Einstein's mistake

 

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A minor mistake of Einstein led to the greatest scientific mistake of the past century

Henk Dorrestijn
hjdorrestijn@outlook.com

January 2018

 Summary

 When developing his theory, Einstein introduced the Lorentz contraction with which he unwittingly revived the aether theory. The concept of curved space must be considered a version of the aether theory. A meticulous repetition of his derivation of the Lorentz transformation formula for time leads to the same result, but Einstein made a mistake in the derivation of the transformation formula ξ = γ (x v.t)  for the X-coordinate. We will point out the mistakes of Einstein and show that these make the Lorentz contraction for moving objects unnecessary. It turns out that all phenomena that up to now have been considered a combination of time delays and the Lorentz contraction can be fully attributed to the time dilation.  
The interpretation of the Lorentz transformation formulas must be revised and the term ‘travelled distance’ must be given a key role. All results regarding the dynamics proposed by Einstein with his general theory of relativity can be acquired easily with the amended special theory of relativity presented here.

 Contents  

  1. Introduction     page 2
  2. Philosophical assumptions        page 3
  3. Clock test in a moving system  page 5
  4. Place and time in the moving system     page 6
  5. The time function         page 7
  6. Calculation of the integration constant  page 9
  7. Consequences of the slower time speed           page 11
  8. And Einstein created the Lorentz contraction   page 14
  9. The place function        page 16
  10. The timeline diagram    page 18
  11. Example: Time and location in moving systems page 20
  12. Discussion       page 22
  13. Literature         page 23  

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1.    Introduction

Einstein's theory of relativity is still standing proud after over a hundred years. However, famous scientists regularly lament that our human mind is incapable of imagining the four-dimensional 'spacetime' that plays an important role in the theory. Consequently, Einstein's genius is highly praised and any scientists that ‘fails to understand’ can consider themselves lucky to be normal. Einstein was an exceptional person.  

The issues scientists are struggling with result from the fact that mainstream science still has one foot in the aether theory. Here is why:
Einstein assumed that the velocity of light has the same consistent value for each observer, regardless of his movement speed relative to the source. This assumption makes aether as a transmitter for the movement of light in space unnecessary. He could use the constant velocity of light to explain the results of Michelson and Morley but also of Fizeau and others.
 
This convinced Einstein that he had dealt the aether theory the death blow, but he unwittingly invited it back in through the back door. When developing his theory, he introduced the contraction of the 'space’ of a moving system. The contraction does not only concern the space but also all physical objects located in this moving space. This is called the Lorentz contraction. This contraction serves as the foundation of the term ‘curved space’.

Einstein first removed the extremely rarefied substance of ‘aether’ from space to subsequently assign physical properties to space itself. He filled space with a substance that was able to contract or expand.
This substance is an extremely rigid material that only measurably vibrates during great cosmic events such as colliding stars. As described by a professor (lit 1): “the stiffness of this spacetime is comparable to that of a thick steel plate” (sic!).

This is simply a new form of the aether theory.

Even though persons interested in the theory of relativity expressed and still express numerous comments about the physical incomprehensibility of the Lorentz contraction, established science considers each critical sound as a base attack on Einstein's genius theory which can only be haughtily contested by pointing out the mathematical complexity of the theory.  

In an article on the Lorentz contraction (lit 2), I set out why the Lorentz contraction must be incorrect on philosophical grounds. In continuation of this conclusion we will indicate in the present article the mistakes Einstein made deriving in 1905  the special theory of relativity (lit 3). These mistakes forced him to introduce the Lorentz contraction to make the theory work.  

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2.   Philosophical principles

 Einstein attempted to provide a philosophical foundation for his theory of relativity multiple times. He referred particularly to the principle of causality (cause and effect) to support his general theory of relativity (lit 4, page 772). This is somewhat sparse for a theory which has dramatically changed our worldview. We will set out some nature-philosophical thoughts that must precede the physical theory in the next article.

We will start with our principle of relativity: there is only one physical reality. We can define this as a consistent description of all observers of a physical event taking place somewhere.

For example, an event such as the toppling of a row of dominoes with the last stone activating a switch which makes a glass water fall, causing a puddle on the floor, will be described in the same manner by each observer. A passing hiker, a train passenger or a person in a roller-coaster, any observer who can freely observe the event from beginning to end will describe the event to an outsider in the same manner, including the puddle. If there is an observer who claims that part of the event has not taken place like the others describe it, he will be asked to explain how the puddle ended up on the floor. If he denies that there is a puddle, he will not be taken seriously as an observer: “Everyone can see that there is a puddle on the floor!”

It is essential for physics that various observers of a physical event give the same description.

However, this agreement also has its downsides. If established science claims to ‘perceive’ that the earth is flat, an observer who disagrees will be ridiculed. He will be required to convince others to adopt his position. He will need to present the facts and bombard them with arguments. Only persons who can do this successfully will contribute to the development of science.
It took Einstein every effort to start his “Dialogue” at the start of the last century (lit 5) to convince the “Criticasters”, as he called them, of his theory of relativity.
 

We will attempt to convince the “Deniers” with the same gusto in this article that based on the physical reality a small improvement to the theory of relativity is necessary. The small improvement – rejecting the Lorentz contraction – has major implications for our ideas of ‘spacetime’, however.  

We will define a physical event as a change during a certain period from a recognizable state into a new recognizable state of a collection of material [1] objects. 'Reconcilability’ means that all eligible observers can see the same distinctive elements of a state.  

