Most curiously, the radius of the observable universe and its Schwarzschild-radius are presently equal to each other or at least near to equal (app. 10(exp26) m). Since the radius of the observable universe expands with light velocity and the Schwarzschild-radius stays constant, constant energy provided, this seems to be a unique, extremely unlikely coincidence. An increase in energy of the universe however, would expand its event horizon too and it can be shown that an increase of one Plank-particle per Plank-time would cause the Schwarzschild-radius to expand with light velocity too. Are there any observations that contradict such an explanation?
Furthermore, an increase in total energy in such an amount would also keep physical constants really constant if they are expressed in values of cosmic parameters. What would be the impacts on main cosmological theories including general relativity, if the total energy of the universe increases by one Plank-particle per Plank-time and if physical constants and cosmic parameters would not be independant from each other (besides the fact that we would not have a singularity at t=0, since E also would equal zero)?
Unfortunately, I´m not a physicist to be able to answer these questions!
This relationship between the presumed Schwarzschild radius of the observable universe and its radius is actually a surprisingly trivial mathematical identity. The critical density (required for a "flat" universe) is given by
ρ = 3H2 / 8πG,
where H is Hubble's constant, G is the gravitational constant. If we assume expansion at a uniform rate (no deceleration/acceleration), the age of the universe is given by 1/H, and the corresponding radius of the observable universe is
R = c / H,
where c is the speed of light.
The volume that is associated with this radius is given by
V = 4πR3 / 3,
and therefore, the mass of the observable universe is
M = Vρ = [3H2 / 8πG][4πR3 / 3] = c3 / 2GH.
The Schwarzschild radius is given by
Rs = 2GM / c2 = 2Gc3 / 2c2GH = c / H,
which is identical to the value of R above.
So no curious coincidence here, the identity is a result of elementary mathematics.
There are, however, two curious coincidences that lead to this identity, which remain to be explained satisfactorily. First, why is the universe spatially flat? (I.e., why is its actual density the same as the critical density?) Second, why is it that although the expansion of the universe slowed down during the first 8 billion years or so, and has been accelerating since, the expansion rate today and the age of the universe are related exactly as if there was no deceleration/acceleration at all, just expansion at a uniform rate?
The first of these questions may be answered, e.g., by inflation, but the second question relates to one of the great unresolved problems of physical cosmology, the cosmological constant problem.
This relationship between the presumed Schwarzschild radius of the observable universe and its radius is actually a surprisingly trivial mathematical identity. The critical density (required for a "flat" universe) is given by
ρ = 3H2 / 8πG,
where H is Hubble's constant, G is the gravitational constant. If we assume expansion at a uniform rate (no deceleration/acceleration), the age of the universe is given by 1/H, and the corresponding radius of the observable universe is
R = c / H,
where c is the speed of light.
The volume that is associated with this radius is given by
V = 4πR3 / 3,
and therefore, the mass of the observable universe is
M = Vρ = [3H2 / 8πG][4πR3 / 3] = c3 / 2GH.
The Schwarzschild radius is given by
Rs = 2GM / c2 = 2Gc3 / 2c2GH = c / H,
which is identical to the value of R above.
So no curious coincidence here, the identity is a result of elementary mathematics.
There are, however, two curious coincidences that lead to this identity, which remain to be explained satisfactorily. First, why is the universe spatially flat? (I.e., why is its actual density the same as the critical density?) Second, why is it that although the expansion of the universe slowed down during the first 8 billion years or so, and has been accelerating since, the expansion rate today and the age of the universe are related exactly as if there was no deceleration/acceleration at all, just expansion at a uniform rate?
The first of these questions may be answered, e.g., by inflation, but the second question relates to one of the great unresolved problems of physical cosmology, the cosmological constant problem.
Dear V.T. Toth,
you are correct, the simple equation R=Rs=c/H is the same that I have found too. But, as you state, the curiosity remains, even if it is formulated in another way. Here is another aspect of the same coincidence. The value of G, the gravitational constant may be expressed as:
G=R5 / 2*t4*E
There seems to be a deeper relationship between space, time, energy and physical constants.
