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Black holes are formed when energy is put in a small region of space, I was wondering if there is a threshold to how much energy should I add in a certain volume of space to get a black hole, I read about the Schwarzschild radius formula that tells us the radius of event horizon of black hole formed by a star of mass $M$. My question is that is there any relation between energy and volume of a region to form a black hole?

Can I use Einsteins Energy-mass equation to find the energy and substitute the value of mass in terms of energy in the formula of Schwarzschild radius so that I have a relation between radius of event horizon and energy contained in it and calculate volume accordingly?

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Yes, you can use $$E=mc^2$$

in combination with $$r_s=\frac{2GM}{c^2}$$

to calculate the energy required to form a Schwarzschild black hole of a given Schwarzschild radius.

However, in General Relativity, gravity isn't just caused by the amount of energy (including rest mass energy) in a region. The Einstein field equations say that the spacetime curvature is due to the stress-energy-momentum tensor.

stress-energy-momentum

So to properly calculate the gravity as a black hole forms we also need to include the effects of pressure, momentum density, momentum flux, and shear stress.

That pressure term is very important. Normally in a star, there's an equilibrium between the gravity that's trying to compress the star, and the pressure due to temperature that's trying to expand the star. When a large star is on the verge of becoming a supernova, the temperature and pressure in the star core become enormous. And because the pressure is so large it increases the gravity, leading to a feedback loop that rapidly collapses the core, initiating the supernova explosion and the formation of the black hole remnant.


I should mention that that Wikipedia article is a little controversial. Please see the answers and comments here for details.

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  • $\begingroup$ the matrix is very useful, thanks for including it here. -NN $\endgroup$ – niels nielsen Jun 27 '20 at 16:41

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