A special physical event is the ‘duration’. This event can be observed on a clock. Because the duration is a physical event, the presence of an object in relation to the duration must be a physical event in itself. A stone lying in the countryside in peace represents a physical event as long you observe it!  

A system is defined as all material points that do not move or experience a different acceleration in relation to each other.

A stationary system is defined as the system from which a moving system is observed.

A moving system is a system with a certain speed (or acceleration) relative to the stationary system.


[1] You may wonder why only material objects are indicated, while numerous forces and fields also play 
a role in physics. This is because the role they play only becomes recognizable based on the effect on the 
material with which they interact.
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Based on symmetrical considerations, we must state that the amount of the speed is mutually equivalent between the stationary system and the moving system.  

The constant velocity of light means that a clock in a moving system – depending on the speed with which the system moves – is slower than the clocks of the observers looking at the moving system. The fascinating thing is that the trailing clock should be moving faster if we stand next to this clock. This means that we have changed systems. We will discuss this extensively later.  

We will always work with identical clocks that if placed next to each other work at the same speed.

Because these identical clocks in the moving system move slower, the events in the moving system take place at a slower time speed. The Lorentz factor (gamma) γ ≥ 1 is linked to this: the time in the moving system runs γ sec/sec slower compared to the stationary system. We limit ourselves to v << c to ensure γ is close to one.  

The physical event – like a full rotation of the large hand of a clock – will take longer according to the clocks of different observers with different velocity's relative to the clock. It is a comparable event but at the moment the observer finds that the long hand of his own clock has made one full rotation, the long hand of the observed, moving clock has not yet reached this point. An event in a moving system takes place at a slower time speed.  

An enlightening concept is the point-event. This is an event taking place at one location at one moment in the system of an observer. A collision or flash of light. A point-event will take place at another moment and another place in the system of an second observer relatively moving related to the first observer.  

If an observer sees an object pass by with a speed of v m/s, the corresponding 'event' after t sec will according to him be a duration of t sec on the clocks in his system and a travelled distance of v.t meters in his system. The clock on the moving object shows a time of t* = t/γ sec at that moment. The covered distance according to the moving system is equal to v.t* = v.t/γ meters. This is ostensibly less than stated by the first observer. How is that possible? We will discuss this later.

What identical event must the (moving) observer located near the object assign to this? He is standing still relative to the object. Nothing happens there besides the ticking of the clock. The duration t* is the only thing the observer can assign to an event.

Because t and t* differ, the observers see different events. The moving observer can call it the same event as the observer in the stationary system only if t* = t. This means that the moving system must move γ.t sec before the observer in that system experiences the identical event.  

The mutual event has a special position. This is the travelled distance of the moving system in the stationary system. This is accompanied by a travelled distance of the stationary system in the moving system. The duration of an identical event must be the same length according to the own clock. We will see that the travelled distance in the stationary system for an identical event according to the moving system must be γ2 greater than the distance travelled in accordance with the stationary system. This surprising result is derived in section 7.

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3.    Clock test in a moving system

 At the start of his article on the special theory of relativity (lit 3, page 894), Einstein emphasised the importance of synchronized, identical clocks. He gave us a method to test whether two clocks, A and B, work at the same speed in a system. The fixed distance between the clocks is AB.
The two clocks will work at the same speed if the duration AB/c  sec for the light to travel one way is equal to the duration  BA/c  sec of the way back. 
This gives us three point events.

The moment tA  (the moment the pulse is sent from A)  and the moment tB on which the pulse is reflected in B, and the moment tA' on which the pulse arrives back at A (fig 1), have the following relationship:

tB tA = tA' tB

or

½(tA' + tA)= tB .                                                                     (1)

 Moment tB takes place on the synchronised clocks exactly halfway between the times of arrival and departure at A.
This clock test may seem self-evident but is of paramount importance for understanding the special theory of relativity. Einstein proved (lit 4, page 902) that a clock on a moving object has a slower time speed.
 

The way Einstein came up with the relativity theory is often highly praised. He mathematically found the coordination-transformations based on the assumption of the constant velocity of light.
He carried out as a thought experiment the clock test with two clocks moving at a constant velocity of v m/s. An example is two clocks in a moving train (fig 1).
A light pulse is sent from A to B and reflected back to A from B. For an observer located in the train, this takes 2.AB/c  sec. This is the time he can see on his clock.
 

The duration is determined by both the observers on the train and the observers on the ground using the clocks of their own system. For the observers on the ground that watched the experiment in the moving system, the light pulse also moves with speed c from A to B and back, but they see point B move with velocity v in the meantime. The distance travelled by the pulse on the way there is greater than AB. We can calculate at what place B must be when the pulse catches up to it but taking the value (c v) m/s for the speed of the pulse relative to points A and B is sufficient. 

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The distance on the way back that must be travelled by the pulse is smaller than AB. To determine the length of time on the way back, we can use the expression (c + v) m/s for the same mathematical reasons.
The analysis in the following sections shows that the duration of the movement to and fro on the clocks for the observers on the ground is greater than the duration on the clocks of the observers on the train! The time speed of the clocks of the observers on the ground must have been faster than that of the clocks of the observers on the train.
So the observed event requires more time[1] according to the clocks of the ground team than to the clocks of the team on the train. You can justify this based on the fact that more than one event takes place for the persons on the ground: the travelling light beam to and fro and the travelled distance of the train.

Time plays an unfathomable role here as both clock teams do not experience any faster or slower passage time, they simply experience the time of the event.

