Special Relativity Experiments short

 Copyright Carl Janssen 2024

I do not want to delete this content or edit it to remove things but I am not going to finish it.  I will copy some of the material into another article.  With some of the sections I do not want removed to make it shorter and then add more stuff

Title : Special Relativity Experiments Short Explanation

Main points of this article - This section might be edited, renamed - moved or removed

A wide multitude of past interferometer experiments asserted that the earth was moving and declared that a certain interference patterns in light waves should occur based on the speed the earth moved.  They found null results in their search for that specific interference pattern, which were unexpected, these null results were explained by length contraction and time dilation.  Now the interferometers were stationary relative to the laboratory they were in, were stationary, relative to the air in the laboratory and were stationary relative to the human looking at them but were said to be moving because the earth was moving.  Although I do not deny the legitimacy of using reference frames in which the earth is moving these results could more easily be explained without using length contraction and without using time dilation because a reference frame existed in which the earth was stationary, the interferometer was stationary, the human looking at the interferometer was stationary and the medium the light propagated in during the experiment was stationary.  Even though everything look stationary to the human doing the experiment they asserted the earth must be moving so it was moving.  Sometimes a medium called Aether was asserted to be moving relative to the interferometer because the earth was asserted to be moving but the interferometer was actually stationary relative to the medium that light propagated in which in many but not necessarily all cases was called Air.  Interferometers can either be enclosed in transparent glass or plastic or open.  If they are closed then the device would to some degree stay stationary relative to the medium that light propagated through inside the device, even if the device is moved.  But if the interferometer is opened then moving the interferometer by gently pushing or gently pulling it or slowly wheeling it would result in the device moving relative to the medium the light propagates through, however it would be moved no where near as fast as the asserted speed of the earth still resulting in a null result not necessarily because of no change but because the result would be much less then expected for the much greater speed that is the asserted speed of the earth usually used by researchers in these experiments when doing calculations.

This will explain a theory to why two different cesium clocks may show a different amount of change in the time displayed in clocks between two events due to Doppler shifting relative to a source of radiation

Suggestions of means to test the theory about Cesium clocks in a very basic way that would require more detail than described in a real life experiment

How to calculate Doppler shifts in ways that apply to a wider variety of situations than generally taught such as when objects are not moving in straight lines or are moving in more than one dimension but which also works for straight line movement and one dimensional problems

That outer space is actually a low density fluid not a perfect vacuum.  That this low density fluid undergoes black body radiation and that people call this radiation the cosmic microwave background.  That this radiation frequency distribution overlaps with frequencies related to Cesium clocks.  That it's Doppler shifted energy emission pattern in a approximately vertical direction of light propagation is a perfect match for the Time Dilation Constant Alpha on a model involving straight travel over a flat surface, with a "flat sky" and approximates to an equation that is a perfect match for alpha on a model involving circular travel over a round surface with a "round sky" presuming the standard radius of the earth.  By "round sky" I mean the boundary where certain properties of the atmosphere change to meet some criteria or another would approximately form the shape of the surface of a sphere.  That light has to propagate a further distance through a medium with at least a certain density in the approximately horizontal directions possibly making it insignificant to the Cesium clock as a source of radiated power in the horizontal direction compared to the vertical direction.

Suggested experiments section - This might be moved toward the end or near the end

Seeing if moving a radiation detection device results in a change in radioactive power near the frequency range of cesium clocks in a manner related to the time dilation constant alpha.  This would be moving the device relative to the source of the cosmic microwave background such as similar to the Hafele-Keating experiment except for using a device to detect the power of radiation at different frequencies instead of a cesium clock.

Seeing if directly applying radiation with a frequency pattern similar to that of a atomic clock and or that of the cosmic microwave background from outside a atomic clock can change the time displayed on a atomic cesium clock such that the change in time for a cesium clock not exposed to such radiation will be different than the change it time for a cesium clock exposed to such radiation between the same two events and with the clocks stationary relative to one another.

The analysis of the Fizeau experiment was wrong because it made predictions based on light traveling through water flowing in the same direction as the light and in a uniform flow but the water was traveling close to perpendicular of the direction expected near the corners and because the flow of water was closer to laminar than uniform.  Fizeau never directly measured the time it takes light to travel but did indirect measurements based on interference patterns.  Several modifications might be made because technology is more advanced now than when Fizeau's original experiment occurred.  

Seeing if water running perpendicular to a laser can displace the laser destination to a different location then where it would land if stationary.  Although there might not be a long enough structure.

Seeing if water as close to uniformly flowing as possible can change the amount of time it takes sound or light to travel.  Modifications of this can be made to make it closer to uniformly flowing than the Fizeau experiment.  Sound might be used because light might travel to fast to measure.  Of course the results would be objected to because of measuring one way speed requires two different clocks..  Water can be made to be closer to uniform by putting a water proof speaker and a water proof microphone inside a much wider source of running water than the Fizzeau experiment which is open where the microphone and speaker are not near the edges, problems can occur if sounds travels through other places than the water.  Sound can be of a specific frequency and a louder intensity than that produced by surroundings so that one can spot when the sound from the speaker first occurred by looking at a wave graph.  Should be tested with water running perpendicular to direction between microphone and speaker as well as two opposite parallel directions.  Not sure if modern electronics are sensitive enough even with sound let alone light.  Not sure if a large enough object can affordably be produced and might be more expensive than the Cesium clock experiments.

This beginning of the article might start here

This article is not a peer reviewed scientific journal article

If you complain that I claim something that should have been footnoted or that someone else might have discovered something before I did I am not going to necessarily disagree.  However, I know of no one else I can think of that has combined the same set of previous existing stuff in exactly the same way or combination that I have at the time of writing this, but that does not mean it has not already been done by someone else.  Just keep in mind that this is not serious scholarship, I am simply trying to get my ideas out as quickly as reasonably possible in a less than serious academic method and I am not submitting this particular article to a peer reviewed journal nor for a Thesis nor for homework.  If you want to use my ideas to do your own research go ahead and I do not even care if you credit me so long as you do not copy my material then copyright it in such a way as to block me from publishing my own writings.  In fact I am hoping that people who can do a better job than me academically, or experimentally are inspired by my ideas to do work related to my ideas in a way that is considered more academically proper to society at large.

What is unique or rare about my theoretical model

The most unique or rare thing I feel this article compared to all Special Relativity Theory variants and Aether or Lorenz Aether Theory variants that I know of is that I do not claim that light travels through a perfect vacuum and I additionally do not claim that light travels through Aether.  There are other theories that are similar to my theory involving something called complete Aether drag except instead of claiming that light is propagating through Aether I would claim that light is traveling through a chemical substance such as for example air or water or something else with a measurable chemical composition, density, pressure, temperature and flow.  I have tried to modify the short version of my theory presented here such that it would be compatible with both complete and partial Aether drag except replacing the Aether with a chemical substance.  In many interferometer experiments the source and observer which I call a receiver are both stationary relative to the medium of propagation for light.  Since all of those relative velocities are zero, I believe both full and partial Aether drag will produce the same result even though full and partial Aether drag would produce different results with a non zero velocity of one of these things relative to another.  It is a funny thing when people claim an interferometer is moving at a speed other than zero, when the person observing the interferometer, the surface of the earth, the medium the light propagates through and the interferometer itself are all stationary relative to one another.  Unfortunately I think my theory will not be so popular because it is not as funny and does not produce as many interesting paradoxes to amuse and entertain science fiction fans as claiming that a stationary object is moving or claiming that you have traveled in time when your clock displays a different number than another c;lock.

https://www.astronomytrek.com/what-direction-do-stars-move-in-the-sky/

Model Section

Difference between frequency of emission and frequency of photons

Hertz are cycles per second

It is said that the amount of energy a photon transfers or a photon has is equal to it's frequency times a constant

A device could admit a 50 hertz photons once every second or could admit a 1 hertz photon once every 0.02 or seconds both would be the same amount of energy per time but the frequency of admission and the frequency of photons would not be the same in both scenarios.  In order to avoid certain complicated things related to this to keep the article short I will simply talk about the amount of energy in the Doppler shift instead of ambiguously talk about frequency without specifying if I am talking about frequency of emission of photons or the frequency of photons.  Such a difference might matter related to this model but for now I am trying to explain things in a short matter.

