This document contains wording terminology errors and sections that are too long or worded badly and I will write another document instead of fixing this one, although some of the material in it maybe used to inspire future documents
Copyright Carl Janssen 2025
Doppler Shifts in three dimensions
What about moving mediums
If both the source of emission and the receiver of emission are described as moving in a straight line in the same direction at the same constant speed v with no acceleration nor rotation in a reference frame in which the medium of propagation is stationary and has a uniform flow of zero
that is equivalent
to the same event described in a reference frame in which the source and receiver are stationary and the medium of propagation is moving in a straight line with no acceleration nor rotation with a uniform flow at a constant speed of v
and vice versa
What about the Fizzeau experiment
The Fizzeau experiment involved laminar flow not uniform flow because water travels slower near the edges of pipers than the center of pipes and because the pipes bent, the laminar flow was not always parallel or anti-parallel or in the same or opposite direction as the direction the light wave or particles were expected to propagate but closer to perpendicular to it or at least off by some angle like 45 degrees at some locations ( how far the angle deviated away from parallel will be different in different locations ) every time a corner was approached
Please skip ahead to the bold section labeled resume reading
TLDR Too Long Don't Read
You the reader can probably skip reading to the next bold section which is on the interferometer experiment.
The following paragraph about the Fizzeau experiment and partial vs complete Aether drag is not important for the three dimensional Doppler calculations I will demonstrate involved when the medium of propagation is stationary relative to the reference frame but I am putting this in as a note right now so I might not forget it later. It is not important in the sense that I did not base my calculations on the Fizzeau experiment. But it is important in that the calculations would be different if I accounted for the factor in the Fizzeau experiment.
The Fizzeau experiment might indicate that from a reference frame in which the source of emission is stationary that light travels at a different speed when water is moving in laminar flow as opposed to when it is stationary from that reference frame. The increase in light speed might not be v as I would predict ( with uniform flow ) from my model but v multiplied by a factor calculated as a function of v in both special relativity and the lorenz aether transformation however not necessarily calculated from velocity in the same context as those theories. I am not really sure about this being correct but I am putting it in as a note to check later because it may undermine my theory which would predict a increase by v instead of v multiplied by a factor. The specific part of my theory it would undermine is how to calculate Doppler shifts in three dimensions as it would give a different result however it would not undermine all the claims or points of my theoretical model.
The reason it would undermine my theory as well as the classical Doppler model is because it would mean that waves would propagate faster in the direction a source is moving which you would figure out when you switch reference frames to one in which the medium of propagation is stationary and the source is moving. In both classical Doppler and in my theory if the source is moving that does not change the speed at which a wave the source emits is propagated through the medium after it is emitted.
I believe that the addition of v multiplied by a factor was called partial Aether drag as opposed to complete Aether drag in which the addition of v without any factor modifying it was used. According to my theory the medium of propagation for the light would be water and not Aether but I believe that my theories calculations would be closer to complete Aether drag theories when the medium of propagation is in uniform flow.
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What about the ( Michelson Morley ? ) interfermoter experiment
When there is a reference frames in which the medium of propagation is traveling with uniform flow at a constant speed v in a straight line with no acceleration nor rotation and the source of emission and the receiver of emission are also traveling in the same direction at the exact same constant speed v in a straight line with no acceleration nor rotation then this is equivalent to another reference frame in which the medium of propagation is stationary and has a uniform flow with a velocity of zero, the source of emission is stationary and has a velocity of zero and the receiver of emission is stationary and has a velocity of zero
In my opinion it was erroneously assumed that the earth was moving at a certain speed and direction which although certainly was true from some reference frame was not true from another reference frame. It was further more erroneously assumed that light was traveling through aether which the earth was moving relative to. My theory assumes that perfect vacuums do not exist. My theory assumes that if light is traveling through water or air or some mixture of chemical compounds than the velocity of the water or air or mixture of chemical compounds it is traveling through should be measured than used instead of making up a speed of Aether because some people argue about whether the earth is traveling and the sun is moving or the sun is traveling and the earth is moving.
In my opinion when the alleged movement of the aether which was set as the speed of the asserted movement of the earth was used to determine the speed at which things that were stationary relative to one another were moving people were shocked to get a null result and concluded that moving object shrunk and time was distorted but they should have made no such conclusion because the medium of propagation, the source of emission and the receiver of emission were all stationary relative to one another so they should have plugged in a velocity of zero instead and got no such result. Even if you shake the interferometer very fast with your hand or put it on top of a car it still moves much slower than the alleged assertion of the speed at which the earth moves and if the medium is contained inside the interfermoter and moves with the interferometer which would be the case for some types of interferometers but not others than the relative movement of the source of emission, the receiver of emission and the medium of propagation would still be zero for one relative to the other.
