# 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?

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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Layton Leach
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.