# Gravitational Refraction of Compact Objects with Quadrupoles

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## Abstract

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## 1. Introduction

## 2. Refraction in a Slightly Deformed Spacetime

#### 2.1. Refractive Index

#### 2.2. Deflection Angle

## 3. Refraction in a Slowly Rotating Spacetime

#### 3.1. Refractive Index

#### 3.2. Deflection Angle

## 4. Analysis of the Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Refractive index as a function of r for different values of q and $m=1$. In the interval $r\in (1,2)$, the refractive index can become negative, depending on the value of q.

**Figure 3.**The refractive index as a function of the radial distance for different values of the specific angular momentum a. Here, we set $m=1$ and $b=10$.

**Figure 4.**Equivalence between the static quadrupole q and the stationary quadrupole a in terms of the radial distance r.

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**MDPI and ACS Style**

Beissen, N.; Utepova, D.; Abishev, M.; Quevedo, H.; Khassanov, M.; Toktarbay, S.
Gravitational Refraction of Compact Objects with Quadrupoles. *Symmetry* **2023**, *15*, 614.
https://doi.org/10.3390/sym15030614

**AMA Style**

Beissen N, Utepova D, Abishev M, Quevedo H, Khassanov M, Toktarbay S.
Gravitational Refraction of Compact Objects with Quadrupoles. *Symmetry*. 2023; 15(3):614.
https://doi.org/10.3390/sym15030614

**Chicago/Turabian Style**

Beissen, Nurzada, Daniya Utepova, Medeu Abishev, Hernando Quevedo, Manas Khassanov, and Saken Toktarbay.
2023. "Gravitational Refraction of Compact Objects with Quadrupoles" *Symmetry* 15, no. 3: 614.
https://doi.org/10.3390/sym15030614