Does the Fizzeau experiment undermine my claims? No because whether partifal Aether drag calculations or complete Aether drag calculations are used when the velocity is set to zero for source, medium and receiver relative to one another the results are the same even though they would be different for non zero velocities.
What about clocks having different changes in time values as predicte\d by special relativity and the Lorenz Aether Transformation?
Change in time displayed on clock = Reaction Rate * Change in Actual Time
Reaction rate definition vagueness TLDR skip until resume reading if this confuses
The use of the term reaction rate is an over simplification since a clock is calibrated based on tracking the progress of a reaction and the relationship between the time displayed and the progress of the reaction that is being tracked might not be linear. By reaction rate I mean the rate at which the time displayed changes per change in actual time not the rate of another reaction which is being tracked to get to the displayed time, I am skipping steps since every clock has a different mechanism and I am skipping the step in which a function is applied to the progress of the reaction to calculate the time to display.
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In my opinion Special Relativity and Lorenz Aether Transformation predicted how far specific types of clocks deviated from displaying the correct time or how badly they were incorrectly calibrated when they started moving relative to a certain thing and not how much time actually went past, many fiction stories involving time travel assume their prediction involves how much time actually passed instead of how wrong the clocks became
If two people start with their watches ( clocks ) sent to the exact same time and go about their day then come back and meet each other and one clock is different than the other, the reasonable thing to assume is not that a different amount of time passed for each person but that the same amount of actual time passed for each person wearing a watch ( clock ) but some physical, chemical or nuclear mechanism caused the clocks to have different reaction rates then what one or both of the clocks were calibrated for and that change in reaction rate resulted in one or both of the clocks being wrong.
I propose that special relativity and the Lorenz aether transformation might have correctly calculated by coincidence how the reaction rate of a specific type of Cesium atomic clock is modified as it moves relative to the surface of the earth or relative to the source of the cosmic microwave background. I would suggest that the cosmic microwave background is not evidence of the big bang as some materialistic atheists and some creationists propose nor is it evidence of creation as some creationists propose nor is the cosmic microwave background related the alleged expansion of the universe but that the cosmic microwave background is simply black body radiation from what is erroneously called the vacuum of outer space which is not a perfect vacuum but a medium of a mixture of chemicals with a specific type, speed and direction of flow that simply has a lower density than air near the surface of the earth. I would suggest that earth is moving relative to this medium and that this medium is also a source of emission of radiation. I would suggest that when clocks on earth move relative to this source of emission the radiation from the cosmic microwave background is Doppler shifted and this effects the reaction rates of a specific type of atomic cesium clock. I would also suggest that a B field may additionally be involved as charges in the clock are moving relative to charges in the medium that is erroneously called the vacuum of outer space.
Although no perfect vacuum exists the distance at which light must travel through a medium of a certain pressure, density and chemical concentration or greater to reach the surface of the earth is longer in the horizontal direction than the vertical direction. This may result in ignoring the movement of the receiver relative to a source of emission parallel and anti-parellel to it's direction of travel but paying greater emphasis on it's movement perpendicular to the source of emission. When I say perpendicular to and parallel to this is an over simplification to decrease the number of words and will hopefully be explained better later.
I plan to compare the what I will call right handed, left handed and symmetric Doppler shift calculations in which there is perpendicular motion to see if they resemble similarity to calculations for factors in the Lorenz Aether Transformation and Special Relativity.
I am very skeptical of the Doppler shift calculations I present matching real life results and the calculations predicted are for a very narrow over simplified situation and may result in different answers for right hand, symmetric and left hand calculations. My intuition is that the faster one moves relative the the source of the cosmic microwave background the more radiation one should receive but my right handed calculations show there should be less radiation when only radiation from perpendicular motion is considered. I am not sure if other people use the phrase right hand in terms of doppler shift calculations or if this is a term I made up. I use this term because it reminds me of the right hand vs left hand vs symmetric derivative calculation in calculus although my calculations are not the same as a derivitive.
