# Metrics with Zero and Almost-Zero Einstein Action in Quantum Gravity

## Abstract

**:**

## 1. Introduction

## 2. Simplified Recipe for the Discretized Path Integral

#### 2.1. Spherically Symmetric Spaces with Constant ${g}_{tt}$

#### 2.2. Discretized Action

#### 2.3. Comparison to the Scalar Field Case and Lorentzian Path Integral

#### 2.4. Back to Gravity: ${S}_{E}$ Is Not Positive, then Sample with ${S}_{E}^{2}$ or $|{S}_{E}|$

## 3. Classical, Exact Zero Modes

## 4. Results of the Quantum Simulations

#### 4.1. Results with Transition Probability $exp(-\beta \Delta |{S}_{E}|)$

## 5. Comparison with Other Approaches

## 6. Conclusions and Outlook

- Exact zero modes (${S}_{E}=0$) with curvature polarization exist in the weak field approximation.
- Exact zero modes without curvature polarization exist also in strong field, but
- Quantum zero modes (${S}_{E}\ll \hslash $) of any strength are predominantly of the polarized kind.

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Values of the action measured during a typical Metropolis run with inverse temperature $\beta ={10}^{13}$. An equilibrium value of the order of ${10}^{-5}$ is attained (all quantities in units such that $\hslash =c=G=1$; interval length $L=1$, number of sub-intervals $N=100$).

**Figure 2.**Components ${S}_{1}$ and ${S}_{2}$ of the action for the same run of Figure 1. ${S}_{1}$ is given by the sum from 1 to $\frac{N}{2}$, i.e., on the left half of the interval. ${S}_{2}$ is given by the sum from $\frac{N}{2}+1$ to N, i.e., on the right half. Both ${S}_{1}$ and ${S}_{2}$ are much larger, in absolute value, than the total action. Since the sign of R is opposite to that of the local S, we conclude that the inner part of the metric has negative curvature and the outer shell has positive curvature. The total integral of the curvature is small and negative.

**Figure 3.**Metric component $A={g}_{rr}$ as a function of the discretized radius $r=\delta h$ ($\delta =L/N$ is the lattice cutoff). The condition at the right boundary is $A=1$.

**Figure 4.**Action contributions from two metric “plateaus” where the metric is approximately constant. The contribution ${S}_{1-plateau}$ comes from the inner region with $30\le h\le 40$. The contribution ${S}_{2-plateau}$ comes from the outer region with $60\le h\le 70$. Note that the sign of ${S}_{1-plateau}$ is opposite to the sign of ${S}_{1}$, and the same holds for ${S}_{2-plateau}$ and ${S}_{2}$.

**Figure 5.**Total action with probability ${e}^{-\beta \left|S\right|}$ and inverse temperature $\beta ={10}^{8}$. It reaches a positive maximum value of the order of ${10}^{-4}$ and then decreases to ${10}^{-9}$.

**Figure 6.**Components ${S}_{1}$ and ${S}_{2}$ of the action for the same run of Figure 5. They are about 8 magnitude orders larger than the total action.

**Figure 7.**Total action with probability ${e}^{-\beta S}$ and inverse temperature $\beta ={10}^{8}$. The system is clearly unstable with respect to negative fluctuations of the action.

**Table 1.**Feynman-Hibbs transition elements computed numerically in the Lorentzian path integral (14) in dependence on the amplitude a of the integration interval. The last column gives the maximum value of the action recorded during the Monte Carlo integration, which runs typically over 10 to 100 million random values of the set ${\varphi}_{1},\dots ,{\varphi}_{50}$.

a | Re$\langle {({\mathit{\varphi}}_{25}-1)}^{2}\rangle $ | Im$\langle {({\mathit{\varphi}}_{25}-1)}^{2}\rangle $ | ${\mathit{S}}_{\mathit{max}}$ |
---|---|---|---|

${10}^{-4}$ | $3.33\times {10}^{-9}$ | $2.78\times {10}^{-17}$ | $5\times {10}^{-6}$ |

${10}^{-2}$ | $3.33\times {10}^{-5}$ | $2.78\times {10}^{-9}$ | $5\times {10}^{-2}$ |

1 | $(4\pm 2)\times {10}^{-1}$ | $(1\pm 2)\times {10}^{-1}$ | $5\times {10}^{2}$ |

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**MDPI and ACS Style**

Modanese, G.
Metrics with Zero and Almost-Zero Einstein Action in Quantum Gravity. *Symmetry* **2019**, *11*, 1288.
https://doi.org/10.3390/sym11101288

**AMA Style**

Modanese G.
Metrics with Zero and Almost-Zero Einstein Action in Quantum Gravity. *Symmetry*. 2019; 11(10):1288.
https://doi.org/10.3390/sym11101288

**Chicago/Turabian Style**

Modanese, Giovanni.
2019. "Metrics with Zero and Almost-Zero Einstein Action in Quantum Gravity" *Symmetry* 11, no. 10: 1288.
https://doi.org/10.3390/sym11101288