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

TL;DR

This paper introduces a new coordinate transformation, CTMPC, that accurately describes time dilation and Doppler effects in curved space-time, extending classical concepts from Minkowski space-time to more general scenarios.

Contribution

The paper proposes the CTMPC transformation, generalizing Lorentz transformations to curved space-time, and analyzes its implications for time dilation and Doppler effects with potential astronomical applications.

Findings

CTMPC correctly relates observer measurements in curved space-time

Doppler effect in curved space-time can broaden and shift frequencies

Results align with clock hypothesis in Minkowski space-time

Abstract

In this paper, we theoretically investigate the time dilation and Doppler effect in curved space-time from the perspective of quantum field theory (QFT). A Coordinate Transformation which Maintains the Period of Clocks is introduced, and such coordinate transformation is named as CTMPC throughout this paper. By analogy with the Lorentz transformation in Minkowski space-time, CTMPC is a correct transformation in curved space-times in a sense that it shows the correct relation between the time measured by the two observers, moreover, Lorentz transformation is just a special case of CTMPC applied in Minkowski space-time. We demonstrate that the Coordinate Transformation which Maintains the Local Metric (CTMLM) is one CTMPC, while the mathematical forms of physics formulas in QFT will be maintained. As applications of CTMLM, the time dilation and Doppler effect with an arbitrary time-dependent relative velocity in curved space-time are analysed. 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 may, in general, broaden the frequency spectrum and, at the meantime, shift the frequencies values. These new findings will deepen our understanding on the nature of space-time and the Doppler effect in curved space-time, they may also provide theoretical guidance in future astronomical observations.