What about gravity and time displayed on clocks and the general theory of relativity
There is no reason that gravity could not influence the reaction rate at which the time displayed in a clock progresses as actual time progresses
What about length changes in special relativity and Lorenz Aether Transformation
Copied and pasted from what was labeled as assumption number 9 at the time it was copied and pasted
If someone uses the change in time displayed on a clock instead of the change in actual time to calculate how far two objects travel away from or towards one another in a straight line at a constant speed then they may get the wrong answer. If the speed is measured correctly but the time is measured incorrectly if the time is measured to take N times as long as it actually does then the distance will be incorrectly calculated to be N times as long as it actually is.
N can be a fraction less than 1 or a positive number greater than 1. If the clock calibration is really messed up and is running backwards N could be a negative number.
What about allegations that the earth is moving or stationary?
There are some reference frames in which the earth is moving and there are some in which the earth is stationary. What is more important to think about is not what is and is not moving but what is moving relative to something else. I suggest that you can look up in the skies and see that the stars are moving relative to the earth and that the earth is also moving relative to the stars. I suspect that outer space is not a perfect vacuum and that we are moving relative to it and moving relative to the earth will effect how much cosmic microwave background radiation is received by cesium atomic clocks which effects how much time they display as passing per actual time that passes via Doppler shift. I also suspect that it is very important to consider that the source, medium of propagation and receiver were stationary relative to one another during the ( Michelson Morley ? ) interferometer experiment and comparing their velocity with the alleged speed at which the earth was moving instead of with one another lead to incorrect conclusions about time dilation and space dilation or constriction.
What about the relativistic Doppler shift vs the standard Doppler shift
If one uses a clock to count how many times a signal is received during a one second time period but a clock incorrectly displays N seconds have passed then the number of signals per second will give the wrong answer and the wrong answer it will give will be the correct answer divided by N. N can be a fraction less than 1 or a positive number greater than 1. If the clock calibration is really messed up and is running backwards N could be a negative number.
Assumptions of model
0 There might be more assumptions not on this list
1 Reference frames describe events, changing what reference frame you describe an event from can not change the physical nature of an event
2 Perfect Vaccums do not exist. The correct medium to describe for the propagation of light is the mixture of chemicals that light is propagating through and not a perfect vacuum nor aether. This mixture of chemicals has a specific type and velocity of flow which unlike a perfect vacuum or aether can be objectively measured and agreed upon by scientists. When the mixture is vacuum pumped the density decreases closer to zero but does not reach zero. The direction, speed and type of flow of the medium influences how light is propagated.
3. During the same event the same clock will display the same time in all reference frames but different clocks can display different times.
Different observers may read different times on the same clock during the same event because of the time it takes for the light on the clock display to reach the observers eye but the time displayed on the same clock is the same for all reference frames for the same event, the different observers who co-exist during the same event at different distances from the same clock would be looking at light emitted during different events even though their observation would be described during the same event.
4. The absolute value of the change in actual time between the same two events is the same for all reference frames but the change in time displayed between two events maybe different for different clocks because some clocks may not be calibrated correctly to account for environmental factors that change their reaction rate due to their relationship with their outside environment. The absolute value is used because sometimes people "count down" instead of "counting up"
5. Events always happen in the same order and changing the reference frame at which a list of events are described in can not change the order in which events occur
6. The length from one part of an object to another part of an object is the same during the same event or in other words it's shape and size are the same for the same event no matter what reference frames the event is described in
7. The distance between two objects is the same during the same event no matter what reference frames the event is described in.
8. The same object can have a different shape and size during a different event in response to a change in pressure distribution upon the object, if there is no change in pressure distribution ( for an object in dynamic equilibrium ? ) then a object can not change shape and size for no reason just because it is moving relative to something. However moving relative to something can change the pressure distribution on a object.
9. If someone uses the change in time displayed on a clock instead of the change in actual time to calculate how far two objects travel away from or towards one another in a straight line at a constant speed then they may get the wrong answer. If the speed is measured correctly but the time is measured incorrectly if the time is measured to take N times as long as it actually does then the distance will be incorrectly calculated to be N times as long as it actually is.
10. There are only three orthogonal spatial translations dimensions in real life, if someone is to claim a fourth dimension that is perpendicular to the other three then they need to take a ruler and measure something in that dimension to prove it.
