#
A Class of 5D Inhomogeneous Models with a Cosmological Constant^{ †}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Methods

- 1.
- The spacelike vector field ${w}^{A}$ is geodesic and vorticity-free, i.e., ${\dot{w}}_{A}={w}_{A;B}{w}^{B}=0={w}_{[A;B]}={\nabla}_{[B}{w}_{A]}$.
- 2.
- The symmetric tensor ${\widehat{h}}_{AB}={\widehat{g}}_{AB}-{w}_{A}{w}_{B}$ is the proper metric of the distribution $\mathbf{w}=\mathbf{0}$ with a well-defined covariant derivative ${D}_{A}$ that satisfies ${D}_{C}{\widehat{h}}_{AB}={\widehat{h}}_{C}^{K}{\widehat{h}}_{A}^{I}{\widehat{h}}_{B}^{J}{\nabla}_{K}{\widehat{h}}_{IJ}=0$.
- 3.
- There exist 4 independent unit vector fields $\left\{\mathbf{u},\mathbf{x},\hspace{0.17em}\mathbf{y},\hspace{0.17em}\mathbf{z}\right\}$ that are hypersurface orthogonal ${u}_{[A}{u}_{B;\hspace{0.17em}C]}={x}_{[A}{x}_{B;\hspace{0.17em}C]}={y}_{[A}{y}_{B;\hspace{0.17em}C]}={z}_{[A}{z}_{B;\hspace{0.17em}C]}=0$. For each pair, e.g., $\{\mathbf{w},\mathbf{x}\}$, the second order symmetric tensor ${\widehat{p}}_{AB}={\widehat{h}}_{AB}-{x}_{A}{x}_{B}$ represents the proper metric of the screen space $\mathbf{S}$, which is defined as $\mathbf{w}\wedge \mathbf{x}=\mathbf{0}$, with associated 3D derivative ${\overline{\nabla}}_{A}$. In particular, ${\overline{\nabla}}_{C}{p}_{AB}={p}_{C}^{K}{p}_{A}^{I}{p}_{B}^{J}{\nabla}_{K}{p}_{IJ}=0$ and $\{{w}^{A},\hspace{0.17em}{x}^{A}\}$ are surface forming, i.e., ${p}_{A}^{K}{\mathbf{L}}_{w}{x}_{K}=0$.
- 4.
- The Weyl tensor of the hypersurfaces w = const. vanishes.

## 3. Results and Discussion

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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## Share and Cite

**MDPI and ACS Style**

Apostolopoulos, P.S.
A Class of 5D Inhomogeneous Models with a Cosmological Constant. *Phys. Sci. Forum* **2023**, *7*, 33.
https://doi.org/10.3390/ECU2023-14064

**AMA Style**

Apostolopoulos PS.
A Class of 5D Inhomogeneous Models with a Cosmological Constant. *Physical Sciences Forum*. 2023; 7(1):33.
https://doi.org/10.3390/ECU2023-14064

**Chicago/Turabian Style**

Apostolopoulos, Pantelis S.
2023. "A Class of 5D Inhomogeneous Models with a Cosmological Constant" *Physical Sciences Forum* 7, no. 1: 33.
https://doi.org/10.3390/ECU2023-14064