Doppler Shift and Conservation of Energy

When a source of energy is Doppler shifted although the power is increased the duration is decreased such that the amount of energy emitted from the source is equal to the amount of energy received by the observer

Calculating Doppler Shift when the reference frame is stationary relative to the medium of light propagation

If the source is stationary relative to the medium of propagation and reference frame and the observer is moving

Step 1 Calculate how much time it would take for the light to get to the observer if the observer was stationary instead of moving

Step 2 Calculate how much time it would take for the light to get to the observer if the observer was moving

Step 3 Take Step 1 divided by step 2 gives the power if the observer is moving divided by the power if both objects are stationary

If the source is moving relative to the medium of propagation and reference frame and the source is moving

Step 1 Calculate how much time it would take for the light to get to the observer if the source was stationary instead of moving

Step 2 Calculate how much time it would take for the light to get to the observer if the source was moving

Step 3 Take Step 1 divided by step 2 gives the power if the source is moving divided by the power if both objects are stationary

Doppler Shift examples

when source or observer moves at speed v in a straight line perpendicular to the direction between source and observer and the other object is stationary and the medium is stationary relative to the reference frame

Step 1

Light travels from ( 0, 0 ) to ( 0, Y )

x coordinate 0 to x coordinate 0

and y coordinate 0 to y coordinate Y = c * t1

speed of light in x direction = 0

speed of light in y direction = c

Step 2 

Light travels from ( 0, 0 ) to ( X, Y )

x coordinate 0 to x coordinate X = v * t2

y coordinate 0 to y coordinate Y = ( ( c ^ 2 - v ^2 ) ^ 0.5 ) * t2

speed of light in x direction = v

speed of light in y direction = ( c ^ 2 - v ^2 ) ^ 0.5

Step 3 

solve for t1 / t2

c * t1 = Y = ( ( c ^ 2 - v ^2 ) ^ 0.5 ) * t2

t1 / t2 = ( ( c ^ 2 - v ^2 ) ^ 0.5 ) / c

t1 / t2 = ( ( c ^ 2 - v ^2 ) / c ^ 2 ) ^ 0.5

t1 / t2 = ( 1 - v ^2 / c ^2 ) ^ 0.5

How does that compare with the Lorentz Factor Alpha in special relativity?

It is a perfect match

Screenshots from Wikipedia article on Lorentz Factor

https://wikimedia.org/api/rest_v1/media/math/render/svg/bce2537b3a8f5029c517044add6f297e01da1b97

https://wikimedia.org/api/rest_v1/media/math/render/svg/5fa55fc5785e8c6fcc7251f8c169aea9a0d2dbfe

https://en.wikipedia.org/wiki/Lorentz_factor

Doppler shift involving a observer on a rotating circle and a stationary source and a medium stationary relative to the frame of referance

Observer is rotating around a axis of ( 0, 0 ) with a radius of R1 starting at ( 0, R1 ) at a tangential speed of v

Stationary source starts at a radius of R2 away from the same axis starting at ( 0, R2 )

Step 1

Coordinates of Stationary Source ( 0, R2 )

Coordinates of Observer ( 0, R1) 

t1 = R2-R1 / c

Step 2

Coordinates of Stationary Source ( 0, R2 )

Using radians 

circular path length / radius = number of radians

x Coordinates of Observer R1*sin( v*t2 / R1 )

y coordinates of Observer R1*cos( v*t2 / R1 ) positive or negative sign depends on direction of rotation

Distance between source and observer

( ( R2 - R1*cos( v*t2 / R1 ) ) ^ 2 + ( R1*sin( v*t2 / R1 ) ) ^ 2 ) ^ 0.5

in a time amount of t2 light travels to a source located at the distance between the source and the observer

c * t2 = ( ( R2 - R1*cos( v*t2 / R1 ) ) ^ 2 + ( R1*sin( v*t2 / R1 ) ) ^ 2 ) ^ 0.5

for theta close to 0 the sin(theta) is approximately equal to theta

when v * t2 / R1 is less than 0.175 or 0.175 radians the error in estimation for sine is less than 0.51%

round cos cos ( v*t2 / R1 ) to 1

round sin ( v* t2 / R1 )  to v*t2 / R1

https://en.wikipedia.org/wiki/Small-angle_approximation

https://en.wikipedia.org/wiki/Skinny_triangle

All equations below are rounded

c * t2 = ( ( R2 - R1 ) ^ 2 + ( R1*v*t2 / R1 ) ^ 2 ) ^ 0.5 approx

c ^ 2 * t2 ^ 2 = ( R2 - R1 ) ^ 2 + ( R1*v*t2 / R1 ) ^ 2 approx

c ^ 2 * t2 ^ 2 - ( R1*v*t2 / R1 ) ^ 2 = ( R2 - R1 ) ^ 2 approx

c ^ 2 * t2 ^ 2 - ( v*t2 ) ^ 2 = ( R2 - R1 ) ^ 2 approx

t2 ^ 2  * ( c ^ 2 - v ^ 2 ) = ( R2 - R1 ) ^ 2 approx

 t2 ^ 2 = ( R2 - R1 ) ^ 2 / ( c ^ 2 - v ^ 2 ) approx

Step 3

t1 = R2-R1 / c

t1 ^ 2 / t2 ^ 2 = ( ( R2-R1 ) / c ) ^ 2 / ( ( R2 - R1 ) ^ 2 / ( c ^ 2 - v ^ 2 ) ) approx

t1 ^ 2 / t2 ^ 2 = ( c ^ 2 - v ^ 2 ) ) * ( ( R2 - R1 ) / c ) ^ 2 / ( R2 - R1 ) ^ 2 approx

t1^2 / t2 ^2 = ( c ^ 2 - v ^ 2 ) / c ^2 approx

t1 / t2 = ( ( c ^ 2 - v ^2 ) / c ^ 2 ) ^ 0.5 approx

t1 / t2 = ( 1 - v ^2 / c ^2 ) ^ 0.5 approx this approximation is a perfect match for special relativity

This could have also been shown to be approximately correct on the basis of the infamous and often misunderstood "8 inches per mile squared" quote if used correctly without the complicated math to show that for all practical purposes it is the same as the previous Doppler shift equation calculated when it comes to an observer moving on the surface of the earth relative to a source vertically above them because the vertical distance traveled would be so small compared with the horizontal distance traveled by the observer for if light takes less than a certain amount of time to propagate from the source to the observer.  "8 inches per mile squared" is an approximation that represents the "drop" but the vertical height of the barrier of water halfway between two objects at sea level is a smaller quantity which I believe I calculated elsewhere to be approximately one fourth that height at 2 inches times the number of miles squared based on the radius of the earth that is stated in various sources.

pop special relativity vs moving medium theory

 Copyright Carl Janssen 2024

Title : Pop Special Relativity versus moving medium theory

What is moving medium theory?