11. There are only three orthogonal rotational dimension in real life. If someone claims there to be a fourth orthogonal rotational dimension then they need to measure the rotation of something in the fourth dimension and give a measurement of the angle to prove it
12. Time is not a spatial dimension.
13. The speed at which one object moves toward or away from another object is the same in all reference frames during the same event but the speed at which an object moves relative to the coordinates of the origin of a reference frame is different in different reference frames.
14A Optional Assumption ( Is this called Complete Aether Drag? Except my theory does not involve Aether )
The speed of light relative to a medium that is isotropic in which it is propagating is the same in all reference frames but the speed of light is not the same in all reference frames relative to the origin of the reference frame. The speed at which light moves once emitted is not effected by the speed at which the source at which it is emitted is moving relative to the medium of propagation but the location at which it is emitted from each time it is emitted is effected by the movement pattern of the source as a function of time.
TLDR skip reading until resume reading
14B Optional Assumption ( Is the called Partial Aether Drag? Except my theory does not involve Aether )
The veloicty at which light moves once emitted from a source is effected by both the velocity of the source and the velocity of the medium but not in a matter of simple addition. Instead it is multiplied by a factor that can be calculated in the Lorenz Aether Transformation times the difference in velocity between the source and the medium plus the velocity of the medium. And I could very well have this math wrong. I do not believe the Fizzeua experiment can be generalized this way because the flow was laminar and sometimes the wrong direction and also my idea about how to interpret the results could be deeply wrong.
I assume 14A instead of 14B when doing doppler shift calculations later and choosing 14B would have given different results.
14C Whether 14A is assumed or 14B is assumed ( Michelson Morley ? ) interferometer experiment should have produced a Null result because the source, the medium and the receiver were stationary relative to one another even if the earth was asserted to be moving. There was no justification for time or length dilation or constriction to explain the null results.
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15 Light emitted from a source never reaches it's receiver before it is emitted
16. The universe is not expanding or shrinking and it is a steady state universe.
Although unproven and not necessarily true I start with this assumptions instead of factoring in the expansion of the universe from some big bang or creation event into the mathematical models of my theory.
Too long don't read TLDR skip until the next resume reading.
red shifts in star light occur because light is traveling through a medium which is a mixture of chemicals from outer space as well as possibly for other reasons and not because space is expanding from a big bang. The farther light travels through a medium the more it is red shifted and different mediums red shift lights a different amount over a different distance. Gravitational red shift might also occur. The father away a light source is the more light it must produce to be seen on earth and the more light it must produce the more mass it is more likely to have and the more mass it is more likely to have the greater the gravitational red shift before the light reaches earth, this is also one of the many reasons why a universe of unlimited volume would not have unlimited brightness to an observer on earth.
The universe is unlimited in size and mass and there is no part of the universe with zero density but any section of the universe with a finite volume has a finite mass. The universe is eternal and is a steady state universe not a universe that came about through a big bang.
If the universe was created by some god or gods then the universe was created in a way that it has the properties of a steady state universe from the moment it came into being some finite amount of actual time ago or the creator god or gods created it to always exist both forwards and backwards in time eternally with steady state properties such that the creator created it to always exist no matter how far back or forward you go in actual time.
If there is no creator god then the universe existed eternally as a steady state universe without being created by a god.
The earth does not experience unlimited brightness from a universe of unlimited size because the farther away a star is the greater amount of light must be emitted to be viewed from earth and because some objects in space or opaque and block light from other objects.
Entropy and thermodynamic laws do not just gradually result in the universe becoming hotter and hotter and too vaguely defined somehow become to disordered for living things to continue to exist in some places or another for eternity because hot objects cool down and emit light, the light can be harvested and used by living things to create things in an ordered manner, entropy stuff does not prove the universe is of a finite age, we do not have to worry about some vaguely defined claims about entropy resulting in extinction of all life. It is not valid to claim the universe could not have existed for eternity because by now entropy would have prevented life from continuing to exist after an eternity of time from some starting point so since we do exist the universe must have started a finite amount of time ago.
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Difference between light frequency and frequency of emissions
A photon with a frequency of 1000 hertz could be emitted once per second or photons with a frequency of 1 hertz could be emitted 1000 times per second. The frequency of a photon is not the same as the frequency at which a photon is emitted.
How to correctly calculate period of emission and frequency of emission
If a photon is sent every second starting at the zero second mark and one simply takes a range of time and counts the number of photons emitted during that range of time then divides it by the range of time then one will get the wrong answer for the frequency of emission unless they are extremely lucky or already know the answer and chose the correct time interval to get the desired answer. Choosing a different time interval to sample photon emissions will give a different nnswer.
