In the discussion of the Unruh effect and Bremsstrahlung (e.g. here) I always come across "zero-energy Rindler photons". What exactly are these? Shouldn't a zero-energy photon correspond to a static electric field?
What you are asking about is in connection to the radiation from a uniformly accelerated charge as observed from different reference frames.
It is very important to understand that there are two basic metrics in connection to your question:
Minkowski, combination of the three dimensional Euclidean space and time in a four dimensional manifold where the spacetime interval is independent of the inertial frame of reference in which they are recorded
Rindler, coordinates of a hyperbolically accelerated reference frame. In SR, a uniformly accelerated particle undergoes hyperbolic motion
Now, your quesion is about zero energy Rindler photons. It can be shown, that the classical rate of particle emission in an inertial frame can be re-produced in a co-accelerating frame, where the rate of emission and absorption can be equivalent to the emission and absorption of zero energy Rindler particles.
Now in your case, the name Rindler comes from this accelerated frame, and they are using photons in the experiment.
Now why zero energy? In the experiment, the computation requires a regulation procedure, because the static source in Rindler coordinates can only excite zero frequency photons. In this case, there is no induced emission, but what happens is that the density of quanta goes to infinity as the frequency goes to zero.
In the present note we show that this result can be related to the computation of the emission rate in the presence of an Unruh-DeWitt detector. We express the emission rate for any value of the energy shift E of the detector and we show that for E going to zero we recover de classical result. We begin discussing the radiative process for the Unruh-DeWitt detector in the inertial reference frame. Then, we compute this same emission rate of a Minkowski scalar photon from the point of view of the co-accelerating observer using the mode expansion of the Green function in terms of the Rindler particles. By this way we can directly express the emission rate of a Minkowski scalar photon as a combination of the Rindler particle emission and absorption rates weighted by thermal factors. We then make the connection with the radiation pattern of a classical source and with the results of [8, 9, 10] by taking a limiting procedure, that we shall call the inert limit. It consists in suppressing the internal structure of the detector by letting the energy of the excited state E go to zero. This follows an observation made by DeWitt  and used later by Kolbenstvedt  in a purely inertial description of the excitation of the DeWitt detector.
Please see here: