Accedeceleration Phenomenon: A Study of Newton's First Law Secrets and Extensions to Newton's-Relativity Gravitational Theories with New Insight into Quantum Mechanics
MohammedA.H.Ali1,2✉Emailhashem@um.edu.my
1Department of Mechanical Engineering, Faculty of EngineeringUniversity of Malaya50603Kuala LumpurMalaysia
2Center of Theoretical and Computational Physics (CTCP), Faculty of ScienceUniversity of Malaya50603Kuala LumpurMalaysia
Mohammed A. H. Ali1,2
1Department of Mechanical Engineering, Faculty of Engineering, University of Malaya, 50603 Kuala Lumpur, Malaysia
2Center of Theoretical and Computational Physics (CTCP), Faculty of Science, University of Malaya, 50603 Kuala Lumpur, Malaysia
Email: hashem@um.edu.my
A
Abstract
A new study on the continuation of inertia moving at a constant speed, as stated in the Newton's first law, called an accedeceleration phenomenon, is introduced in this paper. The accedeceleration can be defined as the movement of the masses by a tiny constant acceleration followed by the same deceleration over a short time period. It accurately interprets how the external forces and energies are generated in a constant speed movement, ensuring the continuity of the movement over time. The paper also introduces a real special relativity that strengthen our knowledge on the current apparent special relativity as a consequence of the accedeceleration phenomenon. The law of equivalence between the constant speed and accedeceleration, together with a mathematical model of accedeceleration has been thoroughly derived. An analysis of Newton's first law utilizing the accedeceleration phenomena aids in the introduction of a missing component in the Newton’s Universal law, where the overall acceleration of the objects on the planets is a vector sum of Newtonian acceleration and accedeceleration. This overall acceleration has been validated through the calculation of the well-known precession of Mercury perihelion, which couldn’t be calculated earlier using Newton’s Universal law, with a value of 43.12”/century. Such a value is the best world-wide close value to the well-measured value of 43.1”/century and even much more accurate than 42.98”/century in general relativity theory. Another validation for real-special relativity has been accomplished through the calculation of GPS time-dilation with a value of 38.7µs, which is almost equal to the current GPS time-dilation of 38.6µs. A depth discussion and comparison with the special and general relativity theories are performed to show the effectiveness of the new theory. This work has been also discussed with the wave-particle duality in quantum mechanics, showing a new insight and potential agreement between classical and quantum mechanics.
Keywords:
Newton’s First Law
constant speed
Accedeceleration phenomenon
Newton’s Universal Law
Special and General Relativity
Quantum mechanics
1. Introduction
Newton's first law is one of the most common physical axioms and well-known knowledge, including both Newton and Einstein theories [1–3]. This law states that, if the object is at rest or moving at constant speed, it will remain at rest or keep moving at constant speed unless it is acted upon by an external force. Hook has recently revised the Newton's first law, based on the original Latin version written by Newton, which states [4, 5]: Everybody perseveres in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by the forces impressed. This study emphasized that Newton has no intention to talk about an imaginary-force-free objects in the Newton's first law rather than to highlight the necessity of the force to disturb the object movement, as there are no objects with free external forces in nature. DiSalle has reported that the misinterpretations of 1st Newton’s law caused the researchers to disagree with special relativity of Einstein’s and Newton's Universal law theories [4, 6]. Smith has highlighted that Newton in his theory is referring to all bodies, not just to the theoretical imaginary force-free ones that are not existed in nature, as in the current understanding of the Newton's first law [4].
There are many researchers who reported such confusion about the Newton's first law in the last century, such as Ernst Mach, Morris Cohen, Philip Frank, Ernst Cassirer, and William Hay [7]. In general, the above-mentioned discussions indicate that the current understanding of the Newton's first law requires further revisions.
It is simply thought, in the current understanding of Newton's first law, that the objects moving at constant speed will keep their movement until infinity as there are no external forces. This analysis is not really true, as such consideration should be valid for all types of movements, not only for the constant velocity in straight lines, as discussed in [4–7]. The other reasons, to rebut such analysis, are:
Following the same reasoning, the accelerated objects will continue to accelerate indefinitely, as there is no external force acting to decelerate them, e.g., throwing the ball with a certain force will cause it to accelerate, and once it starts the acceleration, it will never stop accelerating until infinity.
In accordance with special relativity, if an object approaches the speed of light, its mass will grow, resulting in a decrease in its velocity. If such acceleration-deceleration movement of masses is detectable with high speed, it also happens during the movement at slow speed, but it is not sensible.
De Broglie has found that all matters experience wave-like properties, which are always propagated in the moving direction at a constant velocity. It is yet unclear how such matter waves are generated and why the objects have to oscillate during their movement, i.e., electromagnetic waves are generated by electromagnetic field, but the causes of oscillation in other micro and macro matters are still unknown. The depth analysis of the constant velocity can help to know the relation between oscillation and constant speed in micro and macro matters.
It is thought that the Newton’s first law is applicable when objects are devoid of weight, allowing for unimpeded motion. However, this implies the absence of a rest position, as all objects would perpetually move at an arbitrary velocity, contradicting both rest and constant speed scenarios outlined in Newton’s first law.
The other sorts of object's movements will persist indefinitely in the absence of external forces, such as spring oscillation and fluid movement. There is no consideration for the internal resistance inside such objects.
