How did Einstein’s equations prove the existence of black holes?
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So these equations are pretty “vague” in a way. It didn’t prove they exist. When you have those equations you look for solutions. Physics is often written in differential equations. Like x”=-D×x. This is a differential equation and there are solutions to it. Here we have one solution. Its A×sin(u×t+k) where A and k are free parameters and u²=D. This is the equations that describes the motion of an object on a spring. And that A×sin(-||-) is the function for the position of the object.
Now Einstein’s equations are similar you imput some energy conditions, how much, in what distribution and you can find solutions that satisfy the equations.
The Schwarzschild solution was found by the guy its named after during WWI when he found a letter of Einstein. He tried to find a solution for packing some mass into one point. He found the solution (the solution is the resulting spacetime curvature) and thats it. Later physicists realise an object like that must be black so named it black ball but black hole sounded cooler.
The solution did prove they existed it only stated that a black hole is a solution to the equation. A spinning black hole is also a solution. So is one with charge.
We’d like to think that given sensible energy conditions so less than the energy of the universe and certainly not negative the resulting spacetime curvatures are physically possible. And black holes were the kind of energy conditions that seemed a bit too out there, but they weren’t. Wormholes are a bit out there too its really hard to say.
Some physicists try doing the reverse, engineering a spacetime curvature they want and figuring out the required energy conditions. Like with warp drive attempts which results in requiring negative mass.
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