# The CMB, Preferred Reference System, and Dragging of Light in the Earth Frame

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Basics of the Ether-Drift Experiments

## 3. A Modern Version of Maxwell Calculation

## 4. Dragging of Light as an Irregular Phenomenon

## 5. The Classical Experiments in Gaseous Media

## 6. Experiments in Gases vs. Vacuum and Solid Dielectrics

## 7. Modern Experiments with Optical Resonators

#### 7.1. Basic Aspects of Present Experiments in Vacuum

#### 7.2. A 10^{−9} Refractivity for the Vacuum on the Earth’s
Surface

**a**) and (

**b**) of Figure 10 are physically distinct, but in General Relativity, it is assumed that both observers will measure the same c of Lorentz transformations. A non-zero vacuum refractivity for system (

**b**) can thus be expressed as

**b**) in our Figure 10) could differ at the level of ${10}^{-9}$ from the ideal value c, which is operationally defined with the same apparatus in a truely freely falling frame (panel (

**b**) in our Figure 10). As discussed at the end of Section 6, this ${\u03f5}_{v}\sim {10}^{-9}$ was suggested by the last precise measurements of the velocity of light and, by comparing with Equation (62), could now provide a physical argument for seriously considering the presently observed ${10}^{-15}$ fractional frequency shift of two vacuum optical resonators. Let us therefore take a closer look at the present experiments.

#### 7.3. A Closer Look at Experiments and Numerical Simulations of the Signal

## 8. Summary and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Notes

1 | With very few exceptions, modern textbooks tend to give a negative meaning to the idea of a fundamental state of rest. However, this was the natural perspective for the first derivation of the relativistic effects by Lorentz, Fitzgerald, and Larmor. Over the years, the value of a Lorentzian formulation has been emphasized by many authors, notably by Bell [7]; see Brown’s book [8] for a complete list of references. For more recent work, see also De Abreu and Guerra [9] and Shanahan [10]. |

2 | |

3 | This was well illustrated in Ref. [14]: “Thus, Nonlocality is most naturally incorporated into a theory in which there is a special frame of reference. One possible candidate for this special frame of reference is the one in which the cosmic background radiation is isotropic. However, other than the fact that a realistic interpretation of quantum mechanics requires a preferred frame and the cosmic background radiation provides us with one, there is no readily apparent reason why the two should be linked”. |

4 | Preferred-frame effects are common to many models of dark energy (and/or of dark matter), such as the massive gravity scheme proposed by Rubakov [20], the effective graviton–Higgs mechanism of Ref. [21], or non-local modifications of the Einstein–Hilbert action [22,23,24]. In these cases, one also expects a dependence of the velocity of light on the state of motion of the observer. |

5 | However, a null result in an ideal vacuum can also be deduced [56] without assuming Lorentz transformations, but only from simple assumptions on the choice of the admissible clocks. |

6 | Joos’ optical system was enclosed in a hermetic housing and, as reported by Miller [27,67], it was traditionally believed that his measurements were performed in a partial vacuum. In his article, however, Joos was not clear on this particular aspect. Only when describing his device for fine electromagnetic movements of the mirrors does he refer to the condition of an evacuated apparatus [35]. Instead, Swenson [68,69] declared that Joos’ fringe shifts were finally recorded with optical paths that were placed in a helium bath. Therefore, we followed Swenson’s explicit statements and assumed the presence of gaseous helium at atmospheric pressure. |

7 | In [40], a numerical simulation of the Piccard–Stahel experiment [31] is reported for both the individual sets of 10 rotations of the interferometer and the experimental sessions (12 sets, each set consisting of 10 rotations). Our analysis confirms their idea that the optical path was much shorter than the instruments in the United States, but their measurements were more precise because spurious disturbances were less important. |

8 | One should further restrict light propagation to a small enough region that tidal effects of the external gravitational potential ${U}_{\mathrm{ext}}\left(x\right)$ can be ignored. |

9 | However, the time ${\tau}_{0}$ could also be considerably larger than 1 s, as, for instance, in the cryogenic experiment of [46]. There, the RAV at 1 s was about 10 times larger than the range in Equation (79), but, in the quiet phase between two refills of the refrigerator, ${\sigma}_{A}(\mathsf{\Delta}\nu /{\nu}_{0},\tau )$ was monotonically decreasing from ${\tau}^{-1/2}$ up to ${\tau}_{0}=250$ s, where it reached its minimum value of ${\sigma}_{A}(\mathsf{\Delta}\nu /{\nu}_{0},{\tau}_{0})\sim 5.3\xb7{10}^{-16}$. This is still consistent with the lower bound in Equation (79), so we would tentatively argue that Equation (79) should be replaced by the more general form $5\xb7{10}^{-16}\lesssim \phantom{\rule{3.33333pt}{0ex}}[{\sigma}_{A}(\mathsf{\Delta}\nu /{\nu}_{0}),{\tau}_{0}){]}_{t}\phantom{\rule{3.33333pt}{0ex}}\lesssim 12\xb7{10}^{-16}$ with the same range, but with a ${\tau}_{0}$ that now depends on the experiment. |

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**Figure 2.**The scheme of a modern ether-drift experiment. The light frequencies are first stabilized by coupling the lasers to Fabry–Perot optical resonators. The frequencies ${\nu}_{1}$ and ${\nu}_{2}$ of the signals from the resonators are then compared in the beat note detector, which provides the frequency shift $\mathsf{\Delta}\nu ={\nu}_{1}-{\nu}_{2}$.

