# The Second-Order Correction to the Energy and Momentum in Plane Symmetric Gravitational Waves Like Spacetimes

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## Abstract

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## 1. Introduction

## 2. Time Conformal Plane Symmetric Spacetime

## 3. Second-Order Approximate Noether Symmetry Equation

## 4. Second-Order Approximate Noether Symmetries and Conservation Laws along with Second Order Time Conformal Plane Symmetric Spacetimes

#### 4.1. Five Noether Symmetries and the Corresponding Conservation Laws

#### 4.2. Six Noether Symmetries and the Corresponding Conservation Laws

## 5. Conclusions and Observations

- Second order approximation appears in the classes of five and six Noether symmetries for the perturbed Lagrangian of plane symmetric spacetimes, which are given in Section 4.1 and Section 4.2, respectively.
- Second order approximate Noether symmetries exist for the components of metric of the form ${g}_{\mu \nu}={e}^{{\left(\frac{x}{\alpha}\right)}^{\gamma}}$ for the five Noether symmetries and ${g}_{\mu \nu}={\left(\frac{x}{\alpha}\right)}^{\gamma}$ for six Noether symmetries, which are evident from the plane symmetric metrics given in Equations (19) and (23).
- Flat spacetimes do not admit second order approximation (second-order perturbation) as well as first-order approximation/first-order perturbation); for example, Minkowski spacetime does not admit approximate Noether symmetries. Therefore, the conservation laws do not hold in general relativity globally and they hold in special relativity.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

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**MDPI and ACS Style**

Ali, M.; Ali, F.; Saboor, A.; Ghafar, M.S.; Khan, A.S.
The Second-Order Correction to the Energy and Momentum in Plane Symmetric Gravitational Waves Like Spacetimes. *Symmetry* **2019**, *11*, 220.
https://doi.org/10.3390/sym11020220

**AMA Style**

Ali M, Ali F, Saboor A, Ghafar MS, Khan AS.
The Second-Order Correction to the Energy and Momentum in Plane Symmetric Gravitational Waves Like Spacetimes. *Symmetry*. 2019; 11(2):220.
https://doi.org/10.3390/sym11020220

**Chicago/Turabian Style**

Ali, Mutahir, Farhad Ali, Abdus Saboor, M. Saad Ghafar, and Amir Sultan Khan.
2019. "The Second-Order Correction to the Energy and Momentum in Plane Symmetric Gravitational Waves Like Spacetimes" *Symmetry* 11, no. 2: 220.
https://doi.org/10.3390/sym11020220