Submission Deadline-26th July 2024

July 2024 Issue : Publication Fee: 30$ USD Submit Now

Submission Open

Special Issue of Education: Publication Fee: 30$ USD Submit Now

Time Dark Matter Geometrical Model and Newton’s Law Recovery

Saoussan A. Kallel-Jallouli
406-414
Apr 17, 2024
Physics

Time Dark Matter Geometrical Model and Newton’s Law Recovery

Saoussan A. Kallel-Jallouli

Université de Tunis El Manar, LR03ES04, Faculté des Sciences de Tunis, 2092, Tunis, Tunisia,

DOI: https://doi.org/10.51584/IJRIAS.2024.90337

Received: 23 February 2024; Revised: 13 March 2024; Accepted: 18 March 2024; Published: 17 April 2024

ABSTRACT

Recently, a new geometrical model was proposed to solve the problem of “time”. Moreover, the proposed model offers a solution for the dark matter enigma. In this work, we explain how Newton’s law of rotation is an immediate consequence of the proposed “time” model. Furthermore, a more general formula is required for the acceleration of a particle to be more accurate. This explains our need for the Einstein theory of gravitation.

Keywords: Time, dark matter, velocity, acceleration

PACS number(s): 04.20. Gz, 95.35. +d, 06.30. Gv, 06.30. Gv, 96.50. Pw, 96.50. Pw, 45.50. Dd

INTRODUCTION

The majority of the energy stuff in the cosmos cannot be regular matter, which was one of the fascinating scientific discoveries of the 20th century. It is another type of matter that interacts with classical matter through its gravitational effect. Scientists have called this new stuff “Dark matter” (DM). The type and physical properties of DM are enigmatic (Young 2017). We cannot see DM, but we can know it is there by its gravitational impact on seen matter.

In previous work (Kallel-Jallouli 2018; 2021 a,b,c), Saoussan Kallel proposed the existence of invisible unusual stuff, named “Zaman”, responsible for the variations of “time” by its rotational motion (Kallel-Jallouli, 2023a,b). The proposed model led to the formation of rings, or multiple images of the same object, then, to the self-evident conclusion that “Zaman” exists, and it solves the DM enigma. In this work, using the new definition of “time”, we shall explain how to recover Newton’s law of rotation. Let us begin by recalling the new geometrical model.

THE “ZAMAN” GEOMETRICAL MODEL

Introduction

The concept of time is complex. For many decades, the difficulty of giving a complete account of “time” remained unresolved (DeWitt 1967; Earman 1987; Isham 1993; Kucha ̌r 1999; Merali 2013; Rovelli 1991; Smolin 2014; Wheeler 1968). Most scientists just see time as what clocks measure, so we may as well disregard it (Barbour 2001; Magueijo and Smolin 2019; Smolin 2001). The problem of “time” remained unsolved until the new revolutionary idea proposed by Pr. kallel in 2018 (Kallel-Jallouli 2018). She believed time is not simply a question of order of events (Rovelli 2018), or a real or complex variable. She succeeded, using the Dark Matter strong lensing effect, to demonstrate the existence of an unusual kind of unseen staff called “Zaman” responsible for the variations of “time” by its spin (Kallel-Jallouli 2021).

The idea proposed in the new theory is very interesting and helps scientists obtain a better understanding of our physical universe. We shall see how, using the new physical perception of “time”, things are more rigorous.

A NEW GEOMETRICAL MODEL

1. Definition of U-day (Kallel-Jallouli 2018, 2021, 202)

U will be a spinning sphere (generally called “universe” or “halo”), automatically filled with “Zaman”. Assume U has a solid body rotation and that, in comparison to its non-rotating (inertial) state UI, it takes T units of clock-time, for U, to complete one rotation in the positive direction about its axis. If the rotational speed of U is constant and repeatable, then T is the length of the U-day. One U-day is equal to one full rotation of U in the positive direction around its axis. The U-day’s length is determined using a chosen clock (habitually expressed in seconds). Evidently, the clock is an instrument used just to fix the length T of the physical U-day.