If emissions occur at the 1, 2, 3, 4, 5 and 6 second marks then if one measures that there is one emission occurring between 0.5 seconds and 1.5 seconds one would get the correct frequency by taking 1 emission divided by the 1 second range calculated by 1.5 - 0.5 but if they went between 0.1 and 1.9 and counted 1 emission divided by a 1.8 second range they would get a lower frequency then is actually occurring and if they counted 1 emission in divided by a 0.2 second range between the 0.9 and 1.1 then they would get a higher frequency then is actually occurring. For a larger time interval in which more emissions occurred the error would tend to get smaller and smaller but the probability of selecting a time interval in which there is no error by this method although greater than zero is very low.
Instead to get a better answer one would need to list all the times that a photon is emitted. One would get the period of emission by subtracting the two nearest times one from the other. One could then calculate the frequency of emission by taking the reciprocal of the period of emission. There is a problem with this method is the photons are not emitted at a constant rate.
I define the right handed calculation of the period of emission at a moment in time as the time at which the photon will next be emitted minus the time in which the photon was emitted at the moment in time you want the period for
I define the left handed calculation of the period of emission at a moment in time as the time at which the photon at the moment in time you want the period for minus the time in which the photon was previously admitted
I define the symmetric calculation of the period of emission at a moment in time as the time at which the photon will next be emitted minus the time in which the photon was previously emitted divided by two. Where next and previously are relative to a time when a photon was emitted that you want to calculate the symmetric period of emission for
Periods of reception can be calculated the same way but using the word received instead of emitted
The frequency of emission is the reciprocal of the period of emission
The frequency of reception is the reciprocal of the period of reception
Parameters required to do calculations
1 The medium is stationary and of a uniform flow of zero meters per second relative to the reference frame. If the medium is not stationary but it is of uniform flow then another reference frame can be used in which the medium is stationary to do these calculations. Changing the reference frames will change the coordinates of the source of emission and the receiver of emission. These calculation methods can not be used correctly for mediums with other types of flow than uniform flow, these methods do not work for laminar flow for example.
2 The medium is a isotropic medium in which the speed of light propagation is the same in all directions
3 The source of emission is moving relative to the medium at less than the speed of light propagation for that medium
4 The receiver of emission is moving relative to the medium at less than the speed of light propagation for that medium
5 The receiver of emission is moving relative to the source of emission at less than the speed of light propagation for that medium
6 The period of emission from the source is constant and does not change as a function of time
How to do calculations
Step 1
Having achieved a reference frame for which all the required parameters are met. Express the coordinates of the source and the receiver of the emission as a function of time. If the medium is moving rewrite the problem in a reference frame in which the medium is stationary and express the coordinates of the source and receiver keeping this adjustment in mind.
If in a reference frame in which the medium is moving with uniform flow
The x, y and z components on the velocity of the medium are Vmx, Vmy and Vmz
The velocity components of the source of emission are Vex, Vey, Vez
The velocity components of the receiver are Vrx, Vry and Vrz
Then this can be rewritten in a reference frame in which the medium is stationary as
New Vmx = Old Vmx - Old Vmx = 0
New Vmy = Old Vmy - Old Vmy = 0
New Vmz = Old Vmz - Old Vmz = 0
New Vex = Old Vex - Old Vmx
New Vey = Old Vey - Old Vmy
New Vez = Old Vez - Old Vmz
New Vrx = Old Vrx - Old Vmx
