Why shouldn't photons be able to curve spacetime? I have read this question. It was asked if photons are able to curve spacetime. But if classical electromagnetic fields can curve spacetime (due to the energy contained in the fields contributing to the mass-momentum tensor) why shouldn't photons be able to do this? If the em field curves spacetime then photon fields should be able too. After all, they constitute the classical em field when the number of photons is large.
I mean, what would be the reasons to ask if a photon would not be able to make spacetime curve?
One thing I can think of (thanks to @WolphramJohnny): a photon has a different energy in different frames of reference, so if it curves spacetime in one frame it curves spacetime differently (more or less) in different frames of reference. This holds true also for massive particles but massive particles have a rest mass, while photons have not, so curvature due to mass is not there. Only curvature due to motion, which gives rise to linear frame-drag. Though this can also be said for classical em waves. So photons must emit gravitons (or small spacetime distortions) like a speeding boat emits bow waves (if it speeds. But how can this be different for different frames? In one frame the photon will not even emit gravitons (no energy).
Can we say that photons can't exist in General relativity in the first place (because they are a quantum object)? Can only fermions emit gravitons (because they are quantum objects)? Of course, you can say that we need a quantum theory of gravity (which doesn't necessarily mean that gravitons are involved though; spacetime itself can also be seen as quantized) but the theory has to involve an interaction with a graviton field.
 A: *

*There's no frame where photon does not exist. You would need infinite velocity to drive its energy to zero


*The metric differs in different frames. The tranformation is
\begin{equation}
\tilde{g}_{\mu\nu}= (\tilde{\partial_\mu}x^\alpha)(\tilde{\partial_\nu}x^\beta)g_{\alpha\beta}
\end{equation}
The Lorentz boost applied to the gravitational wave spacetime transforms it... Into a gravitational wave with a boosted momentum.


*The monochromatic classical electromagnetic plane wave deforms spacetime so that it is accompanied by the monochromatic gravitational plane wave (see pp-wave solutions). For more complex electromagnetic fields the nonlinearities start playing their role.


*When you construct the quantum gravity you start with linearized approximation where all interactions are infinitely weak. This gives you a free field theory that you can easily quantize and consider interactions as a perturbation (this works for gravity as long as the energies and non-linearities are sufficiently suppressed and you can neglect the infinite number of terms required for the renormalization) However even then there is some mixing between the gravitational and matter degrees of freedom. So even in such limit you see that the photons are not just electromagnetic waves but they are accompanied by the gravitational waves.


*The current cosmological model works well and it tells us that the chemical composition of the universe is very sensitive to the rate of the universe expansion at tempeeatures of $~MeV$. At this temperatures the significant gravity was caused by the thermal gas of photons. This is not something you can describe well by a classical electormagnetic wave


*No free particle in absence of other particles to interact with emit anything. Unless it decays. The electron doesn't emit photons slowing down. It keeps flying by inertia. The same is for a free photon in empty space. It doesn't emit gravitons. However that doesn't mean that it does not curve spacetime. Electron does not emit photons but it produces the electromagnetic field around itself.
