# How did Einstein “see” in his equations that black holes should exist before they were observed?

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I have some knowledge of calculus and differential equations, but what is it about his equations that jumped out? How did he see his equations and decide that this was a legitimate prediction rather than just some constructed “mathy” noise?

In: Mathematics

He didn’t.

Einstein’s Theory of Relativity is a series of fiendishly complex equations that describe how space curves in the presence of matter (among other things).

Another scientist solved those equations for a particular set of physical conditions (Karl Schwarzschild) and saw that they predicted an object that has the properties of what we now call a black hole. He sent these results to Einstein to see if he had made a mistake, and Einstein said that the math looked correct.

That’s part of what has made relativity a robust theory: It has made multiple predictions beyond what its original author saw, and most of those predictions have aligned with later observations.

He didn’t.

Others (like Karl Schwarzschild) took Einstein’s theory of general relativity and its equations and realised there’s a solution that involves infinite deformation of space time, in particular if you put a lot of mass concentrated in a small volume, you get a region of space time where even light cannot escape.

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The theory of black holes are actually a hundred years older then Einstein. Starting with Newtons theory of gravity you can calculate the escape velocity. Basically this is the integral of the gravitational acceleration from the current distance between the bodies to infinity. Any object with a higher speed will be able to escape the gravity well while any object with less speed will orbit back at some point. The problem is that even light have speed so it is possible for some objects to be dense enough that even light can not go faster then the escape velocity and will therefore not be able to escape.

Einsteins theories of relatively is related to this. He expanded on the concept of a fixed speed of light and combined this with Newtons laws to make these equations fit. And he found out a lot of things, for example that the speed of light were actually a universal speed limit and that nothing can go faster then this. He also basically confirmed the theories of black holes, which at that point had been just cool ideas.

It was actually another physicist, Karl Schwarzschild, who worked on Einsteins field equations around black holes and found a solution to them that could accurately predict the ratio between the size and mass of a black hole. Using just the escape velocity does not quite work as Newtons formulas were not as accurate in these kinds of extreme environments. But Schwarzschild also found that at this radius a lot of the field equations ended up growing to infinity, a singularity. Essentially according to the math there were no space in a black hole and there were infinity space at this specific radius.