# Quantum-Only Metrics in Spherically Symmetric Gravity

## Abstract

**:**

## 1. Introduction

## 2. The Discretized Action

#### Results at the Planck Scale

## 3. Scaling Properties

^{2}.

#### 3.1. Results at “Macroscopic” L

#### 3.2. Scaling in $\beta $ for Microscopic L

#### 3.3. Polarization Pattern

## 4. Discussion, Conclusions

#### 4.1. A 2D Integral with Stationary Phase and Zero Modes

#### 4.2. Extension to Higher Dimension

#### 4.3. Limitations of the Present Approach and Comparison with Other Methods

- (1)
- In [19], the degree of freedom in the metric is a conformal factor, while here it is the component ${g}_{rr}$ in a stationary approximation.
- (2)
- The authors of [19] search for the vacuum state by minimizing the action through a rigorous analytical approach, while we rely on numerical simulations. That is why they interpret the rippled spacetime obtained as one that becomes flat upon averaging over a periodicity volume, i.e., after a purely classical coarse graining. In our discretized model, we interpret the rippled spacetime obtained as an ensemble of purely quantum states with no classical counterpart. We also find, however, that a proper continuum limit is possible only in the low temperature limit (large $\beta $), which takes us back to the classical theory. In this sense there is a qualitative agreement between the two approaches. There might also be a connection between our concept of zero modes of the action and the restricted-space minimization of Section 2 in [19].

#### 4.4. Conclusions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Scaling of the average squared action $\langle {S}^{2}\rangle $ as a function of N. Log-log scale; dots represent values from Table 1, while the dashed line represents a dependence ${N}^{-1}$ and the solid line a dependence ${N}^{-2}$.

**Figure 2.**(

**a**) Polarization pattern for N = 800 sub-intervals. (

**b**) Polarization pattern for N = 6400 sub-intervals.

**Figure 3.**(

**a**) detail of the polarization pattern for N = 6400 sub-intervals, showing 1/4 of the total interval. (

**b**) Showing 1/16 of the total interval.

**Figure 4.**Contributions to the integral of $cos\left[{\u0127}^{-1}({x}^{2}-{y}^{2})\right]$ in the square $\{0\le x\le 1,0\le y\le 1\}$ obtained through an adaptive Monte Carlo integration with inverse temperature $\beta $. Parameters: (

**a**) $\u0127=0.1$, $\beta =1$. (

**b**) $\u0127=0.05$, $\beta =2$. (

**c**) $\u0127=0.1$, $\beta =7.8\times {10}^{-3}$. The contributions near the origin come from the stationary point of the phase, those along the diagonal from the zero mode $y=x$. In (

**b**) the region near the stationary point is smaller because ħ is smaller, but the length of the zero mode is unaffected. In (

**c**) the contributions of the disconnected zero modes $y=x\pm 2\pi \u0127$ also appear, because the temperature is much higher.

**Table 1.**Average values of the return probability $\langle {e}^{-\beta \widehat{S}}\rangle $, the action $\langle \widehat{S}\rangle $ and the squared action $\langle {\widehat{S}}^{2}\rangle $ in dependence on N (number of sub-intervals of $(0,L)$). Here $L=10$ cm. The inverse temperature $\beta $ changes in proportion to N, in order to maintain the factor $\beta /N$ constant. The discretized field components ${A}_{h}$ are randomly increased in the Montecarlo steps by $\pm \epsilon $, with $\epsilon ={10}^{-6}$. For the last two values of N the calculation of $\langle {e}^{-\beta \widehat{S}}\rangle $ (and of $\langle {A}_{h}\rangle $) was omitted in order to speed-up the algorithm and increase the precision in $\langle {\widehat{S}}^{2}\rangle $. See also plot of $\langle {\widehat{S}}^{2}\rangle $ in Figure 1.

N | $\mathit{\beta}$ | MC Steps | $\langle {\mathit{e}}^{-\mathit{\beta}|\widehat{\mathit{S}}|}\rangle $ | $\langle \widehat{\mathit{S}}\rangle $ | $\langle {\widehat{\mathit{S}}}^{2}\rangle $ |
---|---|---|---|---|---|

100 | ${10}^{7}$ | $2\times {10}^{9}$ | 0.17 | $4.3\times {10}^{-9}$ | $2.0\times {10}^{-14}$ |

200 | $2\times {10}^{7}$ | $2\times {10}^{9}$ | 0.11 | $6.4\times {10}^{-9}$ | $5.1\times {10}^{-15}$ |

400 | $4\times {10}^{7}$ | $4\times {10}^{9}$ | 0.022 | $5.4\times {10}^{-9}$ | $1.3\times {10}^{-15}$ |

800 | $8\times {10}^{7}$ | $8\times {10}^{9}$ | 0.024 | $1.4\times {10}^{-9}$ | $3.2\times {10}^{-16}$ |