New Vry = Old Vry - Old Vmy
New Vrz = Old Vrz - Old Vmz
Where new refers to the coordinates in the new reference frame we created in which the medium was described as stationary and old refers to the coordinates in the old reference frame in which the medium was described as moving
Step 2
Set up a differential equation
The distance between the receiver at the time of reception and the source at the of emission is equal to the speed of light propagation for the medium times the time of reception minus the times of emission
Blank ^ 2 = Blank Squared
Xe = X coordinate of source of emission at time of emission
Xr = X coordinate of receiver of emission at time of reception
Note : It is very important to understand that Xe and Xr are spatial coordinates for different time values because it takes time for an emitted photon from a source to travel to the receiver at which it is received
Te = Time of emission
Tr = Time of reception
c = speed of light in that medium
( Xr - Xe ) ^2 + ( Yr - Ye ) ^ 2 + ( Zr - Ze ) ^ 2 = c ^2 * ( Tr- Te ) ^ 2
Plug in equations from the previous step
Step 3
Using the differential equation created in the previous step
Solve for Te as a function of Tr
Solve for Tr as a function of Te
If Tr as a function of Te is at a early time than Te then the result must be excluded or is wrong
There should only be one solution for Te as a function of Tr for each Tr value
There should only be one solution for Tr as a function to Te for each Te value
Step 4
To figure out the period of reception at the time of reception
RPr = Period of Reception right hand
LPr = Period of reception left hand
SPr = Period of reception symmetric
Pe = Period of emission
Tr ( Te ) = Time of reception as a function of time of emission solved in previous step
Te ( Tr ) = Time of emission as a function of time of reception solved in previous step
RPr = Tr ( Pe + Te ( Tr ) ) - Tr
LPr = Tr - Tr ( Te ( Tr) - Pe )
SPr = Tr ( Pe + Te ( Tr ) ) - Tr ( Te ( Tr) - Pe )
Step 5
Take the recipricol of the period to get the frequency
Doppler shift frequency multiplier is frequency of reception divided by frequency of emission
Examples of calculations for parallel, perpendicular and circular movement
one dimensional source and receiver move parallel or anti-parallel to one another at a constant speed with no acceleration and are in a straight line from one another in the direction that they are moving
Step 1 write function of coordinates as a function of time
Vex = Velocity of source of emission
Vrx = Velocity of receiver
Xe(Te) = Position of source of emission as a function of time of emission
Xe0 = Position of source of emission at time of 0 seconds
Xr(Tr) = Position of receiver of emission as a function of time reception
Xr0 = Position of receiver of emission at a time of 0 seconds
Xe(Te) = Xe0 + Vex*Te
Xr(Tr) = Xr0 + Vrx*Tr
Step 2 Set up differential equation
( Xr - Xe ) ^2 + ( Yr - Ye ) ^ 2 + ( Zr - Ze ) ^ 2 = c ^2 * ( Tr - Te ) ^ 2
since this is one dimensional this can be rewritten as
( Xr - Xe ) ^2 = c ^2 * ( Tr - Te ) ^ 2
Xr - Xe = c * ( Tr - Te )
Plug in equations from the previous step
Xe(Te) = Xe0 + Vex*Te
Xr(Tr) = Xr0 + Vrx*Tr
Xr0 + Vrx*Tr - Xe0 - Vex*Te = c*Tr - c*Te
Putting receiver of emission and source of emission type variable on different sides of equation isolating them from one another to make it simpler
c*Te - Vex*Te - Xe0 = c*Tr - Vrx*Tr - Xr0
Step 3 solve for Tr as a function of Te and Te as a function of Tr
Solve for Te as a function of Tr
c*Te - Vex*Te - Xe0 = c*Tr - Vrx*Tr - Xr0
Moving everything that does not contain a multiple of Te to the right side of the equation
c*Te - Vex*Te = c*Tr - Vrx*Tr - Xr0 +Xe0
Putting everything that multiplies by Te inside parenthesis
Te * ( c - Vex ) = c*Tr - Vrx*Tr - Xr0 +Xe0
Solving for Te by dividing by the stuff inside parenthesis from the previous step
Te = ( c*Tr - Vrx*Tr - Xr0 + Xe0 ) / ( c - Vex ) also called Te ( Tr )
Solve for Tr as a function of Te
c*Te - Vex*Te - Xe0 = c*Tr - Vrx*Tr - Xr0
Switching left and right side because usually the variable you want to solve for is written on the left in the final solution so this makes it easier to think about for me
c*Tr - Vrx*Tr - Xr0 = c*Te - Vex*Te - Xe0
Moving everything that does not contain a multiple of Tr to the right side of the equation
c*Tr - Vrx*Tr = c*Te - Vex*Te - Xe0 + Xr0