A partial amount of internal kinetic energy will be consumed for the internal resistance of the moving objects; especially, if the object consists of different materials, where there will be a loss of kinetic energy and velocity due to the frictions between the components of the object.
Even-though there are no external forces disturbing the objects movement, such as air resistance and friction, there will be internal force disturbances due to compression/expansion of particles upon movement.
There exist few objects in nature that move at a constant speed, such as light, the earth's rotation on its axis and around the sun, and objects affected by magnetic and electrical fields, such as electrons and motors. Light, as a great example of movement by constant speed, is actually moving in a wave, which is a kind of accelerating and decelerating with high-frequency times due to the magnetic and electric fields behavior. The rotor in the electrical motors tends to accelerate due to the magnetic field, but an inverse electromotive force (emf) is generated, causing a resistance to decelerate. The same is true for an electron accelerating inside a magnetic field, as Einstein stated in special relativity: when an electron accelerates, its mass increases, forcing it to decelerate [2]. One can conclude that if such acceleration-deceleration movement of masses is discernible at high speeds, it also occurs at low speeds, but it is not sensible. The rise of inertial energy E = mc² leads to an increase in electron mass as light speed remains constant. However, it is unclear how the electron's mass will be physically increased.
It is clear how the light, electron, and motor, in the preceding examples, accelerate and decelerate to continue moving at constant speed. However, when the earth rotates with constant speed around the sun, due to the effect of the gravity of masses in the universe as shown in Fig. 1, it is still unclear how the inertial and gravitational forces are combined to produce the constant velocity. Einstein stated in the general relativity theory that according to the Newton's first law, the geodesic lines caused by curved space-time are actually the straight lines in Riemann geometry, allowing the planets to move with constant speed around the sun [3]. Newton mentioned that the earth is still moving roughly in a straight line with a constant velocity. With some analysis, we can discover that both Einstein and Newton are applying a kind of force to move the earth in its orbit, either directly via Newton's Universal law or indirectly by the space-time curvature as in Einstein, respectively, as shown in Fig. 1. Both Newton's and general relativity theories don’t explicitly explain how the external source of the force or energy is continually generated during the constant speed movement. As the sun's gravitational field forces cause large masses, such as earth, to rotate around it at constant speed, one can wonder if the inertial forces caused by moving masses at constant speed can generate a gravitational field.
Fig. 1
Forces applied on the earth when rotating around the sun: (a) Einstein gravitational field (b) Newton Universal law gravity
The continuous movement of the object with a constant speed in Newton’s first law is still a tricky problem, since there exists an internal source of energy while there is no obvious external source of the energy for such movement, causing widespread confusion in understanding the continuity of such constant movement [4–7]. Thus, Newton has considered that an object moving at constant speed is under null external forces and distinguished the case of constant speed from the acceleration case as in the second law where an external force is applied [1]. Such thought is influencing our understanding of gravity and the planet's constant movement, raising the question of why the earth and all other planets are still moving at a constant speed despite being subjected to external gravitational forces/space-time curving caused by the sun and other planets. However, objects on planets typically accelerate in the presence of gravity, given that gravity is always perpendicular to inertial forces.
The above-mentioned discussions stimulate in-depth investigation into how the continuous movement in Newton's first law is generated and whether the self-generated energy is sufficient to keep objects moving. It appears that there are undiscovered secrets in Newton's first law, where there is a potential for a new theory that could interpret how the internal energy is kept generated. This paper concerns on studying the continuity of the inertia moving at a constant speed as stated in the Newton's first law by introducing a new phenomenon called an accedeceleration. The analysis of the Newton's first law using accedeceleration phenomenon reveals a missing component in the Newtonian Universal law, which will be addressed in sections 2.2 and 2.3. Such analysis also leads to introduce a real special relativity as in section 3.2.2. A depth discussion with special and general relativities, Newton laws and quantum mechanics is provided in section 3.2.1–3.2.4 to show the effectiveness of the proposed theory. The proposed theory is validated by calculating the precession of Mercury's perihelion in section 3.3.1, which couldn’t formerly be computed using Newton's universal law. Another validation is accomplished by calculating the GPS time dilation in section 3.3.2 through the introduced real special relativity.
2. Methodology
The Accedeceleration phenomenon definition and its equivalence of law are given in this section. An experimental work is performed to show where and how this phenomenon is occurred.
2.1 Accedeceleration Phenomenon and Law of Equivalence
One can consider that the constant velocity is just a movement of an object by low constant acceleration followed by the same constant deceleration in a finite small duration, thus the body feels no change. Such a kind of movement is called “accedeceleration", which can be defined as the movement of the masses by a tiny constant acceleration followed by the same deceleration over a short time period. With this assumption, external forces and energies will exist to ensure continuous movement at a constant speed, even if their aggregate is zero.
Let’s compare the constant speed and accedeceleraton systems as shown in Fig. 2:
System 1: mass is moving with constant speed V m/s within t (s) time.
System 2: mass is decelerating and accelerating by a tiny constant deceleration (-a m/s2) and acceleration (a m/s2).
Fig. 2
constant speed (System 1) vs accedeceleration (System 2)
The velocity, distance and work for system 1 are: v1 = v0, D1 = vt, W1 = 0,
The velocity, distance and work and internal energy for system 2 are given in Eqs. 1–5:
As the acceleration and deceleration will have the same rate: v0i = vfk, vfi=v0k, and
, Eq. 2 can be written as in Eq. 3
where n is the frequency of switching between acceleration and deceleration.
