# Shapovalov Wave-Like Spacetimes

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

**Definition**

**1.**

**Definition**

**2.**

**Theorem**

**1.**

**Definition**

**3.**

**Theorem**

**2.**

**Definition**

**4.**

**Definition**

**5.**

**ignored variable**.

## 2. Shapovalov Spaces

**Definition**

**6.**

**isotropic (null)**, if along the coordinate line of this variable the spacetime interval is equal to zero.

**Definition**

**7.**

**Shapovalov space**if in a privileged coordinate system where separation of variables is allowed, there is a nonignored isotropic variable.

**Definition**

**8.**

**conformal Shapovalov space**if there is a nonignored isotropic variable in a privileged coordinate system where separation of variables is allowed.

## 3. Type I Shapovalov Spaces

#### 3.1. Integration of the Eikonal Equation for the Shapovalov Space of Type I

#### 3.2. Integration of the Hamilton–Jacobi Equation of a Test Particle for the Shapovalov Space of Type I

## 4. Type II Shapovalov Spaces

#### 4.1. Type II.A Shapovalov Spaces

#### 4.2. Exact Solution of the Einstein Equations for Shapovalov Spaces of Type II.A

#### 4.3. Integration of the Eikonal Equation for a Shapovalov Space of Type II.A

#### 4.4. Integration of the Hamilton–Jacobi Equation of a Test Particle for a Shapovalov Space of Type II.A

#### 4.5. Type II.B Shapovalov Spaces

#### 4.6. Exact Solution #1 of Einstein’s Equations for II.B Type Shapovalov Space

#### 4.7. Exact Solution #2 of Einstein’s Equations for Shapovalov Space of Type II.B

#### 4.8. Integration of the Eikonal Equation for the Shapovalov Space of Type II.B

#### 4.9. Integration of the Hamilton–Jacobi Equation of Motion of a Test Particle for the Shapovalov Space of Type II.B

## 5. Type III Shapovalov Spaces

#### 5.1. Exact Solution of the Einstein Equations for Shapovalov Spaces Type III

#### 5.2. Integration of the Eikonal Equation for the Shapovalov Space Type III

#### 5.3. Integration of the Hamilton–Jacobi Equation of a Test Particle for Shapovalov Space Type III

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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Osetrin, K.; Osetrin, E.
Shapovalov Wave-Like Spacetimes. *Symmetry* **2020**, *12*, 1372.
https://doi.org/10.3390/sym12081372

**AMA Style**

Osetrin K, Osetrin E.
Shapovalov Wave-Like Spacetimes. *Symmetry*. 2020; 12(8):1372.
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Osetrin, Konstantin, and Evgeny Osetrin.
2020. "Shapovalov Wave-Like Spacetimes" *Symmetry* 12, no. 8: 1372.
https://doi.org/10.3390/sym12081372