Moving Medium Theory is the name I have chosen for an alternative theory to what I call pop special relativity.

Why did I choose the name moving medium theory?

I chose to call it that name for multiple reasons.  

First, because light in this theory always travels through a medium with a chemical composition instead of a perfect vacuum although the medium can be an imperfect vacuum.  In this theory the so called vacuum of outer space is actually a medium which is a low density fluid with a chemical composition and the movement.  In this theory the movement of this fluid relative to other objects effects how much black-body radiation this fluid sends to them in the form of the cosmic microwave background, which can be calculated by Doppler shift equations.  How much radiation from the cosmic microwave background a cesium clock experiences in this theory effects what time it displays.  

Second, because instead of speculating about the velocity of a hypothetical medium called Aether that light travels through, it instead supposes that light is traveling through a medium which can be observed and has a measurable velocity such as for example saying that light might travel through air with a measurable wind speed instead of through Aether.  

Third because the velocity the medium moves at may effect how light propagates in this theory.  Moving Medium theory explains the null results of many inferometer experiments by a means other than length contraction.  In moving medium theory the speed at which the inferometer moves is measured relative to the medium it is in when predicting how light should propagate, where as in Pop Lorenz Aether Theory and Pop Special relativity it is asserted that the earth is moving very fast therefore the inferometer must be moving really fast even though it might appear stationary relative to the person taking the measurements.

What do I mean by pop special relativity

By pop special relativity, I mean special relativity as generally defined by the popular mainstream media, from my point of view, from many years of watching, hearing and reading what has been labeled as both fiction and non-fiction sources.  According to wikipedia "A straw man fallacy (sometimes written as strawman) is the informal fallacy of refuting an argument different from the one actually under discussion, while not recognizing or acknowledging the distinction."

https://en.wikipedia.org/wiki/Straw_man

One might argue that what I call pop special relativity is not the same as genuine theory special relativity and that I am using a straw man to debunk something that is not the real one and only genuine theory of special relativity.  But, I am calling it pop special relativity to avoid such a straw man argument so as not to accidentally misrepresent the one and only genuine theory of special relativity if there is such a thing.  Now my point of view about what is generally displayed to the public might be wrong but adding the phrase "from my point of view" to the phrase pop special relativity just makes this too long to write.

Another way to think about what I mean by pop special relativity is a certain theory I am presenting based on what would happen if certain popular misconceptions about special relativity were assumed to be true but only if you assume what I call pop special relativity does not line up with the one and only genuine theory of special relativity.  If pop special relativity does line up with the one and only genuine theory of relativity than what I call misconceptions are not misconceptions about that theory.

Basically, I am combining about 4 to 6 common mistakes, about the nature of reality, I believe people make when they mention Special Relativity and combining these mistakes into a single theory called Pop Special Relativity, which I will try to debunk because I do not believe it accurately reflects the nature of reality.  In addition to debunking it I will suggest an alternative theory called the Moving Medium Theory to explain some results in some experiments which some people believe support some of their Pop Special Relativity claims, hopefully I will do so without making the same 4 to 6 common mistakes that I attribute to Pop Special Relativity.

This article is not meant to be up to academic standards but just a starting place to get my ideas out in a informal manner

This is not a peer reviewed scientific journal article or homework for a academic institution and although some works are cited, this would not meat the academic requirements for citing works.  I will not cite any particular source to back up my claim that the theory I call Pop Special Relativity is a theory that I believe generally has been presented to the public under the name of Special Relativity.

How does moving Medium Theory Compare to the One and Only genuine Special Theory of Relativity

I do not assert that Moving Medium Theory either agrees with or disagrees with whatever you think is the One and Only Genuine Special Theory of Relativity is, you can judge that for yourself upon looking at the theory presented and perhaps you might judge that it is not complete enough for you to come to a conclusion one way or another.

Credit to conversations with people that can not be cited

One source of inspiration for this idea is someone who told me that they were measuring the speed of sound in a vacuum who claimed that sound does actually travel in a vacuum but it travels really slow.  I do not even know the name of this person so I can not attribute to them and they did not give me this theory but it strongly influenced me to think that light never actually travels in a perfect vacuum.

Another source of inspiration were some conversations I had with someone who died before I came up with this theory or at least this theory at the stage I currently have it in, based on my memory of when I was informed of this person's death.  This person mentioned how a previous researcher did not want to block the flow of air from outdoor windows when doing experiments but in doing so it ruined the necessary temperature regulation of the experiment to know if the source of any different results was from the temperature or from other factors.  This might have influenced me to think about the idea of how moving wind would effect light or sound waves compared to stationary air.

I can not footnote such conversations and they would legally be hearsay, nor do I trust that they would agree with my ideas, nor do I trust that my memory of such conversations would be correct.  Furthermore, I wonder if they would prefer not be associated with my ideas, anyway.

Four or five mistakes that I believe resulted in what I call Pop Special Relativity

First Mistake 

Believing that light travels through a medium called either a Perfect Vacuum or Aether which has a different velocity than the medium that light is actually travelling through.

Second Mistake part one

Assuming the speed of light is the same in all reference frames and does not depend on the velocity of the medium light is traveling through

Second Mistake part two

In the case of the Fizeau experiment they actually did pay attention to the velocity of a medium but incorrectly treated it like the flow was uniform when it was laminar and then incorrectly assumed the flow was going in the same direction as the light propagated when it changed directions to run a different direction than the direction light was propagating in when it got near turns in pipes which in my opinion resulted in different results than Fizeau predicted not because Pop Special Relativity is correct but because he made the wrong assumptions about the velocity of water in his model.  It is important to note that although some people object to the conclusions involving the Fizzeau experiment because of laminar flow.  I am adding a second reason involving the change in direction of water near the turns in the pipes.

https://en.wikipedia.org/wiki/Fizeau_experiment

Third Mistake

Ignoring or denying the legitimacy of a reference frame in which The Earth is stationary when doing interferomoter experiments.

I will give an example of this involving the first special relativity lab experiment I ever remember doing, although I fear my memory maybe greatly wrong, I can not go back and ask the person, it is hard to tell what really happened and what was supplemented in my imagination as a result of later exposure to physics theories, so I will just give something that kind of fits my memory and kinds of fits the kinds of theories and experiments people do whether or not it is perfectly true to the event.  Unfortunately I have never seen such a object when I have tried to find a replication of this on the internet.  I can not really put a foot note of my vague memory, and as I said before this is not a formal academic article.

We were given equations and a piece of paper.  We were to stare at a closed circular object.  I believe it might have been a enclosed light where you could see circular fringe patterns through glass or transparent plastic.

The equations mentioned a velocity.  So I told the instructor that I was confused because the object was not moving and in fact I saw absolutely nothing moving in the entire experiment so I did not know what velocity to use because I did not know what object's velocity I should be measuring since nothing was moving.

He said the fringes should change pattern when they move but they will not change patterns and that I should shake it around and see that it does not change patterns when it was moving.  He did not say this but so that the reader understands, it was predicted that it should change fringe patterns according to some earlier theories and later theories were designed to explain why a multitude of devices like it do not change fringe patterns or do not change fringe patterns as much as predicted.

I believe he said it was a sodium lamp that he got at Home Depot or the name of some other store that was common during that year and that he modified it.

Looking back on this experiment I realized they most likely wanted us to put in the asserted velocity the earth was moving even though the earth was stationary relative to me from my reference frame.

