# Variations of the Energy of Free Particles in the pp-Wave Spacetimes

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Geodesic Equations

## 3. Kinetic Energy and Geodesics of Free Particles

#### 3.1. Non-Normalized Gaussians

#### 3.2. Normalized Gaussians

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**$\Delta K$ as a function of $\lambda $, considering ${H}_{1}$ and the initial conditions I.

**Figure 2.**$\Delta {K}_{N}$ as a function of $\lambda $, considering ${H}_{1}$ and the initial conditions I.

**Figure 3.**$\Delta {K}_{N}$ as a function of $\lambda $, considering ${H}_{2}$ and the initial conditions I.

**Figure 4.**$\Delta {K}_{N}$ as a function of $\lambda $, considering ${H}_{2}$ and the initial conditions II.

**Figure 5.**$K\left(u\right)$, considering ${H}_{1}$, the initial conditions III and $\lambda =7.30247$.

**Figure 6.**$K\left(u\right)$, considering ${H}_{1}$, the initial conditions III and $\lambda =13.6043$.

**Figure 7.**$K\left(u\right)$, considering ${H}_{1}$, the initial conditions III and $\lambda =19.9062$.

**Figure 8.**$K\left(u\right)$, considering ${H}_{1}$, the initial conditions III and $\lambda =26.2081$.

**Figure 15.**$\Delta {K}_{N}$ as a function of $\lambda $, considering ${H}_{3}$ and the initial conditions I.

**Figure 16.**Trajectory of the particle in the x-z plane, considering ${H}_{3}$, the initial conditions I and $\lambda =0.1$.

**Figure 17.**Trajectory of the particle in the x-z plane, considering ${H}_{3}$, the initial conditions I and $\lambda =0.05$.

**Figure 18.**Trajectory of the particle in the x-z plane, considering ${H}_{3}$, the initial conditions I and $\lambda =0.025$.

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**MDPI and ACS Style**

Maluf, J.W.; Da Rocha-Neto, J.F.; Ulhoa, S.C.; Carneiro, F.L.
Variations of the Energy of Free Particles in the pp-Wave Spacetimes. *Universe* **2018**, *4*, 74.
https://doi.org/10.3390/universe4070074

**AMA Style**

Maluf JW, Da Rocha-Neto JF, Ulhoa SC, Carneiro FL.
Variations of the Energy of Free Particles in the pp-Wave Spacetimes. *Universe*. 2018; 4(7):74.
https://doi.org/10.3390/universe4070074

**Chicago/Turabian Style**

Maluf, José Wadih, José Francisco Da Rocha-Neto, Sérgio C. Ulhoa, and Fernando L. Carneiro.
2018. "Variations of the Energy of Free Particles in the pp-Wave Spacetimes" *Universe* 4, no. 7: 74.
https://doi.org/10.3390/universe4070074