<p>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$^{3}$.</p>

<p>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.</p>

<p>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.</p>

<p>Note though that the Schwarzschild solution in General Relativity is <em>static</em>. The universe is definitely not static.</p>

CC BY-SA 3.0
2018-03-25T00:38:27.627Z
Does that mean the Schwarzschild solution in GR is merely a close approximation, given that the accelerating expansion of space due to dark energy applies not just to the universe as a whole but also to smaller regions of space where gravity is the dominant factor?
Chappo Hasn't Forgotten
CC BY-SA 3.0
2018-03-25T03:48:43.517Z
@chappo (a) Yes - the Schwarzschild metric can only be an approximation for real astrophysical black holes - since they have to form. (b) Your point about expansion is incorrect when applied to gravitationally bound structures - they do not experience expansion.
ProfRob
CC BY-SA 3.0
2018-03-25T11:38:49.627Z
I respectfully suggest on point (b) that the error is yours. Gravitationally bound structures do "experience" expansion: they're not somehow exempt from the FLRW metric. It would be more accurate to say that expansion is negligible at all but the very largest scales. For example, at the scale of Earth's orbit, the rate of expansion is a mere 1.2 mm per hour. Nonetheless, *negligible* is not the same as *non-existent*.
Chappo Hasn't Forgotten
CC BY-SA 3.0
2018-03-25T12:57:23.423Z
@chappo No, it is much more complicated than that (and  even if it were not, then 1 km/century, would imply $\dot{a}/a \simeq 10^{-8}$/century, that is completely ruled out by current estimates of $\sim 10^{-14}$/century.) https://arxiv.org/pdf/1108.0246.pdf  .  You could also usefully look at this https://physics.stackexchange.com/questions/70047/can-the-hubble-constant-be-measured-locally/70056#70056
ProfRob
CC BY-SA 3.0
2018-03-26T09:52:07.623Z
Wouldn't your argument as treating the universe we are in as the interior of a black hole? Wouldn't this a self conflicting situation? Or in other words, is the interior of a black hole supposed to have a flat space too? If not than we can take any mass as big as what we think and calculate the S radius. Perhaps is the question that is highly pictorial.  It should be "what is the formula for Schwarzschild radius?".
Alchimista
CC BY-SA 3.0
2018-03-26T09:57:12.123Z
@Alchimista AFAIK there is little evidence that we are inside a black hole. Therefore all I have done is answer the question asked. Birkhoff's theorem says that the Schwarzschild solution is appropriate *outside*  any spherically symmetric mass distribution.
ProfRob
CC BY-SA 3.0
2018-03-26T09:58:27.627Z
Yes I understand. Just a way to get a number to plug in....indeed I commented to the question as well.
Alchimista
CC BY-SA 3.0
2018-03-24T20:44:45.830Z
2018-03-26T09:58:01.260Z
2018-03-26T09:58:01.260Z
ProfRob
<p>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.</p>
<p>Another user said &quot; The universe by definition is everything that exists.&quot;</p>
<p>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.</p>
<p>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&gt; ro (the event horizon radius) the coefficient (signature) of dr^2 is +  and that of dt^2 is -.  But for r&lt;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)</p>
<p>To help wrap one's head around a time-space swap, let me point out that one has to be careful asking a &quot;where&quot; question, as the answer might not be a place but a time. Similarly you have to careful asking a &quot;when&quot; question, as the answer might not be a time, but a place.
Let me give you some examples:</p>
<p>&quot;If we are in a BH, WHERE is the event horizon?&quot;
Answer: 13.5 billion years ago</p>
<p>&quot;What came BEFORE  the Big Bang?&quot;
Answer: &quot;The Outside!&quot;</p>
<p>What fun!</p>

CC BY-SA 4.0
2022-09-26T03:41:08.770Z
It is also interesting to point out that an thinker on the Outside would have no problem conceiving of mass that had already reached the central singularity at r=0 as well as mass that had not got there yet. But to an inside thinker, mass at the central singularity is ahead of him in TIME. It is harder for us conceive of matter that is displaced from us in time, but this cosmological model shows that it can be. I conculde that this is the nature of Dark Energy/ Dark Matter
Paul
CC BY-SA 4.0
2022-09-26T05:28:11.123Z
You can format equations using MathJax. Please see https://physics.stackexchange.com/help/notation & https://math.stackexchange.com/help/notation
PM 2Ring
CC BY-SA 4.0
2022-09-26T05:34:21.553Z
Some questions on our sister site about what happens to coordinates inside the event horizon: https://physics.stackexchange.com/q/370595/123208 & https://physics.stackexchange.com/q/673633/123208
PM 2Ring
CC BY-SA 4.0
2022-09-26T03:21:43.067Z
2022-09-26T03:21:43.067Z
Paul
<p>Before I answer this, it's important to correct a few assumptions:</p>

