# 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
$\begin{array}{}\text{(1)}& E=m{c}^{2}\end{array}$
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
$\begin{array}{}\text{(2)}& {r}_{s}=\frac{2GM}{{c}^{2}}\end{array}$
Using (1) we can then write
$M=\frac{E}{{c}^{2}}$
so that equation (2) becomes
${r}_{s}=\frac{2GE}{{c}^{4}}$
or
$E=\frac{{r}_{s}{c}^{4}}{2G}$
Note that this is still energy, and to get to energy density we need to define the volume which is of course
$V=\frac{4}{3}\pi {r}_{s}^{3}$
so that the energy density is
$ϵ=\frac{3{c}^{4}}{8\pi G{r}_{s}^{2}}$
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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