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SCIENCE CHINA Information Sciences, Volume 62, Issue 8: 082304(2019) https://doi.org/10.1007/s11432-019-9856-y

Variation of a signal in Schwarzschild spacetime

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  • ReceivedJan 21, 2019
  • AcceptedMar 28, 2019
  • PublishedJul 12, 2019

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. 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.


Acknowledgment

This work was supported in part by National Natural Science Foundation of China (Grant Nos. 61421001, 61331021, U1833203).


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  • Figure 1

    (Color online) World lines of (a) a static observer and (b) a moving observer in Schwarzschild spacetime.

  • Figure 2

    (Color online) A spaceship leaving a star and communicating with a transmitter at a fixed position.

  • Figure 3

    (Color online) A spaceship moving around a star with (a) an elliptic, (b) a parabolic, and (c) a hyperbolic trajectory while communicating with a transmitter at a fixed position.

  • Figure 4

    (Color online) A signal propagating from star system A to another faraway star system B.

  • Figure 5

    (Color online) FSPs from gravitational effect (GFSP; top panels) and gravitationally modified Doppler effect (GMDFSP; bottom panels) vs. time for a spaceship moving away from (a), (d) the Sun, (b), (e) a white dwarf, and (c), (f) a neutron star. Results are plotted for different initial positions of the receiver and different transmitter positions.

  • Figure 6

    (Color online) (a) Difference between the Doppler effect $\beta'_1(\tau)$ (special relativity version) and the gravitationally modified Doppler effect $\beta_1(\tau)$ as a spaceship moves away from a neutron star at a fixed Newtonian velocity; (b) time $\tau$ of the moving spaceship vs. Newtonian time $T$; (c) difference between $\tau$ and $T$ vs. $T$. The receiver is initially located at $5R_n$.

  • Figure 7

    (Color online) FFTs of a $10^4$ Hz signal transmitted in (a), (d), (g) the solar system, (b), (e), (h) a white dwarf system, and (c), (f), (i) a neutron star system with or without gravitational effects at $\tau=50$ s (top panels), 150 s (center panels), and 250 s (bottom panels). The transmitters in the solar, white dwarf, and neutron star systems are located at $2R_{\odot}$, $2R_w$, and $2R_n$, respectively. The initial positions of the receivers are $4R_{\odot}$, $4R_w$, and $4R_n$, respectively. The sampling rate is 2.5 times the signal frequency.

  • Figure 8

    (Color online) FSPs of the gravitationally modified Doppler effect ($\beta_1(\tau)$), gravitational effect ($\beta_2(\tau)$), and the whole frequency shift parameter ($\beta(\tau)$), as functions of time. The spaceship moves around (a), (d), (g) the Sun, (b), (e), (h) a white dwarf, and (c), (f), (i) a neutron star with different conic trajectories: an ellipse ($e=0.2$, top panels), a parabola ($e~=1$, center panels) and a hyperbola ($e~=2$, bottom panels) with $a=5R_\odot$, $5R_w$ and $5R_n$, respectively.

  • Figure 9

    (Color online) FFTs of a $10^5$ Hz signal under (a) gravitational effects only $(\beta_2(\tau))$, (b) gravitationally modified Doppler effect only $(\beta_2(\tau))$, and (c) the whole effect $(\beta(\tau))$. The signals are transmitted in the neutron star system from $2R_n$ to the spaceship following an elliptic trajectory with $e=0.7$ and semi-major axis $a=5R_n$. The sampling rate is 2.5 times the signal frequency.

  • Figure 10

    (Color online) FSPs of the gravitationally modified Doppler effect ($\beta_1(\tau)$), gravitational effect ($\beta_2(\tau)$), and the whole frequency shift parameter ($\beta(\tau)$), as functions of time. The signal is received at a position with an elliptic trajectory ($e=0.0167$) in (a), (d), (g) a Sun-like star system, (b), (e), (h) a white dwarf system, and (c), (f), (i) a neutron star system, and is transmitted from a fixed position in another Sun-like star, white dwarf, and neutron star systems.

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