# Signatures of Lorentz Violation in Continuous Gravitational-Wave Spectra of Ellipsoidal Neutron Stars

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## Abstract

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## 1. Introduction

## 2. Theoretical Basics

## 3. Numerical Examples

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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1 | Relativistic corrections are reasonably characterized by the compactness of the body, which is about $0.1$ for a NS. |

**Figure 1.**Euler angles transforming the X–Y–Z inertial frame to the x–y–z body frame. First, rotate the X–Y–Z frame about the Z axis with angle $\alpha $ so that the X-axis aligns with the intersection line $MN$. Then, rotate the just obtained X–Y–Z frame about the line $MN$ with angle $\beta $ so that the Z-axis aligns with the z-axis. Last, rotate the new X–Y–Z frame about the z-axis with angle $\gamma $ so that it overlaps with the x–y–z frame.

**Figure 2.**Illustrations for an example of the Lorentz-violating twofold precession (

**left**) and an example of the Lorentz-invariant free precession (

**right**). The green trajectories trace the tail of the body-frame unit vector ${\widehat{\mathit{e}}}_{z}$ in the inertial frame, while the blue trajectory in the left plot traces the tail of the angular momentum unit vector in the inertial frame. The red arrows are the body-frame unit vectors ${\widehat{\mathit{e}}}_{x},\phantom{\rule{0.166667em}{0ex}}{\widehat{\mathit{e}}}_{y}$ and ${\widehat{\mathit{e}}}_{z}$ at $t=0$, while the blue arrows indicate the angular momentum unit vector at $t=0$. The angular momentum is conserved in free precessions so the blue arrow in the right plot remains unchanged with time. The initial values for both solutions are ${\alpha |}_{t=0}=0$, ${\beta |}_{t=0}\approx 0.762$, ${\gamma |}_{t=0}=\pi /2$, $\dot{\alpha}{|}_{t=0}\approx 1.26$, $\dot{\beta}{|}_{t=0}=0.5$ and $\dot{\gamma}{|}_{t=0}\approx 0.0449$. Time and time derivatives are dimensionless under the time unit ${t}_{c}$ given by Equation (23).

**Figure 3.**GWs from a rigid body undergoing the rotations in Figure 2. The observer receiving the waves has colatitude ${\theta}_{o}=0.8\phantom{\rule{0.166667em}{0ex}}\mathrm{rad}$ and azimuth ${\varphi}_{o}=0$ in the X–Y–Z frame. The geometrized unit of time and distance is the time unit ${t}_{c}$ given by Equation (23).

**Figure 4.**Fourier transformations of the $r{h}_{+}$ waves in Figure 3. The two noticeable peaks at about 1.4 and 2.9 in the left plot are the first and the second harmonics for both twofold precession and free precession. The modulation due to nonaxisymmetry is clearly represented by the adjacent peak at about 2.7 close to the second harmonic for both kinds of motion. However, the barely visible tiny peaks, reflecting modulations due to Lorentz violation, only exist for twofold precession. The right plot, which zooms in on the tiny peak between 4 and 5, demonstrates the point. Note that in the plots the geometrized unit of Fourier amplitude is ${t}_{c}$, and the geometrized unit of angular frequency is $1/{t}_{c}$.

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**MDPI and ACS Style**

Xu, R.; Gao, Y.; Shao, L.
Signatures of Lorentz Violation in Continuous Gravitational-Wave Spectra of Ellipsoidal Neutron Stars. *Galaxies* **2021**, *9*, 12.
https://doi.org/10.3390/galaxies9010012

**AMA Style**

Xu R, Gao Y, Shao L.
Signatures of Lorentz Violation in Continuous Gravitational-Wave Spectra of Ellipsoidal Neutron Stars. *Galaxies*. 2021; 9(1):12.
https://doi.org/10.3390/galaxies9010012

**Chicago/Turabian Style**

Xu, Rui, Yong Gao, and Lijing Shao.
2021. "Signatures of Lorentz Violation in Continuous Gravitational-Wave Spectra of Ellipsoidal Neutron Stars" *Galaxies* 9, no. 1: 12.
https://doi.org/10.3390/galaxies9010012