Galilean transformation equations and relativity.

[rbwinn]

In 1905 Albert Einstein changed perception of relativity by his idea of time dilation, which scientists subsequently claim to have proven by experiment. To quote Einstein, describing two frames of reference, K and K', "An event, wherever it may have taken place, would be fixed in space with respect to K by the three perpendiculars x,y, and z and with regard to time by a time value t. Relative to K', the same event would be fixed in respect of space and time by corresponding values of x',y',z',t'."

Einstein then substitutes the Lorentz equations in place of the Galilean transformation equations, which had been used with Isaac Newton's interpretation of absolute time before the Michelson-Morley experiment to describe relativity.

The problem is, however popular the idea may have become with scientists, it is mathematically incorrect. The correct way to describe relativity is with the Galilean transformation equations, but not with the absolute time interpretation that was formerly used. If a cesium isotope atom in K' has slower transitions than a cesium isotope atom in K, then a second in K' is not the same amount of time as a second in K. You cannot use them as equal amounts of time in equations for relativity. It is like using the rotation of Jupiter and the rotation of the earth as equal amounts of time because one rotation of Jupiter is called a day and one rotation of earth is called a day. The mathematically correct way to describe the different times in K and K' is by using the Galilean transformation equations twice, once for each frame of reference. For frame of reference K we have

 

x'=x-vt

y'=y

z'=z

t'=t

 

What the last equation t'=t means is that t is the time on a clock in K, and t' isequal to t, which means it is also the time on a clock in K. The slower clock in K' is completely ignored in computing relativity from K.

Now we use the Galilean transformation equations to compute relativity from K'. the variable t' has already been used in our equations for K and was defined to be the time of a clock in K. We cannot use it again is our equations for K' We have to use a different variable t2' for time of the slower clock in K'.

 

x=x'+v'(t2')

y=y'

z=z'

t2=t2'

 

The last equation shows that the time of the slower clock in K' is being used for time in both frames of reference. This time the faster clock in K is being completely ignored.

Posting these equations in the newsgroup, sci.physics.relativity resulted only in obscenities, profanity and insults. I am posting them in other forums with the hope that some more rational discussion of them might take place.

 

[Bignose]

For something you claim is mathematically incorrect, it seems to be pretty excellent at making predictions that agree with measurements. Please review http://relativity.livingreviews.org/Articles/lrr-2006-3/

 

and then demonstrate how your equations make even better predictions then we have today. Because in terms of what is rational is that the model that makes the most accurate predictions is favored. If you think your model is better, you need to demonstrate that your model makes even better predictions than the one favored today.

[rbwinn]

Well, that is fairly easy to do because the Lorentz equations predict that a slower clock in K' will show the same speed between frames of reference as a faster clock in K. This is accomplished by means of a length contraction in the frame of reference in motion. What reality shows is that if a clock in K' is slower, then an observer in K' will believe the speed of K relative to K' is faster than an observer in K would believe the speed of K' relative to K to be.

But aside from that, if t'=(t-vx/c^2)/sqrt(1-v^2/c^2) gives a correct representation of the time in K as compared to the time in K', then the correct representation of relativity would be to use the Galilean transformation equations for each rate of time as I have done. All I get from scientists is that this cannot be true because there is no length contraction. So if you want to believe that your car gets shorter every time you step on the accellerator, then it seems to me that you are free to do that. I don't really think it does.

[Bignose]

Well, that is fairly easy to do...

And yet you didn't do it. Where is the comparison between your prediction, the prediction of the current mainstream theory, and measurements?

 

That paper I gave you have 100s of references to predictions and experiments performed. Use your math to make a prediction and show us how it is more accurate than GR.

[rbwinn]

Well, if you won't accept the example I gave, you are not going to accept anything else either, but, as I said, if you want to believe your car gets shorter when you step on the accellerator, you are certainly free to believe that. Here is the problem with your mathematics. Scientists say that a second is a certain number of transitions of a cesium isotope atom, but that if a cesium isotope atom is put in motion, the rate of its transitions slows down. If you set a second in K' equal to a second in K, then you are introducing a length contraction because a second in K' is longer than a second in K. It is no different from saying that one rotation of a planet is a day, and a day on one planet is the same as a day on another. Then you say there has to be a length contraction because Jupiter rotates every ten hours, and earth rotates every 24 hours.