With applying numerical values (m = 5kg, V = V0 = Vf=4m/s, a = 0.01m/s2, t = 6s, t1 = t2 = 3s) in Eqs. 1–5, one can get:
System 1:
In the above-mentioned numerical example, each deceleration and acceleration took 3s, which is quite long; however, it will be insensible if a small acceleration and deceleration are applied oppositely within a shorter time. It behaves similarly to the alternating current when turned on/off within a frequency of 50Hz, making the cut-off impossible to perceive with the naked eye.
As a result, we can come out to the following equivalent systems:
The Accedeceleration process can be imagined as shown in Fig. 3.
Fig. 3
Accedeceleration phenomenon generation with movements in straight (left) and curve (right) paths.
Let’s set the rules of equivalence law between the constant speed and Accedecleration systems as follows:
- The Accedeceleration system must be decelerated and sequentially accelerated by the same tiny rate.
- The start and final speeds in the Accedeceleration system must be equal to the constant speed system
- The total external work and energy remain zero for both systems
- The internal energy is equal in both systems
- The total travelled distance is equal in both systems
The question now is how such an Accedeceleration phenomenon is generated while the masses are moving by constant speed. Section 2.2 will reply to such a question, with more details.
2.2 Experimental work and Modeling of Accedeceleration
This section will go over how “Accedeceleration” can be generated without applying the external forces to an object moving at a constant speed. An experimental work will be performed 1st to show how and where the Accedeceleration is occurred. It will be followed by deriving the mathematical model of the accedeceleration.
2.2.1 Experimental work on Accedeceleration
A
By referring to Fig. 1, one can conclude that the constant speed persists in the presence of a gravitational potential field. That happened because the gravitational potential field can generate a balancing effect on the earth, resulting in the continuation of rotation by constant speed. As gravity is the sole force that allows huge masses to move with constant speed in nature, and according to the Einstein law of equivalence, one can conclude that the inertial force caused by moving masses with constant speed can be converted back into a local gravitational field. The local gravitational field can generate an accedeceleration on the moving object, allowing it to move at a constant speed, with the following harmony:Constant speed
Induced local potential field
Accedeceleration
Constant speed
To find the relationship between the constant speed and accedeceleration, let’s do the following experiment:
- When the car is accelerating, the object will bend the rubber to the back side as in Fig. 4 (b)
- When the car is moving by a constant speed, the object will bend down the rubber in the middle as in Fig. 4 (c) and (d)
- When car is decelerating, the object will bend the rubber to the front side as in Fig. 4 (e)
Note
The measurement is done by moving observer.
(d) Accedeceleration (e) deceleration
Fig. 4
Car moving with different speed
Consider a person pushing the car with a force Fex as in Fig. 4 (a). The car's mass will accelerate upon the force exerted on it and its inner objects at a constant acceleration. The object will react by applying force Fin in the opposite direction of Fex, as illustrated in Fig. 4(b). When reaching time t’, the person releases his hand and leaves the car moving at constant velocity (no exit of external resistances). According to Newton first law, the mass of the object doesn’t require any external force to generate the constant speed, hence Fex=Fin=0. There is a crucial question here: why does the object bend down the rubber as illustrated in Fig. 4 (b) and (c), if Fin=0.
Fig. 5
Local Gravitational Field Generation in the constant speed movement: (a) generation of potential field in x direction (car movement) (b) the result of accedeceleration effect
Special relativity couldn’t interpret what was happening in cases (c) and (d) in Fig. 4 due to the following reasons:
- The special relativity states that there will be an increase of mass while they are speeding up by velocity close to the velocity of light using Lorentz law, which is supposed to decelerate the car and all objects on the car and move them to the front as in case (e) in Fig. 4, rather than moving them down as it really happened.
- It appears that there is a rise in the mass (weight) of objects on the car, which will move them down, while the car’s mass (weight) is unaffected, which contradicts special relativity which asserts that both of the moving car and its carrying objects are subjected to mass increase.
2.2.2 Mathematical Modelling of Accedeceleration
The relation between the constant velocity V and local gravitational potential field
,as in Fig. 5, can be written as in Eq. 6:
In Cartesian coordinate system, Eq. 6 can be written as in Eq. 7:
It is preferred to calculate the potential field in the spherical coordinate system, as shown in Eq. 8:
As the mass moves at constant speed, it is expected to experience a centripetal acceleration in addition to the radial acceleration/deceleration caused by accedeceleration phenomenon. Thus, V varies in all directions, including r, θ, and φ.
On the other hand, the gravitational acceleration depends on the change of the gravitational potential field in the r direction solely, so-called radial acceleration, as determined by Newton, Gauss, and Poisson laws [1, 8, 9]. The relationship between δ and Φ(r) can be expressed in Eq. 9:
By substituting the value of Φ by
from Eq. 9 in Eq. 8, one can get Eq. 10:
Eq. 10 can be simplified into Eq. 11
As the gravitational acceleration is sequentially followed by gravitational deceleration on the opposite side as in Fig. 5, one can further simplify Eq. 11 into Eq. 12:
Where
as stated in Equivalence Law Eq. 2
Only the potential field in the direction of
and
will remain as shown in Fig. 5(b), which is acting perpendicularly to r and rsin(θ).
