Friday 10 July 2020

cosmology - If the observable universe were compressed into a super massive black hole, how big would it be?


I understand only a little of general relativity, but that's why I'm here! :)


Consider the hypothetical situation of some extra-terrestrial intelligence pushing all the mass in the observable universe, every last subatomic particle, into a giant black hole. What is the surface area of such an object? For the sake of argument, let's keep this very idealized. Dark energy, whatever it is, can be captured and thrown into the supermassive black hole; assume there's no Hawking radiation causing it to slowly evaporate, etc.




Answer



About 73% of the energy of the Universe is stored in dark energy - the cosmological constant, most likely - which has a negative pressure numerically equal to the energy density. Because it is a cosmological "constant", this portion of the mass of the Universe cannot be really compressed. So it is a problematic idea to include the dark energy into the "mass that you want to compress".


Only dark matter and visible matter - whose pressure is nearly zero - can really be "compressed". So that would be about $1\times 10^{54}$ for the visible Universe. $2GM/c^2$ for this mass $M$ produces about 150 billion light years which is about 3 times larger than the radius of the observable Universe.


We obtained the result that even if we only count the "particulate" component of the mass of the Universe, we find out that the Universe is actually smaller than the black hole. It is no contradiction because the Universe is not a static system. It would be impossible for the Universe to be kept this small. However, the galaxies are receding from each other which prevents the formation of a nearby event horizon. Instead, the closest horizon in the reality is the "cosmic horizon" - how far we can actually see because of the finite speed of light and finite age of the Universe. It depends on the observer but effectively makes everything beyond this horizon unphysical.


Our Universe, dominated by the dark energy, is already rather close to an empty de Sitter space which is, from many viewpoints, analogous to a black hole except that the interior of the visible part of the de Sitter space is analogous to the exterior of a normal black hole, and the analogy of the interior of a black hole is everything that is behind the cosmic horizon - where we don't see. It is misleading to create the analogy with the static black holes directly because our Universe is not static in the normal cosmological coordinates.


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