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I first want to know what's the spacetime when there is only one photon in "Minkowski" spacetime? The quotation marks mean that when taken the photon's reaction to background into consideration, it's not exactly Minkowski spacetime.

Then I realized that photon is a quantum object and the precise description of my question is what's the exact solution of Einstein-Maxwell equation , which corresponds to the monochromatic plane electromagnetic wave?

So your question is one about the classical field equations?

The metric for a monochromatic plane EM wave is described in an exercise of Morris, Thorne and Wheeler. There are some details here, as well :

https://en.wikipedia.org/wiki/Monochromatic_electromagnetic_plane_wave

In general spacetimes for monochromatic plane waves are included within the study of pp-wave spacetimes.

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