What if the Earth were hollow? A basic look into Newton’s Shell Theorem
One thing that I learned in my Astronomy studies in University was the value of performing what Einstein once named Gedankexperiment, or a thought experiment. This is a strategy that he utilized to think about the consequences of hypothesis that didn’t have a way to be physically tested, like the theory of relativity. Before I studied mathematics, I would be stuck with a lot of these questions that I wish I knew the answer to. What would happen if I’m driving through space at 0.5c and would shine my headlights out front? Would I see the light move at c? What about somebody standing by, would they see the light move at 1.5c? (Quick answer to that is neither of those are correct, I want to get into that in another blog since I love the science of light). Today, I will explore a question that I had as a child and can never wrap my mind around. What would happen if the Earth was hollow?
In order to better visualize this thought experiment, let’s say that a huge celestial being decided to shove a balloon directly into Earth’s core at the exact center of gravity. He’s a celestial being so he’s able to blow the balloon up, let’s pretend that it’s a vacuum inside and will ignore all other influences of gravity and that the surface of the Earth doesn’t expand, let’s keep things reeeeeaaaaally simple. Now we have a hollow ball of Earth, same mass, same surface area, just empty inside. To explore what will happen in this scenario, we want to use Newton’s Shell Theorem.
The equation to find the attraction force between two massive objects is the force equals the product of the universal gravitational force and the two masses, all divided by the distance between the two gravitational centers squared, or in silly text because I don’t know how to subscript:
This is a relatively straight-forward and intuitive answer to us, on the planet’s surface. We won’t be able to tell the difference in terms of the gravitational force, as the center of gravity of the hollow Earth does not change (which is why I mentioned the balloon was placed at the direct center of gravity of the Earth). Even if Earth’s shell were only 1cm thick, the center of gravity will not change, it’ll just be inside of a vacuum, and we will still be standing on the surface enjoying our day.
So we all joked at least once that if we dug a hole deep enough into the Earth, we might end up in Australia somewhere. If you dig a hole in your back yard and stand in it, you’re still having gravity act upon you (albeit a very slight bit stronger now, because your radius to the center of gravity of the Earth has lessened). What if you dig a hole in the hollow Earth and jump in? What happens then?
Starting simply, let’s say you find yourself at the location within the shell at center mass. All of the gravity vectors pulling on you from the outer shell (isotropic, for sake of argument), equals zero, which is easy to visualize. But what happens if you’re not at the center? What if you did jump through on the outside? Will you fall towards the center? The answer with a simple explanation, is no, you’ll stop accelerating once you’re inside the shell, just like if you were in space. The simple explanation, yes you are closer to one side of the shell and the gravitational vectors are stronger in between you and the side you just fell in on. However, there are more gravitational vectors pulling you in the opposite direction (since there are fewer particles pulling you in the direction closer to mass), and the net gravitational force remains zero.
The actual derivation of this was proven by Isaac Newton using calculus. For the sake of not throwing a bunch of equations on this blog in the text, I would suggest checking out Saul Remi Hernandez on YouTube where he explains the derivation of Newton’s Shell Theorem in one of his videos. He explains things in a nice conceptual manner before the derivation as well, which gives non-scientists and non-mathematicians a way to understand the consequences of this hypothesis without having to understand calculus at all. I find explaining science in layman’s terms to be fun, and I hope you enjoyed this little Gedankexperiment with me.