# Quadratic Theory of Gravity with a Scalar Field and Type I Shapovalov Wave Spacetimes

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## Abstract

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## 1. Introduction

## 2. Field Equations of Quadratic Gravity Theory with Scalar Field

## 3. Shapovalov’s Wave Models of Spacetime

## 4. Analysis and Solution of Field Equations

#### 4.1. Variant I. ${{\varphi}_{1}}^{\prime}\ne 0$, $\gamma \left(\varphi \right)\ne 0$

#### 4.1.1. Variant I. A. ${{\varphi}_{1}}^{\prime}\ne 0$, ${\gamma}^{\prime}\left(\varphi \right)\ne 0$

#### 4.1.2. Variant I. B. ${{\varphi}_{1}}^{\prime}\ne 0$, ${\gamma}^{\prime}\left(\varphi \right)=0$ and $\gamma \left(\varphi \right)\ne 0$

#### 4.2. Variant II. ${{\varphi}_{1}}^{\prime}\ne 0$, $\gamma \left(\varphi \right)=0$

#### 4.2.1. Variant II. A. ${{\varphi}_{1}}^{\prime}\ne 0$, $\gamma \left(\varphi \right)=0$, $F({x}^{2},{x}^{3})\ne \mathrm{const}$

#### 4.2.2. Variant II. B. ${{\varphi}_{1}}^{\prime}\ne 0$, $\gamma \left(\varphi \right)=0$, $F({x}^{2},{x}^{3})=\mathrm{const}\ne 0$, ${\sigma}^{\prime}\left(\varphi \right)\ne 0$

#### 4.2.3. Variant II. C. ${{\varphi}_{1}}^{\prime}\ne 0$, $\gamma \left(\varphi \right)=0$, $F({x}^{2},{x}^{3})=\mathrm{const}\ne 0$, ${\xi}^{\prime}\left(\varphi \right)\ne 0$

#### 4.3. Variant III. $\varphi =\varphi \left({x}^{0}\right)$

#### 4.3.1. Variant III. A. $\varphi =\varphi \left({x}^{0}\right)$, $\gamma \left(\varphi \right)=0$

#### 4.3.2. Variant III. B. $\varphi =\varphi \left({x}^{0}\right)$, $\gamma \left(\varphi \right)\ne 0$

## 5. The Eikonal Equation and Trajectories of Test Particles

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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Osetrin, K.; Kirnos, I.; Filippov, A.
Quadratic Theory of Gravity with a Scalar Field and Type I Shapovalov Wave Spacetimes. *Universe* **2022**, *8*, 664.
https://doi.org/10.3390/universe8120664

**AMA Style**

Osetrin K, Kirnos I, Filippov A.
Quadratic Theory of Gravity with a Scalar Field and Type I Shapovalov Wave Spacetimes. *Universe*. 2022; 8(12):664.
https://doi.org/10.3390/universe8120664

**Chicago/Turabian Style**

Osetrin, Konstantin, Ilya Kirnos, and Altair Filippov.
2022. "Quadratic Theory of Gravity with a Scalar Field and Type I Shapovalov Wave Spacetimes" *Universe* 8, no. 12: 664.
https://doi.org/10.3390/universe8120664