Do photons bend spacetime or not?
Classical electromagnetic fields carry energy and momentum and therefore cause spacetime curvature. For example, the EM field around a charged black hole is taken into account when finding the Reissner-Nordstrom and Kerr-Newman metrics.
The question of whether photons cause spacetime curvature is a question about quantum gravity, and we have no accepted theory of quantum gravity. However, we have standard ways of quantizing linear perturbations to a metric, and reputable journals such as Physical Review D have published papers on graviton-mediated photon-photon scattering, such as this one from 2006. If such calculations are no longer mainstream, it is news to me. Given that photons have energy and momentum, it would surprise me if they do not induce curvature.
I also note that the expansion of the "radiation-dominated" early universe was caused by what is generally described as a photon gas and not as a classical electromagnetic field. So the idea that photons bend spacetime is part of mainstream cosmology, such as the standard Lambda-CDM model.
Finally, the idea of a kugelblitz makes no sense to me unless photons bend spacetime.
So in Rennie v. Safesphere, I am on the Rennie side, but I look forward to Safesphere defending his position in a competing answer.
Addendum:
Safesphere declined to answer; in a now-removed comment, he said that knzhou’s answer explains the disagreement. I don’t agree. I disagree with knzhou that “bends spacetime” is vague. It is commonly understood by most physicists to mean “contributes to the energy-momentum tensor on the right side of the Einstein field equations”. And most physicists believe that real photons do exactly this, for the reasons that Ben Crowell and I have stated.
In classical general relativity, electromagnetic fields do bend spacetime. They have a nonvanishing stress-energy tensor, and the Einstein field equations relate the stress-energy to the curvature.
We even have fairly direct experimental proof that electromagnetic fields interact gravitationally in this way, from Cavendish-like experiments. See Kreuzer, Phys. Rev. 169 (1968) 1007, which can be interpreted as confirming the correctness of the coupling of gravity to the pressure components of the stress-energy. For a discussion of Kreuzer and similar tests, including lunar laser ranging, see Will, “The Confrontation between General Relativity and Experiment,” The Kreuzer experiment is discussed in section 4.4.3.
We can also confirm that this holds for electromagnetic waves, not just static fields. One empirical confirmation of this comes from the fact that models of big bang nucleosynthesis (BBN) agree pretty well with observed data on things like the H/He ratio; during the BBN period, cosmological gravity was radiation-dominated.
It would also be extremely problematic if light rays didn't produce gravitational fields, because we have detailed studies confirming that gravitational lensing works as predicted by GR. If the gravitational field of matter affected the momentum of light rays, but not the other way around, then conservation of momentum would be violated. This sort of thing is discussed in section 4.1.1 of Will, and is parametrized by $\gamma$ in the PPN framework. A variety of experiments constrains $\gamma$ to be equal to the GR value to about $10^{-4}$.
There is no reason to think that the situation is any different when the electromagnetic field is quantized. By the correspondence principle, photons have to produce gravitational fields when the conditions are such that the classical theory is a good approximation (coherent states with lots of photons). In the case where the classical theory is invalid, and we really need to talk about photons, the best we can currently do, lacking a real theory of quantum gravity, is semiclassical gravity. Semiclassical gravity works by replacing the stress-energy tensor $T$ in the Einstein field equations with its expectation value $\langle T \rangle$. $\langle T \rangle$ can easily be nonzero.
Imagine an isolated, spherical, and homogeneous spherical body somewhere in outer space and with zero velocity (as seen from a local inertial frame). If we let a parallel bundle (for creating more energy) of a high number of high-energy continuous laser beams (every beam consisting of real photons coherent in space and time; see here) pass this mass on one side, this bundle will, because of the curvature of spacetime around the massive object, change its direction toward the object.
This means that the "outgoing" bundle isn't parallel to the "incoming" one. In other words, the momentum of the bundle (and the photons constituting it) has changed direction. This, in turn, means that the momentum of the massive object has changed also to compensate for the change in momentum of the laser bundle. The only way this massive object can acquire this momentum (the three basic forces are not involved here) is because of a curved spacetime produced by the bundle lasers which consists of real photons.
Without the bundle of photons, the curvature around the mass is spherically symmetric, like Peter A. Schneider wrote rightly in a comment below. The only way for the massive body to acquire momentum is when the curvature of spacetime "surrounding" it is asymmetrical. It's obvious that the laser bundle is responsible for this asymmetry. Which means photons do curve spacetime.
EDIT In the answer given below by Ben Crowell (someone who knows what he's talking about) I read:
It would also be extremely problematic if light rays didn't produce gravitational fields, because we have detailed studies confirming that gravitational lensing works as predicted by GR. If the gravitational field of matter affected the momentum of light rays, but not the other way around, then conservation of momentum would be violated. This sort of thing is discussed in section 4.1.1 of Will and is parametrized by γ in the PPN framework. A variety of experiments constrains γ to be equal to the GR value to about 10−4.
Now I don't care too much about someone's reputation and pointing out: "But the famous Mr. X has said..." but in this case, I find it strange that nobody said that his argument is circular (of which I obviously think it's not). Off course he gave also a lot of other great information but nevertheless...