  4.   Place and time in the moving system

 The relationship between time and place of the own clock in the stationary system and the time and place of the passing clock in the moving system was found by Einstein in the following manner. He used two coordinate systems:
Ø        
the coordinate system in the stationary system with coordinates (x, y, z, t), and
Ø        
the coordinate system of the moving system with coordinates (ξ, η, ζ, τ).

Each point in space is always in both systems at once and a moving object will also always be in both systems simultaneously. Each place where the object may be at any time will have a different set of four coordinates in the two different systems.
The trick is finding formulas that enable you to find the four coordinates (ξ, η, ζ, τ) from the set of coordinates (x, y, z, t).
This is called a coordinate transformation.
 

Because the constant movement takes place along the X-axis, we assume – together with Einstein[2] – without proof that the values of the coordinates in the Y-direction and the Z-direction do not change, η = y and  ζ = z.  

To find place ξ and time τ in the system moving with v , if time t and place x in the stationary system are known, Einstein used the clock test on two clocks A and B in the moving system. We will follow and examine his calculation scrupulously .
When the origin A of the moving system passes origin O in the stationary system (fig 1) a light pulse departs from A to point B. The initial conditions in the stationary system x = 0 and t = 0 and in the moving system ξ = 0 and τ = 0 apply to this moment. Point A is the origin of the moving system and  ξB = AB = ℓ  meters applies to point B.
We will also call the stationary system: system O , and the moving system: system A.
 

At what point of time t does the pulse reach point B according to an observer in the stationary system? Einstein assumed that the (moving) distance AB for the observer in the stationary system may not have the length . He calls this length x'.


[1] The observers in a system can read the clocks of the observers in the other system and see the slower 
movement of the clocks.  

[2]
We exclude any form of Lorentz contraction in those directions in advance. For proof that the Lorentz 
contraction does not exist in the transverse direction, see (lit 6, page 29).

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The following formula applies to place B in the stationary system: xB = x' + v.t.
We also write this as x' = xB
v.t meter[1]. At t = 0, the following applies xB = x' meters.
 

5.  The time function

 In the moving system A, the formula (1) for synchronized clocks will be expressed in the time τ of the system A:

½(τA'+τA) = τB                                                                      (1A)

The time coordinate τ is a function of the coordinates (x, y, z, t) of the stationary system. This makes sense, as we want to know the time at a place in system A if we know the coordinates including the time of that place in system O.
To determine the formula, we must use the time and place of the three point
events in the stationary system. We find the following three points of times:  

Ø         The 1st time in the stationary system when the light pulse left A is tO = 0 sec.

Ø         The 2nd moment on which the pulse arrives at B is equal to:  sec.

Ø         The 3rd time, the return to A', is:  sec.                                (2)

At those times, the three places in the stationary system are also known:  

Ø         The 1st place in the stationary system is the origin O at xO = 0 meters, because the pulse left O at TO = 0.

Ø         The 2nd place is point B after  sec, resulting in   meters, as that was the time the pulse was reflected back in B.

Ø         The 3rd place is point A' after tA' sec: , because this is the moment the pulse arrived back at A.

 
Formula (1A) can be written down as:
 

                  

We can now enter the found coordinates (x, t) of the 3 pointevents:           (3)

 This expression can be found as a total differential if x' is not too great:                       

The meaning of this expression is found in:

 We can now expand the equation (3) using partial differentiations.


[1] When calculating the time function, Einstein takes an – unnecessary – intermediate step by first making 
point x' move along as the origin of a new coordinates system. We don't follow this step.

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The first term in the left part gives:

= .

The term in the right part gives:                                                                      

 =                                              

 When entered in (3), we get:


thus


which means that

   .

This leads to

 If x' is small, we can look at the right term to find the derivative for x in point (0,0):

where
So
 
or
 thus

If we delve into this, we find the solution:     sec.              (4)

In this formula a is an integration constant.
   

6.   Calculating the integration constant.

 We calculate a just like Einstein by considering the behaviour of time in the Y-direction. We will have to calculate this again because of our modified approach.  

We use the clock test, but now for clocks along the Y-axis: 

                                             
½(τA' + τA) = τB                                           (1A)

We let a light pulse move along the Y-axis from point A to point B – located at a distance of y0 – and back. According to the observers in both systems, the distance between A and B is equal to y0 meters. For the observers in the moving system, the pulse moves perpendicularly up and down. For the observers in the stationary system, the light pulse moves in a triangle (fig 2). We can use the formula for the time  sec.

 

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According to the observers in system A, the duration for the way there is τB = y0/c  sec and the same for the way back. The return time is τA'=2y0/c  sec.                                      

 

Looking from system O, the pulse travels in a diagonal line from O to B. The component of the speed in the Y-direction is equal to  m/s. The time the pulse needs to reach point B is  sec. The duration to A' is equally great.

The return time will be  =  sec.                         (5)

So the return time in the Y-direction is γ times as long in the stationary system compared to the moving system. We will use this result to calculate a.  

Note: Here we introduce the Lorentz factor  with γ1  applies.  

By considering the Y-direction, we now have 3 coordinates in the stationary system (x, y, t) and in the moving system (ξ, η, τ).

We will use the initial values t = τ = 0 and x = ξ = 0 and y = η = 0 when the origins of O and A coincided.

The function for the time τ is now τ(x, y, t). Formula (3) will then look as follows:
 

  =                           (3A)

The total differential for the three variables x, y and t will be:

The meaning of this can be found as follows:

The first term in the left part of (3A) can now be written as:

 =       

The right part will be:   

 =      


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When using this  and the fact that τ (0,0,0) = 0   , we will get:

= , thus .

