Local observed time and redshift in curved spacetime

TL;DR

This paper redefines observed frequency and momentum in curved spacetime using local observable scales, providing a new perspective on wave measurement in general relativity, with specific application to the Robertson-Walker universe.

Contribution

It introduces a formalism for defining observed frequency and momentum based on local observable scales in curved spacetime, extending Einstein's original concepts.

Findings

Defines observed frequency as phase change rate relative to local time

Defines observed momentum as phase change rate relative to local spatial scale

Applies the formalism specifically to the Robertson-Walker universe

Abstract

Using the observed time and spatial intervals defined originally by Einstein and the observation frame in the vierbein formalism, we propose that in curved spacetime, for a wave received in laboratories, the observed frequency is the changing rate of the phase of the wave relative to the local observable time scale and the momentum the changing rate of the phase relative to the local observable spatial length scale. The case of Robertson-Walker universe is especially considered.