RELATIVITY IN QUANTUM

Azzam K.I. AlMosallami

Arab Consultants Group, P.O. Box 1067

Gaza, Palestine, Via Israel

Abstract

The concepts, principles and laws on which Einstein built his relativity theory (special and general) is in contrast with the concepts, principles and laws on which the quantum theory is built.

The goal of our work is to get a new thoery that agrees with the concepts, principles and laws of quantum and contains all the experimental measurements of the relativity. On other words, deriving the equations of the relativity which agree with the experimental measurements on the basis of the concepts, principles and laws of quantum (Copenhagen School).

In part 2, section 1, I derive a new formula for Newton’s second law, it expresses a quantized force, and agrees with the concepts, principle, and laws of quantum. In section 2, I also derive the quantized inertial force.

Introduction

When Einstein started building his special relativity theory, he was believed in the objective existance of the phenomenon, where we can find that in the derivation of equations of the relativity. Also, he was believed in the continuity principle and in the causality and determinism laws in the world.

Quantum theory discovered the observer has the main formation of the phenomenon, and that is clear in the definition of Heisenberg to the wave function (1958), where he defined it as " it is a mixture between two things, the first is the reality, and the second is our realizing to this reality."

Einstein was disagreed with this concept to the phenomenon, where Pais said (1979), when he was walking with Einstein, he said " look at the moon, do you believe it is existed because we are looking at it.

Also, quantum theory fosters the discontinuity principle, uncausality and indeterminism laws in the world.

The mathmatical formation of the relativity depends on Rieman’s space with four dimensions, but quantum on Hilbert space with infinite dimensions.

Stapp said (1972) " The Copenhagen School refused understanding the world as the concepts of (space-time), where it considers the relativity theory is inconsistant for understanding the micro world, where quantum theory is formed the basis for understanding this word."

In equations of relativity of Einstein, it can be measured the velocity of a particle and its location at the same time, but the experimental measurements proved contrary of that ( Heisenberg uncertainty principle )

Oppenheimer said " Einstein -in his last years researching- tried proving the inconsistancy of the laws of quantum theory but he failled. After all that Einstein said, he dislikes the quantum theory, especially Heisenberg uncertainty principle."

In our work, we define the reference frame as the frame at which the observer is static, and the inertial frame is that frame which its velocity is constant with time for any inertial frame of reference.

THEOERY

1- POSTULATES OF THE THEORY

1- The speed of light is constant and equals to in any inertial frame of reference , where is the speed of light in vacuum.

2- The speed of light in any frame moving with constant velocity is equal to for any inertial frame of reference, where, whereas does not depend on the direction of the velocity of the moving frame, it depends only on the absolute value of the velocity.

To understand the two potulates, suppose a static observer on the earth surface, in this case the earth surface is considered as a reference frame. If the static observer made an experiment for measuring the speed of light in his reference frame he would find it equals to . Also, if there is a train moving with constant velocity on the earth surface, and one of the static riders of it made an experiment for measuring the speed of light inside his train, in this case the moving train is considered as a reference frame, thus the speed of light that the rider would measure equals to , as for the static observer , and this is what the first postulate includes.

Now, suppose the static observer made an experiment for measuring the speed of light inside the moving train, in this case he would find it equals to , and this is what the second postulate includes. Now If then , that means in quantum the wave function , thus, the probability of getting any information inside the train for the observer approaches zero, where , where is the complex conjugate of .