Of course I could shake the object but no matter how fast I shake it without breaking it, it simply will not move as fast as the earth is asserted to move in such experiments so the change in fringes would only be based on a small fraction of the velocity asserted to be used for fringe calculations.

More over even if I did move the object the medium inside the object which the light propagates through would move right along with the object meaning that there is a 0 difference in velocity between the device and the medium which I suggest would result in no change in the fringes.  There maybe a Doppler shift of the light after it leaves the device on the way to my eyes if I shake it but I would suggest that for all practical purposes the interference pattern of the light inside the closed device should be the same before it leaves the device since the device is not moving relative to the medium inside it through which light propagates while inside it,

Some interferometers unlike this set up do not enclose the medium so they could move relative to the medium but they still would move at a much lower velocity than the asserted speed of the earth that is used however instead of attributing that a zero or much lower speed should have been plugged into calculations supporters of Pop Special Relativity assert that it was correct to plug in the asserted speed of the earth which was much higher and then the lack of significant change in fringes must be attributed to objects shrinking.

I propose ignoring the reference frame in which the earth is stationary has resulted in a wrong idea that objects shrink when you change reference frames in Pop Special Relativity.

Fourth Mistake

Confusing the change in time displayed on a clock with the amount of time that actually passed.  

If someone knows they are traveling at a certain speed but the clock displays that half as much time has passed as has actually passed then if they try to calculate how far they have traveled they will get half of the correct answer.

I would suggest the mistake of confusing the change in time displayed with a clock between two events with the actual change in time between two events further reinforces the belief that objects are different lengths in different reference frames for the same event and that likewise mistakenly believing objects are different lengths and the length between objects is different in different reference frames for the same event further reinforces the incorrect belief that the actual change in time between the same two events is different for different reference frames.  

Some people might object that in One and Only Genuine Special Theory of Relativity the lengths or distances involving actual real world objects do not change but that the only length or distances that change are in the coordinate systems however even if we assume that is true I would suggest that that is not how much of the general public sees it which is why I am comparing this with what I call Pop Special Relativity not with whatever you may think is the One and Only Genuine Special Theory of Relativity.

Potential Fifth Mistake

Possibly confusing the distance objects travel with the distance between objects and the length of objects.  This might have occurred with waves and or with non wave objects.

Time compared in Newtonian Physics vs moving medium theory vs pop special relativity

In Newtonian Physics the actual time an event occurs is the same in all reference frames.

In Newtonian Physics the change in actual time between two events is the same in all reference frames.

In Moving Medium Theory the actual time an event occurs is the same in all reference frames but the time displayed on clocks can be different from one clock to another.  In moving medium theory the time displayed on the same clock is the same for the same event in all reference frames.

In Moving Medium Theory the change in actual time between two events is the same in all reference frames but the change in time displayed on clocks can be different from one clock to another.  In moving medium theory the change in time displayed on the same clock between two events is the same in all reference frames.

In pop special relativity the actual time the same event occurs can be different in different reference frames.

In pop special relativity the change in actual time between two events can be different in different reference frames

Distance objects travel compared in Newtonian Physics vs Moving Medium Theory vs Pop Special Relativity

The distance an objects travels between two events in Newtonian Physics, Moving Medium Theory and Pop Special Relativity can be different in different reference frames.

For example, in both Newtonian Physics and Moving Medium Theory, if the distance between a runner and a race track start line, increases at a speed of v then from one reference frame the runner would not travel any distance at all but the start line would travel a distance of v*t  where as from a second reference frame the start line would not travel any distance at all but the runner would travel a distance of v*t and from a third reference frame they would both travel an equal distance of 0.5*v*t in opposite directions, but in all reference frames including the ones I have not listed, the distance between the object of the runner and the object of the start line would be equal to the same quantity of v*t

Distance between objects and Length of objects compared in Newtonian Physics vs Moving Medium Theory vs Pop Special Relativity

If the shape and size of an object is the same than the length between the same two parts of the same object are the same

In both Newtonian Physics and Moving Medium Theory the shape and size of the same object can be different for two different events.  In both Newtonian Physics and Moving Medium Theory if the distribution of pressure on the parts of an object are different for two different events then the shape and size of the object will be different for those two different events.  

In both Newtonian Physics and Moving Medium Theory if the distribution of pressure is the same on all parts of an object for two different events than the object will have the same shape and size for both events.

In both Newtonian Physics and Moving Medium Theory the shape and size of the same object during the same event is the same in all reference frames.

In both Newtonian Physics and Moving Medium Theory the distance between two objects will be the same in all reference frames for the same event.

In Pop Special Relativity the Shape and size of the the same object during the same event can be different in different reference frames

Velocity of light compared in Moving Medium Theory vs Pop Special Relativity

In Pop Special Relativity the speed of light is the same in all reference frames and for all events but the length of the medium light is moving through shrinks in the direction the medium is moving since the length of objects shrink relative to the direction they are moving in a reference frame.  However the medium does not shrink if it is stationary relative to the reference frame since shrinking of objects does not occur if they are stationary relative to a reference frame in Pop Special Relativity.

Ignoring rotations - If a medium is moving in uniform flow then in Moving Medium Theory the speed of light relative to that medium is the same in all reference frames for the same event and is based on the table for the speed of light for that medium.  The table would list the chemical composition, density, pressure and temperature and not rely on speculative results for a hypothetical Aether.

In moving medium theory the velocity of light is different for the same event in two different reference frames if the velocity of the medium the light is propagating through is different in each of those reference frames for the same medium.

Let's say light would travel from a source at a velocity of A*c in the X direction and B*c in the Y direction where the square root of A squared plus B squared equals one as according to the Pythagorean theorem from a reference frame in which the medium light is propagating through is both stationary and of uniform flow and in which the source is stationary relative to that reference frame.  Now if you are in the same reference frame but the medium is moving in a uniform flow with a velocity of E*c in the x direction and  F*c in the Y direction ( where the square root of E squared plus F squared might or might not equal one ) then in the Moving Medium Theory the light would travel at a velocity of ( A + E )*c in the X direction and ( B + F )*c in the Y direction and the length of the medium would not shrink in the direction that it is moving but in Pop Special Relativity the light would travel at a velocity of A*c in the X direction and B*c in the Y direction and the medium the light is traveling through would shrink in the direction the medium is traveling because in Pop Special Relativity the speed of light is the same in all reference frames

Let's look at another example which is one dimensional.  

Twin twin paradox in a pool with running water

Copyright Carl Janssen 2024

Corrections : Go back and use the phrase "platforms with their connected frames" in some places

Corrections : Many of the paragraphs would make more sense if read in a different order than they are currently written.  This probably should be rewritten with the paragraphs in a different order but since the paragraphs reference other paragraphs the style they are written in would have to change and they can not merely be cut and paste to change the order and still make sense.

Book Title : Misconceptions of space and time in popular understandings of special relativity

Short Chapter Title : Twin twin paradox in a pool with running water

Long Chapter Title : Modified twin twin paradox in a pool or running river instead of outer space to account for a uniformly moving medium

This is a modified version of the twin twin paradox

What do I mean by pop or popular special relativity?

By pop special relativity I mean special relativity as it has been presented on a popular level whether or not that is the genuine theory of special relativity if there is such a things as the one and only correct genuine theory of special relativity.

What is it important to know about pop special relativity in regards to the twin twin paradox?

