Variation of a Signal in Schwarzschild Spacetime

TL;DR

This paper derives a general equation for frequency shift in Schwarzschild spacetime, considering gravitational and Doppler effects, and applies it to various space communication scenarios involving different celestial bodies.

Contribution

It introduces a comprehensive formula for frequency shift that accounts for gravitational and Doppler effects in Schwarzschild spacetime, with practical applications to space communication.

Findings

FSP depends on gravitational and Doppler effects.

FSP can cause signals to become non-bandlimited.

Numerical examples illustrate the theory for different star types.

Abstract

In this paper, the variation of a signal in Schwarzschild spacetime is studied and a general equation for frequency shift parameter (FSP) is presented. It shows that FSP depends on the gravitationally-modified Doppler effects and the gravitational effects of observers. In addition, rates of time of a transmitter and a receiver may be different. When FSP is a function of the time of a receiver, FSP contributed by gravitational effect (GFSP) or gravitationally-modified Doppler effect (GMDFSP) may lead a bandlimited signal to a non-bandlimited signal. Based on the equation, FSP as a function of the time of a receiver is calculated for three scenarios: a) a spaceship moves away from a star with a constant velocity communicating with a transmitter at a fixed position; b) a spaceship moves around a star with different conic trajectories communicating with a transmitter at a fixed position; c) a signal is transmitted at a fixed position in a star system to a receiver moving with an elliptical trajectory in another star system. The studied stars are sun-like star, white dwarf and neutron star, and some numerical examples are presented to illustrate the theory.