# Graviton to Photon Conversion in Curved Space-Time and External Magnetic Field

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## Abstract

**:**

## 1. Introduction

## 2. Metric Perturbations in General Case

## 3. Helicity Decomposition and Choice of Gauge

## 4. Mixing of Metric Perturbation Modes

## 5. Metric Perturbations in Magnetic Field

#### 5.1. Equation in the FLRW Metric

#### 5.2. Corrections to the Energy–Momentum Tensor (EMT)

#### 5.2.1. Corrections to EMT Emerging from the Maxwell Action

#### 5.2.2. Heisenberg–Euler (HE) Lagrangian

**B**and

**E**coming to the well-known relation ${\left(\tilde{F}F\right)}^{2}={\left[-4\phantom{\rule{0.166667em}{0ex}}(\mathbf{E}\xb7\mathbf{B})\right]}^{2}$.

#### 5.2.3. Corrections to EMT Emerging from the HE Action

#### 5.2.4. Summary

#### 5.3. Maxwell Tensor and Cosmic Magnetic and Electric Fields

#### 5.4. Scalar and Tensor Mode Mixing in External Magnetic Field

## 6. Electromagnetic Wave Propagation in External Magnetic Field

#### 6.1. Equation of Motion from the Maxwell Action

#### 6.2. Equation of Motion from the Heisenberg–Euler Action

#### 6.3. Conformal Anomaly Effect

#### 6.4. Plasma Interaction Effects

## 7. Defining $\mathbf{g}$ and $\mathbf{\gamma}$ System of Differential Equations (SoDE)

#### 7.1. Simplification of SoDE for Metric Perturbations

#### 7.2. Simplification of SoDE for Electromagnetic Waves

## 8. Two Examples of Gravitational Wave Directions

#### 8.1. $\mathbf{k}||\mathbf{B}$

#### 8.2. $\mathbf{k}\perp \mathbf{B}$

## 9. System Solution in the Case $\mathbf{k}\perp \mathbf{B}$

#### 9.1. Method of Solution Validation

#### 9.2. Numerical Solution Results for the System $\{{\mathbf{h}}_{\times},\phantom{\rule{0.166667em}{0ex}}{\mathbf{f}}^{\mathbf{x}}\}$

## 10. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

GW | Gravitational Wave |

FLRW space-time | Friedman–LeMaitre–Robertson–Walker space-time |

EMT | Energy–Momentum Tensor |

EMW | Electromagnetic Wave |

HE Lagrangian | Heisenberg–Euler Lagrangian |

SoDE | System of Differential Equations |

EoM | Equation of Motion |

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**Figure 1.**Verification of the numerical solution (blue line) by the analytical solution (red line) for two frequencies ${k}_{1}$ (

**left**) and ${k}_{2}$ (

**right**). (

**a**) without phases; (

**b**) with phases.

**Figure 2.**Absolute difference between the numerical solution and the analytical solution for two frequencies ${k}_{1}$ (

**left**) and ${k}_{2}$ (

**right**).

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**MDPI and ACS Style**

Dolgov, A.D.; Panasenko, L.A.; Bochko, V.A.
Graviton to Photon Conversion in Curved Space-Time and External Magnetic Field. *Universe* **2024**, *10*, 7.
https://doi.org/10.3390/universe10010007

**AMA Style**

Dolgov AD, Panasenko LA, Bochko VA.
Graviton to Photon Conversion in Curved Space-Time and External Magnetic Field. *Universe*. 2024; 10(1):7.
https://doi.org/10.3390/universe10010007

**Chicago/Turabian Style**

Dolgov, Alexander D., Lyubov A. Panasenko, and Vladimir A. Bochko.
2024. "Graviton to Photon Conversion in Curved Space-Time and External Magnetic Field" *Universe* 10, no. 1: 7.
https://doi.org/10.3390/universe10010007