# What the lifetime of a massless black hole of photons?

Let a bundle of photons very concentrated in a very small area so that the space-time is curved as a black hole, and the photons can't escape: this is what I call a massless black hole of photons.

Question: what the lifetime of a massless black hole of photons (with its initial energy as parameter)?

• – Kyle Kanos Aug 20 '15 at 15:17
• While photons are individually massless, a collection of photons taken as a system generally have a mass (the only exception comes when they all have parallel momentum), and if they form a black hole they certainly have a mass. – dmckee --- ex-moderator kitten Aug 20 '15 at 15:48
• @AcidJazz E=mc^2 also works in the other direction: m=E/c^2 – user2338816 Aug 20 '15 at 17:54
• @user2338816 yeah, I started kicking myself 5 mins afterwards, next time I count to ten before rushing in, thanks very much for reply – user81619 Aug 20 '15 at 17:59
• @user2338816 That's the equation for rest mass; it does not work for photons. – Blackbody Blacklight Aug 21 '15 at 3:16

While photons can in principle form a black hole, the black hole will not be massless. The mass of the black hole will be related to the energy of the photons that went into it by Einstein's famous equation $E = mc^2$.

The black hole will be a regular black hole, and classically it has an infinite lifetime. Once you include Hawking radiation the black hole will have a finite lifetime, but that lifetime will be no different to a black hole formed by the equivalent amount of massive particles.

In General Relativity we usually make no distinction between mass and energy. The mass/energy is described by an object called the stress-energy tensor, and this is usually written as an energy density. To write the tensor we convert mass to energy using $E = mc^2$.

The mass of a black hole is a surprisingly elusive concept. As it happens there has recently been a question on just this subject here. The Schwarzschild metric is actually a vacuum solution i.e. it describes a spacetime that has no matter or energy present. You cannot point to any place and say "aha, here is the mass!". What we normally think of as the mass of the black hole is more precisely the ADM mass, which is a property of the spacetime geometry as a whole.

• Isn't the ADM mass more properly a property of an asymptotic end? And I think people do have an intuitive concept of mass beyond something that merely works for asymptotically flat spacetimes. Because the black bole didn't really have the outside geometry change as it formed. The curvature outside is the relic curvature from the collapsing material. – Timaeus Aug 20 '15 at 16:43
• You cannot point to any place and say "aha, here is the mass!" -- Just as you couldn't do the same for the original system of gravitationally interacting photons. In fact this sort of concept of properties only defined for a system predates GR -- you can't locate the potential energy of a system of interacting particles, nor can you pinpoint where the entropy of a gas resides. – user10851 Aug 20 '15 at 18:50
• @ChrisWhite: all true of course, though I still think GR brings a new dimension to the meaning of unintuitive. – John Rennie Aug 20 '15 at 19:25

To someone outside it looks the same as a regular, massive black hole.

Classically the lifetime is the lifetime of the universe. It might merge with another black hole. It might last forever. It might meet a singularity in a big crunch if the whole universe contracts to a singularity.

If you are worried that it can't decay by Hawking radiation because an infalling negative energy virtual antielectron has no real electron inside to annihilate with then don't worry. It can interact with photons inside and virtual electrons inside and there can be a whole chain of interactions between a positron coming out of one side of the black hole and an electron coming out the other side with a whole spacelike chain (not temporal chain) of interactions with the photons inside. It will still look like a thermal outpouring of particles.

Any problems with firewalls or information are the same for a regular massive black hole and one with photons inside.

• I don't understand if about Hawking radiation, there is a difference between regular massive black hole and photons black hole. Is there a (energy) lower bound for such a black hole and what would be its lifetime? – Sebastien Palcoux Aug 20 '15 at 16:36
• @SébastienPalcoux You said you got the photons into a small region. So they are a black hole. Are you asking if there is a limit to how small a. Lack hole you can make by concentrating photons? That's a separate question and you should ask it as such. I'm saying it didn't matter what was inside the lifetime is the same whether it is photons, neutrinous, highs bosons, or electrons and quarks. Just have more and it lasts longer or it lasts forever if classical. – Timaeus Aug 20 '15 at 16:47