# Black hole radius and universe

The Schwarzschild equation for a black hole's event horizon is rsh = 2GM/c2 or (1.48 x 10-27 m/kg) x M. Thus, the ratio of mass to radius is 6.7 x 1026 kg/m for all black holes.
If the mass and radius of the universe are calculated as follows,
Mass of the gravitationally connected universe, MU = c3/2GH = 9.25 x 1052 kg
Hubble distance (radius of universe), RU = c/H = 1.38 x 1026 m
where H = 2.18 x 10-18 /sec (67.15 km/sec/Mpc)
Then the ratio holds true. Two questions. One, assuming a finite universe, do we live in a black hole? Two, when the universe was much smaller (say 12 billion years ago after atoms formed), the mass to radius ratio had to be much larger, how is this interpreted?

Bandersnatch

Last edited by a moderator:
Thanks. No, the search results did not include FAQ. However, one FAQ I just read raised another question.
"How does conservation of energy work in general relativity, and how does this apply to cosmology? What is the total mass-energy of the universe?
One can also estimate certain quantities such as the sum of the rest masses of all the hydrogen atoms in the observable universe, which is something like 10^54 kg. Such an estimate is not the same thing as the total mass-energy of the observable universe (which can't even be defined)."

My question is: assuming a finite universe (a volume) and using the critical density equation, why does this not provide an estimate of the total mass/energy in the universe?

phinds
Gold Member
2021 Award
"How does conservation of energy work in general relativity

There IS no conservation of energy in GR.

WannabeNewton
"How does conservation of energy work in general relativity, and how does this apply to cosmology?
Conservation of energy is well defined in GR when one has a time-like killing field i.e. if the space-time is stationary. The total mass energy of a space-time is well defined if the space-time is asymptotically flat. Expanding cosmologies are neither of these things.

phinds
Gold Member
2021 Award
Conservation of energy is well defined in GR when one has a time-like killing field i.e. if the space-time is stationary. The total mass energy of a space-time is well defined if the space-time is asymptotically flat. Expanding cosmologies are neither of these things.

Good point. I should have said that energy conservation does not apply in the existing universe, not that it was broadly non-existent in GR.

The quote was from one of the FAQ. I did not make this clear. My question is on the last part. Quoting:
"One can also estimate certain quantities such as the sum of the rest masses of all the hydrogen atoms in the observable universe, which is something like 10^54 kg. Such an estimate is not the same thing as the total mass-energy of the observable universe (which can't even be defined)." end of quote.
I think the mass/energy can be defined with the critical density equation and an estimated radius based on the observable universe. Thanks

marcus
Gold Member
Dearly Missed
Hi Jim, I found the quote in the FAQ:
There was also an earlier (March 11, 2011) post by Ben Crowell which looks identical

It's not entirely clear to me why Ben says the total mass-energy of the observable universe cannot even be defined.

Let's quote the whole paragraph, so as not to take it too much out of context:

"One can also estimate certain quantities such as the sum of the rest masses of all the hydrogen atoms in the observable universe, which is something like 10^54 kg. Such an estimate is not the same thing as the total mass-energy of the observable universe (which can't even be defined). It is not the mass-energy measured by any observer in any particular state of motion, and it is not conserved."

Personally I don't see why you can't define it. The "observable universe" is definable in a practical approximate way although its extent changes over time. I suppose it could be defined as the spherical region around an observer at CMB rest which encompasses all the material he can in principal have gotten some type of signal from, even if he doesn't yet have the technology to detect every type of signal.
We have a pretty good idea of the RADIUS (which is gradually increasing as time goes on--even discounting expansion, more and more material is actually or potentially "heard from" and gets included.)

If you know the concept of "comoving distance" (the proper or freezeframe distance at a definite modernday epoch e.g. Jan 1, 2010, so that expansion doesn't affect it) then you can say the comoving radius of the observable region is increasing (as light etc from more and more distant stuff comes in) and it is now roughly 46 billion ly.

We have a pretty good idea of the mass-energy DENSITY within this comoving radius of 46 Gly, centered at where Solar System currently is. (say from CMB stationary observer standpoint, which is often adopted in this kind of talk.)

So I personally don't see why the mass energy inside that sphere (say including ordinary+dark matter plus radiation) is not DEFINABLE.

But I think the important point the FAQ is making is that it is not CONSERVED. For two reasons:

1. the observable region is not a fixed region, as time goes on it comprises more and more stuff. So it would be crazy to expect energy to be conserved when you aren't even talking about the same region, from year to year.
Percentagewise the change is so tiny as to be ignored but still, it's not a fixed region.

2. even in a fixed region, with say some definite fixed comoving radius, you wouldn't expect total mass-energy to apply because the geometry is changing, distances are increasing over time. So the basic assumption of the energy conservation theorem (by Noether 1915) is not met. Expanding distance is especially hard on radiation because it affects a photon's wavelength. If you double a photon's wavelength, you cut its energy by half.

Google "energy not conserved in expanding universe" to get an essay by Sean Carroll which gives what I think is a fair balanced nontechnical discussion. Also check our own PF FAQ. And it's great if you find stuff that you don't think is clear or consistent for some reason and want to give feedback! I'll make a thread in case there's more from you or others.

extra info: google "Emmy Noether" She was 33 when she proved her conservation theorem. It explains why (in which circumstances, under which assumptions) quantities like energy, momentum, and rotational momentum are invariant. http://en.wikipedia.org/wiki/Emmy_Noether
(gnarly technical account: http://en.wikipedia.org/wiki/Noether's_theorem )

Last edited:
• 1 person
timmdeeg
Gold Member
Two questions. One, assuming a finite universe, do we live in a black hole? Two, when the universe was much smaller (say 12 billion years ago after atoms formed), the mass to radius ratio had to be much larger, how is this interpreted?

I think this idea is not consistent with the redshift data. If we lived in a black hole, we should see anisotropy regarding cosmological distances.

I think this idea is not consistent with the redshift data. If we lived in a black hole, we should see anisotropy regarding cosmological distances.

I agree, for one BH's feed according to supply of material. If we were inside a BH we should be able to measure the feeding rates. Its one aspect I disagreed on when I studied Poplowskii's model involving torsion. Which as far as I studied it, had a consistent inflow of energy.

For the OP the model I'm referring to is on this page

http://www.nikodempoplawski.com/publications.html [Broken]