2. Definition of U-time inside U

Let us choose the spherical coordinate system (r,θ,φ), given by figure 1

Figure 1. A. Spherical Coordinates. B. Isotime-discs (Kallel-Jallouli 2021)

U-time will be the same for a solid body rotation over each semidisc limited by the axis of rotation and a meridian (figure 1).

Figure 2. U-time inside U

Let us select the isotime-disc enclosed in the plane (Oxz) as the semidisc of U-time 0 (t_u≡0). It is called the U-prime meridian (figure 2). Any certain point P in U_I with coordinates (r, θ, φ) concurrently indicates the space position (r, θ, φ) and the U-time variation in relation to the U-prime meridian, provided for the first day by (Kallel-Jallouli 2021b, c, d):

(2.1)

This is how internal U-time varies, in one U-day. For the nth day, U-time is provided by:

(2.2)

The value of U-time inside UI is t_u

3. U-time difference between two points inside the UI.

The following formula is used to determine the U-time difference between two sites inside of U for the same day based on the difference in longitude:

(2.3)

4. A particular example: the classical time

When 𝑇=24 hours, then

(2.5)

The classical variation of time is the variation of U-time:

(2.6)

Remarks.

1. In this classical case,

* 360 degrees corresponds to one U-day of length 24 hours

* 180 degrees corresponds to half a U-day of length 12 hours

* 15 degrees corresponds to 1 hour

Let us call this “universe”, the classical “universe”, and the corresponding U-time, the classical time.

2. if T>24 hours, then ∆t_u>∆tc, and U-time is stretched, with respect to the classical time. If T<24 hours, then ∆t_u<∆tc and U-time is shrunken with respect to the classical time.

Actual physical rules inside of U must be based on U-time, which is tied to the actual spin of U, rather than clock time, which has no physical relation with the U-spin.

NEWTON’S LAWS FOR ROTATION

1. U-celerity

Let ‘s consider a spinning spherical halo U_o with rigid body rotation, with constant angular velocity Ω_o (remember that angular velocity is related to the length of the U_o-day by: Ω₀=2π/T_o ), about a fixed axis (z-axis) passing through its centre O, relative to its nonrotating inertial frame of reference. According to Newton’s first law of rotation, U_o will continue to rotate at the same angular velocity unless it experiences an external torque. If a tracer particle P is placed inside U_o, at a distance r from the origin, then it follows circular orbits around the central axis with tangential velocity given by Newton’s law of rotation:

(3.1)

Now, we seek to retreeve this formula using our revised concept of U-time.:

Theorem 1

The U_o-velocity of P given by (using our new definition of U_o-time):

(3.2)

satisfies the relation:

(3.3)

Proof

P will rotate in a circle of fixed radius (rsinφ). This circle has the U_o-axis of rotation as the axis. Note that if the particle completes one revolution, θ will not become zero again, but 2π rad, corresponding to one U_o-day. Consider the particle to be at point P_i, which corresponds to angular position θ_i, at U_o-time t_i. Once a U_o-time ∆t_Uo passed, the particle changed to point P_f, which represents the angular position θ_f, at U_o-time t_f. Its angular displacement given by ∆θ=θ_f-θ_i is positive for counterclockwise rotation (increasing θ). The U_o-time it takes for the particle to pass from P_i to P_f must be based on the difference in “time” between the two points, so, using relation (2.4), we must have:

By definition (3.2) of the U₀-radial velocity of a particle P inside U_o,

Using relation (3.4), we can write, with O’ the projection of P on the z-axis:

Finally, we can deduce:

(3.5)

This means the equality between the known Newton’s tangential velocity V_0N given by (3.1), and the

U_o-velocity V_Uogiven by (3.2) (based on U_o-spin) of a test particle living inside U_o.