1600 | $16\times {10}^{7}$ | $8\times {10}^{9}$ | 0.055 | $3.0\times {10}^{-10}$ | $7.8\times {10}^{-17}$ |

3200 | $32\times {10}^{7}$ | $8\times {10}^{9}$ | 0.14 | $1.2\times {10}^{-9}$ | $2.3\times {10}^{-17}$ |

6400 | $64\times {10}^{7}$ | $8\times {10}^{9}$ | 0.27 | $1.5\times {10}^{-9}$ | $9.9\times {10}^{-18}$ |

12,800 | $128\times {10}^{7}$ | $16\times {10}^{9}$ | 0.25 | $8.7\times {10}^{-10}$ | $2.8\times {10}^{-18}$ |

25,600 | $256\times {10}^{7}$ | $16\times {10}^{9}$ | 0.29 | $3.4\times {10}^{-10}$ | $5.8\times {10}^{-19}$ |

204,800 | $2048\times {10}^{7}$ | $16\times {10}^{10}$ | $3.3\times {10}^{-11}$ | $5.3\times {10}^{-21}$ | |

409,600 | $4096\times {10}^{7}$ | $16\times {10}^{10}$ | $1.5\times {10}^{-11}$ | $1.3\times {10}^{-21}$ |

**Table 2.**Scaling of $\langle {e}^{-\beta \widehat{S}}\rangle $, $\langle \widehat{S}\rangle $ and $\langle {\widehat{S}}^{2}\rangle $ in dependence on $\beta $ with N fixed, $L=10$, $\epsilon ={10}^{-6}$. Note the decrease of the average return probability $\langle {e}^{-\beta \widehat{S}}\rangle $, coherent with the role of the inverse temperature $\beta $ in the thermalization process.

$\mathit{\beta}$ | N | MC Steps | $\langle {\mathit{e}}^{-\mathit{\beta}|\widehat{\mathit{S}}|}\rangle $ | $\langle \widehat{\mathit{S}}\rangle $ | $\langle {\widehat{\mathit{S}}}^{2}\rangle $ |
---|---|---|---|---|---|

$128\times {10}^{7}$ | 12,800 | $16\times {10}^{9}$ | 0.25 | $8.7\times {10}^{-10}$ | $2.8\times {10}^{-18}$ |

$256\times {10}^{7}$ | 12,800 | $16\times {10}^{9}$ | 0.16 | $5.0\times {10}^{-10}$ | $8.2\times {10}^{-19}$ |

$512\times {10}^{7}$ | 12,800 | $16\times {10}^{9}$ | 0.095 | $3.3\times {10}^{-10}$ | $2.8\times {10}^{-19}$ |

$1024\times {10}^{7}$ | 12,800 | $16\times {10}^{9}$ | 0.055 | $2.2\times {10}^{-10}$ | $1.0\times {10}^{-19}$ |

**Table 3.**Scaling of $\langle {e}^{-\beta \widehat{S}}\rangle $, $\langle \widehat{S}\rangle $ and $\langle {\widehat{S}}^{2}\rangle $ in dependence on $\beta $ with N fixed, $L={10}^{-13}$ cm (“microscopic scale”), $\epsilon ={10}^{-6}$.

$\mathit{\beta}$ | N | MC Steps | $\langle {\mathit{e}}^{-\mathit{\beta}|\widehat{\mathit{S}}|}\rangle $ | $\langle \widehat{\mathit{S}}\rangle $ | $\langle {\widehat{\mathit{S}}}^{2}\rangle $ |
---|---|---|---|---|---|

$128\times {10}^{21}$ | 12,800 | $16\times {10}^{9}$ | 0.25 | $8.8\times {10}^{-24}$ | $2.9\times {10}^{-46}$ |

$256\times {10}^{21}$ | 12,800 | $16\times {10}^{9}$ | 0.15 | $4.9\times {10}^{-24}$ | $8.0\times {10}^{-47}$ |

$512\times {10}^{21}$ | 12,800 | $16\times {10}^{9}$ | 0.093 | $3.3\times {10}^{-24}$ | $2.8\times {10}^{-47}$ |

$1024\times {10}^{21}$ | 12,800 | $16\times {10}^{9}$ | 0.053 | $2.2\times {10}^{-24}$ | $1.0\times {10}^{-47}$ |

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Modanese, G.
Quantum-Only Metrics in Spherically Symmetric Gravity. *Quantum Rep.* **2020**, *2*, 314-325.
https://doi.org/10.3390/quantum2020021

**AMA Style**

Modanese G.
Quantum-Only Metrics in Spherically Symmetric Gravity. *Quantum Reports*. 2020; 2(2):314-325.
https://doi.org/10.3390/quantum2020021

**Chicago/Turabian Style**

Modanese, Giovanni.
2020. "Quantum-Only Metrics in Spherically Symmetric Gravity" *Quantum Reports* 2, no. 2: 314-325.
https://doi.org/10.3390/quantum2020021