# Vacuum Energy, the Casimir Effect, and Newton’s Non-Constant

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## Abstract

**:**

## 1. Introduction

#### 1.1. Vacuum Energy and the Cosmological Constant (Problem)

#### 1.2. Vacuum Energy in the Casimir Effect

#### 1.3. Hypotheses

#### 1.4. Scale Dependence, Quantum Fluctuations, and Scale Setting

- In this approach, the principle of equivalence as discussed in the literature is replaced by general covariance of the Einstein tensor ${\nabla}^{\mu}{G}_{\mu \nu}=0$. This implies that the covariant derivative of the entire right-hand side of Equation (11) vanishes. Thus, a conservation of each individual term, such as ${\nabla}^{\mu}{T}_{\mu \nu}=0$, is not strictly required anymore.
- If the Casimir energy density ${T}_{C}^{\mu \nu}$ contributes to the matter energy density ${T}^{\mu \nu}={T}_{M}^{\mu \nu}+{T}_{C}^{\mu \nu}$, then the corresponding Lagrangian must have a shift symmetry $\mathcal{L}={\mathcal{L}}_{M}+{\mathcal{L}}_{C}$. Interestingly, such a shift symmetry can be used as an additional condition, which allows one to directly solve Equation (11). The condition arising from this symmetry applied to Equation (12) is the so-called null energy condition. We introduced and investigated this condition in [87,88,96,99].

## 2. Results

#### 2.1. Gravitational Field Equations with Weak Curvature and Weak SD

- The ‘original’ mass density ${\rho}_{M}/{c}^{2}$, which is typically determined with electromagnetic forces, gauged in the absence of the Casimir effect. The integral over this density is the mass we find in the absence of $\mathrm{\Delta}G$: ${M}_{2}={\int}_{{V}_{2}}\phantom{\rule{-0.166667em}{0ex}}{\mathrm{d}}^{3}x\phantom{\rule{0.166667em}{0ex}}{\rho}_{M}\left(x\right)/{c}^{2}$. However, this is not the density that allows one to define a force for the gravitational acceleration from Equation (20) since it does not fulfil Newton’s third law.
- The apparent gravitational mass density (18), which appears in the gravitational potential (17). The gravitational force caused by one object on another object ${\overrightarrow{F}}_{12}$ must obey Newton’s third law ${\overrightarrow{F}}_{12}=-{\overrightarrow{F}}_{21}$. In the WG-WSD limit, this is only guaranteed for the apparent gravitational mass density ${\tilde{\rho}}_{M}/{c}^{2}$, for which this quantity must appear in the definition of gravitational force.

#### 2.2. Scale Setting

#### 2.2.1. Density-Induced Scale-Setting

#### 2.2.2. Explicitly Covariant Scale-Setting

#### 2.3. Perspectives for Experimental Tests

## 3. Discussion and Conclusions

#### 3.1. Why So Strong?

- (i)
- Comparable order of ${\rho}_{{\mathrm{\Lambda}}_{0}}$ and ${\rho}_{C}$.At experimental scales of $a\approx 5\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{10}^{-5}\phantom{\rule{3.33333pt}{0ex}}$m, the cosmological energy density given by Equation (1) and the Casimir energy density are of the same order of magnitude. This means that relative corrections in ${\rho}_{\mathrm{\Lambda}}$ are not necessarily small, unless $\left|\alpha \right|\ll 1$. This is usually not the case in estimates like Equation (40).
- (ii)
- Link ${\rho}_{\mathrm{\Lambda}}\leftrightarrow \mathrm{\Lambda}\left(k\right)\leftrightarrow G\left(k\right)$.In our discussion, $\mathrm{\Lambda}\left(k\right)$ and $G\left(k\right)$ are linked through Equations (5) and (9). Therefore, sizable corrections to ${\rho}_{\mathrm{\Lambda}}$ can imply sizable corrections to $G\left(k\right)$, and, thus, to the Newton potential. This link is usually not considered when quantum corrections to the Newton potential are discussed [121].
- (iii)
- Dynamical $\mathrm{\Delta}{t}_{\mu \nu}$.The modifications in $G\left(k\right)$ enter gravitational field Equation (13) twofold: first as a “static” modification of the prefactor of the usual matter energy density ${\rho}_{M}$ and second as a “dynamical” contribution through $\mathrm{\Delta}{t}_{\mu \nu}$. In the WG-WSD limit in Equation (17), the latter contribution dominates over the former (more intuitive) contribution. To check the validity of the corresponding expansion (14), we can ignore the leading contribution to Equation (17) and recalculate the sensitivity (37) with the “static” $\mathrm{\Delta}G/{G}_{0}$ term. This would result in a loss of sensitivity of 35 orders of magnitude. The same is true for conventional estimates like the one of Ref. [121], where no such ‘dynamical’ effect is considered.
- (iv)
- Small skin depth.Since the $\mathrm{\Delta}{t}_{\mu \nu}$ contribution contains two spatial derivatives, the exponential form of ${\rho}_{Q}$ in Equation (35) leads to an additional enhancement near the boundary, and generally for ${\delta}_{c}\ll 1\phantom{\rule{0.166667em}{0ex}}$m. As mentioned, the experimental and theoretical uncertainty for this model may result in errors of several orders of magnitude. The skin effect is a particular feature of the experimental Casimir setup, for which it does not enter conventional estimates [121].

