The Mass of Everything

There is a paper in the American Journal of Physics that got some news attention in mid-October 2023.  The version that I first saw excitingly implied that it provided evidence that the universe is a black hole, which naturally caught my eye. (Anton Petrov also put out a video about it.)

 

This is the image that was being heralded:

 

Up in the top right corner, we can see the Hubble radius, which is the age of the universe times the speed of light (which is also the speed of light divided by the Hubble parameter).  Above that and to the left is a region marked as “forbidden by gravity”, basically indicating that anything in this region of the chart would be denser than the densest of black holes (a non-rotating black hole).

 

We will return to that, but the first thing that leapt out to me about this chart was the fact that atoms, the Covid virus, an unspecified bacterium, an unspecified flea, the (average) human, an unspecified whale, planets, moons, the Earth, main sequence stars and the Sun are all in a straight line.  That line seems to have the gradient log₁₀M/log₁₀r≈3 with little or no offset.  This makes eminent sense since there’s an established relationship between the mass of something and its volume, mediated by its density.  What might not be so immediately obvious is the question of the offset.

 

Let us introduce the density as ρ, and note that we are talking about a radius, such that the associated volume is 4πr³/3, so M=ρ.4πr³/3= (ρ.4π/3).r³ and therefore

 

log₁₀M=log₁₀((ρ.4π/3).r³)=log₁₀(ρ.4π/3)+log₁₀(r³)=3log₁₀r+log₁₀(ρ.4π/3)

 

If there were little or no offset, this would imply that log₁₀(ρ.4π/3)≈0 and so ρ.4π/3≈1, or ρ≈0.25.  However, in the paper, it’s noted that the line is consistent with the density of water (1g/cm³).  This represents an offset of log₁₀(4π/3)=0.6.

 

I looked at the background data (provided in a zip file) and noted that they estimated the radius of spherical humans, blue whales and fleas using the following process – To get the radius, take the length, divide by 2, that gives you a "ball", then divide by 2 again because we want the radius, not the diameter of the ball.  The figures that they used were (where yellow fill means that the values are received, and white means they are calculated, noting that the virion here is a Covid virus particle):

 

I went into a bit more effort and got (using the same colour coding):

 

Even though the virion and human densities is very wrong from their estimate, and the whale was somewhat wrong, there was very little effect on the relationship:

 

 

Note that the blue line in my chart represents black holes of different mass and the thin red line is log₁₀M=3log₁₀r+0.6 with the dots near it coming from the table above.  The other dots, left to right are, the Milky Way galaxy, the current Hubble radius and the “observable universe” (inflated radius of 46.5 ly).

 

We’ll get back to the “observable universe” later.

 

The second thing that leapt out at me was that there are other apparent lines on the chart, albeit shorter. These lines have the same gradient but a different intercept with log₁₀M=0.  There’s a vague line created by globular clusters, a more distinct line for galaxies and clusters of galaxies, and then a line for super-clusters right up against critical density (the density of the Hubble radius).  The offset for galaxies seems to imply a density in the order of about 10^-25g/cm³.  Note that there are voids illustrated that are below the critical density, which makes sense given that if some areas are higher than the critical density overall for a larger region, then some areas must be lower than this density – and these would be voids.

 

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So … the observable universe.  According to calculations performed by Ned Wright (although he might not be the original), the observable universe has a radius of about 47 billion light years – if one includes the assumption of accelerating expansion from about 5 billion years ago, as required to get the appearance, today, of constant expansion at the speed of light since 13.787 billion years ago, due to the disturbances caused by a period of inflation and two different periods of slower expansion. In The Problem(s) with the Standard Cosmological Model, I explained why I have issues with that explanation but if we accept that the observable universe is that big and note that the argument for that radius is based on the assumption of a critical density which is pretty much the density that we observe, then we have a problem.

 

The large orange dot in my chart above shows where the observable universe plots to, given its density and its radius (and thus its volume).  It can be seen to sit above the black hole line and thus in the “forbidden by gravity” zone, or (a little less clearly):

 

 

And, if you read Ned Wright further, you will note that he goes on to say that the universe, as a whole, is more than 20 times the volume of the observable universe.  This is a low-end estimate if space.com is to be believed, given that they report a measurement of 7 trillion light years across, or 3.5 trillion light years as a radius.  They also suggest that another possible figure is 10²³ light years, but this is clearly muddled since the universe would have expanded faster than the speed of light during any inflation event (not at the speed of light).  Plotting these three values as well, maintaining the same critical density (which is an assumption that is common to all three), we get:

 

So, something seems wrong here.  It does make me wonder, given that the paper says “we plot all the composite objects in the Universe: protons, atoms, life forms, asteroids, moons, planets, stars, galaxies, galaxy clusters, giant voids, and the Universe itself”, they don’t plot the universe itself, not even the “observable universe” … unless the authors’ view is that the Hubble radius is the radius of the universe.

 

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One issue raised by various people, including Anton Petrov, is that for the universe to be (inside) a black hole then there could be nothing outside of it, which introduces the issue of how the density would suddenly plummet to zero at the boundary.  Generally, it is thought that the density outside of the Hubble radius is the same as inside (because we are not privileged, our part of space in not special) – and this is the basis on which I chart the “observable universe” above the black hole line.

 

Note that Anton at one point misrepresents the Hubble radius as the radius of the “observable universe”.  The Hubble radius is 13.787 billion light years, not 93 billion light years.

 

Note also that, just after 11:00, he says that when a black hole gets big enough you can technically go inside it and feel nothing, so I am not convinced by the argument that outside the Hubble radius would have to have zero density for us to be in a black hole.

 

If the region inside the Hubble radius constitutes a black hole (which it does, since it has the radius, mass and density of a black hole of its radius, mass and density), then it’s possible to have black holes inside of black holes, since there’s a supermassive black hole at the centre of many galaxies and probably many other, smaller black holes in other locations.  Consider then the possibility that the Hubble radius itself is inside another, larger black hole.

 

Say, for example, that there’s a greater black hole the size of the “observable universe” at 93 billion light years.  What would its density be?  The calculation for the density is ρ=3c²/8Gπr², which for 93 billion light years is 8.3×10^-31g/cm³, or a little under one tenth the critical density of the universe of ρ_c=9.4×10^-30g/cm³ while having a volume that is 38 times greater.

 

Then consider the nesting of many effective black holes between the Hubble radius and the radius of the “observable universe” and think of one that is only slightly larger than the Hubble radius – 15 billion light years.  The density of that black hole would be 8.0×10^-30g/cm³ or about 85% as dense as where we find ourselves, with a total volume that is 30% greater.

 

So then the question is, how dense would it have to be beyond the Hubble radius to still constitute a black hole at 15 billion light years out?  I estimate that it would be in the order of 30% of the critical density.  And for a 93 billion light year radius, the average would be 6%.  The bottom line is that density doesn’t need to suddenly drop to exactly zero, even though we do introduce an apparent problem with privilege since the region inside the Hubble radius is special (due to being towards the centre of a zone that is of higher density than the surrounding area).

 

This is not, however, the only solution.  The other solution, more consistent with the notion of FUGE, and the implication of the paper’s writers, is that there’s nothing outside of the Hubble radius – not even empty space.  The problem with that is that it also seems to indicate that we are in a privileged location, because we appear to be at the centre of such a universe - unless the geometry is such that all points within appear to be central.