If a black hole is created from light, can this black hole then move at the speed of light?

Question

Imagine we managed to squeeze light into a very tiny region of space so that the energy concentration at that point becomes a black hole. Can this black hole then move at the speed of light?

Answer 1

No. I assume you're thinking that a black hole made from light would have a zero rest mass and could therefore travel at the speed of light. However the rest mass of any black hole is due not only to the mass that went into it but also the energy (e.g. photons) that went into it. The increase in mass due to the energy is given by Einstein's famous equation $E = mc^2$.

So if we create the black hole from purely mass $m$ the rest mass of the black hole is just $m$.

If we create the black hole from purely energy $E$, e.g. from photons with a total energy $E$, then the rest mass of the black hole is $E/c^2$.

Or for completeness we could use a mixture of mass $m$ and energy $E$ in which case the rest mass would be $m + E/c^2$.

So a black hole made from just photons would not have a zero rest mass and therefore could not travel at the speed of light.

This conversion of photons to a mass isn't unique to a black hole. For example suppose we start with a hydrogen atom in the ground state, $1s$, and let it absorb a 10.2eV photon to excite it to the $2p$ state. This would increase the mass by $10.2\textrm{eV}/c^2$ i.e. even though the photon is massless absorbing it increases the mass of the hydrogen atom. As a general rule mass is not a conserved quantity either in special or general relativity.

Answer 2

No. Nothing with non-zero mass can move at lightspeed. A kugelblitz would behave just like a more conventional black hole formed from matter, due to the no-hair theorem.