#
Spherically Symmetric C^{3} Matching in General Relativity

## Abstract

**:**

## 1. Introduction

## 2. Eigenvalues of the Riemann Curvature Tensor

**L**and

**M**are symmetric and trace free,

## 3. ${C}^{3}$ Matching

- (i)
- Define the matching surface $\Sigma $ by means of the matching radius ${r}_{match}$, defined as$${r}_{match}\in [{r}_{rep},\infty )\phantom{\rule{4pt}{0ex}},\phantom{\rule{1.em}{0ex}}\mathrm{with}\phantom{\rule{1.em}{0ex}}{r}_{rep}=\mathrm{max}\{{r}_{l}\}\phantom{\rule{4pt}{0ex}},\phantom{\rule{1.em}{0ex}}\frac{\mathit{\partial}{\lambda}_{i}^{+}}{\mathit{\partial}r}{|}_{r={r}_{l}}=0\phantom{\rule{4pt}{0ex}}.$$This means that the repulsion radius is determined by the location of the first extremum that is found when approaching the source of gravity from infinity.
- (ii)
- Perform the matching of the spacetimes $({M}^{+},{g}_{\mu \nu}^{+})$ and $({M}^{-},{g}_{\mu \nu}^{-})$ at $\Sigma $ by imposing the conditions$${\lambda}_{i}^{+}{|}_{\Sigma}={\lambda}_{i}^{-}{|}_{\Sigma}\phantom{\rule{1.em}{0ex}}\forall i\phantom{\rule{4pt}{0ex}}.$$

## 4. The Spherically Symmetric Matching

## 5. Final Remarks and Perspectives

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Quevedo, H.
Spherically Symmetric *C*^{3} Matching in General Relativity. *Universe* **2023**, *9*, 419.
https://doi.org/10.3390/universe9090419

**AMA Style**

Quevedo H.
Spherically Symmetric *C*^{3} Matching in General Relativity. *Universe*. 2023; 9(9):419.
https://doi.org/10.3390/universe9090419

**Chicago/Turabian Style**

Quevedo, Hernando.
2023. "Spherically Symmetric *C*^{3} Matching in General Relativity" *Universe* 9, no. 9: 419.
https://doi.org/10.3390/universe9090419