gravity - What is the reasoning behind the idea that light cannot escape from a black hole?

According to the definition, light cannot escape from a black hole.

How did scientists deduce that light cannot escape from a black hole?

Answer

Scientists asked the question "How does a body of arbitrary mass affect spacetime around it?" To answer this question, they took Einstein's General Relativity and applied it to the description of a spherically symmetric spacetime (meaning you can rotate any way you like and it looks the same) centered on a body of arbitrary mass, $M$.

I'll spare you the dirty details of this calculation, but what they found it that spacetime can be interpreted (loosely) as "falling" towards a gravitating body. They also found that the direction of an object's motion through time for any path that led away from the gravitating body was given by the sign (positive or negative) of

$$1-\frac{2GM}{c^2R}$$ where $G$ is Newton's Gravitational constant, $c$ is the speed of light, and $R$ is the radius of the body. What does this mean? If the sign is positive, that means that paths away from the body move forward in time (meaning we can follow them, since we only move forward). If the sign is negative, paths leading away from the body go backward in time, which means the only way to move away from the body is to go back in time; alternatively, it means any path you choose to take would lead towards the body. If the above expression comes out to be zero, then in GR that means that it is a path light would take. This means that only light could follow paths leading away from the body if the expression gives $0$ and not even light could move away if it is negative.

This is how we came to deduce the existence of a black hole. If $1-\frac{2GM}{c^2R}<0$, nothing can escape and if it equals zero, only light can escape. So they said, that means any body of mass $M$ that has a radius $R\le\frac{2GM}{c^2}$ would have a gravitational pull so strong that not even light could enter it and escape afterwards. Because this means that no light could ever come from such a body (and therefore it would look black), and because things fall into it but don't come back out (like a hole), they dubbed it a Black Hole