I am a bit confused about this situation: according to general relativity, when two masses orbit one another, they emit graviational waves, which carry away certain energy. For example, check out this lecture notes. However, each mass (assuming it is a point mass) follows a geodesic, and so it is not accelerating in their local frame. So each mass individually should not produce radiation, at least when seen from their local frames. So what is going on here? Do they emit gravitational radiation or not? I would like to understand this from a more matehmatical point of view (what equations to use), and also conceptually (what exaclty is going on wrong with the reasoning).
Interestingly, I found that Rovelli has a similar question in his new introductory book to GR but he does not answer it.
Any advise would be very helpful!