2-TIME IN OUR RELATIVITY

( 2.1 ) Suppose a train at rest and a static observer , on the earth surface. The length of the train is . If one of the riders of the train sent a ray of light along the length the train. Thus the time required to the ray of light to pass the length of the train for the static observer and the rider is ,where

( 2.1.1 )

Now, suppose the train moved with constant velocity and then the rider sent a ray of light along the length of his train during the motion. If the static observer catches his clock and desired computing the time required to the ray of light to pass the length of the moving train. According to the second postulate, the speed of light inside the moving train is relative to the static observer, where . Thus the time required to the ray of light to pass the length of the moving train is for the observer, where

From the second postulate, we proposed, does not depend on the direction of transmitting the ray of light comparing to the direction of the velocity of the train. Also, the equation above is in contrast with the Lorentz transformation equations. Lorentz transformation equations built on the concepts of continuity, causality, and determinism, but, in our work we believe in the discontinuity, uncausality and indeterminism. The measurement that is taken in the equation above is taken from a wave function, and to get another measurement we must get another wave function ... , and vise versa and those wave functions are unrelated. Also, Lorentz transfomation equations proposed that we can measure the velocity of the train and its location at the same time, and that is in contrast with the uncertainty principle of Heisenberg.

From equation ( 2.1.1 ), we get

Then

Thus

( 2.1.2 )

Where is the time required to the ray of light to pass the length of the train when it is at rest.

Equation ( 2.1.2 ) means, the time separation of any event that happens in any moving frame with constant velocity is bigger than the rest time separation, (if the same event happens when the frame at rest) for any frame of reference.

( 2.2 ) Now, suppose one of the riders of the moving train catches his clock inside the train and he desires measuring the time required to the ray of light to pass the length of his train during the motion. According to equation ( 2.1.2 ), the time separation for any event which happens inside the train is bigger when it is moving than when it is at rest for the reference frame of the earth surface. And because the motion of the clock is an event inside the train, thus its movement will be slower when the train is moving than when it is at rest for the reference frame of the earth surface. Thus, the clock of the rider will be slower than the clock of the static observer. And, if we assumed, both the observer and the rider will agree on the beginning of the event and ending it inside the moving train, thus, if the observer computes by his clock the time for the ray of light to pass the length of the moving train, then the rider will compute the time , where

Where

Since from equation ( 2.1.2 )

Thus we get

Thus, we can write equation ( 2.1.2 ) as

( 2.2.1 )

According to equation ( 2.2.1 ), the speed of light for the moving rider according to his clock is , where

( 2.2.2 )

Equation ( 2.2.1 ) leads us to the first postulate of the theory, the slowing of the speed of light for any frame moving with constant velocity , leads to slowing of time in that frame (movement of clocks). Thus, the speed of light for all frames of reference is the same and equals to C.

Thus, in this case we have

 .

( 2.3 ) Suppose the static observer desires comparing the motion of clock of the moving rider with the motion of his clock. According to equation ( 2.1.2 ), and, because the motion of the clock of the rider is an event inside the moving train, thus, the clock will be slower when the train is moving than when it is at rest for the observer. Thus, if the observer looks at his clock and computes the time, in this moment he will find that, the clock of the rider computes the time where

( 2.4 ) Now, suppose the rider of the moving train desires using the clock of the static observer for computing the time required to the ray of light to pass the length of his train. The time which will be measured by the static observer via his clock is where

If we consider the rider is moving with constant velocity to the right, then the clock of the observer is moving with the same velocity to the left relative to the rider, in this case, the rider’s frame is considered as a reference frame and the clock as a frame moving with constant velocity for him. Thus, according to the preceding discussion, the clock will be slower for the rider than the observer for the reference frame of the earth surface. Thus, if the observer computes the time by his clock, in this moment the rider will compute the time by the same clock [ or by his clock inside the train as we have seen in ( 2.2 ) ], where

Suppose, the length of the train is , and its speed is . If the clock computes by where , then the time required to the ray of light to pass the length of the moving train for the static observer is , where

And

Then

Thus, the static observer will compute via his clock for the ray of light to pass the length of the moving train. For the rider, the time is where

So, the rider will compute for the ray of light to pass the length of his train. Both, the observer and the rider will agree on the beginning of the event and ending it, and when both used the same clock to compute the time separation to this event , the clock was slower for the rider than the observer. So, when the observer received to the time separation, in this moment the rider received only the first of the clock, where we can consider the rider lives in the past of the oberver of the earth surface.

