# Pure-Connection Gravity and Anisotropic Singularities

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## Abstract

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## 1. Introduction

## 2. Pure-Connection Gravity and Its Modification

#### 2.1. Eddington–Schrödinger Theory

#### 2.2. The Chiral Plebański Formulation of GR

#### 2.3. Modified Gravity

#### 2.4. Pure-Connection Formulation

## 3. Black-Hole Solution

## 4. Bianchi I Cosmology

#### 4.1. Behavior in GR

#### 4.2. Behavior in Modified Gravity

## 5. Discussion and Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

GR | General Relativity |

DOF | Degrees of Freedom |

## References

- Starobinsky, A.A. A New Type of Isotropic Cosmological Models Without Singularity. Phys. Lett. B
**1980**, 91, 99–102. [Google Scholar] [CrossRef] - Ade, P.A.R.; Aghanim, N.; Arnaud, M.; Arroja, F.; Ashdown, M.; Aumont, J.; Baccigalupi, C.; Ballardini, M.; Banday, A.J.; Barreiro, R.B.; et al. Planck 2015 results. XX. Constraints on inflation. Astron. Astrophys.
**2016**, 594, A20. [Google Scholar] [CrossRef] - Plebański, J.F. On the separation of Einsteinian substructures. J. Math. Phys.
**1977**, 18, 2511–2520. [Google Scholar] [CrossRef] - Bengtsson, I. The Cosmological constants. Phys. Lett. B
**1991**, 254, 55–60. [Google Scholar] [CrossRef] - Krasnov, K. Renormalizable non-metric quantum gravity? arXiv, 2006; arXiv:hep-th/0611182. [Google Scholar]
- Eddington, A.S. The Mathematical Theory of Relativity; Cambridge University Press: Cambridge, UK, 1920; Chapter 7. [Google Scholar]
- Schrödinger, E. Space-Time Structure; Cambridge University Press: Cambridge, UK, 1950; Chapter 12. [Google Scholar]
- Capovilla, R.; Jacobson, T.; Dell, J.; Mason, L. Self-dual 2-forms and gravity. Class. Quantum Gravity
**1991**, 8, 41–57. [Google Scholar] [CrossRef] - Krasnov, K. Plebański formulation of general relativity: A practical introduction. Gen. Relativ. Gravit.
**2011**, 43, 1–15. [Google Scholar] [CrossRef] - Urbantke, H. On integrability properties of Su(2) Yang–Mills fields. I. Infinitesimal part. J. Math. Phys.
**1984**, 25, 2321–2324. [Google Scholar] [CrossRef] - Krasnov, K. On deformations of Ashtekar’s constraint algebra. Phys. Rev. Lett.
**2008**, 100, 081102. [Google Scholar] [CrossRef] [PubMed] - Krasnov, K. Pure connection action principle for general relativity. Phys. Rev. Lett.
**2011**, 106, 251103. [Google Scholar] [CrossRef] [PubMed] - Krasnov, K.; Shtanov, Y. Non-metric gravity: II. Spherically symmetric solution, missing mass and redshifts of quasars. Class. Quantum Gravity
**2008**, 25, 025002. [Google Scholar] [CrossRef] - Krasnov, K.; Shtanov, Y. Halos of modified gravity. Int. J. Mod. Phys. D
**2009**, 17, 2555–2562. [Google Scholar] [CrossRef] - Herfray, Y.; Krasnov, K.; Shtanov, Y. Anisotropic singularities in chiral modified gravity. Class. Quantum Gravity
**2016**, 33, 235001. [Google Scholar] [CrossRef] - Krasnov, K. Spontaneous symmetry breaking and gravity. Phys. Rev. D
**2012**, 85, 125023. [Google Scholar] [CrossRef]

**Figure 1.**Conformal diagram of the spherically symmetric solution with the Lambda-function (11) and with the Schwarzschild radius ${r}_{g}(\infty )$ > ${r}_{c}$ (left image) and ${r}_{g}(\infty )$ < ${r}_{c}$ (right image), with ${r}_{c}$ given in Equation (21). Different regions are numbered in such a way that the coordinate t is timelike in the odd regions, and spacelike in the even ones. The flow of time is vertical in the white regions and changes to horizontal in the grey regions. The boundaries between grey and white regions are places where metric changes signature and, therefore, becomes singular. Thick dashed lines indicate the true singularity, where r = ∞ and $\psi $ = ∞. The configuration on the left image extends periodically and indefinitely upward and downward. We are living in one of the regions of type I; the asymptotic spatial infinity in this region is denoted by ${i}^{0}$, and the future and past null infinities are denoted by ${J}^{+}$ and ${J}^{-}$, respectively.

**Figure 2.**Behavior of the variables ${X}_{i}$ given by Equation (30) in general relativity. Each variable ${X}_{i}$ has a pole at respective $\tau $ = ${\tau}_{i}$. These are just special points of the solution, with all connection and metric components remaining finite. Furthermore, all variables ${X}_{i}$ have common poles at $\tau $ = ±1. These are the singular points for both the connection and the metric. The region “between” two Kasner singularities describes a solution with negative $\mathsf{\Lambda}$, while the two outer regions describe solutions with $\mathsf{\Lambda}$ > 0.

**Figure 3.**Behavior of the variables ${X}_{i}$ given by Equation (35) in the modified theory (11) with $\alpha {\mathsf{\Lambda}}_{0}=\phantom{\rule{3.33333pt}{0ex}}5\phantom{\rule{3.33333pt}{0ex}}\times {10}^{-3}$. As in general relativity (see Figure 2), each variable ${X}_{i}$ has a pole at respective $\tau ={\tau}_{i}$. However, the common poles at $\tau $ = ±1 disappear, and the variables ${X}_{i}$ become regular at these points.

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**MDPI and ACS Style**

Krasnov, K.; Shtanov, Y.
Pure-Connection Gravity and Anisotropic Singularities. *Universe* **2018**, *4*, 12.
https://doi.org/10.3390/universe4010012

**AMA Style**

Krasnov K, Shtanov Y.
Pure-Connection Gravity and Anisotropic Singularities. *Universe*. 2018; 4(1):12.
https://doi.org/10.3390/universe4010012

**Chicago/Turabian Style**

Krasnov, Kirill, and Yuri Shtanov.
2018. "Pure-Connection Gravity and Anisotropic Singularities" *Universe* 4, no. 1: 12.
https://doi.org/10.3390/universe4010012