In pop special relativity it is assumed that objects moving relative to a reference frame are shrunk in the direction they are moving relative to the length that would be measured in another reference frame in which they are stationary and the higher the speed the object moves in a reference frame the more it is shrunk in that reference frame.  Whether or not certain objects are shrunk in certain reference frames results in different travel times in different reference frames when comparing the two events which are the start of travel at one destination and the end of travel at another destination.  There is more to pop special relativity than this but that is what I shall discuss in regards to the terms of the twin twin paradox.

What is the twin twin paradox

The twin twin paradox is the idea that if a identical twin flew in a spaceship away from earth to a destination then came back to earth they would turn out to be younger than the twin that stayed on earth when they come back.

The claim is that from the point of view of the twin that stays on earth the twin moving away has the size of their space ship shrunk in the direction that it is moving but the distance to their destination does not change based on the speed the spaceship is moving away from and then coming back to the twin on earth.

From the point of view of the twin in the spaceship, the earth is moving away from the spaceship and then the earth is moving towards the spaceship, but the spaceship is stationary, this results in a decreased distance between the destination and earth.  The ship is not traveling toward the destination but the destination is traveling toward then away from the ship.  This is presuming they are traveling in a straight line going both there and back so that the shrinking is in the direction between earth and the destination.  Since the distance is shrunk and since the speed of the spaceship relative to the twin on earth is the same from both points of view, it seems to take less time for the twin in the spaceship to complete the trip from their point of view.

A implied medium is necessary to tell which twin is younger in order to know what is and is not shrunk

I would suggest that if space is assumed to be a perfect vacuum then you can not actually tell which twin would be younger than the other twin without establishing a preferred reference frame.  This is because if there are no objects other than the earth and the spaceship then the spaceship moving away from the earth and the earth staying still is no different than the earth moving away from the spaceship and the spaceship staying still since you do not have a medium between the earth and the destination the spaceship is going to with a known velocity to determine if it is shrunk or not and by how much as a function of it's velocity.  However, since no perfect vacuum exists and they are actually travelling through a medium no preferred reference frame is required to decide which twin will be younger.  

Misconceptions about the nature of the medium that is unfortunately called the vacuum of outer space in pop science

I will suggest that instead of thinking about the medium that pop scientists and some pop science fiction writers call the vacuum of outer space as either aether and or a perfect vacuum this medium should instead be thought of as a fluid such as a plasma, gas, liquid or other state of matter with a chemical composition just as air is a fluid called gas with a chemical composition.  In some regions this fluid could have a low enough density and or low enough pressure to be called a imperfect vacuum but even in those regions the density and pressure would both be greater than zero and this fluid would still have a velocity and chemical composition.  I am presuming this substance is a fluid and not a crystalline solid because solid objects can move through it although I do not deny that this assumption could be wrong if it were a solid that could be deformed and which some light could pass through.

Please do not ignore known mediums while paying attention to hypothetical mediums

If someone absolutely must insist on thinking of aether then one can assume that aether has exactly the same velocity as a section of this fluid at the location in which one wishes to measure the velocity of aether.  But I really think it is better not to invoke the velocity of some hypothetical aether at a location when the velocity of some actual fluid with a actual chemical composition can be measured at that location.  I will suggest in other chapters that a series of absurd mistakes were made when ignoring the chemical composition, pressure, density and velocity of actual mediums that exist and actually have known methods of measurement in order to pay attention to a hypothetical medium called aether which one hypothesized a velocity of and hypothesized how to measure.  Even though mediums such as air or water, or hydrogen or sodium or cesium or various other chemicals existed at locations and had known velocities at those locations, people in past historical experiments, instead choose another velocity for a hypothetical medium of propagation at the same location called aether which in my opinion resulted in absurd results.  I am not denying the possibility that aether might exist but simply saying not to ignore mediums that are known to exist in the name of aether.

Making the case that a medium is necessary in order to resolve the special relativity twin twin paradox thought experiment through another thought experiment

I will try to make the case that what is important is not the reference frame or the reference frame alone but the medium, using a modified version where someone is swimming between two platforms with a body of water that can be stationary or moving relative to the platforms starting with the assumptions of pop special relativity as best as I understand it.

By left and right I do not mean rotation to the left or right but the direction of moving in a straight line to the right or left which is sometimes called translation instead of rotation

This is a thought experiment and I do not believe it can be done in real life exactly as described in the experiment because the uniform flow of water and air as described could not be achieved since the objects involved would distort the flow

Default scenario to which other scenarios will be compared

Imagine there are two platforms with water and air and ground between them.  Each platforms is connected to the ground below it with a frame with gaps allowing water to go through the frame looking much like some guard towers in movies, cartoons and video games or like the frame of a transmission tower.

https://pixabay.com/photos/transmission-tower-electric-tower-3387882/

https://web.archive.org/web/20210816031520/https://pixabay.com/photos/transmission-tower-electric-tower-3387882/

Image produced using combination of these three online programs

Data on the geogerbra table was used to make picture above but is not relevant or used in the physics equations or scenarios.  Screenshot was taken of section of geogebra graph.  Colors were filled in using Pixlr and text was added using imgflip 

One of the meme templates or images called "horizontal line" was used on imgflip which was added to draw a straight line that was resized and which was modified with imgflip draw to form a double arrow with a thickness of 10 and black color

https://imgflip.com/i/9b80wo

https://imgflip.com/i/9b3i2j

https://imgflip.com/i/9b3p1f

https://pixlr.com/editor/

https://www.geogebra.org/geometry

The platforms are a distance L0 between them when the water and air and ground is stationary relative to the platforms from a reference frame in which the platforms are stationary.  You could measure the length of the section of ground between the two frames attached to the ground or the section of water between the platforms or platform frames or the section of air between the two platforms and any and all of those measurements would result in a length of L0 when the water, air, ground, frames and platforms are stationary relative to one another and stationary in this reference frame.

I will now compare different scenarios.  In all scenarios the air between the platforms will be stationary relative to the platform frames and the ground between the platform frames will be stationary relative to the platform frames and the platforms will be stationary relative to the platform frames.  But in some of these scenarios the water will move through the platform frames with uniform flow and move relative to the platform frames where as in other scenarios the water will be stationary relative to the platform frame with uniform flow if you can call it flow when it is stationary.  In all the scenarios the air will be assumed to be stationary relative the the platform frames with uniform flow.  

Realistically as I already mentioned earlier the flow of the water and air could not be uniform when the water is moving relative to the platform frames because the flow of the water would be distorted by the objects around it and the flow of the air would be distorted by the flow of the water but this is a thought experiment not a real experiment.

The two twins have separate stop watches that both display the same time of 0 seconds at the event of the start of each scenario but may display different times at different later events in the scenario

alpha = ( 1 - ( v ^ 2) / ( c ^ 2 ) ) ^ 0.5

https://wikimedia.org/api/rest_v1/media/math/render/svg/bce2537b3a8f5029c517044add6f297e01da1b97

https://wikimedia.org/api/rest_v1/media/math/render/svg/5fa55fc5785e8c6fcc7251f8c169aea9a0d2dbfe

https://en.wikipedia.org/wiki/Lorentz_factor

Scenario 1

Twin twin paradox in which the water is stationary relative to the platform

One twin called the waiting twin stays on the left platform the other twin called the swimming twin swims from the left platform to the right platform and then back to the left platform again 

Scenario 1A 

From the point of view of the swimming twin in which the swimming twin is stationary and the left platform is moving away from and then toward the swimming twin at a speed of v

Because the water, ground and air are moving relative to the twin the section of those mediums between the two platforms all shrink to change length from L0 to L0 times alpha

Length of mediums between platforms for swimming twin = L0 * alpha

Time for platform to go away and then back to swimming twin = 2 * L0 * alpha / v

Scenario 1B

From the point of view of the waiting twin in which the swimming twin is moving away from and then back to the platform a a speed of v

Because the water, ground and air are stationary relative to the waiting twin they do not shrink from the viewpoint of the waiting twin

Length of mediums between platforms for the waiting twin = L0

Time for swimming twin to go away and then back to platform for the waiting twin = 2 * L0 / v

Scenario 2

There is a waiting twin on the left platform stationary relative to the left and right platforms and platform frames and the ground and air.

