How much energy can be added to a small volume of space? Perhaps like the focal point of a very high power femtosecond laser for a short time, or are there other examples like the insides of neutron stars that might be the highest possible energy density? Is there any fundamental limit?

Hunter Shah 2022-10-16 Answered
How much energy can be added to a small volume of space? Perhaps like the focal point of a very high power femtosecond laser for a short time, or are there other examples like the insides of neutron stars that might be the highest possible energy density? Is there any fundamental limit?
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Answers (1)

Layton Leach
Answered 2022-10-17 Author has 15 answers
There is a limit to how much energy that can be contained in a finite volume, after which the energy density becomes so high that the region collapses into a black hole.
We also know that matter and energy are equivalent according to the Einstein equation
(1) E = m c 2
So if we can determine the greatest amount of matter that can fit into a volume just before it collapses into a black hole, the corresponding energy should also indicate the greatest energy confined in the volume just before it becomes a black hole.
The maximum amount of matter, mass M, that can be contained in a given volume before it collapses into a black hole, is given by the Schwarzschild radius
(2) r s = 2 G M c 2
Using (1) we can then write
M = E c 2
so that equation (2) becomes
r s = 2 G E c 4
or
E = r s c 4 2 G
Note that this is still energy, and to get to energy density we need to define the volume which is of course
V = 4 3 π r s 3
so that the energy density is
ϵ = 3 c 4 8 π G r s 2
This computation is based on not much more than the equivalence of matter and energy. It represents a bound on the maximum amount of matter, and therefore energy, in a spherical volume of radius rs before the volume containing the matter collapses into a singularity, which of course has no properly defined volume.
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