ramparts":3l0crvtj said:
Here's the calculation:
http://www.wolframalpha.com/input/?i=2* ... 289.9+x+10^-30+gm%2Fcm^3+*+4%2F3+*+pi+*+%2813.7+billion+light+years%29^3%29%2F%28speed+of+light%29^2
I'm not sure where you went wrong, although you didn't get the right value of the mass, so it's probably somewhere around there.
The answer is 1.3*10^26 meters, or about 1.4 times the Hubble length (the size of the observable universe). Actually, WolframAlpha seems to have an incorrect value of the Hubble length, but regardless, the answer is about 14 billion light years, so the answer is larger than the size of the observable universe.
Remember this isn't a
really meaningful calculation, it's just saying that if you took our observable universe and neglected gravitational effects from elsewhere in the universe (that is, took our observable universe as a standalone sphere), it wouldn't be a black hole.
Thanks ramparts,
I was unable to follow your link but I rechecked my calculation using WolframAlpha (a
very useful site, by the way):
Using 4.4 x 10^26 meters as the radius of the presently observable universe I got a volume of 3.568 x 10^80 m^3. this value is smaller by a factor of ten than my original calculation (mistake #1). The WolframAlpha web site indicates that this corrected number is equal to 1.0005 x volume of the observable universe so I feel pretty good about it.
I then took the Wikipedia value for the average density of the universe (9.9 x 10^-30 gm/cm^3) and converted it to kg/m^3. This comes out to 9.9 x 10^-27 kg/m^3. This number is 10^15 larger than my original calculation (
big mistake #2). Interestingly, the WolframAlpha site indicates that this corrected value in kg/m^3 is ~10 times larger than the average density of the universe. Upon closer inspection I found that the WolframAlpha site is using the same g/cm^3 value as Wikipedia. By my reasoning the average density in kg/cm^3 should be 1/1000 that given for gm/cm^3 and this would give me 9.9 x 10^-33 kg/cm^3. Since 1m=100cm, 1m^3 should equal (100 cm)^3 or 1 x10^6 cm^3. Using this conversion factor, I come up with an average density of 9.9 x 10^-27 kg/m^3 - which is the same calculated value that the WolframAlpha site produced. I don't know why the site is saying that this is ~10 times the value of the density of the universe.
Taking these corrected results, the WolframAlpha site calculates that the mass of the universe is ~5.5 x 10^54 kg. Here, again, the site indicates that this is ~120 times the mass of the universe - which it cites as being ~3 x 10^55 gm (~3 x 10^52 kg).
Taking the above calculated value for the mass contained in the observable universe and plugging it into the formula for the Swarzschild radius of an uncharged, non-rotating black hole I get 5.245 x 10^27 m (~554.4 Billion LY, or about 6 times the 93 billion LY diameter of the observable universe - according to Wikipedia - that I started with). Since this final calculated value is a radius, the diameter of the "event horizon" of the mass in the observable universe would be twice this, or 12 times the diameter of the observable universe.
If I use the WolframAlpha cited value for the mass of the universe (3 x 10^52 kg) I get 4.455x10^25 m (4.709 billion LY, or about 1/20 the diameter of the observable universe. Again, since this calculated value is a radius, the diameter of the "event horizon" of the mass in the observable universe would be about 1/10 that of the presently observable universe.
The meaning of these numbers perplexs me. Whether my calculated value for the Schwarzschild radius of the mass contained in the observable universe (~6 times the diameter of the observable universe) or the Schwarzschild radius of the mass contained in the observable universe (~1/20 the diameter of the observable universe) according to the WolframAlpha site is more probable, I'm left scratching my head.
In the first case (my calculation), it seems that an event horizon for the mass contained in the observable universe already exists at a distance (from Earth) of 554.4 billion LY. If there is mass beyond our observable universe, it will serve only to shrink this distance.
It's my understanding that the reason our present-day observable universe is described as having a diameter of ~93 billion LY is that objects (stars and such) beyond 46.5 billion LY from us are moving at superluminal velocities due to the expansion of the intervening space. The light from these objects will never reach us. If the expansion of space is accelerating then the diameter of our observable universe will shrink accordingly. For my present calculation, however, there appears to be sufficient mass in our observable universe to create an event horizon.
If this is the case, then my question would be: If we're living in a black hole with an event horizon, can the expansion of space push matter and energy past the event horizon to the "other side"?
In the second case (WolframAlpa's value for the mass of the universe), it would seem that the mass contained in our present-day observable universe would have to be concentrated in a volume within ~4.7 billion LY from where Earth now is in order to form a black hole with an event horizon. According to the currently accepted cosmological model, this was the case at some point in the past. Since our present-day observable universe stretches beyond this limit by a factor of ten, the answer to the question I posed above seems to be that, yes, the expansion of space can push matter and energy beyond an event horizon.
All of this may seem like idle speculation, but to me it seems that the rules that apply to black holes within our universe should also apply to our universe in total. If there's a mechanism by which matter and energy can be forced out of the event horizon of a black hole encompassing all the matter in our observable universe, then there must be a lower limit of mass within which this mechanism can't prevail against gravity. This seems likely since there's ample evidence that black holes do, in fact, exist in our universe. My question then becomes: Is there a certain maximum mass beyond which a black hole cannot exist due to the expansion of space contained within its event horizon?
Chris