A Simple Question

 Here’s a simple question, although getting to the question is quite so simple.

 

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There is a radius associated with a non-rotating, spherical black hole, called the Schwarzschild radius, which can be shown to be the radius at which light, travelling in a vacuum, cannot escape.  This is given by r_S=2GM_S/c², where G is the gravitational constant, M_S is the mass of the (Schwarzschild) black hole and c is the speed of light.  There is, therefore, a radius to mass ratio such that M_S = r_S.c²/2G. 

 

Such a black hole, by definition is spherical, and the volume of it can be given by V_S = 4/3.πr_S³.  This gives us the opportunity to calculate the density of a Schwarzschild black hole – given by

 

ρ_S = M_S/V_S = 3.c²/(8π.G.r_S²)

 

Interestingly, the density of a Schwarzschild black hole, therefore, is inversely proportional to the square of its radius (and also to the square of its mass, because the mass and radius are directly proportional).  The larger and more massive a black hole is, the less dense it is. Note that the gravitational time dilation equation is:

 

 

Note that a critical value of r, at which this equation becomes t_dilated = 0, is r=2GM/c².

 

You could wonder, then, what the density of an enormous black hole would be.  Let’s say one with a radius of 13.77 billion light years (the distance that light could have travelled in the time since the beginning of the universe, assuming flat space and something with uniform rectilinear motion that it was travelling relative to).  13.77 billion light years is equal to 13.03x10²⁶ m.  So, we have:

 

r_S = 1.303x10²⁶ m

c = 299792458 m/s

G = 6.6743x10^-11 m³/kg/s²

π = 3.14159 (-ish)

 

Plugging these in, we get a density of 9.47x10^-27 kg/m³ – or 9.47x10^-24 g/m³ – or 9.47x10^-30 g/cm³.  This is precisely the critical density of the universe (that should be of little surprise, since ρ_c = 3H²/(8π.G), where H is the Hubble parameter, which happens to be the inverse of the age of the universe, so in our equations above H = 1/(age of the universe) = c/r_S).

 

So: my simple question is this, why is it that the density of our universe is pretty much precisely (within the bounds of experimental uncertainty) the density of a Schwarzschild black hole that is precisely the size that a universe would be after 13.77 billion years, if it expanded at the speed of light?

 

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A slightly more complicated question is as follows.

 

Noting that, for any volume with a radius r_S, the density of that volume is critical when ρ_S is given by ρ_S = 3.c²/(8π.G.r_S²) – and the “criticality” thus only gets worse when the radius is greater, how can it be that the (observable) universe is 46.508 billion light years in radius and, according to WMAP measurements, 9.9x10^-27 kg/m³ in density, when a Schwarzschild black hole of that radius would have a density of 8.3x10^-28 kg/m³ (or about 1/12 of what we have)?