How can the Schwarzschild radius of the universe be 13.7 billion light years?

Question

So i was reading about Schwarzschild radius on Wiki and I found a interesting thing written there link.

• It says that the S. radius of the universe is as big as the size of the universe?

• How is this possible?

• Since most the universe is empty space shouldn't the S. radius of our universe be significantly smaller then 13.7 light years?

Answer 1

Firstly we should note that the universe as a whole is not described by the Schwarzschild metric, so the Schwarzschild radius of the universe is a meaningless concept. However if you take the mass of the observable universe you could ask what the Schwarzschild radius of a black hole of this mass is.

For a mass $M$ the Schwarzschild radius is:

$$ r_s = \frac{2GM}{c^2} \tag{1} $$

If the radius of the observable universe is $R$, and the density is $\rho$, then the mass is:

$$ M = \tfrac{4}{3}\pi R^3 \rho $$

and we can substitute in equation (1) to get:

$$ r_s = \frac{8G}{3c^2} \pi R^3 \rho \tag{2} $$

Now we believe that the density of the universe is the critical density, and from the FLRW metric with some hair pulling we can obtain a value for the critical density:

$$ \rho_c = \frac{3H^2}{8\pi G} $$

And we can substitute for $\rho$ in equation (2) to get:

$$ r_s = \frac{H^2}{c^2} R^3 \tag{3} $$

Now, Hubble's law tells that the velocity of a distant object is related to its distance $r$ by:

$$ v \approx Hr $$

and since the edge of the universe, $r_e$, is where the recession velocity is $c$ we get:

$$ r_e \approx \frac{c}{H} $$

and substituting this in equation (3) gives;

$$ r_s = \frac{1}{r_e^2} R^3 \tag{4} $$

If $r_e = R$ then we'd be left with $r_s = R$ and we'd have shown that the Schwarzschild radius of the mass of the observable universe is equal to it's radius. Sadly it doesn't quite work. The dimension $R$ is the current size of the observable universe, which is around 46.6 billion light years, while the size used in Hubble's law, $r_e$, is the current apparent size 13.7 billion light years.

If I take equation (3) and put in $R$ = 46.6 billion light years and $H$ = 68 km/sec/megaParsec I get $r_s$ to be around 500 billion light years or a lot larger than the size of the observable universe.

Answer 2

Yes, it is. It could be so because the density of a black hole decreases with its squared radius (as does the critical density of a flat universe, by the way). The bigger the S. radius, the lower the density, and there are known supermassive black holes in galactic nuclei that should have an average density lower than the density of the air...