The Underlying Mechanisms of Time Dilation and Doppler Effect in Curved Space-Time

In this paper, we investigate the time dilation and Doppler effect in curved space-time from the perspective of quantum field theory (QFT). A coordinate transformation keeping the local metric values unchanged between an original coordinate and another new coordinate system is introduced. We demonstrate that the mathematical forms of physics formulas in QFT are the same in these two coordinate systems. As applications of the coordinate transformation between these two coordinate systems, the time dilation and Doppler effect with an arbitrary time-dependent relative velocity in curved space-time are analyzed. For Minkowski space-time, the time dilation and Doppler effect agree with the clock hypothesis. For curved space-time, we show that even if the emitted wave has a narrow frequency range, the Doppler effect will, in general, broaden the frequency spectrum and, at the meantime, shift the frequencies values. At last, the Doppler-shifted frequency value will be given under certain conditions and approximations. These new findings will deepen our understanding on Doppler effect in curved space-time and may also provide theoretical guidance in future astronomical observations.