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An electron in orbit around a gravitating body is accelerating. In Classical terms, inwards toward the centre of the body at a rate of (v^2)/r. From my distant position I see that electron accelerating due to the force of gravity exerted by the body. If it is accelerating will it emit radiation? If so will it's orbit decay? If it's orbit decays then if you are orbiting with the electron I will see the electron orbit decay but you stay in your orbit. Now, the fact that we are different observers cannot account for two different events. From your co-orbit perspective why would the electron move away from you as you are both in free fall and following the same geodesic?
Maybe I don't fully understand the question. An electron falling toward a star will emit radiation to all observers except an observer falling with the electron. An electron orbiting a star will emit radiation to a far away observer stationary with respect to the star but to an observer at the center of the star.
An accelerated charged particle emits radiation with respect to a stationary observer but if the observer is accelerated along with the particle does the observer still detect the particle emitting radiation?.
The connection between an accelerated charge and one at rest in a (weak) gravitational field is discussed in accordance with the principle of equivalence . For that purpose, the fields produced by a freely falling charge and a supported one (i.e., at rest in a gravitational field) are transformed to the rest-frame of the observer, who may be similarly supported or freely falling. A nonvanishing energy flux is found only if the charge is freely and the observer supported. or vice versa. This agrees with previously established results.