I know that if a star collapses into a volume with radius less or equal to the Schwarzschild radius $r_s=\frac{2GM}{c^2}$ then a black hole is created and it has the same mass of the star that gave it origin. But is there a way to calculate the mass of a black hole without knowing the volume of the star?
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2 Answers
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[a black hole] has the same mass of the star that gave it origin
No, not really. A stellar-mass black hole is formed after a star with mass around 20 times that of our sun collapses due to lack of core fusion and by some (as of yet) unknown process, rebounds and explodes. This explosion, a Type II supernova, kicks off something like 90% of its matter. Thus, the black hole would be something around the 5 solar mass range.
is there a way to calculate the mass of a black hole
Well you can't actually observe black holes in free space (because you can't see them), you need to infer it by using something around it (either a star or gas cloud) and computing the mass from Kepler's laws.
answered Apr 21, 2014 at 20:44
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$\begingroup$ Thanks. I heard that the mass can be also calculated using GR is that true? $\endgroup$ Commented Apr 21, 2014 at 20:47
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$\begingroup$ The Schwarzschild radius is derived from a particular metric for the GR field equations. Outside of that, I'm not sure what you mean. $\endgroup$ Commented Apr 21, 2014 at 20:50
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$\begingroup$ I heard that it has something to do with space-time bent but I may be wrong $\endgroup$ Commented Apr 21, 2014 at 20:54
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1$\begingroup$ @Peterix GR (General Relativity) describes gravity as space-time curvature, so black holes curve space and time around them so it means as you get near to black hole time will pass slower and slower and someone who is watching you will never see you crossing the event horizon. Sample image that shows curvature of space around black hole: encrypted-tbn2.gstatic.com/… , or just google "black hole curvature of space" and open images tab $\endgroup$– user21420Commented Apr 21, 2014 at 21:02
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1$\begingroup$ @Peterix: yes, someone far from a black hole can measure small variations in the gravitational force that the y observe and infer the mass and spin of the black hole. $\endgroup$ Commented Apr 21, 2014 at 22:10
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|[a black hole] has the same mass of the star that gave it origin
as Kyle figured out, the mass of the black hole correlates with the star, but in general a star loses much mass in a supernova.
Furthermore, there are many more ways to create a black hole, like colloding neutron stars in theory, even at the LHC/Cern, colliding nuclei could lead to black holes. So they can be large, or very small.
To calculate the mass of a black hole: well, that's a bit tricky, because you can fully describe a black hole by the parameters: 1) mass, 2) electric charge, 3) rotation momentum.
So the question is: how to calculate the mass from what? What you could do: measure the Schwarzschild radius or mesaure the gravity field of the black hole to derive it's mass. But there is no real way to "calculate" the mass of a black hole. Its like if you want to calculate the amount of water in a bucket. It just depends on the amount of water in the bucket.
About your comment on GR: At the current stage of phsics, the general relativity theory is the only theory, that can describe black holes. They only exist in this theory.