#### 3.2. Interpretation

- (iv)
- (iii)
- The result (37) would not hold if the $\mathrm{\Delta}{t}_{\mu \nu}$ term was absent from modified field Equation (11). However, since this term is needed to restore diffeomorphism invariance, it can not be ‘just absent’. It could, however, be replaced by a less minimal extension of GR, in which case Equation (37) would have to be recalculated for the particular non-minimal model.
- (ii)
- It could happen that $\mathrm{\Lambda}\left(k\right)$ is only very weakly linked to $G\left(k\right)$ in the infrared. In our parametrization, this possibility is contemplated for parameter points with ${C}_{1}\ll {C}_{1}-{C}_{3}$. From the SD perspective, this is would be an unusual scenario since, in typical benchmark scenarios, this is not the case (see Table A1).
- (i)
- Finally, except for scenarios (iv)–(ii), there remains the possibility that Equation (37) provides a window into the unknown relation between the quantum world and cosmology/gravity. Due to the strength of the boundary (37), this will most likely, and under the assumption that WG-WSD represents a valid approach, allow experiments to gain insight on the CCP (2). This scenario will be discussed in the next subsection.

#### 3.3. Back to the CCP

#### 3.4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Parameters in Asymptotic Safety

**Table A1.**Parameters of the three benchmark scenarios for renormalization group results; ${B}_{1}$ corresponds to pure gravity, ${B}_{2}$ is for the particle content of QED, and ${B}_{3}$ considers the particle content of the Standard Model.

${\mathit{B}}_{1}$ | ${\mathit{B}}_{2}$ | ${\mathit{B}}_{3}$ | |
---|---|---|---|

${N}_{S}$ | 0 | 0 | 4 |

${N}_{D}$ | 0 | 1 | 12 |

${N}_{V}$ | 0 | 1 | 12 |

${C}_{1}$ | $-15/\left(16\pi \right)$ | $-4/\pi $ | $-11/\left(2\pi \right)$ |

${C}_{3}$ | $-15/\left(16\pi \right)$ | $-3/\left(2\pi \right)$ | $-3/\pi $ |

${C}_{1}/({C}_{1}-{C}_{3})$ | ∞ | $1.6$ | $2.2$ |

## Notes

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2 |

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**Figure 1.**Schematic sketch of an assumed exponential attenuation of ${\rho}_{C}\left(z\right)$ over the skin depth inside of a material, plotted from the middle between two plates at $z=0$ towards the plate in the region with $z>0$; the region between the plates is labeled (i), the skin of the plate is labeled (ii), and the region with effectively vanishing ${\rho}_{C}$ inside the plate is labeled (iii). The configuration is symmetric under $z\leftrightarrow -z$.

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**MDPI and ACS Style**

Koch, B.; Käding, C.; Pitschmann, M.; Sedmik, R.I.P.
Vacuum Energy, the Casimir Effect, and Newton’s Non-Constant. *Universe* **2023**, *9*, 476.
https://doi.org/10.3390/universe9110476

**AMA Style**

Koch B, Käding C, Pitschmann M, Sedmik RIP.
Vacuum Energy, the Casimir Effect, and Newton’s Non-Constant. *Universe*. 2023; 9(11):476.
https://doi.org/10.3390/universe9110476

**Chicago/Turabian Style**

Koch, Benjamin, Christian Käding, Mario Pitschmann, and René I. P. Sedmik.
2023. "Vacuum Energy, the Casimir Effect, and Newton’s Non-Constant" *Universe* 9, no. 11: 476.
https://doi.org/10.3390/universe9110476