THE MATHEMATICAL RELATIONS BETWEEN
DOPPLER EFFECT AND SPECIAL RELATIVITY

Han Erim

January 1, 2010

In this publication, I aimed at showing the mathematical relations of Doppler Effect with Special Relativity. Three significant terms related to Special Relativity could be directly explained through Doppler Effect. These terms are Space Deformation, Time Dilation and Simultaneousness.

In order to understand how these relations are formed, it is by all means that how (c+v) (c-v) mathematics which is the basic mathematics of Special Relativity is formed and the effects of “Ghost and Spring” that takes an important place in Alice Law are required to be known. It is because of the fact that all these effects which are included in the scope of Special Relativity occur and are observed on the Ghost.

SPACE DEFORMATION

I showed in detail how space deformation is materialized and how to calculate it in my work titled Special Relativity and Space Deformation . Now, I present the relation of space deformation with Doppler Effect.

As it is seen in the work of Space Deformation, the following equation exists.(1)

(1)

The frequency deviation on an electromagnetic wave which Doppler Effect is observed is given with the following equation. Reference “Doppler Effect and Special Relativity” (2)

(2)

By utilizing (1) and (2) equations, the relation between the frequency deviation on electromagnetic wave and space deformation is easily established (3)

(3)

TIME DILATION

I carried out a way as follows in order to put forward the relation of Doppler Effect with Time Dilation;

Based on my work titled “Special Relativity and Time Synchronization ”, I rewrote the source codes of 1st animation in a short cut which have been written according to the rules of (c+v) (c-v) mathematics in such a way that it would give identical result. By this way, I calculated how the tick intervals of the clock located in the GHOST have changed. I compared the tick intervals which I have derived with the tick intervals of the clock located in SPRING.

As it is seen from the calculations, the ratio of tick intervals at both clocks to each other is equal to c/(c±v) ratio. (Here, c is speed of light constant and v value is the speed difference between observer and Spring). Since the clock of spring and of observer work equal, the same ratio is valid between the clock of observer and of ghost.

After all, the effect which we will observe in the moving clocks depends on c/(c±v) ratio. (4)

(4)

At the same time, c/(c±v) ratio indicates the frequency change on electromagnetic waves so that the relation of Time Dilation with Doppler Effect emerges in this way (5).

(5)

Except for ActionScript, you may know a different programming language. Thus, I put the source codes of animation here, as well. If you know Flash CS3 ActionScript 3.0, you can download the source codes as “.fla” file from here .

 /* ============================= This source code is under GNU GENERAL PUBLIC LICENSE Copyright © 2009 Han Erim  ALICE LAW http://www.aliceinphysics.com ============================= */ var speed:int=-2; var relativeSpeed:int=-10; var rot:int=5; var springRotation:Number=2; var ghostRotation:Number=0; var speedOfLight:int=10; var distance:Number=0; var arrivalTime:Number; var sign:int= 1; var rememberSpring:Number=0; var rememberGhost:Number=0; addEventListener(Event.ENTER_FRAME,action); function action(e:Event):void { observer.x<=-400 || observer.x>=400 ? speed=-1*speed : "" ; //c-v observer.x >0 && speed<0 ? relativeSpeed= -Math.abs(speed) : ""; observer.x <0 && speed>0 ? relativeSpeed= -Math.abs(speed) : ""; //c+v observer.x >0 && speed>0 ? relativeSpeed= Math.abs(speed) : ""; observer.x <0 && speed<0 ? relativeSpeed= Math.abs(speed) : ""; alo.text=String(relativeSpeed); observer.x>0 ? observer.bottom.scaleX=1 : observer.bottom.scaleX=-1; //===== observer.x+=speed; distance=spring.x-observer.x; arrivalTime=distance/speedOfLight; observer.x>0 && relativeSpeed>0 ? sign=-1 : ""; observer.x>0 && relativeSpeed<0 ? sign=1 : ""; observer.x<0 && relativeSpeed>0 ? sign=-1 : ""; observer.x<0 && relativeSpeed<0 ? sign=1 : ""; ghost.x= sign*Math.abs(relativeSpeed)*arrivalTime; //========== observer.x>0 && relativeSpeed<0 ? sign=1 : ""; observer.x<0 && relativeSpeed>0 ? sign=-1 : ""; observer.x<0 && relativeSpeed<0 ? sign=-1 : ""; observer.x>0 && relativeSpeed>0 ? sign=1 : ""; springRotation+=rot; spring.second.rotation=springRotation; spring_timeInterval.text=String(springRotation-rememberSpring); rememberSpring=springRotation; ghostRotation=springRotation+sign*arrivalTime*rot; ghost.second.rotation=ghostRotation; ghost_timeInterval.text=String(ghostRotation-rememberGhost); rememberGhost=ghostRotation; proportion1.text=String(Number(spring_timeInterval.text)/Number(ghost_timeInterval.text)); observer.x>0 && relativeSpeed>0 ? sign=-1 : ""; observer.x>0 && relativeSpeed<0 ? sign=1 : ""; observer.x<0 && relativeSpeed>0 ? sign=-1 : ""; observer.x<0 && relativeSpeed<0 ? sign=1 : ""; proportion2.text=String(speedOfLight/(sign*Math.abs(relativeSpeed)+speedOfLight)); }

SIMULTANEOUSNESS AND SPEED OF PERCEPTION

SPEED OF PERCEPTION OF EVENTS AND SIMULTANEOUSNESS

Our speed of perception of events is a measure of which speed the actions on GHOST has occurred for us. The operating speed of the clock on Ghost is, at the same time, a measure of speed of our perception of events on Ghost.

As a matter of course, the smaller the tick intervals of the clock on Ghost is, the higher the speed of our perception of advancing events on the Ghost will be. Speed of perception and Time Dilation are inversely proportional to each other.

In order to show the equation, I will refer to my publication titled “Simultaneousness Concept and Special Relativity ”. As seen in the animation, three reference systems start to watch telecast at the same time. However, as seen again, flow rate of telecast is different for each reference system.
We know that signals from television station will reach each of the three reference systems with the speed of c. Consequently, the following equations exist between reference systems.

d=c.t=(c-v).t1=(c+v).t2

Here, the values of t, t1 and t2 are time values which show the flow rates of telecast and the sequencing is t1>t>t2. We can calculate the flow rate of telecast for the passenger on airplanes by making use of that equation.(6)

(6)

ERIM EQUATIONS AND ITS RESULTS

Our precision measuring instruments can perceive even rather small Doppler Effect. At the same time, this means to observation and measurement of Special Relativity.

It is really an important point to understand that the wavelength and frequency changes, i.e. the Doppler Effects and the effects of Special Relativity occur simultaneously. I feel the pride and pleasure of being the first person who discovered this truth. I wanted to name these equations with my surname.

The equations above is registered under Notary Public 37 of Beyoglu – Istanbul – Republic of Turkey on 29/December/2009 with no: 35035, by Han Erim.

In these equations, we can use the values belonging to our reference system or the values of any reference system which are stationary with respect to our own reference system for ds, ts , ƒs  , λ values.

Establish: December 2001