In any event, if you are as good at mathematics as you claim to be, you should be able to see that every prediction of General Relativity you cite is also predicted by the Galilean transformation equations as I use them. The only difference is that instead of there being a length contraction, 186,000 mi/sec is not the fastest speed in the universe.

The real problem is that I am talking about relativity, and scientists want to talk about wave mechanics, which was put together by James Clerk Maxwell and others using the Galilean transformation equations and absolute time. Scientists were unable to adapt Maxwell's equations without keeping velocity the same in both frames of reference the way it is with absolute time, which is done in both Special and General Relativity. I am sure Newton and Maxwell could have done the mathematics to account for different rates of time, but scientists of today cannot, so we are left with endless conversations about experiments.

So I'll tell you what. You find an experiment that the Galilean transformation equations as I have used them cannot describe.

[Sensei]

Well, if you won't accept the example I gave, you are not going to accept anything else either, but, as I said, if you want to believe your car gets shorter when you step on the accellerator, you are certainly free to believe that. Here is the problem with your mathematics. Scientists say that a second is a certain number of transitions of a cesium isotope atom, but that if a cesium isotope atom is put in motion, the rate of its transitions slows down. If you set a second in K' equal to a second in K, then you are introducing a length contraction because a second in K' is longer than a second in K. It is no different from saying that one rotation of a planet is a day, and a day on one planet is the same as a day on another. Then you say there has to be a length contraction because Jupiter rotates every ten hours, and earth rotates every 24 hours.

Caesium was chosen in 1967, because of great precision. Before that there was used other ways to calculate 1 second f.e. 1/86400 of Earth's day.

 

In any event, if you are as good at mathematics as you claim to be, you should be able to see that every prediction of General Relativity you cite is also predicted by the Galilean transformation equations as I use them. The only difference is that instead of there being a length contraction, 186,000 mi/sec is not the fastest speed in the universe.

If light speed is not constant, but variable, then airplane with v=1000 km/h emitting light in direction of flight should give us v= ~300,000 km/s + 1000 km/h/3600s/h=~300,000.277(7) km/s

Instead there is Relativistic Doppler Effect observed, instead of greater speed.

XX century new modern physics was developed to explain this observation.

 

[Bignose]

Well, if you won't accept the example I gave, you are not going to accept anything else either, but, as I said, if you want to believe your car gets shorter when you step on the accellerator, you are certainly free to believe that. Here is the problem with your mathematics...

rbwinn,

 

it's not a question of what I believe or not. Did you read anything in the link I posted? It shows many, many experiments that have agreed really quite well with GR.

 

You know what I will accept? If you take an example out of that paper, and post a graph with the prediction from GR, the prediction from your math, and the measured values. If you can demonstrate that your predictions are more accurate than those of GR, then you've got something. That paper makes it easy, in a lot of cases it has already plotted 2 of those 3 things I am looking for.

 

Ultimately, in my heart, I am a very practical person. I will use the model that makes the best predictions. GR is that at this moment, since you've posted no good evidence to think otherwise.

 

And yes, science does have 'endless conversations about experiments' because, well, that's what science is. Making predictions that agree with experiments. If you are looking for 'truth' or something like that, you are looking for more philosophy than science. Sure, most people naturally have some overlapping interests. But ultimately, agreement with experiment is what is most meaningful scientifically.

 

Look, I know that the above request is difficult. I don't expect it immediately. What I am trying to get you to do is show us that your idea has scientific merit. I am asking you to take the opportunity to make your argument stronger.

 

That is, asking me what I personally believe doesn't make your argument stronger. Showing that your idea makes even better predictions than what is mainstream today will make your argument very, very strong. Here's hoping you chose the latter...