The conclusion is that , which means that τ = constant in a direction perpendicular to the

 movement direction. This in contrast to the time along the X-axis, which is linear.
 

For a light pulse shot into the Y-direction, the following applies: η = c.τ meters. When we use the function  found for τ, we will get:  meters.

We can enter t and x, since we know that point y0 is reached at  sec, so the place will be  meters. With these results we find: .

But η must at that moment be equal to y0, which is η = y0. If we use the formula and divide by y0, we get:

This results in a = γ and with (4) the following time formula:  sec.        (6)  

This is the well-known Lorentz transformation formula for time τ in the moving system presented by Einstein in his article on the special theory of relativity (lit  3, page 902).

It means that if at place x in the stationary system at the time t on the clocks, a clock in the moving system which currently passes that place will show the stated time τ if the initial values are met.
A random clock in the moving system which on time t = 0 is set to the same time as the clocks in the stationary system and thus shows time τ = 0 will on moment t reach place
x = v.t and the time will correspondingly show the value:

 sec.

This shows that each clock in the moving system runs slower than the clocks in the stationary system by factor γ. The time in the moving system moves slower by factor γ. The atoms move slower, cells in plants develop slower, each identical event moving with the moving system develops slower than in the stationary system according to the observers in that system.
 

A particular consequence of this is that two events occurring simultaneously at two different places in the stationary system cannot be observed simultaneously by an observer in a moving system.
In time formula , this is expressed using the term  sec.

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7.   Consequences of the slower time speed

 According to the observers in the stationary system, the time speed in the moving system is slower by the factor gamma  compared to their own system.

What does this mean?
An event in the moving system that lasts T sec on the clock in the moving system lasts γT sec in the stationary system. The clock in the stationary system will appear to move faster. This is naturally hard to understand. We must remember that these measurements also coincide with a movement. The moving system is not standing still! Thanks to the movement, the observers in the moving system may actually see that the clocks in the stationary system are the slower clocks. We will explain this in the next sections.
 

If point A of a moving system, a train for example, after t sec travelled a distance of = v.t meters from origin O to point P in the stationary system, the clocks in the stationary system will show the time t sec, of course, but the clock of A will show the time t/γ, as this clock is moving slower. In the stationary system, duration t sec may concern a ball being thrown up and falling down. If a ball was simultaneously thrown up in system A, the ball would not have yet returned when A passes point P. The ball will be back when the clock in A also shows time t. The ball will have taken γ times as long. Point A will have moved up to γℓ meters in the stationary system.

For reasons of symmetry, it is clear that the velocity of the systems relative to each other is the same. The distance travelled once the clock shows t sec seems to differ, however.
This difference is caused by the ‘definition’ of the stationary system. This gives the stationary system another role in the story than the moving system. A so-called privileged position. In this position the clocks in the stationary system are moving faster than the clocks in the moving system. This seems to introduce asymmetry for the distance travelled by the moving system in the stationary system after t sec shown on the clock in the stationary system differs from the distance travelled by the moving system in the stationary system after t sec on the clock in the moving system. However, this consideration is not symmetrical.
We can return to the symmetry by looking at the distance travelled in the moving system by the stationary system after t sec on the clocks in the moving system. This is exactly the same as the distance travelled by the moving system in the stationary system after t sec on the clock in the stationary system.
It is like looking through a looking glass: if you look from the one side, things on the other side appear larger, but if you look from the other side, this side is larger.

How can we describe this for the moving systems?
The solution can be found in the fact that looking from the stationary system, the unit of time in the moving system is a factor γ larger than in the stationary system[1]. This is why the clock shows fewer seconds for the same time. We make use of a time unit which is γ times greater than the second. This unit only applies in a system that actually moves, meaning that it is the time unit that must be used in the stationary system to describe the physical events in the moving system.
When expressing time in that system, I want to call it the Lorentz time. The accompanying unit can be called the ‘Lorentz second’, abbreviated to Lsec.

The following applies: 1 Lsec = γ sec.


[1] The question 'why?' is beyond the scope of this article
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The time speed in the moving system has decreased, but that is the only thing we can say about the moving system. The velocity for the system remains v m/s.
However, if we want to show the velocity (= distance divided by time) in Lorentz time, the length unit must also become γ times as great. This is the length unit that must be used for the travelled distance in the direction of movement. 
A distance travelled in that unit leads to a γ times greater distance in the stationary system.
 

The greater length unit could be called the ‘Lorentz meter’, abbreviated to Lmeter.
The following applies: 1 Lmeter = γ meter.

This results in the following for the velocity: v Lmeter/Lsec = v m/s.  So the velocity remains mutually the same.
The distance travelled by the moving system will be:

v
x t (Lmeter/Lsec).Lsec = v.t Lmeter.

The distance travelled in this unit could be called the Lorentz distance.

The following applies: v.t Lmeter = γ.vt meter.

On the time t/γ Lsec for the distance travelled by the moving system in the stationary system we find as expected v.t meter.
This complies with our physical intuition.
 

For our research, it is important to get a deeper understanding of the meaning of the identical event of a travelled distance in both systems. For this identical event the duration in the moving system expressed in Lorentz seconds will be just as great as the duration in the stationary system in normal seconds. Because both the Lorentz sec and the Lorentz meter are γ times as great as the second and the meter, the identical travelled distance (= product of velocity and time) after t  Lsec is γ2 times as great for the moving system in the stationary system than the travelled distance according to the observers in the stationary system after t sec.  

