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At what mass does the light from stars ( I am talking about stars and not black holes) fail to escape the star's gravity? Is it the same (minimum) mass required for an object to be called a black hole?

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It is not about mass, but about density. Any mass, packed densely enough, will have an event horizon outside itself, which would make it into a black hole. This even goes for extremely small masses (I don't know if there is a theoretical lower limit due to quantum effects). – Thriveth Jan 28 '14 at 23:13
Thanks! Is there a name for the lower density limit? – zerosofthezeta Jan 28 '14 at 23:16
Related, probably duplicate:… – Brandon Enright Jan 28 '14 at 23:49
@Thriveth your density comment is really no more accurate than assuming it needs a certain mass. You can form a black hole with any density you want given you have enough mass and enough volume. – Brandon Enright Jan 28 '14 at 23:50
@BrandonEnright But we agree that for a given mass, there i one, and just one, critical density, right? I don't really think I wrote anything else. – Thriveth Jan 29 '14 at 0:12

As long as you can neglect quantum effects, any mass $M$ compressed to a sphere with radius less than its Schwarzschild radius, $$ R_\mathrm{S} = \frac{2GM}{c^2}, $$ will form a black hole. Now one should be careful about what the mass is, given that compressing an object can change its (gravitational, not baryonic) mass, but that's a minor point. For concreteness, if the Sun were compressed to a radius of less than $3\ \mathrm{km}$, it would be a black hole.

You can easily calculate a critical density1 at which a uniformly dense sphere will have a radius smaller than $R_\mathrm{S}$. There isn't really another name for this, and "critical density" works pretty well, since it's analogous to the same term used to describe the density of the universe where it is neither positively nor negatively curved.

Finally, if light cannot escape due to gravity, the object is by definition a black hole. There is no non-black hole equivalent.

1 A term used everywhere in physics.

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