2. U-acceleration

For the test particle P, Newton’s acceleration using Uc-time is given by:

By applying relation (3.1), we obtain

(3.6)

The tangential component:

(3.7)

The radial component:

(3.8)

Let us denote: dt₀≡dt_(U_o ). If the spherical halo U_o rotates with constant angular velocity Ω₀, then, based on our new definition of U_o-time, a tracer particle P will have an U_o– acceleration defined by:

Since, for constant angular velocity, we have (dΩ₀)/(dt₀ )=0, then we conclude that

(3.9)

We find the same relation (3.8) of the radial acceleration given by Newton’s law.

If Ω_0 varies with time, then, U_o-time inside U_ov varies. We will have

Theorem 2

The U_o-acceleration of a test particle P inside U_o defined by

(3.10)

satisfies:

(3.11)

In the particular case T_Uo=T_c, we obtain the equality between the U_o-acceleration and Newtonian’s acceleration:

a₀ (r)=a_N (r)

Proof

Using our U_o-time definition, U_o-acceleration is given by:

Using (3.3), we get

(3.12)

This is the same formula (3.6) given by Newton, where the clock time t_c is replaced by the new physical U_o-time t_(U_o ). However, we do not have equality of the two quantities (3.6) and (3.12). More precisely, using (3.4), we obtain the immediate relation

(3.13)

Formula (3.12) can then be written as:

(3.14)

Or equivalently, the relation (3.11).

Here, this U_o-acceleration (3.14) is a generalization of the approximation formula (3.6) given by Newton. However, we do not have equality of the U_o-acceleration with the Newtonian acceleration, in the case Ω_c≠Ω_0.

CONCLUSION

Finally, the tangential velocity formula (3.3) applies inside any “halo”, without the need to care about U-time inside U (or U spin). Using any time definition leads to the same formula (3.3). So, the choice of “international atomic Time” TAI (or classical time, as measured by the atomic clocks) by the International Commission of Time, which itself was part of the International Astronomical Union (IAU), in 1967, could be explained. Contrariwise, if we need to apply the U_o-acceleration formula (3.12), it is very important to know the U_o-spin, since the first term of our formula (3.11) (3.14) depends clearly on the length T of the U_o-day (or U_o-spin).

The tangential U_0-acceleration

(4.1)

depends on the length T_(U_o) of the day. Its modulus is greater for a shorter length of the day. Since we live inside a “halo” with an increasing length of the day (Gross 2007), Newtonian’s classical acceleration formula, based on classical clock time, is no longer accurate. We have an increasing term “T_Uo ” that will have very interesting impacts on physical, geological, and astronomical sciences.

The tangential force created by this tangential acceleration, on a particle of mass m, will be given by:

(4.2)

For an increasing rate of rotation Ω₀, this tangential force will push ahead any particle moving inside a spinning “halo” in the positive direction (Ω₀ and the z-axis have the same orientation). The particle will then spiral outward. In the case Ω₀ is decreasing, corresponding to an increasing length of the U_o-day, this force will try to push back any test particle. The particle will spiral inward. This will have very interesting applications, especially in physics and astronomy.

Finally, if we hope to obtain accurate laws of nature that can apply at all levels and do not change either when the length of the day changes, we must use in our laws the physical U-time (U-days) related to the spin of U, and not the classical clock time, with no physical relation. Variations in U-time can be measured through longitudinal variations (3.4). A chosen clock serves only to determine the length T of the U-day. If T≈24 hours, then, in the “halo” U, our Newtonian classical laws apply. If T≉24 hours, we just have to use the generalized relation (3.11) for acceleration. We have just to understand that the physical “time” (U-time) inside a “halo” has a feeble chance to coincide with classical time. Therefore, Newtonian’s acceleration has a feeble chance to be accurate. Our need for the Einstein theory of gravity becomes then understandable. Farther the particle is from the axis of rotation of U_o, the greater the error. The heavier the particle is, the greater the error.

The relation (4.2) can also be written as:

(4.3)

For a light particle, not far from the axis of rotation, (4.3) can be neglected provided that d ln ⁡Ω₀/dt_c is small enough. Newtonian’s theory remains then a good approximation. Finally, we want to mention that the U solid body rotation is a very special case. The more general differential rotation case will be studied elsewhere [Kallel-Jallouli 2024a, b].

