# Schwarzschild Radius of the Universe

According to the Wiki on the Rs, the Rs of the observable universe is 13.7BLY.
https://en.wikipedia.org/wiki/Schwarzschild_radius (The observable universe's mass has a Schwarzschild radius of approximately 13.7 billion light-years.[7][8])

The reference for this statement is:

https://arxiv.org/abs/1008.0933 and the Encyclopedia of Distances

Can someone please explain this to me... Is this simply because to get into the non-observable portion of the universe, you have to go faster than the speed of light?

In this paper, the author begins by defining the radius of the observable universe as the radius of the Hubble sphere $$r_{HS}=\frac{c}{H_0}$$, where $$H_0$$ is the Hubble constant. He then assumes that the universe is a homogeneous and isotropic collection of matter with density $$\rho\approx \rho_c$$, where $$\rho_c=\frac{3H^2}{8\pi G}$$ is the critical density of the universe at which the curvature of space is zero.

Since he assumed that the universe is homogeneous and isotropic, the author uses the classical definition of density $$\rho=\frac{3M}{4\pi r_{HS}^3}$$, where $$M$$ is the total mass of the observable universe, and with a bit of algebraic manipulation comes up with $$r_{HS}=\frac{2GM}{c^2}$$. The author then asserts that $$r_{HS}$$ is the Schwarzschild radius of the universe, because what he came up with looks like the formula for a Schwarzschild radius.

This is where the big problem is: the conditions that the author assumed in the beginning are not compatible with the conditions that admit the definition of a Schwarzschild radius. The Schwarzschild solution of the Einstein field equations requires that all of the mass of the universe is concentrated in a physical singularity at $$r=0$$, and the rest is vacuum. The author assumes essentially the exact opposite: that the mass of the universe is as spread out as possible, so that none of it is concentrated anywhere, there is no vacuum, and the universe has uniform density. As such, calling this a Schwarzschild radius doesn't really make sense, as it has nothing to do with the Schwarzschild solution besides sharing a superficial similarity in how we express their definitions. Just because he calls it a Schwarzschild radius doesn't mean that it is one.

The moral of the story: though finding similar expressions in different contexts can often be a useful tool to guide intuition, it doesn't actually prove any connection, and isn't a substitute for an actual proof.

• probably_someone is still being kind ... the paper's author does not seem to understand even the basics of Einstein's formulation of general relativity ... the OP should just ignore this paper – Paul Young Apr 7 at 21:15
• It did not make any sense to me either which is why I posted the question. Thanks for the confirmation... – Rick Apr 8 at 1:06
• There may be a reason why, out of the 14 papers the author has written, only 3 are published, and that in some unknown journals and with no citations. Of the remaining 11 non-refereed papers, 1/3 of the citations are self-citations… – pela Apr 8 at 12:47
• @pela Indeed, there are many red flags which would make one suspicious of this paper's conclusions. The intent of this answer is that the argument can be debunked on its own merits, regardless of origin. – probably_someone Apr 8 at 12:59
• Yes, it's a great answer! :) – pela Apr 9 at 4:42

There are two separate issues here.

The Schwarzschild radius for a black hole, is calculated based on some very specific assumptions. For example, they won't be valid when space is rapidly expanding.

They also may not be valid on a huge scale, such as galactic clusters, where "dark energy" or expansion are factors, but we don't know enough about those to be sure of all their effects. But intuitively, it seems likely that our usual equations for collapsing objects wouldn't apply (or would require major modifications) if we can't assume a locally uniform spacetime metric, so they would apply on small scales up to local galaxies but probably not to large regions of space where expansion varies, or times of extreme expansion, or to the universe as a whole.

That's why the initial universe, though very dense, didn't re-collapse. The equations that determine when collapse happens, which we can apply in the current universe, are based on assumptions and approximations that just wouldn't be valid in the conditions of the early universe.

# Observable universe

The other issue going on, is the observability and horizon of our universe, which is for a completely different reason. Special relativity is the principle / natural law which says that nothing can travel (no known type of signal can propagate) faster than the speed of light. But special relativity applies to propagation within spacetime. In the very early universe (and much less nowadays) spacetime itself was expanding. This wasn't an expansion like we are used to. It was a change to the actual geometry of spacetime itself. As such, it didn't have a limit to its speed. It happened a tiny fraction of a second after the Big Bang. ** Suddenly, points in space that were "close together" (in some sense) became trillions of trillions of times more distant in a very short time. Wikipedia says the expansion was of the order of 10^26 in linear dimensions, or 10^78 in volume terms.

In "intuitive" rather than precise scientific terms, points that might have been reachable by light from each other in moments suddenly found themselves so distant that light needed immense amounts of time to travel between them.

If two such points suddenly found themselves much more than 13.7 billion light years apart (due to expansion), then there wouldn't have been time for light from one point to reach the other, even in the 13.7 billion years since that huge expansion. So they literally would not be observable now, because signals couldn't reach us in any way from them. ** Hence this means there's a practical "radius" or limit to what we can hope to observe, set by the speed of light itself - called the observable universe.

** We could in theory observe some of these distant objects from times before that expansion, when they weren't located outside the observable universe, but the expansion occurred in the first 10^-32 or so of a second, when the universe was so energy intense that we can't really hope to ever observe anything from that era.

• Please note that the radius of the observable universe is currently thought to be about 46 billion light years (not 13.7). – D. Halsey Apr 8 at 13:20
• Irrelevant to this answer, which is much more narrow: "If two such points suddenly found themselves much more than 13.7 billion light years apart (due to expansion [in the first fraction of a second]), they wouldn't be observable." You don't need to know the observable universe's present-day apparent size for that statement, so I didn't bring that into it. The statement given is enough to convey the idea of why there is going to be a limit to the observable universe, without discussing expansion since that time. The latter just changes the amount, not the fact or principle. – Stilez Apr 8 at 19:58