In this example we find when both the rider and the observer used the same clock each one creates his clock to get his reading, and that is in contrast with the objective existance of the phenomenon, where in our example we obtain, the observer has the main formation of the phenomenon as in Copenhagen School concepts.

( 2.5 ) Now, suppose train A at rest, its length is , also there are train B moving with constant velocity and a static observer on the earth surface. Now, both the static observer and the rider of train B will measure the time required to the ray of light to pass the length of the static train A. For the observer, the measured time according to his clock is where

For the rider of train B, since train A is moving with constant velocity -, thus the speed of light inside it comparing to the reference frame of the static observer is , thus the rider should been computing the time for the event where

Where, is the time separation of the event when the train of the rider is static.

Because the rider’s clock is slow during the motion for the reference frame of the earth surface [ as we have seen in ( 2.2 ) ], thus, the rider will compute the time , where

( 2.5.1 )

Equation ( 2.5.1 ) means, both the rider of the moving train B and the static observer will measure the same time separation to the ray of light to pass the length of the static train A, that leads us to, the measured speed of light is the same for each one inside the static train A and it is equal to . Thus, we can write equation ( 2.5.1 ) as

If both the static observer and the rider of the moving train B agree with the time required to the ray of light to pass the length of the static train A, then, they will be different in the beginning of the event and ending it.

Let us assume both the observer and rider will agree on the beginning of the event, in the condition of

= 0 at = 0

= 0.87C at > 0

Where, at , before transmitting the ray of light, the velocity of train B of the rider was equal to zero, and after transmitting the ray of light, the velocity of the train was equal to 0.87C ( in this case, for simplicity we neglect the effect of acceleration ) . In this condition, the static observer and the rider of the moving train B will be agreed on the beginning of transmitting the ray of light inside the static train A, and different in ending it.

If the length of the static train A is , thus, the time required to the ray of light to pass the length of the static train A for the static observer is

For the rider of the moving train B is from equation ( 2.2.2 )

Because the time ( clock ) in the frame of the moving train B is slower than the time ( clock ) of the static observer for the reference frame of the earth surface, then, the ray of light will arrive to the end of the static train A faster for the observer than the rider. Thus, if the observer secures that, the ray of light arrived to the end of the train, in this moment the rider secures that the ray of light arrived to the middle of the train. Where, if the observer secures that the ray of light cut the distance , in this moment the rider will secure that the ray of light cut the distance , where, , also we get ,and . Now, if the observer looks at the clock of the rider, he will secure that the clock of the rider computes only in the moment that his clock computes ,where . But, if the rider looks at the clock of the observer he will secure that the clock of the observer computes only , as in his clock, while the observer secures that his clock computes .

( 2.6 ) Now, if the rider of the moving train desires using the clock of the static observer in the condition of

at = 0

at 0 < £ 4 sec.

at > 4 sec.

Where, is the reading of the static observer from his clock.

We can draw versus for the reference frame of the earth surface as in figure ( 2.6.1 ), where, is the reading of the rider from the clock of the static observer.

From figure ( 2.6.1 ), we find two straight lines, the first for 0 < £ 4 sec. ,

its slope = 0.5, and the second is for > 4 sec., its slope = 1

We find from the figure, the seconds between 24 sec. would not be received by the rider, where, when the train of the rider stopped at 4 sec., he found that the observer was reading the seconds at sec., while his last reading was equal to 2 sec. . That means, the events which were done by the static observer between 24 sec. were not be received for the rider of the moving train.

Figure ( 2.6.1 ): versus .

From the figure we get, the observer has the main formation of the phenomenon, where each one creates his clock during the motion, this is in constrast with the objective existance of the phenomenon.