Then Eq. 12 can be written as in Eq. 13
As
and
, one can write Eq. 13 as in Eq. 14:
Since the movement of the car is in the x direction, with a certain angle of
, Eq. 14 can be further simplified into Eq. 15
With a constant angular velocity, Eq. 15 can be written as in Eq. 16:
in Eq. 16 is the remained centripetal acceleration after applying the accedeceleration to the moving system as in Fig. 5(b) which causes to bend the rubber desk down in Fig. 4 (c) and (d).
From Eqs. 15 and 16, one can see that:
r can’t be zero since no acceleration is generated in this situation. V can be altered from zero to the velocity of light, but it can’t reach infinity. If the car's maximum speed is considered to be the speed of light, the gravitational acceleration
will reach infinity when k = 0, as shown in Eq. 17.
In Eqs. 8–16, r refers to the effective radial distance for the moving zone, where the local gravitational field is still active. When r is increased, the effective radial distance should decrease as the generated acceleration decreases, beginning with the zone beside the moving object's surface and progressing to the non-moving zones. To reduce the influence of gravity from the moving object's surface (r0) till the radius of the non-moving zone, a non-linear function α(r) within the range of α(r)=[1 ε] (ε is a small value near zero) is required, as in Eq. 18:
The exact value of the range's termination ε must be experimentally determined, as the exact local potential field disappearance is unknown, even for the Newtonian gravitational field and Einstein's time-space curving. The vanishing point on a small scale can be empirically identified; however, it is difficult to determine it in a large-scale environment, such as the sun and planets. Two non-linear functions can meet the above-mentioned criteria of
in Eq. 18 as illustrated in Eq. 19 and Eq. 20, and as shown in Fig. 6.
1.
changes in a cosine wave at the effective zone as in Eq. 19:
2.
changes exponentially in the effective zone as in Eq. 20:
Fig. 6
Relation between the effective radial distance
and r
The gravitational accedeceleration can be then written for the transnational movement as in Eq. 21:
However it can be written for the rotational movement as in Eq. 22 :
3. Experiment Results, Discussion and Validation of Accedeceleration
The results of experiment for accedeceleration of the moving car, as shown in Fig. 4 (d), are presented and compared with constant speed system as shown in Fig. 4 (c) as in section 3.1. A depth discussion with special and general relativities theories, Newton’s laws and quantum mechanics is performed to show the effectiveness of the proposed theory as in sections 3.2.1–3.2.4. The proposed theory is validated by calculating the precession of Mercury's perihelion, which couldn’t formerly be computed using Newton's Universal law as in section 3.3.1. Another validation for the real special relativity has been provided through the calculation of GPS time-dilation as in section 3.3.2.
3.1 Results of Experiment
The calculations of the distance, external and internal works when utilizing the accedeceleration phenomenon to analyze what happened to the car up on moving at constant speed as shown in Fig. 4 (c) and (d), are presented in Eqs. 23–26, which are identical the constant speed system.
The external work generated from this accedeceleration on the moving car is zero as it passes through the center of gravity of the car s = 0, as in Eq. 23:
The internal work for the masses when moving by constant speed can be written in Eq. 24:
where n is the number of all potential field lines.
As expressed in Eq. 5,
and if
and V < < C, Eq. 24 can be approximated into Eq. 25:
To find the distance D, one can use Eq. 3 with approximation of V < < C to calculate the distance as in Eq. 26:
Let’s apply all rules that have been stated in the equivalence law in section 2.1 on the derived accedeceleration equations as follows:
- The accedeceleration system is decelerated and sequentially accelerated by the same tiny rate as in Eq. 12.
- The start and final speeds in accedeceleration system are equal to the constant speed system as in Eq. 12.
- The total external work and energy remain zero for accedeceleration system as in Eq. 23.
- The internal work is equal in both constant speed and accedeceleration systems as in Eq. 25.
- The total travelled distance in accedeceleration system is similar constant speed system as in Eq. 26
The total gravitational acceleration during the constant velocity can be written in Eq. 27:
is too small when the constant angular rotational speed is small, such as on the Earth 10− 5 rad/s, Sun 10− 7 rad/s, and other planets; thus it can be neglected. However, with a high constant speed for the moving objects, it is very effective and sensible. This indicates that one can create a sensible gravitational field on small scale of moving objects by increasing their rotational speeds and diameters.
Eqs. 21 and 27 will be validated in section 3.3.1.
3.2 Discussion with Special and General Relativity Theories, and Newton’s Laws
3.2.1 Discussion with Special Relativity
The accedeceleration phenomenon and special relativity are completely consistent, particularly as the mass increases as speed increases. According to special relativity, an object will decelerate as its mass increases when it approaches the speed of light. Similarly, in accedeceleration phenomenon, the constant velocity is simply a movement of object in an acceleration followed by deceleration in a short time period.
The principle of special relativity's work has been considered as a source of much confusion for the past few decades because it relies on calculating the time dilation and dimension contraction by imagination analysis between two inertia frames of reference (one moving at a constant speed and the other is stationary) for the purpose of monitoring light without an any physical difference between such frames. In this work, we presented the physical changes of the moving frame in comparison to the stationary frame.
