If a black hole has a mass of a universe what will be the volume of it?

Question

Will it suck the entire universe in? What will the black hole look like to us, assuming we do not immediately get sucked in?

Answer 1

According to measurements of the cosmic microwave background the universe is geometrically flat - which means that the mass/energy density of the universe is close to the "critical value" of $\sim 10^{-26}$ kg/m$^{3}$.

The radius of the observable universe is 46.6 billion light years, so the mass/energy contained within it is equivalent to $3.6\times 10^{54}$ kg.

The Schwarzschild radius of a black hole is $2GM/c^2$. If the mass/energy of the universe is spherically symmetric then its Schwarzschild radius is 560 billion light years and thus larger than the observable universe.

Note though that the Schwarzschild solution in General Relativity is static. The universe is definitely not static.

Answer 2

Before I answer this, it's important to correct a few assumptions:

(1) we can sit outside a universe-as-black-hole. This is impossible: since the universe includes everything that exists, then by definition we must be within it, so we can't look at it from the "outside".

(2) a black hole "sucks" matter into it. It doesn't, any more than a large star "sucks" matter. If the Sun somehow collapsed and became a black hole (it can't, this is just a thought experiment), all the planets would continue to orbit pretty much as usual, since the Sun's mass wouldn't have changed.

Now, to your core question:

If a black hole has a mass of a universe what will be the volume of it?

The Schwarzschild radius is the radius defining the event horizon of a Schwarzschild black hole. If we take the mass of the observable universe as roughly $10^{53}$ kg, then using the formula $$R=\frac{2GM}{c^2}$$ the Schwarzschild radius of this mass is 15.7 billion light years [NB: by comparison, the comoving distance to the edge of the observable universe is about 46.6 billion light years]. The volume is then easily calculated as 1.6 x $10^{31}$ cubic light years or roughly $10^{79}$ $m^3$.

For comparison, this is less than 4% of the volume of the observable universe.

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EDIT:

Wikipedia's "quick facts" on the observable universe give the mass as $10^{53}$ kg, but the body of the article contains the following qualification:

The mass of the observable Universe is often quoted as $10^{50}$ tonnes or $10^{53}$ kg. In this context, mass refers to ordinary matter and includes the interstellar medium (ISM) and the intergalactic medium (IGM). However, it excludes dark matter and dark energy. This quoted value for the mass of ordinary matter in the Universe can be estimated based on critical density. The calculations are for the observable universe only as the volume of the whole is unknown and may be infinite.

My calculations are based on the mass of ordinary matter in the observable universe, representing 4.9% of the total "mass/energy" derived from the observed critical density and volume. Rob's answer includes dark matter (26.8% of total mass/energy) and dark energy (68.3% of total mass/energy). Both answers are thought experiments, since it's not possible to have a black hole with the mass of the universe within our universe.

In a comment on the main question, userTLK makes an additional valid point that "the escape velocity at [a black hole's] event horizon is c. Dark energy might make that impossible. The vast distance and stretching of space and red-shifting of distant objects might make universe sized black holes impossible."

Answer 3

There are a lot of nice questions and nice answers here. I want to comment on the points that stood out to me most. One user estimated the size and mass of the universe and concluded that the Schwartzschild radius for that mass is larger than the estimated size of the universe. I also get that same result. If the universe is within its own Schwartzschild radius, then we are INSIDE a gigantic BH.

Another user said " The universe by definition is everything that exists."

Well, yes, until we realize there is more to the universe than we thought. For example, if we are in a BH, there is an inside and an outside. The inside is our universe, the outside is something else.

Moreover, if you look at the Schwartzschild metric, the coefficient of dr^2 is 1/(1-(ro/r)^2) and the coefficient of dt^2 is -(1-(ro/r)^2). For r> ro (the event horizon radius) the coefficient (signature) of dr^2 is + and that of dt^2 is -. But for r<ro, i.e. inside, the signs reverse. That means what was regarded as the spatial dimension r outside has become timelike inside and what was t outside has become spacelike inside. We therefore have to re-evaluate what kind of physics can go on in the inside using r as time and t as space. Actually you look for a tensor transformation of the interior Schwartzschild metric from (r,t) to (r',t') that makes the coefficient of dt'^2 inside equal to -1. You will find then that the coeffocient of dr'^2 is a function a(t') which describes an expanding univetse with hyperinflation at t'=0 corresponding to r=r0 ( the Big Bang = the event horizon) , decelerating to a plateau phase ( that we are likely just past) and then accelerating again to a Big Rip at some finite value of t' corresponding to r=0 (the central singularity)

To help wrap one's head around a time-space swap, let me point out that one has to be careful asking a "where" question, as the answer might not be a place but a time. Similarly you have to careful asking a "when" question, as the answer might not be a time, but a place. Let me give you some examples:

"If we are in a BH, WHERE is the event horizon?" Answer: 13.5 billion years ago

"What came BEFORE the Big Bang?" Answer: "The Outside!"

What fun!