Black Holes 3/17/19

By Richard E. Bleil

Let’s see if I can pull this blog off. What I am going to try to do here is walk you through a calculation that struck me a few nights ago, and share my thought process with you. A lot of this information will be from web searches. This is a “back of the envelop” calculation. What I want to know is how close the atoms would be in a black hole.

To do this, I will begin with the assumption that the primary component of a black hole is hydrogen. This is fairly well known (and easily verified with a web search). Hydrogen is the simplest element, containing only one proton and one electrons. There are three known isotopes, the most common of which has no neutrons. the second most abundant having one neutron, and the other with two neutrons. It’s probably the simplicity that makes it so abundant. By one estimate, hydrogen is around 92% of all elements making up the universe. We will also assume that the black hole is spherical. We can do this because, geometrically, the sphere is the easiest shape, and is therefore the most commonly occurring shape in the universe for large bodies as well (sorry, flat-earthers).

The radius of a black hole has been estimated to be (the Schwarzchild radius) to be about 3,000 meters. This gives (using the formula for the volume of a sphere) the black hole a volume of V=(4/3)*pi*r^3 of 1.13×10^11 m^3.

The estimated mass of a black hole is assumed to be between 105 and 1010 times the mass of the sun. The average, then, should be around 557.5 times the mass of the sun, which is approximately 1.99×10^30 kg, or 1.99×10^33 g. Now, the formula mass of hydrogen (from the periodic chart) is about 1.008 g/mol. By dividing the mass of the sun by the formula mass of hydrogen, we see that our sun has 1097×10^33 moles of hydrogen. We know from Avogadro’s Number that there are 6.022×10^23 atoms/mol, so multiplying the moles of hydrogen by Avogadro’s Number, and we get the estimate of 1.19×10^57 hydrogen atoms in the Earth’s sun. The black hole is about 557.5 times more massive, which means we have approximately 6.63×10^59 hydrogen atoms in a black hole. This is 663,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 atoms, but who’s counting?

Sorry about that pause. I know how annoying it is to be reading an article, but then have to wait because the author suddenly stops typing, but I had to take care of some laundry.

Now, if we take the volume of our black hole, 1.13×10^11 m^3, and divide by the number of hydrogen atoms, 6.63×10^59, we find that the average volume afforded each hydrogen atom is roughly 1.70×10^-49 m^3. Already I can see this will be interesting.

The astute reader might argue that this would be the volume if it were cubes, but if hydrogen atoms are spherical, then we need to multiply by about 0.70 since only about 70% of the cube would be occupied by the sphere. While this is true, I’m not so interested in the volume of the hydrogen atoms, as the closeness. Packed in this tightly, I believe it is better to keep the cubic volume, as the length of one side of that cube would be about the average distance between two atoms, nucleus to nucleus. So, taking the cube root of this value ,we see that the distance between two atoms is roughly 5.54×10^-17 m, or 5.54×10^-5 pm. This is 0.0000554 pm, which is fascinating, because the diameter of a hydrogen atom is taken to be about 120 pm. This means that the hydrogen atom is squeeze in to roughly 1/2,200,000 its desired size!!!!!!!!!!!!!

Here’s something else to blow your mind. 5.54×10^-5 pm is roughly 0.0554 fm (I realize these units of distance are unfamiliar to most of my readers; this is why I am providing comparisons). Recall that the most common hydrogen is 1 proton and 1 electron. The proton’s radius is taken to be abut 1 fm, so it’s diameter is about 2 fm. The distance between hydrogen atoms in a black hole is 9 times SMALLER than diameter of a proton. This means the hydrogens are 9 times smaller than their own nucleus!!!!!

Does this strike you as impossible? Conditions in a black hole so extreme that the atoms are compacted so tightly that even their nucleus has to be reduced in size? Yup, me too. I thought this would be an interesting calculation (and I really did run through this real time as I was writing this post), and it certainly didn’t let me down. So how is this possible? Could the estimates of a black hole mass, or volume have been in error?

Sure, it’s possible, but it is more likely that there is a different set of physical laws at work. Just as we discovered new laws for subatomic physics (quantum theory) because the laws of classical physics break down, no doubt in extreme conditions such as black holes, the classical laws of physics again no longer apply. In fact, I doubt that the hydrogen atoms exist as independent atoms at all, perhaps behaving more like a Bose-Einstein condensate (a mass so cold that the atoms no longer occupy individual regions of space). The conditions of black holes make no sense to us because we are trying to apply our laws of physics to understanding them. Clearly this is impossible. This kind of makes me want to research the topic further! I may start with the physics of the Bose-Einstein condensate to see if what we know about this unusual state can be applied to black hole physics as well. Or, maybe you will beat me to it!

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