# Is Dark Matter a Misinterpretation of a Perspective Effect?

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Some Basic Structures for ${\mathbf{\mathbb{R}}}^{n}$

#### 2.1. Algebraic Structures

#### 2.1.1. Affine and Vectorial Structure

#### 2.1.2. Euclidean Structures

#### 2.1.3. Minkowski Structure

#### 2.2. Topological Structure of ${\mathbb{R}}^{n}$

#### 2.3. Differentiable Structure of ${\mathbb{R}}^{n}$

#### 2.3.1. Differential of a Function: Tangent Vectors at a Point

#### 2.3.2. Tangent Vector Fields on ${\mathbb{R}}^{n}$

#### 2.3.3. Covariant Derivation

#### 2.3.4. Flat Covariant Derivation

#### 2.3.5. Covariant Derivation along a Curve: Parallel Transport

#### 2.3.6. Acceleration

#### 2.4. Riemannian Structures on ${\mathbb{R}}^{n}$

#### 2.4.1. Levi-Civita Connection

#### 2.4.2. Some Riemannian Metrics

#### 2.5. Pseudo-Riemannian Structure on ${\mathbb{R}}^{4}$

## 3. Framework for the $\mathbf{\kappa}$-Model

#### 3.1. $\kappa $-Structure on a Riemannian Metric

#### 3.2. $\kappa $-Structure on the Euclidean Metric

#### 3.2.1. Speed Fields

#### 3.2.2. $\kappa $-Uniform Straight Lines

#### 3.2.3. Velocity Fields and Covariant Accelerations

#### 3.2.4. Laser Distance

#### 3.2.5. Circular Motions

#### 3.2.6. a-Uniform and $\kappa $-Uniform Circular Motions

#### 3.3. $\kappa $-Structure on a Minkowski Metric

#### 3.3.1. Quadrivelocities

#### 3.3.2. Changing Co-Ordinate Systems

#### 3.3.3. Observation of a Far-Away Geodesic Worldline

## 4. Applications

#### 4.1. What a Sitting Observer Sees: Size and Measurement

`a`(Figure 6). When arriving at

`b`, the particle emits a photon in the direction of observer

`A`. Then, observer

`A`sees the particle at position

`b’`. They measure the spectroscopic velocity ${\kappa}_{B}\parallel \dot{\sigma}\parallel $ (shown in orange). Likewise, observer

`B`sees the particle starting from

`a`at position

`a’`and measures the spectroscopic velocity ${\kappa}_{A}\parallel \dot{\sigma}\parallel $.

**-aberration**

`ab`taken in the base (the real trajectory of the particle, P) is represented by a multiplet of parallel straight lines

`ab’`,

`a’b`, …, with each of these lines being attached to a real observer. Let us note that the real trajectory of the particle, P, in the base space, does not seem to be parallel to the corresponding multiplet of its images in the bundle. In fact, there is no reason why this should be so. The base space is linked to the bundle by a projection, which can diversely tilt by any small portion of a trajectory, even though this tilt is fictitious. Moreover, the base space is not accessible to the real observers present in the bundle. Then, an unreflected comparison of a vector in the base to the “same” vector in the bundle makes no sense regarding its apparent direction. In the base, the vector $\dot{\sigma}$ (denoted by $\dot{{\sigma}_{b}}$) is constant for a free particle, whereas in the bundle, it is $\kappa \dot{\sigma}$ that is a constant vector. In the sheet of observer

`A`, we have the radial (spectroscopic) and (apparent) tangential components of the velocity, respectively, as seen by this observer (${d}_{A}$ = 10 AU)

**Overlapping images.**

#### 4.2. The Circular Motion of a Test Mass $m=1$ around a Motionless Mass M

#### 4.2.1. Without the $\kappa $-Effect

#### 4.2.2. With $\kappa $-Effect

#### 4.3. An Analysis of the Spiral Substructure

#### 4.4. Galaxy Clusters

#### 4.5. The Bullet Cluster

#### 4.6. Translation of an Extended Object and the $\kappa $-Effect

#### 4.7. The Relativistic Extension

#### 4.7.1. Without Gravity

#### 4.7.2. With Weak Gravity

**Without the $\mathit{\kappa}$-effect**

**With the $\mathit{\kappa}$-effect**

**Gravitational Tide**

**Two “paradoxes” that emerge from the motion of free particles**

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Circles with same radii and centers in ${\mathbb{R}}_{\mu}^{3},{\mathbb{R}}_{\nu}^{3}$, and ${\mathbb{R}}_{\lambda}^{3}$ with $\lambda >\mu >\nu $.

