# The Quantization of Gravity: The Quantization of the Full Einstein Equations

## Abstract

**:**

Contents | ||

1 | Introduction................................................................................................................................................................................................................................. | 1 |

2 | Quantizing the Full Einstein Equations....................................................................................................................................................................................... | 8 |

3 | Spatial Eigenfunctions............................................................................................................................................................................................................... | 11 |

4 | Temporal Eigenfunctions: The Case 3 ≤ n ≤ 16........................................................................................................................................................................ | 15 |

5 | Temporal Eigenfunctions: The Case n ≥ 17............................................................................................................................................................................... | 17 |

5.1 Treating Λ as an Eigenvalue............................................................................................................................................................................................ | 17 | |

5.2 Treating Λ as a Fixed Cosmological Constant................................................................................................................................................................ | 24 | |

6 | Trace Class Estimates for e^{−β^H0}................................................................................................................................................................................................ | 29 |

7 | Conclusions................................................................................................................................................................................................................................ | 32 |

8 | References.................................................................................................................................................................................................................................. | 33 |

## 1. Introduction

**Theorem**

**1.**

**Remark**

**1.**

**Theorem**

**2.**

**Lemma**

**1.**

**Remark**

**2.**

**Remark**

**3.**

## 2. Quantizing the Full Einstein Equations

## 3. Spatial Eigenfunctions

**Lemma**

**2.**

**Proof.**

**Remark**

**4.**

**Theorem**

**3.**

**Theorem**

**4.**

## 4. Temporal Eigenfunctions: The Case $\mathbf{3}\le \mathbf{n}\le \mathbf{16}$

**Theorem**

**5.**

## 5. Temporal Eigenfunctions: The Case $\mathbf{n}\ge \mathbf{17}$

#### 5.1. Treating $\Lambda $ as an Eigenvalue

**Definition**

**1.**

**Definition**

**2.**

**Proposition**

**1.**

**Proof.**

**Lemma**

**3.**

**Theorem**

**6.**

**Proof.**

**Lemma**

**4.**

**Proof.**

**Theorem**

**7.**

**Remark**

**5.**

**Lemma**

**5.**

**Remark**

**6.**

**Theorem**

**8.**

**Lemma**

**6.**

**Proof.**

#### 5.2. Treating $\Lambda $ as a Fixed Cosmological Constant

**Lemma**

**7.**

**Corollary**

**1.**

**Theorem**

**9.**

**Remark**

**7.**

**Lemma**

**8.**

**Remark**

**8.**

**Lemma**

**9.**

**Remark**

**9.**

## 6. Trace Class Estimates for ${e}^{{-\beta}^{^}{H}_{0}}$

**Remark**

**10.**

**Lemma**

**10.**

**Proof.**

**Lemma**

**11.**

**Proof.**

**Theorem**

**10.**

**Proof.**

**Theorem**

**11.**

## 7. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Arnowitt, R.; Deser, S.; Misner, C.W. The dynamics of general relativity. In Gravitation: An Introduction to Current Research; Witten, L., Ed.; John Wiley: New York, NY, USA, 1962; pp. 227–265. [Google Scholar]
- DeWitt, B.S. Quantum Theory of Gravity. I. The Canonical Theory. Phys. Rev.