<p>(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".</p>

<p>(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.</p>

<p>Now, to your core question:</p>

<blockquote>
  <p>If a black hole has a mass of a universe what will be the volume of
  it?</p>
</blockquote>

<p>The <a href="https://en.wikipedia.org/wiki/Schwarzschild_radius" rel="nofollow noreferrer">Schwarzschild radius</a> is the radius defining the event horizon of a Schwarzschild black hole. If we take the mass of the <a href="https://en.wikipedia.org/wiki/Observable_universe" rel="nofollow noreferrer">observable universe</a> 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 $10^{79}$ $m^3$. </p>

<p>For comparison, this is less than 4% of the volume of the observable universe.</p>

<hr>

<p>EDIT:</p>

<p>Wikipedia's "quick facts" on the observable universe give the mass as $10^{53}$ kg, but <a href="https://en.wikipedia.org/wiki/Observable_universe#Mass_of_ordinary_matter" rel="nofollow noreferrer">the body of the article</a> contains the following qualification:</p>

<blockquote>
  <p>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.</p>
</blockquote>

<p>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 <em>within our universe</em>. </p>

<p>In a <a href="https://astronomy.stackexchange.com/questions/25603/if-a-black-hole-has-a-mass-of-a-universe-what-will-be-the-volume-of-it#comment44017_25603">comment on the main question</a>, userTLK makes an additional valid point that "the escape velocity at [a black hole's] event horizon is <em>c</em>. 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." </p>

CC BY-SA 3.0
2018-03-24T08:58:50.573Z
You haven't answered the core of the question, which is to justify the figure of $10^{53}$ kg. I think it may be considerably in error.
ProfRob
CC BY-SA 3.0
2018-03-24T10:48:24.017Z
I took the mass of observable Universe from Wikipedia, and a similar mass and radius were given in an answer in PhysicsSE. But Rob, if you’re aware of a more reputable source please let me know (and the mass quoted).
Chappo Hasn't Forgotten
CC BY-SA 3.0
2018-03-24T10:54:35.147Z
You need to read your sources more carefully. Wikipedia does not say that $10^{53}$ kg is the "mass of the observable universe".
ProfRob
CC BY-SA 3.0
2018-03-24T13:04:10.543Z
Ah yes, I see that it gives 10^53 kg in several places as “mass of the observable universe” but then qualifies this (later) as the mass of “ordinary matter”. Thanks for the correction. I’ll amend my answer accordingly.
Chappo Hasn't Forgotten
CC BY-SA 3.0
2018-03-25T00:26:04.117Z
@RobJeffries: I also relied on James K's answer [here](https://astronomy.stackexchange.com/a/16563), which says "The mass of the observable universe is of the order 10^53kg, mostly comprised of dark energy and dark matter. A black hole with that mass would have a radius of 2MG/c^2=10^26 m". This error must have flown under your radar?
Chappo Hasn't Forgotten
CC BY-SA 3.0
2018-03-24T05:06:58.130Z
2018-03-25T00:17:21.127Z
2018-03-25T00:17:21.127Z
Chappo Hasn't Forgotten
<p>Will it suck the entire universe in? What will the black hole look like to us, assuming we do not immediately get sucked in?</p>

CC BY-SA 3.0
2018-03-24T07:15:06.533Z
A requirement of a black hole is that the escape velocity at it'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.   (just throwing that out there).
userLTK
CC BY-SA 3.0
2018-03-26T09:56:20.793Z
It wouldn't be better to consider how the formula for the Schwartzschikd radius looks like?  As it is your question seem to suggest implications such as "can be our universe the interior of a black hole" and the like.  And even conceptual and or semantic trouble as noted by a user.
Alchimista
CC BY-SA 3.0
2018-03-26T09:56:41.770Z
It wouldn't be better to consider how the formula for the Schwartzschikd radius looks like?  As it is your question seem to suggest implications such as "can be our universe the interior of a black hole" and the like.  And even conceptual and or semantic trouble as noted by a user.
Alchimista
<black-hole><supermassive-black-hole>
If a black hole has a mass of a universe what will be the volume of it?
GroxTheProgrammer
# If a black hole has a mass of a universe what will be the volume of it?

(Asked by: username_0 on 2018-03-23)

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

**Tags:** black-hole, supermassive-black-hole

> **Comments:**
> **username_1**: A requirement of a black hole is that the escape velocity at it'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.   (just throwing that out there).  
> **username_2**: It wouldn't be better to consider how the formula for the Schwartzschikd radius looks like?  As it is your question seem to suggest implications such as "can be our universe the interior of a black hole" and the like.  And even conceptual and or semantic trouble as noted by a user.  
> **username_2**: It wouldn't be better to consider how the formula for the Schwartzschikd radius looks like?  As it is your question seem to suggest implications such as "can be our universe the interior of a black hole" and the like.  And even conceptual and or semantic trouble as noted by a user.  