**Figure 3.**The observable velocity measured in various experiments reported by Miller [27].

**Figure 4.**The fringe shifts reported by Joos [35]. The scale corresponds to 1/1000 of a wavelength.

**Figure 5.**Some second-harmonic fits to Joos’ data. The figure is taken from [40].

**Figure 6.**Joos’ second-harmonic amplitudes in units of ${10}^{-3}$. The vertical band between the two lines corresponds to the range $(1.4\pm 0.8)\xb7{10}^{-3}$. The figure is taken from [37].

**Figure 7.**Joos’ second-harmonic amplitudes (in units of ${10}^{-3}$ (black dots)) are compared with a single simulation (red diamonds) at the same sidereal times as those in Joos’ observations. Two fifth-order polynomial fits to the two sets of values are also shown. The figure is taken from [37].

**Figure 8.**Joos’ second-harmonic amplitudes (in units of ${10}^{-3}$ (black dots)) are now compared with a simulation where one averages ten measurements that were performed on 10 consecutive days at the same sidereal times as those of Joos’ observations (red diamonds). The changes in the averages observed by varying the parameters of the simulation were summarized into a central value and a symmetric error. The figure is taken from [37].

**Figure 9.**The experimental frequency shift reported in Figure 9a of [47] (courtesy of Optics Communications). The black dots give the instantaneous signal, the red dots give the average of the signal over 1640 sequences. For a laser frequency ${\nu}_{0}=2.82\xb7{10}^{14}$ Hz, $\mathsf{\Delta}\nu =\pm 1$ Hz corresponds to a fractional value $\mathsf{\Delta}\nu /{\nu}_{0}$ of about $\pm 3.5\xb7{10}^{-15}$.

**Figure 10.**An intuitive visualization of two physically distinct situations. In case (

**b**), a heavy mass M is carried on board of a freely falling system. Unlike in the ideal case (

**a**), the mass M could introduce a vacuum refractivity so that ${c}_{\gamma}\ne c$.

**Figure 11.**For ${\u03f5}_{v}$ as in Equation (67) and $z=2$, we report in units of ${10}^{-15}$ two typical sets of 45 s for the two functions $2C\left(t\right)$ and $2S\left(t\right)$ of Equation (69). The two sets belong to the same random sequence and refer to two sidereal times that differ by 6 h. The boundaries of the stochastic velocity components in Equations (36) and (37) are controlled by ${(V,\alpha ,\gamma )}_{\mathrm{CMB}}$ through Equations (30) and (40). For a laser frequency of $2.8\xb7{10}^{14}$ Hz [51], the range $\pm 3.5\xb7{10}^{-15}$ corresponds to a typical frequency shift $\mathsf{\Delta}\nu $ in the range of $\pm 1$ Hz, as in our Figure 9.

**Figure 12.**We report two typical sets of 2000 s for our basic white-noise (WN) signal and its colored version, which was obtained with a Fourier transform of the spectral amplitude from [25].

**Figure 13.**We report the Allan variance for the fractional frequency shift obtained from simulations of sequences of 2000 s for our basic white-noise (WN) signal and for its colored version, which was obtained with a Fourier transform of the spectral amplitude from [25]. The direct experimental results of [25] for the non-rotating setup are also shown.

**Table 1.**The second-harmonic amplitudes for the six experimental sessions of the Michelson–Morley experiment. The table is taken from [37].

SESSION | ${\mathit{A}}_{2}^{\mathbf{EXP}}$ |
---|---|

8 July (noon) | $0.010\pm 0.005$ |

9 July (noon) | $0.015\pm 0.005$ |

11 July (noon) | $0.025\pm 0.005$ |

8 July (evening) | $0.014\pm 0.005$ |

9 July (evening) | $0.011\pm 0.005$ |

12 July (evening) | $0.024\pm 0.005$ |

**Table 2.**The average second-harmonic amplitudes of classical ether-drift experiments. These were extracted from the original papers by averaging the amplitudes of the individual observations and assuming the direction of the local drift to be completely random (i.e., no vector averaging of different sessions). These experimental values were then compared with the full statistical average in Equation (46) for a projection of the velocity of 250 km/s $\le \tilde{v}\left(t\right)\le $ 370 km/s and refractivities of $\u03f5=2.8\xb7{10}^{-4}$ for air and $\u03f5=3.3\xb7{10}^{-5}$ for gaseous helium. The experimental value for the Morley–Miller experiment was taken from the observed velocities reported in Miller’s Figure 4 and, here, our Figure 3. The experimental value for the Michelson–Pease–Pearson experiment refers to the only known session for which the fringe shifts were reported explicitly [34] and where the optical path was still fifty-five feet. The symbol $\pm \dots .$ means that the experimental uncertainty cannot be determined from the available information.