**Figure 3.**Deformation of the “future-cone” at e, with $\kappa \left(\overline{a}\right)\phantom{\rule{-0.166667em}{0ex}}<\phantom{\rule{-0.166667em}{0ex}}\kappa (\overline{b})\phantom{\rule{-0.166667em}{0ex}}<\phantom{\rule{-0.166667em}{0ex}}\kappa \left(\overline{c}\right)$ the “space component”, ${\sigma}^{s}$, is represented in dimension 1.

**Figure 4.**Discrepancy in size measurement: ${d}_{a}(a,a+U)={d}_{b}(b,b+V)$, and ${d}_{a}(a,a+U)\ne {d}_{a}(b,b+V)$.

**Figure 6.**Two observers (A and B) separated by a very large distance of ≫1 pc. Very close to each of them, a universal “atom” that is small in size, ∼ 10 AU, is represented. The dashed straight line

`ab’`(respectively,

`a’b`) is a geodesic of the sheet of observer

`A`(respectively,

`B`) equipped with ${\kappa}_{A}$ (respectively, ${\kappa}_{B}$). Let us note that the norm of vector $\dot{\sigma}$ is a constant in the base, but this norm varies by $\frac{1}{\kappa}$ in the respective sheets of the observers. On the other hand, the curve

`ab"`displayed in blue is a geodesic of the bundle equipped with the variable $\kappa $ (Let the action for a free particle be $S=\int \kappa \left(\sigma \right)\sqrt{{\left({\displaystyle \frac{d\sigma}{ds}}\right)}^{2}}ds$ By applying Hamilton’s principle, we find the geodesic equation (curve displayed in blue in Figure 6). $\frac{d}{ds}}\left(\kappa {\displaystyle \frac{d\sigma}{ds}}\right)-\nabla \kappa =0$ ).

**Figure 7.**(

**a**) Images obtained without the $\kappa $-effect; the annulus is represented in yellow; the central disk is represented in red. (

**b**) Images obtained using the $kappa$-effect; the emitter in the annulus sees observer a three times closer. The gap in the image is a consequence of $\kappa $ discontinuity at the border between the annulus and the central disk. (

**c**) Images obtained with $kappa$-effect; the emitter in the central disk sees a three times closer. The image of the central disk overlaps the image of the annulus; the overlapping region is colored orange.

**Figure 9.**Galaxy rotation velocity profiles. (

**a**) The Milky Way in the vicinity of the Sun; (

**b**) M33; (

**c**) NGC 1560; (

**d**) NGC6946. For more details, see [2].

**Figure 10.**A model example: A well-developed conservative spiral substructure seen in the bundle, as opposed to its counterpart existing in the base (a tightly coiled spiral).

**Figure 11.**A conservative grand design spiral produced by a numerical simulation in the $\kappa $-model framework (for more details, see [2]). The elapsed time is given in the unit of 100 Myr. The Newtonian equivalent would be a much tighter spiral with a larger number of turns.

**Figure 12.**A side-on galaxy, as seen by a terrestrial observer (

**a**) in contrast to its compact counterpart existing in the base (

**b**). In this illustrative example, the terrestrial observer measures the density $\rho (r,z)=exp(-r-(5+r\left)\right|z\left|\right)$.

**Figure 13.**The short dashed blue curve is the Newtonian dynamic mass; the dashed-dotted cyan curve is the MOND dynamic mass. The dynamic mass for the $\kappa $-model is displayed as the amber curve (dynamic mass with a constant temperature T = 8.38 keV) and green curve (assuming a non-isothermal temperature profile). The long, red dashed curve is the ICM (intracluster medium) gas mass derived from X-ray observations (for more details, see [3]).

**Figure 14.**A comparison between the gravitational lensing diagrams resulting from the application of both dark matter and $\kappa $-model paradigms in the case of the Bullet Cluster (for more details, see [3]).

**Figure 18.**Apparent variation of a disk of constant size, as seen from the point of view of a terrestrial observer. The real size is the disk shown in gray.

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**MDPI and ACS Style**

Pascoli, G.; Pernas, L.
Is Dark Matter a Misinterpretation of a Perspective Effect? *Symmetry* **2024**, *16*, 937.
https://doi.org/10.3390/sym16070937

**AMA Style**

Pascoli G, Pernas L.
Is Dark Matter a Misinterpretation of a Perspective Effect? *Symmetry*. 2024; 16(7):937.
https://doi.org/10.3390/sym16070937

**Chicago/Turabian Style**

Pascoli, Gianni, and Louis Pernas.
2024. "Is Dark Matter a Misinterpretation of a Perspective Effect?" *Symmetry* 16, no. 7: 937.
https://doi.org/10.3390/sym16070937