Let’s divide the understanding of the special relativity theory into two types:
(a) Apparent special relativity: In his work, Einstein provided a clear explanation of apparent special relativity [2]. Nonetheless, there is some misunderstanding over the work's underlying idea. For instance, there are two analyses available when examining time dilation for two observers, one of whom is stationary and the other traveling at a constant speed:
Based on the moving observer, the time dilation is on the side of the stationary observer, as he perceived himself static and the observer at rest is moving far away from him; however, based on the stationary observer, the time dilation is on the moving observer's side, as he perceived the moving observer is moving far away from him. The same goes for the twin paradox, which was discussed a lot with Einstein during his life [10, 11].
The reason for this misconception is that apparent relativity is depending on the relative location and visual view, which are typically true for both sides. Unfortunately, it doesn't show any physical differences between the static and moving observers that would make them easier to tell apart.
(b) Real Special Relativity: The accedeceleration phenomenon supports the special relativity and presents the physical changes between moving and static objects, where the moving object is subjected to a generation of local centripetal acceleration that attracts the object to the center of gravity, due to accedecleration phenomenon. Everything near to the moving object will be perpendicularly attracted in a certain degree to the moving object by constant speed. This causes a time dilation for a moving object, as the time needed for traveling in the perpendicular direction of x (car movement) will be shorter than those objects on the stationary observer side.
Let’s model the time dilation and relativistic mass in real special relativity as follows:
(1) Time dilation: In the apparent special relativity, two Galilean inertial frames are considered, one for the moving object and the other one for the stationary observer. Such consideration is deemed valid as the earth's gravity will be the same for both of the frames. As the accedeceleration is equivalent to the constant speed, one can still use Galilean inertial frames without any issues.
♣ Time Dilation in Rotational and Orbital Movement:
Such a movement is considered as non-inertial or accelerating frames motion, where the current special relativity can be locally implemented with assuming that the curving is not sensible, however, with real special relativity, the moving and stationary objects are subjected to
and
gravitational accelerations, respectively. Thus, the real special relativity can be analyzed with accedeceleration as follows:
The time for both the stationary observer and the rotating objects can be written with accedeceleration as in Eqs. 28–30:
The time for stationary observer can be written as in Eq. 28:
The time for rotating object as in Eq. 29:
The relation between t and t’ can be written as in Eq. 30:
where t, t’ are the times for stationary observer and moving objects, respectively.
This time dilation calculation is suitable for orbital movement of the planets and electrons.
This Equation will be validated for GPS Time dilation in section 3.3.2.
♣ Time Dilation in Linear Movement
The linear movement with stationary and moving inertial frames has been well studied in the current special relativity. However, when analyzing the constant velocity movement with accedeceleration phenomenon, there will exist a local gravitational field, which doesn’t affect to the inertial frames concept as discussed earlier.
With applying the real special relativity in the experiment of projecting a light source on a moving vehicle with a constant velocity V0 and observing it by the moving and stationary frames.
The distance of light travelling in the moving frame with existence of accedeceleration is:
, where sign (-) will be given to light when it is moving far from the center of gravity and sign (+) will be given when it is moving towards center of gravity.
The vehicle is moving in the presence of accedeceleration by velocity equal to:
as explained in the section 2.
The stationary observer can observe the distance of light travelling as:
The time dilation is accordingly calculated using Eq. 31
31
The border
indicates that not all objects on the moving vehicle have the same time dilation, but based on their location and movement from/to the center of gravity, they can form a relativity of relativity.
The real special relativity as in Eq. 31 can be reduced to apparent special relativity when
.
For the dimension contraction, the border
in Eq. 26 indicates the expansion/contraction of dimension in the moving frame compared to the static frame.
(2) Relativistic Mass: Let’s discuss here deeply how the mass will be increased during the speed increasing of the vehicle, which was a subject of confusion in the apparent special relativity.
Einstein in the clause 10 at ref. [2] has considered a slow accelerated electron to be moving at a constant speed, allowing him to apply the special relativity theory to the inertia frame of reference at the rest and moving frames as illustrated in Eq. 32:
Where x, y, z belongs to the frame at rest and x’, y’, z’ belongs to moving frame.
is the Lorentz transformation
. Later Einstein unified the solutions with
for all equations in Eq. 32.
In Eq. 32, Einstein relies on the apparent special relativity, where the kinetic inertia energy of the electron
increases rapidly when approaching to the light speed, but due to the light speed being constant, the mass of the object will be increased upon speeding up by the light velocity. It is yet unclear how such the relativistic mass will be physically grown in the apparent relativity.
In real relativity, the generated centripetal acceleration according to Eq. 21 is perpendicular to the movement direction, which will become infinite when the car velocity is approaching the speed of light. This will result in sticking the side objects located in the effective acceleration zone on the moving object's center of gravity by a force greater than objects welding. Even yet, such an infinite centripetal force will be able to grip nearby atoms and other tiny objects if the examination is conducted in a vacuum medium.
According to Einstein's apparent relativity as in Eq. 32, the relationship between the static and moving mass is illustrated in Eq. 33.
33
According to real relativity, the relation between the static and moving mass is illustrated in Eq. 34.
Where m1, m2, …, and mn are the masses of objects that are located in the effective zone of centripetal acceleration of the moving object (mc) due to accedeceleration.