Experiment | Gas | ${\mathit{A}}_{2}^{\mathbf{EXP}}$ | $\frac{2\mathit{D}}{\mathit{\lambda}}$ | ${\langle {\mathit{A}}_{2}\left(\mathit{t}\right)\rangle}_{\mathbf{stat}}$ |
---|---|---|---|---|

Michelson (1881) | air | $(7.8\pm \dots .)\xb7{10}^{-3}$ | $4\xb7{10}^{6}$ | $(0.7\pm 0.2)\xb7{10}^{-3}$ |

Michelson–Morley (1887) | air | $(1.6\pm 0.6)\xb7{10}^{-2}$ | $4\xb7{10}^{7}$ | $(0.7\pm 0.2)\xb7{10}^{-2}$ |

Morley–Miller (1902–1905) | air | $(4.0\pm 2.0)\xb7{10}^{-2}$ | $1.12\xb7{10}^{8}$ | $(2.0\pm 0.7)\xb7{10}^{-2}$ |

Miller (1921–1926) | air | $(4.4\pm 2.2)\xb7{10}^{-2}$ | $1.12\xb7{10}^{8}$ | $(2.0\pm 0.7)\xb7{10}^{-2}$ |

Tomaschek (1924) | air | $(1.0\pm 0.6)\xb7{10}^{-2}$ | $3\xb7{10}^{7}$ | $(0.5\pm 0.2)\xb7{10}^{-2}$ |

Kennedy (1926) | helium | $<0.002$ | $7\xb7{10}^{6}$ | $(1.4\pm 0.5)\xb7{10}^{-4}$ |

Illingworth (1927) | helium | $(2.2\pm 1.7)\xb7{10}^{-4}$ | $7\xb7{10}^{6}$ | $(1.4\pm 0.5)\xb7{10}^{-4}$ |

Piccard–Stahel (1928) | air | $(2.8\pm 1.5)\xb7{10}^{-3}$ | $1.28\xb7{10}^{7}$ | $(2.2\pm 0.8)\xb7{10}^{-3}$ |

Mich.–Pease–Pearson (1929) | air | $(0.6\pm \dots .)\xb7{10}^{-2}$ | $5.8\xb7{10}^{7}$ | $(1.0\pm 0.4)\xb7{10}^{-2}$ |

Joos (1930) | helium | $(1.4\pm 0.8)\xb7{10}^{-3}$ | $7.5\xb7{10}^{7}$ | $(1.5\pm 0.6)\xb7{10}^{-3}$ |

**Table 3.**The average second-harmonic amplitude observed in various classical ether-drift experiments and the resulting temperature differences (in mK) from Equation (53).

Experiment | Gas | ${\mathit{A}}_{2}^{\mathbf{EXP}}$ | $\frac{2\mathit{D}}{\mathit{\lambda}}$ | $|\mathbf{\Delta}{\mathit{T}}^{\mathbf{gas}}\left(\mathit{\theta}\right)|$ |
---|---|---|---|---|

Michelson–Morley (1887) | air | $(1.6\pm 0.6)\xb7{10}^{-2}$ | $4\xb7{10}^{7}$ | $0.40\pm 0.15$ |

Miller (1925–1926) | air | $(4.4\pm 2.2)\xb7{10}^{-2}$ | $1.12\xb7{10}^{8}$ | $0.39\pm 0.20$ |

Illingworth (1927) | helium | $(2.2\pm 1.7)\xb7{10}^{-4}$ | $7\xb7{10}^{6}$ | $0.29\pm 0.22$ |

Tomaschek (1924) | air | $(1.0\pm 0.6)\xb7{10}^{-2}$ | $3\xb7{10}^{7}$ | $0.33\pm 0.20$ |

Piccard–Stahel (1928) | air | $(2.8\pm 1.5)\xb7{10}^{-3}$ | $1.28\xb7{10}^{7}$ | $0.22\pm 0.12$ |

Joos (1930) | helium | $(1.4\pm 0.8)\xb7{10}^{-3}$ | $7.5\xb7{10}^{7}$ | $0.17\pm 0.10$ |

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**MDPI and ACS Style**

Consoli, M.; Pluchino, A.
The CMB, Preferred Reference System, and Dragging of Light in the Earth Frame. *Universe* **2021**, *7*, 311.
https://doi.org/10.3390/universe7080311

**AMA Style**

Consoli M, Pluchino A.
The CMB, Preferred Reference System, and Dragging of Light in the Earth Frame. *Universe*. 2021; 7(8):311.
https://doi.org/10.3390/universe7080311

**Chicago/Turabian Style**

Consoli, Maurizio, and Alessandro Pluchino.
2021. "The CMB, Preferred Reference System, and Dragging of Light in the Earth Frame" *Universe* 7, no. 8: 311.
https://doi.org/10.3390/universe7080311