I read that a spin 0 graviton can't couple to traceless energy, such as a photon, in the introduction in the book The Feynman Lectures On Gravitation. Why is this true? What does it mean for a particle's energy to be "traceless"?
This has been my reference all along.
A spin 0 graviton can only couple to the trace of energy, and not to traceless energy.
You manifestly have a metric tnesor in the equation of motion for photons, and a spacetime containing photons manifestly has a nonzero stress-energy tensor, so you don't have to do any spin gynmastics to couple photons to gravitons, it's already there in the classical theory before you even quantize.
The answer of Bob Bee here discusses zero spin gravitons.
Going into quantum field theories, one has to define the number of forces one is going to unite, and the couplings and gauge bosons , i.e. the force carriers. Whether there is a coupling between a photon, the gauge boson for the electromagnetic interaction, and the graviton ( whatever spin, limited up to 2 as said in the link above) is a decision for the model, and if the data validate the model .
For example there exists the possible triple vertex W+W-Z which is not allowed in the standard model of particle physics , so if measured will show a need for extending the model. Above the symmetry breaking energies all these particles have zero mass.
One can hypothesize a model where a zero spin graviton has a coupling with these zero mass gauge bosons,which include the photon, and then check if there is any evidence in data and observations. ( a bit hard with the coupling of the gravitational forces being so small). It is the smallness of the gravitational coupling that is the problem.
The main stream physics research models go in the direction of quantizing gravity (a goal not yet reached) in the framework of General relativity, which has to have a spin two graviton, and of course it couples to the gauge bosons, even with zero mass which happens above the symmetry breaking energies, as the massive W and Z are massless there.
Thus a photon and a graviton do have an interaction, as expected , in an appropriate quantized gravitational theory.