Water moves at a speed of v relative to the platform frames and then moves at a speed of v in the opposite direction relative to the platform frame resulting in the floating twin traveling from the left platform to the right platform and then back to the left platform again

There floating twin will not exert effort to swim but merely float from the left platform to the right platform and then back to the left platform again.  From the reference frame in which the water is moving the floating twin floats at the same velocity that the water is moving at.  From the reference frame in which the water is stationary, the floating twin is stationary and  the left platform moves away from and then back towards the floating twin.

Scenario 2A

Scenario 2B

Does the geometry of scenario 2 show absurdity in the pop special relativity idea of length contraction?

One might argue that is does show absurdity because the length of the material of water between the two platforms is shorter than the length of the material of ground or air between the two platforms with their connected frames in a way that is geometrically impossible.  One might counter argue that actually when the water is shrunk it is not only the water between the platform frames that is shrunk but also the water to the left of the left platform frame and to the right of the right platform frame resulting in the water to the left and the water to the right filling in the gaps.  The problem with this argument is it would change the travel distance from L0 * Alpha back to L0 in order to fix the geometric problem which would result in changing the travel time from L0 * Alpha / v to L0 / v which would negate the idea of the different clocks having different times displayed because of different travel times in different reference frames which is so core to pop special relativity that one might argue that it would negate or at least partially negate pop special relativity.

Length of objects, distance between objects and distance traveled are not the same thing.  Comparing and contrasting them in Newtonian Physics versus Special Relativity

I would suggest a different solution that there is no length contraction in reality but that the clocks display different times not because of different travel differences but because they are running at different reaction rates.  I would also like to point out that the distance traveled is not the same as the length of objects or the distance between objects.  

It is important to note that objects traveling different distances in different reference frames is not unique to special and general relativity but also occurs in Newtonian Physics.  However in Newtonian Physics the length of objects and the distance between objects is the same for all reference frames for the same event where as in pop special relativity it is not.  

For example in Newtonian Physics, if the distance between a runner and a race track start line, increases at a speed of v then from one reference frame the runner would not travel any distance at all but the start line would travel a distance of v*t  where as from a second reference frame the start line would not travel any distance at all but the runner would travel a distance of v*t and from a third reference frame they would both travel an equal distance of 0.5*v*t in opposite directions, but in all reference frames including the ones I have not listed, the distance between the object of the runner and the object of the start line would be equal to the same quantity of v*t which would not necessarily be the case in pop special relativity

Reaction rates and different displayed values on clocks

Actual Time

Actual time is a quantity related to the order of events and is not the same as the time displayed on clocks.  The time displayed on clocks is a way for people to estimate the actual time.  I am not claiming that the earth revolves around the sun in a perfect circle at a constant speed but I will pretend it does so from some frame of reference in which the sun is stationary to make things simpler to explain this concept.

Let's suppose there was an event called the start of a calendar in which people were to track how far the earth travels in that specific reference frame described above.  How far it traveled could be measured in terms of the length of the arc it traveled.  How far it traveled could alternatively be measured in terms of an angle that is equal to a constant times the length of the arc it traveled in divided by the radius of the circular path it travels in.  At the date of the event of the start of the calendar it will be said that it travels progressed a distance of 0 meters or 0 degrees.

The amount of actual time between the date of any event and the date of the start of the Calendar would be equal to a constant times the how far the earth travel has progressed as measured in that reference frame since the event of the start of the calendar.  

The actual time between any two events could be got by taking the actual time between each event and the date of the start of the Calendar and then subtracting each of those actual times from the other actual time.

Now in this case the change in actual time between any two events is equal to a constant times the distance a certain object traveled in a certain reference frame between those two events but this is only because the object is traveling at a constant speed in that reference frame, this is only theoretical and does not apply to real life clocks as real life clocks are designed to estimate that actual time but can not get the actual time correct.  

It would be a foolish mistake to assume the change in display time in flawed clocks between any two events is always the same as the change in the actual time between those two events.  If two people started different stopwatches at the start of the same event and stopped the stopwatch at the end of the same event, it would be absurd to assume a different amount of actual time passed for each person between the two events.  Now maybe they did not really start and or stop the stopwatches at the same time as each other, or maybe the stopwatches were flawed and although both estimated the actual time, the time they displayed was not a perfect representation of the actual time and they both had a different quantity of error.

| Theoretical Error | = | Display Time - Theoretical Actual Time |

I call this theoretical actual time since although I propose actual time exists we can not know it perfectly but only estimate it.  We might say one clock has a better estimate than another clock and sort of get an idea of the error of one clock compared to a better clock but we can not know the error between a clock and a perfect clock

Length, distance between objects and distance traveled are all different than time

Even though in the thought experiment example I gave I said that the change in actual time is equal to a constant times the change in a distance a specific object traveled in a specific reference frame, I am not claiming that time is a fourth spatial event, nor am I claiming that space is the same thing as time nor that time can be measured in terms of length nor that length can be measured in time nor that length and time have the same units.  They distance traveled needs to be multiplied by a constant because length is not the same as time and because length and time have different units.  Although length, distance between objects and distance traveled all have the same units they are different things as I already mentioned.

Actual Time and Order of Events.  The order of events in is the same in all reference frames but some reference frames describe some events differently.

If the actual time of each event was listed as measured relative to some other event such as a start of a calendar then the order of events always occurs such that events which occurred with a actual time that has a lower quantity always happen before events which occur with a actual time that has a higher quantity.  This should be so obvious and I should not even have to say or write it but in pop special relativity I will suggest some people have confused the display time with the actual time and allowed the order of events to be scrambled using phrases like non-simultaneous and even claimed that some events do not occur in one reference frame but occur in another reference frame.  I am not denying that some people might observe some events at different times but I am talking about when the events actually happen in each reference frame.

Academic study or back to the fiction

Movies like "back to the future" in my opinion seem to use the equations of pop special relativity that I would argue predict the change in the time displayed on clocks to claim you can travel back in time to the past or forward in time to the future by confusing the time displayed on clocks with the actual time.

I would suspect that most people even most medical students and medical doctors would actually prefer to watch a entertaining actor who plays a doctor on television who is not an actual doctor than listen to a actual medical doctor who is not an actor give a boring lecture about anatomy and biochemistry.  In the same way I would also suspect that most people even people with PhDs would rather watch a entertaining fiction show about time travel than do a boring study about how much the displayed time on one clocks changes compared to the display time on another clock between two events.  

Some of these people who claim to have PhDs in Physics also show up on actor profiles for fiction shows and I would suggest that even when they are supposed to be doing their non-fiction shows they maybe tempted to say what is more entertaining like they are in their fiction shows than to tell the boring truth.  I would also suspect that the line between fiction and non fiction has been blurred as many people go into science fields after being entertained by science fiction which may in some cases but not all cases cloud their judgment.