If for example origin A after t sec travelled a distance in the stationary system of ℓ = v.t meter, the clock of A will show a duration of τ = t/γ. If shortly after the clock of A shows numerically the time t, the duration has become γ times as great. The origin A will at that time have travelled a distance in Lmeters that is γ times as great:  γ.v.t  Lmeter.

Because  γ.v.t Lmeter is equal to γ2.v.t  meters, A will be located at place γ2.v.t  meter in the stationary system.

The clocks in the stationary system will show at that moment the time γ2.t sec.
So if we write down  for γ2, which is valid for v << c, we can tell that A for the identical event must travel a distance in the stationary system which is  meters longer than the distance that has been travelled at time t in the stationary system[1] according to the observers.
 

To summarise, these are the main three findings:

Ø         The duration of an event in our stationary  system is on our clocks just as large as for the identical event in the moving system on the clocks of that system.


[1] This principle also applies if the time velocity has changed by an acceleration field. This can easily 
explain the bending of light and the perihelion precession of Mercury.

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Ø         The duration of an event at a fixed place in the moving system is on our clocks always γ times as great as on the clocks in the moving system itself.

Ø         The moving system must travel a γ2 times greater distance in the stationary system to travel the identical distance travelled by the stationary system.  

For clarification: Someone can be home in 10 minutes if he leaves now. He lingers and has only 9 minutes left. He decides to take faster steps to get home on time. He adjusts the rhythm of his steps by factor 10/9th and he gets home at the time he would have if he had left on time. Thanks to his greater speed, his steps also increase by 10/9th. As a result, he travels not just the needed distance during these 9 minutes but goes even further. After these 9 minutes, he has travelled 10/9th times the distance. While he would usually travel 9/10th part of the distance to his home in 9 minutes, he has now travelled 10/9th this distance. The travelled distance has become (10/9)2 as great. A quadratic increase.  

Note: As for the slower Lorentz time for the entire moving system, the Lorentz distance is also a property of the entire moving system. This concerns the distance travelled by the entire system expressed in Lmeters. The system itself remains undistorted, as we will show in the next section, and can be considered a rigid object whose dimensions are expressed in meters.  

In the critical article (lit 7) on the Lorentz contraction, “the γ2 greater distance” is used as the baseline, together with the influence of the acceleration of a system on the time speed. This leads to a transparent explanation of:  

Ø       the Ehrenfest paradox with its centrifugal acceleration,

Ø       the bending of light in a gravity field, and

Ø       the derivation of the perihelion precession of Mercury due to its velocity in the gravity field of the sun.

 The last two examples were used by Einstein to demonstrate the strength of the general theory of relativity. 
In the mentioned article, we show that the theory of relativity without Lorentz contraction and better understanding of the influence of the time speed lead to the same or even better results in a clear manner.

 

8.   And Einstein created the Lorentz contraction

 Since we can now determine the time in the moving system using the time and place in the stationary system, we wonder whether we can calculate place ξ along the X-axis in the moving system.
To this end, we study a moving system consisting of a train, for example, in which both the X-direction and the Y-direction a length of ℓ have been plotted. Light pulses are sent in both directions simultaneously that according to the observers in the moving system arrive back at their origin A at the same time after
τA' = 2ℓ/c  sec on their clocks. These are Lorentz seconds according to the observers in the stationary system.

The clocks in the stationary system must show a time of tA' =2γℓ/c  sec.

We distinguish between origin A and the place A' because this place has changed xA' from the perspective of the stationary system. The coordinate ξA = 0 always applies to origin A.

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The time of return in A' can be calculated from the stationary system using time formula . To do this, we must know the value in the stationary system of t in point A' at 

the moment of return. If we know the time t, the value of the place x of A’ will follow automatically from formula x = v.t meter.
Because in the stationary system based on the principle of physical reality the light pulses will meet also at the same time just like in the moving system, the value of t can be determined in two ways, using the way of the light pulse along the Y-axis (fig 2) and using the way of the light pulse along the X-axis.
 

Using the Y-direction, the time tA'  upon arrival according to an observer in the stationary system is:

The place of A' in the stationary system will be:    meters.

It follows      sec in that case.

This matches the value that can be seen by the observers in the moving system.  

The time tA' upon arrival is harder to calculate using the X-direction.  

Ø         Einstein calculated time tA' in the following manner. He proposed to call the length of in the stationary system x' as long it was unknown. He determined the return time tA' using:
     sec.                                                     (7)

He used ℓ = γ.x' to match results (5) and (7). This leads to the conclusion that x' is smaller than . This automatically leads to the conclusion that a moving object becomes shorter along the direction of movement.

Ø         However, this is a hasty conclusion based on incorrect physical calculation.  

The first mistake made by Einstein was caused because he proposed calling length in the stationary system x' (lit 3 p.900). He treated however this length x' not as a length in the stationary system, but as a length in the moving system as he let it move with speed v. If he would have been consistent, he would have multiplied the length again by factor x'/ℓ etc. This means that he gained nothing from giving the length another name.  

He made a second mistake because he did not use the time formula (6) he had derived just now. With this formula it is easy to calculate the time of the point B' in the stationary system where the light pulse is reflected.
As follows (fig 3). In the moving system the light pulse arrives the point B at the time
τB = ℓ/c. The location of B is   in the moving system, so the place and the time are known quantities of  the point B in this system. With the time formula we calculate the time of the same point but now situated in the stationary system. This point is indicated by B'. The resulting time tB' will have the meaning of the time duration ΔtB' for the light pulse to travel from O to B'. By using the time formula we have to realize that we transform back from the

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moving system to the stationary system and then the velocity of the system O is negative relative to the system A, so v  m/s. We find:  

ΔtB'= tB'=     = =       sec.

