Black holes and Relativity

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According to Special Relativity, objects moving fast have more mass than objects at rest. So what if there is an object having a mass density near that required to create a black hole. Now if it were moving fast enough as well, it would gain enough mass density to become black hole. But to the observer moving in its frame of reference it would not. How can an object both be a black hole and not be a black hole? To an observer moving fast enough, every object would seem to be a black how, right?
 

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To an observer moving fast enough, every object would seem to be a black hole, right?
Not right.
A black hole forms or not according to how large the rest mass is, not the relativistic mass.
 
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According to Special Relativity, objects moving fast have more mass than objects at rest.
This statement is common, but wrong. Moving objects have more relativistic mass, but their rest mass is unchanged. Modern usage (i.e. pretty much any paper in the last half century) is that "mass" means rest mass, and relativistic mass is pretty much never used. It's just total energy divided by ##c^2## anyway, and calling it mass leads to confusion of the type you have here.

Pop sci is more concerned wigh sounding cool and less concerned with clear communication, so has yet to catch up with that.
How can an object both be a black hole and not be a black hole?
It can't. Gravity in relativity does not depend on relativistic mass, and people assuming it is basically Newtonian gravity with relativistic mass plugged in is one more reason to abandon the concept. The source term for gravity in relativity is the stress-energy tensor, which includes mass and other terms related to momentum and pressure. But they don't simply add to the mass in any naive way.
 

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