( 3.1 ) Now, let’s go back to the rider of moving train and the static observer, both, will make an experiment for measuring the velocity of the moving train. They appointed two pylons and will measure the time required for the train to pass the distance between the two pylons. Assume, both, will agree on the beginning of this event. Now, the measured time by the static observer to the train to pass the distance is , according to his clock, thus, the measured velocity for him is where

The equation above is not in contrast with Heisenberg uncertainty principle, where and are not measured at the same time, where when the observer measured the distance precisely for the train, he predicted by the equation above the velocity of the train at the distance was equal to .

When the observer secures that the train arrived to the second pylon, in this moment - during the motion - the rider of the moving train does not secure that his train arrived to the second pylon, where, he secures that the train is still arriving to the second pylon. Now, if the static observer computes the time to the train to arrive to the second pylon, in this moment the rider will compute the time , also, if the observer secures the train travels the distance , in this moment the rider will secure that his train travels the distance ,( as we shall see in ( 3.4 ), figure ( 3.4.1 ) ). In this case, the measured velocity of the moving train for the rider is , where

Thus, both the static observer and the rider of the moving train will agree on the measured velocity of the moving train during the motion and they will measure the real velocity .

( 3.2 ) Now, suppose another static train A and inside it a static clock. Thus, as we have seen in ( 2.4 ) the rider of the moving train B secures that the motion of the clock of the static train A is analogous to his clock motion, where the time that he will measure it by his clock is equal to the time that he will measure it by the clock of the static train A. Also, the static observer secures that the motion of the clock of the static train A is analogous to his clock motion. Now, if the train A moved with constant velocity between the two pylons, then, as we have seen in ( 3.1 ), both the rider of the moving train B and the static observer will be agreed at the time separation for train A to pass the distance between the two pylons, then, they will agree at the measured velocity of train A, where each one will measure the velocity of the train equals to . If a ray of light is sent along the length of the moving train A, then, the measured time by the static observer for the ray of light to pass the length of the train is , where

Also, the measured time by the rider of the moving train B for the ray of light to pass the length of the moving train A is , where

Where, is the measured time by the rider of the moving train B for the ray of light to pass the length of train A when train A is static.

Since, both, the static observer and the rider of the moving train B are agreed at the time separation for the ray of light to pass the length of train A when it is at rest as from equation ( 2.5.1 ), then we get

Thus

In this example we have seen the motion of train B of the rider did not affect to the calculations of the rider, where the calculations of the rider were similar to the calculations of the static observer, but the motion of train B made the rider to get these calculations slower than the static observer.

( 3.3 ) Now, suppose a ball is moving with constant velocity on the earth surface. The rider of the moving train and the static observer will be agreed on the measured velocity of the ball, where both will measure the velocity equals to . If this ball entered inside the moving train of the rider, then, the rider will compute the time for the ball to pass the length of his moving train, and the static observer will compute the time where . Now, both the rider and the observer will agree at the beginning of this event and ending it, but, they will be different in the time separation of the event. In this case, the measured velocity of the ball for the rider inside his moving train is where

And, the velocity of the moving ball inside the moving train for the static observer is , where

Thus

From equation ( 3.3.1 ), the motion of the ball inside the moving train of the rider will be slower than outside for the static observer and the measured velocity of the ball inside the train will be less for the static observer than the rider.

, ³ 10

This condition illustrates the velocity of the moving train of the rider comparing to the distance that is travelled by the train for the static observer. The graph of versus as in figure ( 3.4.1 ), where, is the distance which is travelled by the train for the rider. From figure ( 3.4.1 ), we find that, if the observer secures that the train travels , the rider will secure that his train had just travelled the distance , and after that the train stopped. Now, if the rider disamouts from his train and compues the distance that is passed by his train, he will find it equals to not , and he will be taken aback, that he transformed from to suddenly. Where he will be secured that the distance 510 is not passed by his train, but the static observer secures that the train passed this distance.

Figure ( 3.4.1 ): versus , the slope of the straight line is equal to 0.5.