 

To find the time duration for the way back from B' to A' in the stationary system we start at the moment that point B is passing the origin O of the stationary system (fig 4). The clocks in B and O have been set to the initial values t = τ = 0. Now the point B has become the origin of the moving system. At the moment τ = ℓ/c the light pulse reaches the point A at the location . We can with the time and the location of point A in the moving system calculate the duration ΔtA' for the return journey of the pulse in the stationary system:

=   sec

 

The total duration back and forth in the stationary system gives the time in point A':  

tA' = ΔtB' + ΔtA' =
+ =  sec.

 

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This is an important result because it is the same point of time as for the returning pulse in the Ydirection. 
Remark that our principle of relativity is satisfied. The length between A and B however has to keep the same value in the stationary system as in the moving system.  

If we had followed the assumption of Einstein for the length in the moving system and had chosen the value x'  for the length between A and B the result would have been:

tA =     sec.   

Then we find two different times (5) and (8) for the returning light pulse:

Ø         tA ' =   = along the Y-axis and

Ø    tA '  sec along the X-axis.

These times must be the same because the light pulses arrive back at A' at the same time.

This leads to the same important result  x' =   

This means that the interpretation of Einstein that an object in a moving system would shrink due to the Lorentz contraction is incorrect. The dimensions of an object in the moving system are the same as the dimensions of the same object in the stationary system.  

Ø       The travelled distances are affected by the time speed, 
the dimensions of the objects are not.

 

   9.  The place function

The Lorentz transformation formula for time  calculates the time τ of a clock in 

the moving system which just passes place x on moment t in the stationary system. The calculation provides the precise time which the observers in the moving system can observe on their clock. This time is slower than the time in the stationary system. This is confirmed if we return the clock to the stationary system (like in the twin paradox). This clock will have moved slower.
The Lorentz transformation formula for the time is beyond all doubt.

Einstein derived the Lorentz transformation formula ξ = γ(x v.t) meter for the place in the moving system. Place ξ in the moving system can be found for a point located at time t on place x in the stationary system.
For the origin of the moving system, the following applies: x = v.t, which means that ξ = 0.

This is correct: the origin remains the origin, but this is the only thing that is correct!

This formula leads to another value for the length of an object than the observers themselves notice in the moving system. A stick with a length of Δx = ℓ meter between x1 = 0 and x2 = ℓ meter in the moving system becomes by the transformation formula the length ξ = γ(x2 x1) = γ(0) = γ  meter. The stick stretches out with a factor γ. The observers in the  moving system are not in the ability to determine the greater length while their unitstick also has stretched out with the factor γ to a lengthunit of γ meter. So they measure a length of while the observers in the stationary system have calculated a length of γℓ meter for the moving stick. If we retrieve the moving stick, it will shrink to its original length of meter.
This is the way it has to be understood according to Einstein. The stretching and shrinking of

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the stick by these transformations form together an unverifiable, 'spooky' physical phenomenon. Einstein makes an appeal to your trustfulness: the stick has been stretched  but we cannot prove it by measurement. In this way the phenomenon of the Lorentz contraction of a moving object deserves no place in physics. 
Conclusion: The Lorentz transformation formula for place is not applicable  for the determination of the length of an object in the moving system. 

The question is how we can transform the length from stationary to moving system.
We will find the solution by implicating the travelled distance in the problem.  

A travelled distance is another physical quantity than a length. A travelled distance the product of speed and time   is a physical event, closely related to the time. A length is not a physical event, it is a property of an object.  

In section 7 we proposed as the unit for the travelled distance the Lorentz meter.
This transformed unit is γ meter. This is a greater unit than the unit of 1 meter. We have reserved this greater unit for the travelled distance. The transformation of the length unit results in the unit for the travelled distance.  

This is incorrect for a transformation. We cannot transform apples into oranges. This is why we must take a travelled distance as the basis in the stationary system.  

How can we turn length into a travelled distance?
If we have two points x1 and x2 in the stationary system, we can describe their mutual distance
Δx= x2 x1 in the X-direction as the products of speed and time in the following manner as the difference in travelled distance: Δx = x2 x1 = v.t2 v. t1

Here t2 and t1 are the times needed by the origin of the moving system to reach the points when started from the origin of the stationary system.

We can now transform the travelled distance Δx and we can notice that the transformation of a travelled distance is a transformation of the time. 
This can be addressed using the time formula .

So  and  apply too.

The transformation of the length Δx= x2 x1  will become the transformation of Δx = (v.t2 v.t1)  meter to Δξ = (v.τ2 v.τ1)  Lmeter for the travelled distance:
=  Lmeter.
 

The resulting travelled distance in Lorentz meters in the moving system has a length of  = (x2 x1) meter. So the transformed length has in the moving system the same value as in the stationary system. This is what the observers in the moving system can actually see.

This interpretation takes away one shortage in the theory of Einstein that is the observers in the moving system do observe the time calculated with the Lorentz transformation formula but don't observe the calculated  length. So if we interpret the length as a travelled distance the theory fits.

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Conclusion: the transformation ξ = γ(x vt)  of a place in the stationary system into a place in the moving system has to be considered as a transformation of a travelled distance which is expressed in Lorentz meter. 
The Lorentz meter in the stationary system is 1 meter because γ = 0 and in the moving system it is γ  meter.

