Can point masses following geodesics and orbiting one another emit gravitational radiation? 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).
I know that a similar question was asked here but I do not really understand the answer given there and that thread seems to be dead.
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!
 A: 
(...) according to general relativity, when two masses orbit one another, they emit gravitational waves, which carry away certain energy.
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 mathematical point of view (what equations to use), and also conceptually (what exactly is going on wrong with the reasoning).
(...)

It is the system of orbiting masses that emits gravitational waves and not the point masses themselves. They follow geodesics of the system spacetime which is solution of Einstein field equations for the two-body problem. Contrary to Kepler’s problem (Newton’s gravity) there is no stationary solution there. The spacetime of a multi-body system is always transient and includes radiation (see https://en.wikipedia.org/wiki/Two-body_problem_in_general_relativity).
The geodesics in such a system are by the definition transient, too. Therefore, after some time, the gravitating bodies will crash together, due to energy loss caused by the emission of gravitational waves. By the way, this effect was the first indirect evidence for gravitational waves discovered by Russell Alan Hulse and Joseph Hooton Taylor, Jr. in 1974 for that they earned 1993 Nobel Prize in Physics. In the case of the Sun Earth system these gravitational waves are very weak. The energy loss is about 200 watts and thus not really detectable (see https://www.quora.com/Does-a-freely-falling-point-particle-emit-gravitational-waves).
