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 · Accepted Answer

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.

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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