 

10.  The timeline diagram  

Without the Lorentz contraction, we get a better idea of the behaviour of the time in a moving system. We will stick to a moving system carrying out a constant linear movement in this article.

If the origin A of the moving system passes origin O of the stationary system under the known initial values, A will at time t be at the place x = v.t meters. Place and time in the point A are then known in the stationary system, which means that the time of the moving clock A can be calculated using the time formula (6):

Lsec.

Because γ >1, time τ is smaller than time t. The clock in the moving system is moving slower than the clock in the stationary system. However, the clock in the stationary system must also move slower than the clock in the moving system. This seems to go beyond our imagination. The graphs used in the literature to determine and explain this do not address the basic problem, the issue of clocks moving slower relative to each other.

Because this is a problem for laymen to understand and for scholars to explain, we will show a new manner of visualization which may be helpful.  We will use a timeline diagram (fig 5) based on the time formula .  

We make use of the movement of the clocks relative to each other. We have system O and system A. These have a velocity of v relative to each other. The system moves along the X-axis. Because of the fact that we do not need to consider the Lorentz contraction, the entire length of the objects (e.g. trains) is along the X-axis. The time relative to the other system is plotted compared to the time of the own system. According to term , this results in a rising or declining line.

We compare the time of the own system with the time of the other system. This gives us two sloping lines symmetrical with the X-axis.

 

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A sloping line represents the time on the synchronised clocks of a system at a certain point in time. We call it the timeline of the system at that point in time. The timeline of system O moves from the lower left to the upper right. The timeline of system A moves from the upper left to the lower right. BA represents the timeline of the moving train B*A (in which all clocks are synchronised) moving along the X-axis.
In the figure, point A passes point O and the clocks have been set to the initial values

 
t = τ  = 0  at those points.
System O can represent the timeline of a railway track, a stationary system, but can also represent the timeline of another train, a moving system, which moves with a different velocity along the X-axis relative to the first train.
The vertical distance between the timelines indicates the time difference shown on the clocks located at place x at the same place that belongs to different systems. If you are in a system and see the clock in the other system show a later time (meaning that the timeline will be higher at that place), the clock in this first system must be behind when looking from the later system (meaning that the line will be lower at that place).

 Ø         The theory of relativity is a concrete theory. Observers from different systems will see the same event at certain places at a time belonging to that place! The principle of relativity.

 Naturally, the timelines have the same angle (positive or negative) relative to the horizontal X-axis. The angle is determined by the speed of the systems relative to each other. The angles have been exaggerated in the figures for the sake of clarity. The angle between the two timelines is almost  sec/meter.  

Figure 5 shows the situation of the moment when origin A of system A passes origin O of system O. System A moves right relative to system O.
The timeline diagram is a dynamic diagram. We can move point A from O to point P in system O. Then we get fig 6 . We will show the timeline again for the moment A passes point P. Point A has moved along the X-axis.
The time of A is now below the timeline at place P while its time was first (fig 5) equal to the timeline of system O at point O and the time of point O is now significantly below the timeline of system A at the location of the point Q. We can see that both clocks A and O are behind compared to the other system.

 

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The convenience of the timeline diagram is that you can mentally move the timelines to understand the time differences. The time differences in simple situations can be determined directly using some geometry.
For example, when A passed point O (fig 5), the clocks showed t = τ = 0. At that moment, point B showed a time of τ t =
      sec above the timeline of system O, with  

representing the length between A to B.

If point A has moved after t seconds (fig 6) to point P at place x, we can see the time of A has moved below the timeline of system O by t τ   =         sec. The time of point B will, in that case, stay above the timeline of system O by t τ       +     sec.

Note: In point x, the time difference Δt or Δτ shown on the clocks of the stationary system respectively the moving system does not depend on the system from which it is observed. Seen from the stationary system, this difference Δt represents γ as many seconds as the Δτ seen from the moving system in Lorentz seconds.
 

11.     Example: Time and location in the moving system

 Two trains with the same length with timelines OP and AB pass with a relative speed of v m/s. Points O and A at the front of the trains pass at t0  = tA  =  0 . Because of the synchronicity of the clocks all observers will find the time t = 0  on their own clock at that moment. Observing the clocks in the other train they will notice a linear path with place according to the timelines. The timeline diagram (fig 7) shows the initial situation of the trains with dotted lines.

Ø       Both systems are moving, so it is curious to make a difference between a system with 'normal' clocks and a system with 'slower' clocks. That is why we will use in both systems the symbol t  for the time. 

 Observer P at the back of train OP at point +ℓ meter will pass observer A at the front of the other train after t = ℓ/v sec.
The observer at clock P can give the observer at clock A a high five. This is important because this makes clear that the encounter is a physical reality, a point event. Besides the fact that each observer will be able to see the physical event of the hands meeting, they will also observe the time on the clock of both P and A. This marks the event for all observers.
The clock of P shows a time of  tP = ℓ/v  sec and the clock of A shows a time of 

t
A
= tP sec.  

Based on symmetrical reasons, we may put that according to a neutral observer points B and O on the other side of the trains also pass each other at that time at a distance of meters between the point where A and P meet.  

Ø         The neutral observer moves in such a way that he sees the trains pass in opposite directions at the same speed of about ½v m/s.

Ø         Without Lorentz contraction there is no doubt about the length meter.  

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The trains have been represented with solid lines in the timeline diagram. The trains themselves move as projections of the timelines along the X-axis but have not been shown because we want to focus on the times.