The real special relativity in Eq. 34 indicates that there is a possibility to reach and surpass the velocity of light if the effect of the accedecelerations' centripetal forces of the objects in the effective zone on the moving masses can be continuously removed, which is very hard to be eliminated by current technologies. i.e., this kind of attraction between the moving mass and the objects in the effective zone is comparable to powerful welding, and it must be immediately removed while the mass is continually moving at a constant speed (accedeceleration).
3.2.2 Discussion with general relativity and Kerr Metrics
The general relativity field equation consists of 4×4 tensors with 10 independent PDE equations in each tensor that correlate the space-time curving with the momentum-stress-energy. There are several solutions for the Einstein general relativity field equation under special conditions such as symmetry, rotating, non-rotating, weak field, and charge. The common solutions are SchwarzChild and Kerr metrics, where the masses in the former are considered as symmetry, spherical, and non-rotating; however, in the latter, the symmetry and spherical with rotation are considered, which leads to the dragging of frames and the gravitomagnetic field.
The Schwarzschild solution of the Einstein field equation can be approximated into Newton's Universal law; however, the Kerr solution can be approximated into the Newton-Accedeceleration as follows:
- Schwarzchild solution Vs Newtonian Law: The Lagrangian relativistic energy of the geodesic system in Schwartzchild can be written in Eq. 35:
35
As the potential field is independent from the time and changes only with r, Eq. 35 can be written as Eq. 36:
36
The Lagrange relativistic equation for Schwartzchild metrics is written as in Eq. 37:
where the derivation of components of Eq. 45 are:
Eq. 45 can be simplified into Eq. 38:
Eq. 46 can be written as in Eq. 39
With multiplying both sides by
in Eq. 39, one can get Eq. 40
When Φ<<c, Eq. 40 can be written as Eq. 41:
With considering
<<c2, Eq. 41 can be written as Eq. 42:
Eq. 42 is exactly Newton universal law.
- Kerr solution vs Newton-Accedeceleration: The solution of Kerr metrics is given in Eq. 43:
43
It can be written for the slowly rotating body as in Eq. 44 [12]:
44
Where J1, J2, J3,… Jn are related to higher order of J.
As a Shwartzchild Metrics can be approximated to Newton’s universal law as in Eq. 42, the acceleration in Kerr metrics can be approximated into Eq. 45:
Eq. 45 is almost identical to the Newton and Accedeceleration gravitational fields derived in Eq. 21. The current Kerr metrics analysis of angular speed shows a frame dragging near the rotating object, which is a twisting of space-time near the masses; however, no consideration is given to local gravity/space-time curvature caused by rotational speed. The dragging frames cause prograde precession of a rotational object's rotation axis in its orbit around revolving planets; yet, they slow down rotation in the retrograde direction [13]. According to acceleration and Einstein's general relativity, the high rotational speed will cause an increase in local gravity/curving near the rotated masses.
3.2.3 Discussion with Newton's Laws of Motions
It has been demonstrated earlier that the constant speed is simply an accedeceleration in space-time. Thus, accedeceleration will replace the constant speed in the Newton’s first law and both Newton's first and second laws can be merged as follows:
Masses at rest or movement states are due to external forces with three types of movement:
I. If the external acceleration and deceleration are equal and simultaneously occurred, the object will be in its rest position.
II. If the external acceleration and deceleration are equally and sequentially happened (accedeceleration), the object will move at a constant speed.
III. If the acceleration and deceleration are unequal, the object will move according to the difference between them.
Table 1 shows a comparison between Newton’s laws with and without Accedeceleration.
Without Accedeceleration | With Accedeceleration | |
|---|---|---|
Newton’s First Law | For objects at rest position or moving by constant speed | Both Newton’s first and second laws will be merged in one law |
Newton’s Second Law | For object that are accelerated | |
Newton’s Universal Law | g = 9.81 m/s2 |
3.2.4 New Insight and Discussion with Quantum Mechanics
De Broglie has found that all matters have wave-like properties, similar to the light-based electromagnetic waves [14], but it is still unclear how such matter waves are generated and why the objects have to oscillate during their movement, i.e., electromagnetic waves are generated by electromagnetic field, but the causes of oscillation in other matters are still unknown. Such ambiguous reasons are well addressed by Accedeceleration theory, which states that all objects experience acceleration followed by deceleration or vice versa to form a constant speed, through a local gravitational field. Thus, the lighter objects, such as an electron, accelerate and decelerate consecutively while moving at a constant velocity, resulting in a wave-like fluctuation. The object oscillates during accedeceleration movement using De Broglie equation
[14], which serves as the fundamental wavelength equation basis in the modern quantum mechanics, including Shrodinger [15] and Dirac [16] wave equations.
According to accedeceleration theory, the oscillation of objects will only stop once it merely accelerates or decelerates along its movement, which acts as an observer effect, that collapses the wave in the double slit experiment.
Let’s study the De Broglie Equation of the electron when it is accelerated as follows:
The wavelength of electron that moves by constant speed can be determined using De Broglie equation as in Eq. 46:
Where h = 6.626 x 10⁻³⁴ kgm2/s is max Planck constant, v is electron speed,
is the Lorentz factor in apparent special relativity or accedecleration factor in orbital movement
in real special relativity.
When the electron exhibits a merely acceleration or deceleration, the velocity of electron can be written as:
. It is aimed to find the relation a between the acceleration and wavelength
. Hence, Eq. 46 can be written as in Eq. 47.
