CoafOS's Shortform

It's evening, the sun is set. A man walks up to a scholar:
"Scholar, the sun rose yesterday and today morning. Will it rise again tomorrow?"
"Man, I don't know, it's kinda dark right now. Have you heard about the no free lunch theorem?"

One of my favourite Gettier-like problems is about black holes.

Say you have a very dense star. It is so dense, that the gravitational force on its surface is capable of pulling back even the particles of its light, leaving only a black hole in the sky. How large can it be with a given $M$ mass?

It's an easy exercise using Newtonian mechanics. Take a light particle with mass $m$. Its gravitational energy at a distance $R$ is $-G\frac{mM}{R}$, and its kinetic energy is $\frac{1}{2}m{c}^2$ at the start. If the total energy is negative, then the path of the light particles will stay within a boundary. Therefore, the answer to the question is $R=\frac{2GM}{c^2}$, if the object is smaller than this, then it will be a black hole.

Of course, for that dense objects, Newtonian predictions break down. We should care about curved spacetime and use general relativity in our calculations. The answer (to my knowledge) is the Schwarzschild radius, which is $R=\frac{2GM}{c^2}$.