Observers O and B will give each other a high five simultaneously with observers A and P according to the neutral observer. Symmetry determines that clock B must show a time of
tB = ℓ/v
sec and clock O a time of  to = tB  sec.

We can see a range of different times at that moment[1]:

Ø         on clocks P and A respectively tP = ℓ/v  sec and  sec and

Ø         on clocks B and O respectively tB = ℓ/v sec and  sec.

In system O we see the times sec and tP = ℓ/v  sec and in the system A we see the times   sec and tB = ℓ/v sec at a moment that is simultaneously for the neutral observer. The question is how to interpret: two indications of time in one system. The meaning is that we do not observe the point at the front and the point at the back of the moving train at the same moment in their system. If we look at the back point the train as a whole because there is no Lorentz contraction has travelled already somewhat further than when looking at the front point. It looks like the train has shortened.  

It is apparent from the times that in system A, observers believe that the meetings in A and B do not take place simultaneously and the same applies to points O and P in the system O. Observers in the system A believe that the meeting at B takes place later, with a difference of tB tA = ℓ/v –(1/γ).ℓ/v ≈ ½(v2/c2).ℓ/v sec. In that time duration B will have travelled the distance ½(v2/c2).ℓ meter in the direction of A (and P).
In system P they believe that the meeting at O took place sooner with the same time difference. So at that moment O should also be
½(v2/c2).ℓ meter closer to P (and A).
In this way B and O were able to give each other a high five.
The conclusion is that we can describe this problem clear and convincing with the time speed alone. Once again it appears that the concept of  Lorentz contraction has been a superfluous  assumption.   


[1] The time at which point S is halfway TS =  sec
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12. Discussion

 The modifications to the special theory of relativity presented here show that the theory contained a minor imperfection that has not been resolved after a hundred years. It must be noted that the problem was tucked away deep into some physics that many considered part of the theory of relativity which could not be humanly comprehended.
However, we were able to find it by meticulously repeating the derivation presented by Einstein in his article of 1905.
We discovered that Einstein made an assumption concerning a modified length in the stationary system which he did not use in this system but rather, erroneously, in the moving system. This led to a result in which he introduced the Lorentz contraction for the length of an object in the moving system to gain the theory closely
reasoned.  

By determining the length of a distance travelled by a ray of light in the moving system, we were able to show however that the Lorentz contraction is not necessary to achieve the correct result.  

To increase the intelligibility of the theory, we have shown that the Lorentz transformation for the place in the movement direction does not concern a transformation of the length but a transformation of a travelled distance. The dimensions of the elements that make up the moving system are not included. We indicate how we find the place of a point in the moving system. The result of the place gives the same positive confirmation by the observers in the moving system as the transformation of the time.  

It is interesting that an identical event of a travelled distance by the moving system leads to a γ2 greater distance in the stationary system before the same event is completed. The factor γ must be considered the factor by which time is slower due to velocity or an acceleration field compared to a stationary system. This result can be easily applied to the bending of light along a mass and when calculating the perihelion precession of Mercury. Its clarity gave us an improved formula for the perihelion precession (lit 7). Our results should be the same as given by Einstein's theory while the dilatation of time with the factor γ together with the shortening of the length of a distance with the factor γ will lead to the same greater identical distance with the factor γ2, but physically this part of Einstein's theory is wrong.  

These results are an important step for science as the aether theory can now be definitely put aside. Einstein did show that aether was not necessary to explain the behaviour of light in space, but he immediately returned to aether for the bending of light and the behaviour of Mercury. He did not call it aether this time, but rather ‘curved space’. It should be clear that assigning new properties to space results in a new version of the aether theory. Recently measured gravitational waves have incorrectly been described as 'ripples in space': they are actually ripples in time speed.

Our new insights have consequences for our view of the universe. The supposed phenomenon of the Lorentz contraction has resulted in an imaginative world-view. Space is assigned properties based on which the direction of an object or a ray of light is directed by the curvature of space itself. The condition of space is in this theory determined by the mass distribution at the actual locations. If you remove the mass, space returns to its form in relaxed state.

 

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Since we have now shown that the idea of the Lorentz contraction is based on an error made by Einstein the foundation of the term ‘curved space’ disappears.
This may feel as something has been lost.
The arising gap in the cosmological theories however can be filled up if one understands the dynamic interaction between objects and fields in space as a result of the gradient of the time speed at a specific location. Then the time speed differences can be considered as the physical cause of the movements in space.

 

  13Literature

 

1                    Lang H. de 2016 "Rimpels in de ruimte", Ned. Tijdschrift voor Natuurkunde 82, p. 141

2                     Dorrestijn H J 2016 "Een Kritische Blik op Einsteins Relativiteitstheorie", Filosofie 2, p.44; Antwerpen Garant Uitgevers

3                     Einstein A 1905 "Zur Elektrodynamik bewegter Körper", Annalen der Physik 17, p.891

4                     Einstein A 1916 "Die Grundlage der Allgemeinen Relativitätstheorie", Die Annalen der Physik 49, p. 822

5                    Einstein A 1918 "Dialog über Einwände gegen die Relativitätstheorie", Die Naturwissenschaften 48, p.697

6                    Dorrestijn H J 2015 Op het Spoor van de Tijd (Den Haag ISBN 9789087595289) p.135

7                    Dorrestijn H J 2018 "The Lorentz contraction as an artefact in the Theory of Relativity" (refer to http://www.einsteingenootschap.nl/)

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