Putting everything that multiplies by Tr inside parenthesis
Tr * ( c - Vrx ) = c*Te - Vex*Te - Xe0 + Xr0
Solving for Tr by dividing by the stuff inside parenthesis from the previous step
Tr = ( c*Te - Vex*Te - Xe0 + Xr0 ) / ( c - Vrx ) also called Tr ( Te )
Step 4 figure out the period of reception ( Pr ) at the time of reception
Tr ( Te ) = Time of reception as a function of time of emission solved in previous step
Te ( Tr ) = Time of emission as a function of time of reception solved in previous step
Pe = Period of emission
Right Hand RPr = Tr ( Pe + Te ( Tr ) ) - Tr
Left Hand LPr = Tr - Tr ( Te ( Tr) - Pe )
Symmetric SPr = Tr ( Pe + Te ( Tr ) ) - Tr ( Te ( Tr) - Pe )
Te ( Tr ) = ( c*Tr - Vrx*Tr - Xr0 + Xe0 ) / ( c - Vex )
Tr ( Te ) = ( c*Te - Vex*Te - Xe0 + Xr0 ) / ( c - Vrx )
Tr ( Te + Pe ) = ( c* [ Te + Pe ]- Vex* [ Te + Pe ] - Xe0 + Xr0 ) / ( c - Vrx )
RPr = Tr ( Te + Pr ) - Tr = ( c*Pe - Vex * Pe ) / ( c - Vrx )
Tr ( Te - Pe ) = ( c* [ Te - Pe ] - Vex* [ Te - Pe ] Te - Xe0 + Xr0 ) / ( c - Vrx )
- [- Pe ] = + Pe
0 - ( -1 * -1 ) = 0 - 1
0 - ( -Vex * - Pe ) = 0 - Vex * Pe
LPr = Tr - Tr ( Te ( Tr) - Pe ) = ( c*Pe - Vex * Pe ) / ( c - Vrx )
SPr = LPr = RPr whether left hand, right hand or symmetric calculation the period of reception is the same in this case
Pr = ( c*Pe - Vex * Pe ) / ( c - Vrx )
A bunch of stuff cancels out when the two functions are subtracted from one another because everything that is not multiplied by Pe appears the same on both functions, I did not want to write out something so long so just skipped to the final answer
Step 5
Take the recipricol of the period to get the frequency
Doppler shift frequency multiplier is frequency of reception ( Fr ) divided by frequency of emission ( Fe )
Fr = 1 / Pr = ( c - Vrx ) / ( c*Pe - Vex * Pe )
Fe = 1 / Pe
Fr / Fe = Fr / ( 1 / Pe ) = Fr * Pe = Pe * Fr
Fr / Fe = Pe * ( c - Vrx ) / ( c*Pe - Vex * Pe )
Fr / Fe = c - Vrx / ( c - Vex )
matches Wikipedia if the correct plus or minus sign is chosen and the equivalent variable names in wikipedia which mean the same thing but use different symbols are chosen. It is concerning that the positive or negative sign in my original equations had meaning based on the context of the initial positions of the source of emission and the receiver of emission and their velocities so you can tell if they are going towards are away from each other based on the context in the original equations but is missing in the final equation
https://en.wikipedia.org/wiki/Doppler_effect
Perpendicular straight line Doppler shifts
The receiver is moving in a straight line at a constant speed and at the position where it is desired to measure the period of reception the direction the receiver is moving is perpendicular to the direction between the receiver and the source of emission. The source of emission is stationary.
Step 1 Express the Coordinates of the Source of Emission and the Receiver of Emission as a function of time
Ye = Position of source of emission in Y direction
Yr = position of receiver of emission in Y direction = 0
K = Ye - Yr = I am calling it K as a reminder that it is a constant not a variable
Xr0 = position of receiver at a time of 0 seconds = 0 meters
Xr ( Tr ) = position of receiver at time of reception = V*Tr
Step 2 Differential Equation based on Coordinate function in previous step
( Xr - Xe ) ^2 + ( Yr - Ye ) ^ 2 + ( Zr - Ze ) ^ 2 = c ^2 * ( Tr - Te ) ^ 2
( Yr - Ye ) ^ 2 = K^2
( Zr - Ze ) ^ 2 = 0
( Xr - Xe ) ^2 = ( V*Tr ) ^ 2
( V*Tr ) ^ 2 + K ^ 2 = c ^2 * ( Tr - Te ) ^ 2
( Tr - Te ) ^ 2 = Tr ^2 - 2 * Tr * Te + Te ^ 2
( V*Tr ) ^ 2 + K ^ 2 = c ^2 * ( Tr ^2 - 2 * Tr * Te + Te ^ 2 )
Put it into a form in which Tr or Te can be solved in terms of the other using the quadratic equation
0 = c^2 * ( Tr ^2 - 2 * Tr * Te + Te ^ 2 ) - [ ( V*Tr ) ^ 2 + K ^ 2 ]
0 = ( c^2 - V^2 ) * Tr^2 + 2 * c^2 * Te * Tr + c^2 * Te^2 - K ^ 2
Solve for Tr
A = c^2 - V^2
B = 2 * c^2 * Te
C = c^2 * Te^2 - K ^ 2
sqrt ( blank ) = square root of blank
Tr = ( - B + or sqrt ( B ^ 2 - 4 * A * C ) ) / 2 * A
B ^ 2 = 4 * c ^ 4 * Te ^ 2
4 * A * C = 4 * ( c^2 - V^2 ) * ( c^2 * Te^2 - K ^ 2 )
4 * A * C = 4 * c ^ 4 * Te ^ 2 - 4 * V ^ 2 * c ^ 2 * Te ^ 2 - 4 * ( c ^ 2 - v ^2 ) * K ^ 2
B ^ 2 - 4 * A * C = 4 * V ^ 2 * c ^ 2 * Te ^ 2 + 4 * ( c ^ 2 - v ^2 ) * K ^ 2
2 * A = 2 * ( c^2 - V^2)