---

## 3 answers

### Answer by username_3 on 2018-03-24. Score: 2 | Accepted answer

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.

> **Comments:**
> **username_4**: Does that mean the Schwarzschild solution in GR is merely a close approximation, given that the accelerating expansion of space due to dark energy applies not just to the universe as a whole but also to smaller regions of space where gravity is the dominant factor?  
> **username_3**: @username_4 (a) Yes - the Schwarzschild metric can only be an approximation for real astrophysical black holes - since they have to form. (b) Your point about expansion is incorrect when applied to gravitationally bound structures - they do not experience expansion.  
> **username_4**: I respectfully suggest on point (b) that the error is yours. Gravitationally bound structures do "experience" expansion: they're not somehow exempt from the FLRW metric. It would be more accurate to say that expansion is negligible at all but the very largest scales. For example, at the scale of Earth's orbit, the rate of expansion is a mere 1.2 mm per hour. Nonetheless, *negligible* is not the same as *non-existent*.  
> **username_3**: @username_4 No, it is much more complicated than that (and  even if it were not, then 1 km/century, would imply $\dot{a}/a \simeq 10^{-8}$/century, that is completely ruled out by current estimates of $\sim 10^{-14}$/century.) https://arxiv.org/pdf/1108.0246.pdf  .  You could also usefully look at this https://physics.stackexchange.com/questions/70047/can-the-hubble-constant-be-measured-locally/70056#70056  
> **username_2**: Wouldn't your argument as treating the universe we are in as the interior of a black hole? Wouldn't this a self conflicting situation? Or in other words, is the interior of a black hole supposed to have a flat space too? If not than we can take any mass as big as what we think and calculate the S radius. Perhaps is the question that is highly pictorial.  It should be "what is the formula for Schwarzschild radius?".  
> **username_3**: @username_2 AFAIK there is little evidence that we are inside a black hole. Therefore all I have done is answer the question asked. Birkhoff's theorem says that the Schwarzschild solution is appropriate *outside*  any spherically symmetric mass distribution.  
> **username_2**: Yes I understand. Just a way to get a number to plug in....indeed I commented to the question as well.  

---

### Answer by username_5 on 2022-09-26. Score: 2

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!

> **Comments:**
> **username_5**: It is also interesting to point out that an thinker on the Outside would have no problem conceiving of mass that had already reached the central singularity at r=0 as well as mass that had not got there yet. But to an inside thinker, mass at the central singularity is ahead of him in TIME. It is harder for us conceive of matter that is displaced from us in time, but this cosmological model shows that it can be. I conculde that this is the nature of Dark Energy/ Dark Matter  
> **username_6**: You can format equations using MathJax. Please see https://physics.stackexchange.com/help/notation & https://math.stackexchange.com/help/notation  
> **username_6**: Some questions on our sister site about what happens to coordinates inside the event horizon: https://physics.stackexchange.com/q/370595/123208 & https://physics.stackexchange.com/q/673633/123208  

---

### Answer by username_4 on 2018-03-24. Score: 1

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](https://en.wikipedia.org/wiki/Schwarzschild_radius) is the radius defining the event horizon of a Schwarzschild black hole. If we take the mass of the [observable universe](https://en.wikipedia.org/wiki/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.

---

EDIT:

Wikipedia's "quick facts" on the observable universe give the mass as $10^{53}$ kg, but [the body of the article](https://en.wikipedia.org/wiki/Observable_universe#Mass_of_ordinary_matter) 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](https://astronomy.stackexchange.com/questions/25603/if-a-black-hole-has-a-mass-of-a-universe-what-will-be-the-volume-of-it#comment44017_25603), 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."

> **Comments:**
> **username_3**: You haven't answered the core of the question, which is to justify the figure of $10^{53}$ kg. I think it may be considerably in error.  
> **username_4**: I took the mass of observable Universe from Wikipedia, and a similar mass and radius were given in an answer in PhysicsSE. But Rob, if you’re aware of a more reputable source please let me know (and the mass quoted).  
> **username_3**: You need to read your sources more carefully. Wikipedia does not say that $10^{53}$ kg is the "mass of the observable universe".  
> **username_4**: Ah yes, I see that it gives 10^53 kg in several places as “mass of the observable universe” but then qualifies this (later) as the mass of “ordinary matter”. Thanks for the correction. I’ll amend my answer accordingly.  
> **username_4**: @RobJeffries: I also relied on James K's answer [here](https://astronomy.stackexchange.com/a/16563), which says "The mass of the observable universe is of the order 10^53kg, mostly comprised of dark energy and dark matter. A black hole with that mass would have a radius of 2MG/c^2=10^26 m". This error must have flown under your radar?  

---

