general relativity - 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.

Answer

The answer would appear to be "Yes", at least in theory. A "kugelblitz" is a concentration of light so intense that it forms an event horizon and becomes a Black Hole according to general relativity. It would be a BH whose original mass-energy had been in the form of light rather than matter.