Solve for Te
A = c^2
B = 2 * c^2 * Tr
C = ( c^2 - V^2 ) * Tr^2 - K ^ 2
Step 3 Using the differential equation created in the previous step, Solve for Te as a function of Tr & Solve for Tr as a function of Te
Graph, image, picture or text comparing black body radiation with cosmic microwave background from wikipedia
Cesium Atomic Clock frequency compared with cosmic microwave background frequency range from wikipeida
Comparison of perpendicular Calculation with factors in Lorenz Aether Transformation and Special Relativity with Doppler shifting multipliers for Cesium Frequency assuming perpendicular propagation from the boundary of "outer space" and Earth's atmosphere ( The Carmen Line ? )
Experiment
Measure cosmic microwave background in an object that is moving and see if a function to predict it can be made and if that function gives reliable predictions in a retest
Take a stationary atomic clock and add radiation which replicates the added radiation equivalent to that from moving at a certain speed minus the radiation it receives when standing still
Remember going West vs East should modify radiation amount in the opposite direction up until the point at which one is stationary relative to the source of radiation from the cosmic microwave background at which point going faster in the same direction will change whether radiation is increased or decreased
One might not be able to subtract radiation so just choose a signal that replicates the radiation at a speed and direction in which the radiation is decreased
See if the radiation added to the stationary clock which is equivalent to the change in radiation from moving replicates similar changes in the display time to a moving clock that moved in such a manner
It might not entirely give the same results if the B field also effects the results
And an important question is if there is a statistical interaction effect between B field and radiation equivalent to the cosmic microwave background on the clock time
One might also try to replicate a similar B field to the B field that would be created by moving relative to the medium of outer space that is erroneously called the vacuum of outer space but I am not sure if that is possible
Since atomic clocks are very expensive the general public will never know if such results are actually true whether they support or oppose the guesses I am making, until atomic clocks and fast transportation methods of such clocks is affordable to the general public. They can choose to trust the reports of others but can not have first hand knowledge for them self. As such I believe this article may help people in the distant future but is only speculation for now.
Other experimental ideas that would be desirable but most likely impossible with reasonable resources and our current level of technology
It would be good to do an experiment on how uniformly flowing water effects the speed of light but any objects placed would change the flow of water and I think the atomic clock experiment would be more realistic. Fizzeua did a similar experiment to what I would like to test in that he sent light through pipes with flowing water but the pipes bent and also caused laminar flow making the experiment problematic. He also did not measure speed directly which I would like but measured interference patterns. I do not see it as realistic to have a large enough medium with uniform flow for light to propagate without a bent path in which we can test the speed of something that moves so fast without bending the path. It might however be possible to do such a experiment with sound since sound travels so much slower than light. Of course people will then say that applies to sound but not light.
Changes will be made and another document written including some of the points in this article but shortened for the most part and edited although some new things might also be added that are not in this article
The use of the term differential equation is incorrect so I will change the name to Master Equation and hope the term Master Equation is not already used to mean something else. I will replace the e with s for source of emission
( Xr - Xe ) ^2 + ( Yr - Ye ) ^ 2 + ( Zr - Ze ) ^ 2 = c ^2 * ( Tr- Te ) ^ 2
Master Equation
( Xr - Xs ) ^2 + ( Yr - Ys ) ^ 2 + ( Zr - Zs ) ^ 2 = c ^2 * ( Tr- Ts ) ^ 2