ACKNOWLEDGEMENT

The paper is not currently submitted to other journals and it will not be submitted to other journals during the reviewing process.

DATA AVAILABILITY

No new data were generated or analyzed in support of this research.

COMPETING INTERESTS

There is no conflict of interest. The author has no relevant financial or non-financial interests to disclose.

FUNDING

The author declares that no funds, grants, or other support were received during the preparation of the manuscript.

REFERENCES

Barbour, J. B., The End of Time: The Next Revolution in Physics. Oxford University Press (2001)
DeWitt, B. S. Quantum theory of gravity. I. The canonical theory. Phys. Rev., 160 (5), 1113.‏ (1967)
Earman, J., World Enough and Spacetime: Absolute vs. Relational Theories of Space and Time. Cambridge, MA: MIT Press; Norton, J. D. (1987)
Gross, R. S., Earth Roation Variations – Long Period, in Physical Geodesy, edited by T. A. Herring, Treatise on Geophysics, Vol. 11, Elsevier, Amsterdam, (2007).
Isham, C. J., In: Integrable Systems, Quantum Groups and Quantum Field Theories, edited by L. A. Ibort and M. A. Rodríguez (Kluwer, Dordrecht), grqc/9210011 (1993)
Kallel-Jallouli, S., Discover the big bang and the secrets of our universe. A new theory. CreateSpace Independent Publishing Platform, (2018)
Kallel-Jallouli, S., The mystery of time I. A new model to solve the enigma of Dark Matter, Internationl Kindle, Independent Publishing Platform (2021a)
Kallel-Jallouli, S., A New Solution for Dark Matter and a Better Understanding of Quantum Mechanics, J Pur Appl Math, 5(4): 32-37 (2021b)
Kallel-Jallouli, S., The Mystery of Time: A New Solution for Dark Matter and a Better Understanding of Quantum Mechanics, J. Phys. Math.12, 3 (2021c)
Kallel-Jallouli, S. 2023a, A new geometrical time model to explain the discrepancy between theoretical and measured velocities, ResearchGate DOI: 10.13140/RG.2.2.18508.39048
Kallel-Jallouli, S., 2023b, No more twin paradox, ResearchGate, DOI: 10.13140/RG.2.2.12760.88329
Kallel-Jallouli S, 2024a, Keplerian motion inside an isolated Dark Matter halo, ResearchGate: DOI: 10.13140/RG.2.2.30102.32320
Kallel-Jallouli S, 2024b, Phase shift of the Keplerian orbits inside an isolated Dark Matter halo, ResearchGate: DOI: 10.13140/RG.2.2.15244.49287
Kucha ̌r, K. V., In The Arguments of Time. ed. J. Butterfield, Oxford University Press, Oxford (1999)
Magueijo, J., and Smolin, L., A Universe that does not know the time, Universe 5 (3) 84 (2019)
Merali, Z., The origins of space and time: many researchers believe that physics will not be complete until it can explain not just the behaviour of space and time, but where these entities come from. Nature, 500 (7464), 516-520 (2013)
Rovelli, C., What is observable in classical and quantum gravity?. Class. and Quantum Gravity, 8 (2), 297 (1991)
Rovelli, C., The Order of Time. Allen Lane publisher, (2018)
Smolin, L., Time, laws, and the future of cosmology. Phys. Today, 67 (3), 38.‏ (2014)
Smolin, L., The present moment in quantum cosmology: Challenges to the arguments for the elimination of time. (arXiv:gr-qc/0104097v1) (2001)
Wheeler, J. A., Lectures in Mathematics and Physics. in: Battelle Rencontres, edited by C. DeWitt and J. A. Wheeler (Benjamin, New York) (1968)
Young, B. L., A survey of dark matter and related topics in cosmology. Front. of Phys., 12 (2), 121201 (2017).

Article Statistics

Track views and downloads to measure the impact and reach of your article.

PDF Downloads

398 views

Metrics

PlumX

Altmetrics