http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?bibcode=1956MNRAS.116..678G&db_key=AST&page_ind=0&plate_select=NO&data_type=GIF&type=SCREEN_GIF&classic=YES
C. Gilbert was able to calculate the following scenario in 1956.
We have a standard FLRW universe with a perfectly uniform dust density ϱ, except that at the origin of the spatial coordinates there is an empty, vacuum sphere of a radius r, and at the origin there is a spherically symmetric object with a Schwarzschild mass m.
######<---------●--------->######
uniform m a
density ϱ
C. Gilbert says that there is a "hole" in the uniform mass distribution.
Earlier, A. Einstein and E. G. Straus had studied this problem.
https://en.wikipedia.org/wiki/Friedmann–Lemaître–Robertson–Walker_metric
The FLRW model has a scale factor R(t) (in Wikipedia this is a(t)), and reduced circumference polar coordinates with a space curvature 1 / R_0^2 (in Wikipedia this is k).
C. Gilbert finds out that if
m = 4/3 π a^3 ϱ,
then we can glue the Schwarzschild solution in the spherical hole smoothly to the FLRW solution in the uniform density zone.
The formula above looks sensible in the case where the expansion rate of the universe is zero, the dust density very small, and the spatial metric is flat. Then we can start from a uniform FLRW solution and can collect the dust from the hole to the center. The mass-energy of the central mass becomes approximately the right.
A spatially flat universe expanding at a constant rate
However, if the universe is expanding rapidly, the collector first needs to stop the dust in the hole from expanding, for example, using a spring system attached to individual dust particles.
That will give extra energy to the collector.
Does the collector need to spend that extra energy when he pulls the dust particle to the center?
If the spatial metric is flat and grows as
constant * t,
then the collector feels a force F(r) pulling particles at a distance r away from him. In the newtonian approximation, the force is proportional to r^2, and we can define a potential V(r), such that
F(r) = dV(r) / dr.
Thus, in this simple, newtonian case, the collector will spend exactly the kinetic energy which he harvested from a dust particle, to win the potential difference V(r) between the particle position and the center.
However, in most cases the universe is not expanding at a constant rate. The expansion slows down because of the gravity of the matter, or may speed up because of a cosmological constant.
Can the collector get the right m at the center?
It is not at all clear that the collector can collect the right mass-energy m at the center, so that the solution of C. Gilbert would be satisfied.
What if we cannot satisfy C. Gilberts solution? If general relativity has any solution at all for the collection process, there must happen something which either prevents the collection process, or which magically puts the right mass-energy to the center, so that C. Gilbert's solution is satisfied.
From our universe we know that some kind of collection of mass has happened. The collection is not spherically symmetric, though.
What could prevent the collection from happening? If the dust distribution is infinitely rigid, then we cannot collect dust. That would be a very strange solution to the problem.
Is there some magical process in general relativity which might provide the extra energy needed to stop the dust in the hole from expanding? That seems unlikely.
We conclude that general relativity probably is not compatible with the elementary process of collecting dust to form a star. That is a major blow to general relativity as a theory of gravity. The problem again seems to be the excessive strictness of the Einstein equations.
The simulations with the FLRW model and galaxy formation probably use a newtonian gravity approximation glued on top of the FLRW model. According to the paper of Räsänen et al. in our previous blog post, the simulations differ by a factor of 2 or so from our empirical observations.
Have the people running the simulations checked if there is any way of making a solution of general relativity from the approximation? A combination of FLRW and Newton is not guaranteed to produce anything like a solution of general relativity.
Let us look at the literature, what other people have written about this.
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