Many people just find it more fun to believe the predictions involve actual time and time travel occurs than to believe the equations involve predicting how wrong the value displayed on clocks is compared to the actual time and that there is no time travel involved.  Someone for example who was working on a physics bachelor degree and eventually graduated with a bachelor's degree in something told me that it is just more fun to believe in time travel so he wants to believe it.

I am not saying that it is wrong to be entertained by science fiction nor am I denying that science fiction may inspire people to do experiments that are scientifically sound but I am simply suggesting that some people in academic positions may have been inspired by science fiction to do work in their academic field in a way that is not scientifically sound.  Writing research proposals involving far fetched claims that are more entertaining but have less scientifically sound methodology might entertain third party individuals who grant the funding for research more than more scientifically sound experiments that are more boring.  

I am not saying that there is anything morally wrong with entertainment nor that entertainment reduces the quality of experiments but simply that a problem may occur when people sacrifice sound research and good ethics for entertainment and that I suspect that pop special relativity has more to do with what makes entertaining fiction than to do with what makes good physics.

But that is fiction not academics

If you watch a show that is labeled as educational and or non fiction about special or general relativity they often have people who are claimed to have actual PhDs in Physics or ( in a field with a different name that might be in the physics department in one university but a separate department or degree in another University )

When these people who are claimed to have PhDs are interviewed instead of saying the equations are not about traveling in time but about predicting the value displayed on clocks they spend a lot of time talking about time traveling paradoxes or how people might theoretically find a way to travel through time which I suspect the audience simply finds more entertaining.

I suspect that fiction on the media has influenced the general worldview of pop special relativity shared by many people more strongly than the history of physics experiments.

Describing something from a different perspective can not change the physical nature of an object. 

Changing what reference frames you describe an event in does not change the nature of the event, it can not change the length of objects during that event, it can not change the distance between objects during that event and it can not change the shape and size of objects during that event, it can not change any other physical property of objects during that event I have not listed,  it can only change how that event is described mathematically or how the physical properties of objects involved in the event are mathematically described as being during that event.  Describing two different events from a different perspective can not change the amount of actual time between those two events.  In fact describing an event from a different reference frame not only fails to change the actual time but does not even change the time displayed on a clock, because a clock is a physical object and the physical configuration of the clock that displays the time can not be changed simply because it is described from a different reference frame.

Progress

Each type of clock has a mechanism that is different.  Depending on the mechanism progress could for example mean how much sand is on the bottom of a hourglass, the position of a shadow on a sundial or the percent composition of an element involved in radiometric dating.  Progress could be the number of times light is emitted from a object and reflected off a mirror to return back to the same object in the thought experiment clocks of special relativity.  If a person counted the number of times the sun rose then progress could be the number of sunrises the person counted.

Physical or Chemical Reaction Rate and Display time reaction rate

Physical or Chemical Reaction rate is the rate that progress changes over actual time this could be for example how much volume of sand moves down a hour glass per actual time and in such a case the progress would be the volume of sand in a section of an hour glass.  Display time reaction rate is the rate at which the display time on the clock changes per actual time.  If the clock is undergoing a physical change then it is called physical reaction rate but if it is a chemical change then it is called a chemical reaction rate.

Notation on reaction rate equations that are listed next

 

A different function is used to to get display time from actual time than to get display time from progress so I wrote out the word function in entirety instead of f or two different letters because people might not associate another letter with meaning function

I might switch the phrases display time and time displayed on clock they mean the same thing but one sounds better than the other to me in different context or phrases

Display time reaction rate

Final Time displayed on clock - Initial Time displayed on clock = Change in Time displayed on clock

Change in Time displayed on clock = Display time reaction rate * Actual change in time

Display time reaction rate = d Time displayed on clock / d Actual time

Time displayed on clock = function ( Actual time )

Change in Time displayed on clock = function ( Initial Actual time ) - function ( Final Actual time )

Physical or Chemical reaction rate

Time displayed on clock = function ( progress )

Change in Time displayed on clock = function ( final progress ) - function ( initial progress )  

Final progress = Initial progress + Change in progress

Change in progress = Physical or Chemical reaction rate * Actual change in time

d Progress / d Actual time = Physical or Chemical reaction rate

What does reaction rate have to do with two clocks displaying different times in pop special relativity

I would suggest that the so called time dilation in pop special relativity is a result of the reaction rate on cesium atomic clocks changing when moving at different velocities relative to the source of the cosmic microwave background which is simply black body radiation from a fluid in the so called vacuum of outer space.  I would suggest that the radiation is maximized when the cesium atomic clock is stationary relative to this fluid which requires moving at a certain speed in a westward direction and going at a different speed results in more radiation.  

I would suggest that the multiplier of this radiation is exactly equal to the constant Alpha due to the equations for Doppler shifting for a source traveling perpendicular to a source of radiation being exactly equal to the constant Alpha in pop special relativity.  This answer is problematic because it ignores radiation from the fluid of the so called vacuum of outer space from other directions and other sources.  

This might be because the distance of air or the atmosphere that is at least a certain density or higher is longer vertically than horizontally.  If you take two circles each with a different radius but the same center and pick any point on the smaller circle and draw line segments that reach the other circle, different line segments will have different lengths giving reason to believe my claim about radiation having to travel through a longer distance through air of a certain density or greater horizontally than vertically.  The radiation from other directions might be so greatly reduced that it can be ignored.

I also find this answer problematic because it ignores radiation from the sun, from the ground and water on the earth and from man made sources of radiation.  I would suggest that the radiation from the cosmic microwave background happens to be measured close to the frequency involved with cesium in atomic clocks and so if it gives off more radiation in this specific range than other sources than it might influence it more.  

In addition to these other sources of radiation that I am ignoring there is the issue of B fields which might influence some clocks when they move relative to charges or sources of electricity which could also potentially change the reaction rate of clocks as a function of their velocity relative to certain objects.

Although I have not mathematically explained this I will explain this more in other chapters

Where did the idea of the length of objects changing in reference frames come from

Probability that a value selected will be the maximum value in a uniform distribution

Copyright Carl Janssen 2024

Warning I am writing this to think about different possibilities and do not claim that the information presented is true

Title : Probability that a value selected will be the maximum value in a uniform distribution

When a sample size of n is collected with replacement from a population with a discrete uniform distribution including m possible values the probability that maximum value in the population is not sampled is

[ ( m - 1 ) / m ] ^ n

This is not the same as the probability of a value being selected that is higher than the value in the original sample with a sample size of n if the population is sampled from a second time with a sample size of 1

Example to explain why

Let us say the population has a uniform distribution of all whole numbers starting at 1 and ending at 10

Highest value in sample, missing higher numbers, Probability that a second sample with a sample size of 1 would achieve a higher result than the first value

10, no higher numbers, probability 0

9, higher numbers 10, probability 10%

8, higher numbers 8, 9 probability 20%

7, higher numbers 7, 8, 9 probability 30%

2, higher numbers 3, 4, 5, 6, 7, 8, 9, 10 probability 80%

1, higher numbers 2, 3, 4, 5, 6, 7, 8, 9, 10 probability 90%

10 - x, higher numbers 10 -x, 10 - x + 1, 10 - x + 2... 10-2 , 10-1 , 10 , probability x*10% 

Although this enables someone to estimate the probability that upon a second sample with a sample size of 1 there will be a value selected that is higher than the first sample if the highest value in the first sample is known it does not give the probability that there will be a higher value in the second sample that is greater than highest value in the first sample if the test in the first sample has not yet been done and the highest value in the first sample is not yet known.  