Turn out a space, place it in a 4d(in)/3d(out) black hole and get the scale-invariant fractal multiverse in which flatness is normal within system integrity and initial collapse to black hole corresponds to inflation phase. Singularity front corresponds to the eternal now for contained universe (evolving matter, life, intellect). Mind is a fractal.
I somewhat agree with Mohamed El Naschie :)
Michael: It occurred to me that I should also add another important comment: the metric of a Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetime is very different from the Schwarzschild metric. So whereas the Schwarzschild radius makes sense when we investigate a spherically symmetric, static vacuum solution, it loses any physical meaning when the solution we are looking at is a homogeneous, matter filled, spatially homogeneous and isotropic, time-dependent metric. Conflating the two is generally not a good idea, even if one could come up with some contrived scheme to do so.
Let me phrase things somewhat differently from what I have seen among the answers. 1. Independent of the detailed cosmic evolution, the real mystery is that the total mass-energy density (including dark energy) is presently close to the critical density, determined by the present value of Hubble parameter H_0. If the extension of the observable universe at the present time is measured by the Hubble scale c/H_0, and its mass is defined by M= (4\pi/3)(c/H_0)^3 \rho_{crit}, then one obtains immediately c/H_0=2GM/c^2 = Schwarzschild radius for the mass M. Obviously, this strict equality depends on some reasonable conventions (extension, volume), and should be regarded as the result of an estimate.
2. However, we do not understand at all why the the present mass-energy density, including the vacuum energy density (dark energy), is so tiny. For example, any estimate of the contribution from the Higgs condensate (not really calculable) is many many orders of magnitude larger.
Basically, that is what I wanted to say. But maybe the solution for this "special universe" is, that energy, time, space and physical constants are not independent from each other.
Michael:- V. T. Toth presented you with some mathematical gymnastics, however that is all it is, as it is based upon demonstrably false premises. The so-called 'Schwarzschild radius' is not the radius of anything; in fact, it is not even a distance, let alone a radius, in the so-called 'Schwarzschild' metric, and so it has no relevance whatsoever to either the alleged 'black hole' or the alleged 'radius' of the Universe or the 'observable universe'.
Dear Stephen,
Frankly, your last comments are really strange. First of all, for any estimated mass M of a system, the quantity GM/c^2 ('gravitational radius') has the dimension of a length. If it is comparable with the dimension of the system under consideration, then GR becomes important. If it is much smaller, Newtonian gravity is a good approximation. This is, for instance relevant for the justification of N-body simulations for structure formation within the Newtonian framework. In our case, where the gravitational length scale is comparable to the Hubble scale, it is clear that a GR-treatment is essential,
Physics is full of such examples. The following amusing one I learned from Vicky Weisskopf, when I was a young fellow at CERN and he was there General Director: How high can the highest mountains on a planet be, and how does this hight depend on the mass of the planet? Simple considerations based on fundamental physics give a satisfactory answer. Here is another one: Derive an estimate of the Chandrasekhar limit for white dwarfs, using special relativity and the Pauli principle, without ever solving a differential equation. Estimate the radius of a typical white dwarf and show that its structure can be described accurately by Newtonian gravity.
There are countless examples of this kind in all fields of physics which help to arrive at a first understanding. It is important to train and cultivate this kind of analysis.
Dear Norbert,
I reiterate that the 'Schwarzschild radius' is neither the radius nor even a distance in any related spacetime metric. Since you contend otherwise, on what basis do you maintain that the 'Schwarzschild radius' is the radius of anything for the corresponding metric, and what do you say it is the radius of?
Dear Stephen,
Is the Planck mass the mass of an object you know? That it defines an important mass scale no body doubts. Or would you say it is not even a mann?
Dear Norbert,
You have not answered the questions. Please try again.
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