Where a is acceleration/deceleration value,
is the time of the acting force to accelerate the wave source. which can be written in terms of periodic time as
where T is the periodic time of the electron wave, n is a factor which can take values bigger or smaller than 1, based on the acting time of acceleration.
The relation between the acceleration and the accelerator voltage needed for accelerating the electron demonstrates that the electron is always associated with high acceleration values as in Eq. 48, (e.g. if EV = 1v and d = 100cm and
, the acceleration is: a=
m/s2)
Where EV is the voltage of accelerator, d is the length of the accelerator, m is mass of electron m = 9.11×10− 31 kg, e is the charge of an electron e = 1.6×10− 19C.
Eq. 47 can be written as in Eq. 49:
As De Broglie depends on equations:
and
, (E is the electron energy, m is the electron mass and Vp is the phase velocity of electron), one can write
. The phase velocity can be expressed for relativistic motion as
Eq. 49 can be then written as in Eq. 50:
With cross-multiplication, Eq. 50 can be written as Eq. 51:
By solving Eq. 51, one can get the values of
in acceleration mode as:
and in deceleration mode as:
where v0, C and h are constants and only the variables a and n change in this solution.
In the small values of acceleration, the solutions of
in Eq. 51 are:
With considering n = 1 (it means that the acting time of the acceleration/deceleration will be equal to the periodic time of the wave, thus wave wavelength will be compressed into half of its value):
. Such a solution will be resulted in a high values of
in a level of 1014 m in the opposite direction of the wave spreading.
By considering n = 1 and changing the acceleration between 0.1 and 1014 m/s2, one can get two solutions for
as shown in Fig. 8:
(a) 1st solution of
Fig. 8
Relation between Electron wavelength
and acceleration (a m/s2) in the ranges 0.1 m/s2 and 1.8×1014 m/s2
The 1st solution with acceleration values ranging from 0.1 to 1.8×1015 m/s2 has three ranges of positive wavelength λ. In the 1st range (0.1–1.2×109 m/s2), the wavelength is almost equal to 0, while in the 2nd range (1.2×109-1.8×1011 m/s2), the wavelength varies between 5.2×10− 11 and 3.3×10− 11 m. In the 3rd range (1.8×1011-1.8×1015m/s2), the wave length is roughly 3.3×10− 11 m.
In the 2nd solution, the values of wavelength are negative with high values in the level of 1015 m within the 1st range of acceleration 0.1–1.2×109 m/s2, however, it becomes in the levels of 105m and 102 m with the 2nd range 1.2×109-1.6×1011 m/s2 and 3rd range 1.8×1011-1.8×1015 m/s2, respectively.
Similarly, when deceleration is acting with values located between − 1.8×1015 m/s2 and − 0.1 m/s2, Fig. 8 is inversely obtained.
In the 1st range, where there is a small acceleration, both of wavelength solutions show that the wavelength is either too small or too large, causing the wave to collapse and the electron to behave like a particle. According to Dirac wave solutions for anti-matter [16], the previous two wavelength solutions of
in the 2nd and 3rd ranges of acceleration can be interpreted as the existence of a wave (with a positive solution of
) and an anti-wave (with a negative solution of
) that can cause the wave to collapse, creating a unique probability of electron location and moving the electron as a particle when it merely accelerates or decelerates, as exactly occurs with the observer effect in the double slit experiment.
3.3 Validation of Accedeceleration
It is necessary to validate the accedeceleration phenomenon by calculating certain well- measured values in the cosmos and see whether it will give accurate results or not. We chose two common measured values for comparison, namely Mercury Perihelion Procession and GPS Time dilation, as follows:
3.3.1 Validation 1: Calculation of Mercury Perihelion Procession using Accedeceleration
The Mercury orbit has a procession of 43.1'' per century upon reaching the perihelion position [17], which can’t be calculated using Newton's Universal law. Einstein provided the 1st evidence of general relativity by calculating the precession of Mercury's perihelion using the Schwartzchild solution [3]. Christiana claimed that the Newtonian approach to calculating the Mercury perihelion procession was incorrect. In his method, he considers the relative gravitational acceleration between Mercury and the Sun at the perihelion point, rather than just the gravitational acceleration of the Sun [18].
Three main Newtonian methods can be used to calculate the perihelion precession of Mercury, namely, circular orbit approximation, Newton harmonic oscillator, and Kepler’s third law [18], where all will give the same results. The circular orbit approximation method is used in this work to calculate the Mercury perihelion precession, as follows:
With considering the equilibrium forces in the Mercury’s orbit, one can write Eq. 52:
Where G is gravity constant, M mass of sun, m mass of Mercury, r is the distance between the center of sun and Mercury, V is the linear speed of Mercury.
