# Can light gravitationally affect itself?

Consider a electromagnetic wave in a vacuum. From my understanding of general relativity,

1. The wave has momentum, and thus generates a gravitational field in all directions.
2. The gravitational field propagates at speed $c$, same as the wave.

What does the gravitational field look like in the direction of the wave?

I am particularly interested because it seems like the field would affect the wave itself. This would seemingly cause the light to be redshifted, eventually vanishing due to this interaction. Furthermore, this would violate conservation of energy, as the 'drained' momentum of the wave has no natural place to go.

Ultimately, it is not clear under what conditions the field and the wave could interact.

To summarize, my questions are:

1. Can an electromagnetic wave gravitationally affect itself?
2. If so, is this problematic?

Sorry for the long question. Ideas and edits are welcome.

EDIT: The original question focused on a 'self-interaction' of light that only make sense when talking about photons. However, the question now focuses on the classical perspective, so what counts as light interacting with itself is more open to interpretation.

• The title uses the word "photon," and the first sentence insists on "a single photon," and yet there is nothing in the question that has anything to do with quantum mechanics. The actual question is purely classical, and the answer is yes. These are called "electrovac solutions," en.wikipedia.org/wiki/Electrovacuum_solution – Ben Crowell Nov 9 '14 at 21:24
• @BenCrowell Thank you for the link. As for the photon distinction, is the problem that I am mixing in QM where it does not belong? What would I change to fix this? – Jacob Maibach Nov 9 '14 at 23:20
• is the problem that I am mixing in QM where it does not belong? IMO, yes. What would I change to fix this? Maybe change every instance of "photon" to "electromagnetic wave" or "ray of light." It sounds like what you want is the Tolman-Ehrenfest-Podolsky metric for pencil beam of light. – Ben Crowell Nov 9 '14 at 23:25
• Ray of light is a bad term. Rays are just geometrical tools pointing at the energy transfer direction of the electromagnetic wave. An electromagnetic wave is composed out of zillion of photons, at the underlying quantum mechanical level. The wedding of general relativity and quantum mechanics has not yet been decided upon. It is a frontier of research. The answer by Dirk is in the classical framework of general relativity. I think you should use "electromagnetic wave" to be precise classically. – anna v Nov 12 '14 at 9:22