I always struggled with math, so I just don’t get how an astrophysicist could look at some numbers and symbols and say, “Aha, it’s proof of black holes!”
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Math didn’t by itself prove the existence of black holes. What it did was show that black holes were *possible* in that the laws of physics allows them to exist, and gave us some idea of what we might see if we were looking at one. Then, armed with that knowledge, we could look at the sky and say “hey, that thing looks like how the math predicts a black hole would look.”
In other words, math didn’t prove the existence of black holes, but it told us what to look for.
A simple example that might illustrate this is escape velocity. If you look at the escape velocity equation, you’ll see
v = sqrt (2GM/R)
G is just some constant, so the only things that matter are M, R, and v. This equation is saying that if you are distance R from some mass M, your speed needs to be v for you to get away from that mass, otherwise you’re stuck either orbiting it or falling towards it.
Imagine you’re an astronomer armed with this equation, and you’ve got a bit of curiosity. You ask yourself, what happens if I keep R fixed but start increasing M? You’re staying at the same distance from the mass, but you’re making it more massive, essentially increasing its density. If you increase its density enough, you might find that v is now larger than the speed of light.
So, the math allows for some hypothetical object that’s so dense that not even light is fast enough to escape. You don’t know if it’s physically possible for something to be that dense, but the math (or at least, this particular equation) allows it to exist.
By leveraging more complicated equations, astronomers could learn a lot more about what black holes *should* be like without ever having observed one
Math and Physics are developed by observing what you see, trying to make a rule that describes what you see, and then testing that rule in other situations to make sure it holds up (basically the “scientific method.”).
Math showed the existence of black holes from two directions. One direction was that we observed planets and stars breaking existing rules, by having strange orbits or by stretching and visually warping in strange ways. These anomalies _would_ match existing rules, only if there was an extremely dark, and extremely dense object there that we couldn’t see.
The other direction is that we had existing rules and formulas that allowed the existence of such objects, but in order for the equations to work in a way that matched our observations, the object would have to be infinitely dense and it’s gravitational pull would be so strong not even light could escape.
We started with rules developed using the scientific method, but we also have also been using the scientific method in our steps to figure out the black hole: we observed something weird, we came up with rules to match it, and now all that was left was to test these rules. This is where things like observing gravetational waves and attempts to photograph a black hole come in, as they allow us to see if our predictions were correct.
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