**1967**, 160, 1113–1148. [Google Scholar] [CrossRef] - Gerhardt, C. Quantum cosmological Friedman models with an initial singularity. Class. Quantum Gravity
**2009**, 26, 015001. [Google Scholar] [CrossRef] - Kiefer, C.; Sandhöfer, B. Quantum cosmology. Z. Für Naturforschung A
**2022**, 77, 543–559. [Google Scholar] [CrossRef] - Misner, C.W. Quantum Cosmology. I. Phys. Rev.
**1969**, 186, 1319–1327. [Google Scholar] [CrossRef] - Moniz, P.V. (Ed.) Quantum Cosmology; MDPI: Basel, Switzerland, 2022. [Google Scholar] [CrossRef]
- Unruh, W.G. Unimodular theory of canonical quantum gravity. Phys. Rev. D
**1989**, 40, 1048–1052. [Google Scholar] [CrossRef] - Kiefer, C. Quantum Gravity, 2nd ed.; International Series of Monographs on Physics; Oxford University Press: Oxford, UK, 2007. [Google Scholar]
- Thiemann, T. Modern canonical quantum general relativity. In Cambridge Monographs on Mathematical Physics; With a foreword by Chris Isham; Cambridge University Press: Cambridge, UK, 2007. [Google Scholar]
- Gerhardt, C. The quantization of gravity in globally hyperbolic spacetimes. Adv. Theor. Math. Phys.
**2013**, 17, 1357–1391. [Google Scholar] [CrossRef] - Gerhardt, C. The quantization of gravity. Adv. Theor. Math. Phys.
**2018**, 22, 709–757. [Google Scholar] [CrossRef] - Gerhardt, C. The Quantization of Gravity, 1st ed.; Fundamental Theories of Physics; Springer: Cham, Switzerland, 2018; Volume 194. [Google Scholar] [CrossRef]
- Jorgenson, J.; Lang, S. Spherical Inversion on SLn(R); Springer: New York, NY, USA, 2001. [Google Scholar] [CrossRef]
- Gerhardt, C. The quantization of gravity: Quantization of the Hamilton equations. Universe
**2021**, 7, 91. [Google Scholar] [CrossRef] - Gerhardt, C. A unified quantization of gravity and other fundamental forces of nature. Universe
**2022**, 8, 404. [Google Scholar] [CrossRef] - Dacorogna, B.; Moser, J. On a partial differential equation involving the jacobian determinant. Ann. De L’I.H.P. Anal. Non Linéaire
**1990**, 7, 1–26. [Google Scholar] - Helgason, S. Geometric analysis on symmetric spaces. Math. Surv. Monogr.
**1994**, 39. [Google Scholar] [CrossRef] - Gerhardt, C. A unified quantization of gravity and other fundamental forces of nature implies a big bang on the quantum level. 2023. Available online: https://www.researchgate.net/publication/367523922_A_UNIFIED_QUANTIZATION_OF_GRAVITY_AND_OTHER_FUNDAMENTAL_FORCES_OF_NATURE_IMPLIES_A_BIG_BANG_ON_THE_QUANTUM_LEVEL?channel=doi&linkId=63d66bb064fc860638f89b35&showFulltext=true (accessed on 15 August 2023).
- Courant, R.; Hilbert, D. Methoden der mathematischen Physik. I; Dritte Auflage, Heidelberger Taschenbücher, Band 30; Springer-Verlag: Berlin, Germany, 1968. [Google Scholar] [CrossRef]
- Gerhardt, C. Partial Differential Equations II, Lecture Notes, University of Heidelberg. 2013. Available online: http://www.math.uni-heidelberg.de/studinfo/gerhardt/PDE2.pdf (accessed on 15 August 2023).
- Gerhardt, C. The quantization of a black hole. arXiv
**2016**, arXiv:1608.08209. [Google Scholar] - Gerhardt, C. The quantization of a Kerr-AdS black hole. Adv. Math. Phys.
**2018**, 2018, 4328312. [Google Scholar] [CrossRef]

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Gerhardt, C.
The Quantization of Gravity: The Quantization of the Full Einstein Equations. *Symmetry* **2023**, *15*, 1599.
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Gerhardt C.
The Quantization of Gravity: The Quantization of the Full Einstein Equations. *Symmetry*. 2023; 15(8):1599.
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**Chicago/Turabian Style**

Gerhardt, Claus.
2023. "The Quantization of Gravity: The Quantization of the Full Einstein Equations" *Symmetry* 15, no. 8: 1599.
https://doi.org/10.3390/sym15081599