Do photons have mass if they are bent by gravity?

No. It’s a massless force carrier of matter, via the electric force (or the electro-magnetic force to be broader). It’s bent by “space-time”, which is an effect of matter on a dimensional field shared by these things. Matter influences curvature of space; light flows through space. Notably, light only bends through matter’s space-time effect. When it interacts with matter itself, it reacts with the orbiting electrons, then a new energy state is reached which re-emits a new photon. This photon travels through space-time, until it reaches matter again to react as previously mentioned. Otherwise, it’s bent indirectly by matter.

Not to take it lightly, but light “lenses”(bends) in the space-time curvature, not by it’s mass (which is zero), but by it’s innate connection to the field that actual massive bodies influence.

To get heavier (TLDR): light and gravity are bound only by space-time, not mass. Photons are massless, pure energy.

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