Such a thing would be desirable to correct problems with the empirical distribution function that mistakenly assigns a 0% probability of numbers being lower than the lowest number sampled and a 0% probability of numbers being higher than the highest number sampled which is not necessarily true.  

However such a correction would be more difficult because we do not necessarily know if the population that will be sampled by the empirical distribution function has a uniform distribution, if it is discrete or continuous or what it's range is and this method only works to estimate it when the population is discrete and has a known range, unless further things are done to modify things to take into account what is unknown beyond this starting point.

https://en.wikipedia.org/wiki/Empirical_distribution_function

Example

population only has values of whole numbers 1 through 4 and m = 4

sum of

( 4 - 4 ) / 4 * probability that highest number is 4 for sample size of n

( 4 - 3 ) / 4 * probability that highest number is 3 for sample size of n

( 4 - 2 ) / 4 * probability that highest number is 2 for sample size of n

( 4 - 1 ) / 4 * probability that highest number is 1 for sample size of n

equals

probability of a value being selected that is higher than the value in the original sample with a sample size of n if the population is sampled from a second time with a sample size of 1

Probability that the maximum value sampled from the population is not the same as the maximum value in the population

[ ( 4 - 1 ) / 4 ] ^ n = ( 3 / 4 ) ^ n

Probability that highest number is 4 for sample size of n

1 - [ ( m - 1 ) / m ] ^ n

1 - [ ( 4 - 1 ) / 4 ] ^ n

1 - ( 3 / 4 ) ^ n

1 - ( probability that a single number is 3 or lower ) ^ n

Probability that highest number is 3

1 - probability that highest number is 4 - ( probability that a single number is 2 or lower ) ^ n

1 - ( 1- ( 3 / 4 ) ^ n ) - ( 2 / 4 ) ^ n

( 3 / 4 ) ^ n - ( 2 / 4 ) ^ n

Probability that highest number is 2

1 - probability that highest number is 4 - probability that highest number is 3 - ( probability that a single number is 1 ) ^ n

( 2 / 4 ) ^ n - ( 1 / 4 ) ^ n

Probability that a single number is 1

( 1 / 4 ) ^ n

Sum of probabilities to check if they add up to 1

1 - ( 3 / 4 ) ^ n + ( 3 / 4 ) ^ n - ( 2 / 4 ) ^ n + ( 2 / 4 ) ^ n - ( 1 / 4 ) ^ n + ( 1 / 4 ) ^ n = 1

Probability that a second sample with a sample size of 1 will give a higher result than the highest value of a first sample with a sample size of n

Sum of

( 4 - 4 ) / 4 * [ 1 - ( 3 / 4 ) ^ n ]

( 4 - 3 ) / 4 * [ ( 3 / 4 ) ^ n - ( 2 / 4 ) ^ n ]

( 4 - 2 ) / 4 * [ ( 2 / 4 ) ^ n - ( 1 / 4 ) ^ n ]

( 4 - 1 ) / 4 * ( 1 / 4 ) ^ n

equals

Probability that a second sample with a sample size of 1 will give a higher result than the highest value of a first sample with a sample size of n

equals

[ 1 / 4 ] * [ ( 3 / 4 ) ^ n + ( 2 ^ 4 ) ^ n + ( 1 / 4 ) ^ n ]

I removed the exponents so this is not the same but shows the idea of how parts cancel out or technically it is the same when the exponents are 1 but they cancel out in the same way with other values for the exponents.

0+1*( 3 / 4 ) - 1*(2/4)+2*(2/4)-2*(1/4)+3*(1/4)=3/4+2/4+1/4

when n = 1 this is at it's highest for all permitted n values as n => 1

3+2+1 = 6

6 / 4 = 1.5

0 < 1.5 / 4 = 3 / 8 < 0.5 < 1 

if it was greater than 0.5 it would be a problem because we have to consider symmetry of testing for how often a sample size of 1 in the second sample is lower than the lowest value of the first sample

Checking results

Sum of products

Value, Frequency, probability of next sample being higher, product

1, 1/4, 3/4, 3/16

2, 1/4, 2/4, 2/16

3, 1/4, 1/4, 1/16

4, 1/4, 0, 0

Equals

6 / 16 = 3 / 8

Final Answer

From i = 1 to i = m - 1 where i changes by 1

take the sum of

[ 1 / m ] * ( i / m ) ^ n

Check to make sure this never exceeds 1 / 2 for m  => 2 it is at max when n = 1 for n => 1

when n = 1

From i = 1 to i = m - 1 where i changes by 1

take the sum of 

[ 1 / m ] * ( i / m )

equals

[ ( m - 1 ) * m / 2 ] / [ m * m ] = ( m - 1 ) / ( 2 m ) = 0.5 - ( 1 / 2 m ) < 0.5

approaches 0.5 from below as m approaches positive infinity

This value of approaching one half from below makes sense because it should be split into two parts one above the sample value and the other below the sample value except for a very narrow third part of the single sample value and on average there is symmetry and the portion excluded gets smaller and smaller as sample size gets larger and larger

1 + 2= 3 = [ 2 * 2 + 2 ] / 2 = ( 2 * 3 ) / 2

1 + 2 + 3 = 6 = [ 3 *3 + 3 ] / 2 = ( 3 * 4 ) / 2

1 + 2 + 3 + 4 = 10 = [ 4 * 4 + 4 ] / 2 = ( 4 * 5 ) / 2

For other values of n and m

when n = 1

From i = 1 to i = m - 1 where i changes by 1

take the sum of 

[ 1 / m ] * ( i / m )^n

when n = 2

From i = 1 to i = m - 1 where i changes by 1

take the sum of 

[ 1 / m ] * ( i / m )^n

https://calculator-online.net/riemann-sum-calculator/

when n = 3

From i = 1 to i = m - 1 where i changes by 1

take the sum of 

[ 1 / m ] * ( i / m )^n

when n = 3

From i = 1 to i = m - 1 where i changes by 1

take the sum of 

[ 1 / m ] * ( i / m )^n

when n = 4

From i = 1 to i = m - 1 where i changes by 1

take the sum of 

[ 1 / m ] * ( i / m )^n

when n = 9

From i = 1 to i = m - 1 where i changes by 1

take the sum of 

[ 1 / m ] * ( i / m )^n

when n = 19

From i = 1 to i = m - 1 where i changes by 1

take the sum of 

[ 1 / m ] * ( i / m )^n

when n = 24

From i = 1 to i = m - 1 where i changes by 1

take the sum of 

[ 1 / m ] * ( i / m )^n

when n = 49

From i = 1 to i = m - 1 where i changes by 1

take the sum of 

[ 1 / m ] * ( i / m )^n

when n = 99

From i = 1 to i = m - 1 where i changes by 1

take the sum of 

[ 1 / m ] * ( i / m )^n

Computer estimated results very close to 1 / ( n + 1 ) for m = 250

Right and Midpoint Riemann sum overestimated for n = 1 so used left Riemann sums primarily as defined by however the computer system did it

when n = 2 as m increased the calculated result increased and got closer to 1 / ( n +1 ) from below

This was close to previous assumption by dividing sample values into sections that form the gaps between nearest neighbors and assuming equal probability of falling into each gap range and having either n-1 or n+1 gaps depending on if you consider the area above the highest selected value and below the lowest selected value to be a gap or not.  Although I guessed that I did not have a formal mathematical way to show my guess was reasonable before doing this.

Only tested for m up to 250 as it seemed to slow down or not work well for m of 1000 the one time I tried it

250 is nearest factor of 1000 that is below 256 or 255 so it was chosen