The initial period T0 can be calculated as in Eq. 53:
The equilibrium of the Newton-Accedeceleration Equation in perihelion position is written in Eq. 54:
Eq. 55 can be simplified into Eq. 55:
can be written as in Eq. 56:
is then calculated as in Eq. 57:
With simplifying Eq. 58, one can get Eq. 58:
The difference
can be calculated using Eq. 59:
The accedeceleration of the Sun, when the Mercury is rotating around it, can be calculated using Eq. 21:
The coefficient in Eq. 59 have the following values:
Gravity constant G = 6.67×10− 11
, sun radius r0 = rs=696340km, distance from sun to mercury in perihelion: r = 46×106km, sun linear velocity: Vs = 1997m/s, sun angular velocity:
8×10−6
, Ms=1.99×1030 kg, C=299792458
. . By substituting the exact values of these coefficients in Eq. 59, one can get can exact value of perihelion precession as follows:
- If
(Mercury orbit is on the sun’s surface) and r=r0
is considered in the calculation of
, one can get:
- If
and r = r0
are considered in the calculation of
as in Eq. 19, one can get:
and
=
43.12’’/century
- If
and r=r0
are considered in the calculation of
as in Eq. 20, one can get:
and
=3.
29.01’’/century
One can see that the value
gives the best over-world closet value to the well-measured value 43.1’’/century [17] which is more accurate than the value of 42.98”/century calculated via general relativity.
3.3.2 Validation 2: Calculation of GPS time dilation
The time dilation of GPS satellite moving by 3874 m/s using special relativity can be calculated as in Eq. 60:
→
(60)
The time dilation of GPS using general relativity can be calculated as in Eq. 61:
61
where rEarth=6,358 km, rsatellites=20,184km, rGPS_orbit = 26,541km. Substituting these in the above equation, with MEarth= 5.974×1024 kg.
The total time dilation for GPS is the difference between general and special time dilation [19]:
= 45850–7210 = 38640 ns = 38.6 µs/day
Let’s calculate time dilation using real special relativity in Eq. 30 via accedeceleration in orbital movement as in Eq. 62:
As the GPS satellite is in free fall and their orbital velocity depends on the earth gravitational force
, its accedeceleration
depends on the earth velocity and the radius between the GPS orbit and the earth.
is calculated using Eq. 22:
, r0=rEarth=6,358 km, v=vEarth=1600km/h, g = 9.81m/s2and
=7.2×10−5 rad/s.
→
(62)
is calculated based on the time dilation t’ and the day at GPS side, thus it has negative sign, indicating that the time dilation on the earth is lower than GPS.
As
is occurred only on the half time period as in Eqs. 12 and 24, GPS gains a time dilation
equal to:
One can see that real special relativity gives accurate results for GPS time dilation as it is mixing the effect of constant speed (accedeceleration) and gravity, i.e. mixing of special and general relativity effects.
4 Limitations and Future works
4.1 Limitation of this work:
The limitations and future works of this study can be summarized as follows:
- The accedeceleration is valid to describe constant velocity with tiny constant acceleration followed by tiny constant deceleration. However, when large acceleration and deceleration are used, the equivalence values between accedeceleration and constant velocity are not well matched. The accedecleration gravitational field acceleration has
always low acceleration value as long as the velocity is not exceeding the velocity of light.
- The negative value of wavelength resulted from Eq. 51 couldn’t be interpreted well in this study. It has been analyzed as anti-wave effect, which needs a further discussion and confirmation through experiments.
4.2 Future works
The proposed future works for a rigorous validation of the proposed theory are:
- More experiments on the total gravitational acceleration (Universal Newton law g and Accedeceleration
), need to be conducted to validate the proposed theory.
- A detailed comparison between the real special relativity and the apparent special relativity in various scenarios should be conducted.
- The proposed quantum mechanics insight should be experimentally tested on the wave-particle duality with the three above-mentioned quantum accelerations cases as in Fig. 8.
5 Conclusion
This paper focuses on the study of a new phenomenon, called accedeceleration, for a deep understanding of the Newton's first law. The equivalence law between constant speed and acceledeceleration with mathematical models has been introduced. The accedeceleration phenomenon has been validated by calculating Mercury's perihelion procession with the best accurate up-to-date value of 43.12’’/century. A real special relativity has been introduced as a result of the proposed theory, which is then compared with the current apparent special relativity. The findings of this theory can be summarized as follows: (1) It introduces a new analysis and understanding of the external work generation at the Newton's first law; (2) It adds a new local gravitational acceleration for moving masses with constant speed, which will be added to the acceleration generated by Newton's Universal law to be used effectively with rotating masses; (3) The paper also introduces a real special relativity that strengthen our knowledge on the current apparent special relativity as a consequence of the accedeceleration phenomenon; and Lastly, the new theory gives a new insight into quantum mechanics that can lead to an agreement between Newtonian and quantum mechanics. As a future work, a more rigorous experiments on the total gravitational acceleration (Universal Newton law g and Accedeceleration
), real special relativity and wave-particle duality in the presence of accedeceleration, need to be conducted.
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Funding
This research was funded by University of Malaya and Ministry of High Education-Malaysia via Fundamental Research Grant Scheme (FRGS/1/2023/TK10/UM/02/3).
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Acknowledgement
Authors would like to thank University of Malaya and Ministry of High Education-Malaysia for supporting this work via Fundamental Research Grant Scheme (FRGS/1/2023/TK10/UM/02/3).
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Data Availability
The datasets generated during this study are available from the corresponding author, [Mohammed A. H. Ali], upon reasonable request via [hashem@um.edu.my].
Declarations
Conflict of interest
The authors have no relevant financial or non-financial interests to disclose.
Author Contributions Statement
All activities have been done by Mohammed A. H. Ali.
A
Author Contribution
All activities have been done by Mohammed A. H. Ali.
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