rate despite the fact that no matter is present there. In the former regions the metric remains always strongly distorted by the presence of the near black hole and approaches an isolated Schwarzschild metric instead. However as the number of black holes diverges the faraway region asymptotically takes up the whole spacetime. Quite counterintuitively, we find that the more black holes we add into the model, the more empty the spacetime becomes. I also show that the backreaction asymptotically vanishes in these models.

I then consider a more complicated situation when the universe in question is populated by close pairs of black holes instead of single, isolated ones. It turns out that the continuum limit exists the same way as before, but due to the nonlinear interaction between the constituents of the pairs the backreaction does not vanish even as we pass to the limit. This is important as it shows that the mere existence of the continuum limit does not guarantee the vanishing of the backreaction without additional assumptions about the matter distribution. The results are illustrated by numerical analysis of the first 6 configurations of the sequence in question.

The paper presents a new way of approaching the backreaction problem by proving rigorous estimates of the backreation terms. It also demonstrates new mathematical techniques, drawing from measure theory and quasi-Monte Carlo integration theory, which may find broader application in the study of the continuum limit and backreaction problem.

*Read the full article: Class. Quantum Grav. 31 085002*

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