Variation of a signal in Schwarzschild spacetime

In this paper, the variation of a signal in Schwarzschild spacetime is studied and a general equation for frequency shift parameter (FSP) is presented. The FSP is found to depend on the gravitationally modified Doppler effects and the gravitational effects of observers. In addition, the time rates of a transmitter and receiver may differ. When the FSP is a function of the receiver time, the FSP contributed through the gravitational effect (GFSP) or the gravitationally modified Doppler effect (GMDFSP) may convert a bandlimited signal into a non-bandlimited signal. Using the general equation, the FSP as a function of receiver time is calculated in three scenarios: (a) a spaceship leaving a star at constant velocity communicating with a transmitter at a fixed position; (b) a spaceship moving around a star with different conic trajectories communicating with a transmitter at a fixed position; and (c) a signal transmitted from a fixed position in a star system to a receiver following an elliptic trajectory in another star system. The studied stars are a Sun-like star, a white dwarf, and a neutron star. The theory is illustrated with numerical examples.