Edited October 12, 2013 by Bignose

[rbwinn]

What we need to do is go back to Einstein's description of how he was going to solve this with the Lorentz equations. There are a few mathematical problems with what he did, but if it is good enough for mathematicians and scientists, for the present time it is good enough for me.

"Aided by the following illustration, we can readily see that, in accordance with the Lorentz transformation, the law of the transmission of light in vacuo is satisfied both for the reference-body K and for the reference-body K'. A light signal is sent along the positive x axis, and this light stimulus advances according to the equation x=ct.

, i.e., with the velocity c. According to the equations of the Lorentz transformation, this simple relation between x and t involves a relation between x' and t'. In point of fact, if we substitute for x the value ct in the first and fourth equations of the Lorentz equations, we obtain x'=(c-v)t/sqrt(1-v^2/c^2), t'=(1-v/c)t/sqrt(1-v^2/c^2), from which, by division, the expression x'=ct' immediately folows If referred to the system K', the propagation of light takes place according to this equation" (A. Einstein)

So let's apply the same logic to the Galilean transformation equations as I used them.

 

x=ct

x'=c(t2')

 

x'=x-vt

c(t2')=ct-vt

t2'=t-vt/c=(t-vct/c^2) =(t-vx/c^2)

 

So if we take the speed of light to be the same in both frames of reference, the values for x' and t2' are (x-vt) and (t-vx/c^2), which are nothing more than x' and t' in the Lorentz equations without the length contraction introduced by 1/sqrt(1-v^2/c^2). Notice that the resolution of the speed of light being the same in both frames of reference comes from the Galilean transformation part of the Lorentz equations, not from the length contraction.

That is why I say that anything you can do with the Lorentz equations, I can do with the Galilean transformation equations without the inconvenience of a length contraction.

Well, I hope we have put Special Relativity to rest. General Relativity is a different kind of problem. What I see wrong with it immediately is the same thing that made Special Relativity wrong. The speed of K' relative to K measured by an observer in K is the same as the speed of the speed of K relative to K' measured by an observer in K'. However, if we run the experiment, the transitions of the cesium isotope atoms in K' are slower, meaning that in reality, the observer in K' would see a faster speed between the two frames of reference. I don't know a lot about General Relativity. I know it is based on a length contraction. If so, I would compare it to Ptolemaic astronomy, which gave very accurate answers for positions of the sun, planets, and stars using a solar system in which the sun orbited the earth , and the planets went around the earth in epicycles. Some scientists were still using Ptolemaic astronomy long after Copernicus decided the planets were orbiting the sun. I would expect that there will still be scientists using General Relativity centuries from now, but it does not interest me because I can see it is wrong without even learning more about it.

[swansont]

Notice that the resolution of the speed of light being the same in both frames of reference comes from the Galilean transformation part of the Lorentz equations, not from the length contraction.

 

 

No, that's not the case. There is no transformation of c in SR, because c is postulated to be invariant. The Lorentz transforms stem from that postulate. It is inconsistent with Galilean transforms.

[rbwinn]

Well, you can say it is inconsistent. You can say anything whatsoever. What you need to do is prove what you say. I do exactly what Einstein did except I am not using two different rates of time in the same equation.

 

x=ct

x'=c(t2')

x'=x-vt

c(t2')=ct-vt

t2'=(t-vx/c^2)

 

c=x/t = x'/t2' = (x-vt)/(t-vx/c^2) = (x-vt)gamma/(t-vx/c^2)gamma

 

So go ahead and show the inconsistency. I have no need for the length contraction introduced by gamma.

[swansont]

Well, you can say it is inconsistent. You can say anything whatsoever. What you need to do is prove what you say. I do exactly what Einstein did except I am not using two different rates of time in the same equation.

I don't understand what you mean by this. Einstein doesn't set the two times to be equal. But you have to have the two in the same equation as part of the transform, which tells you one in terms of the other.

[rbwinn]

Well, it is fairly simple. If you have a length of time you call a second in K and a different length of time you call a second in K', if you use them in the same transformation equation you are going to have a length contraction in one frame of reference or the other.

The Lorentz equations say the time coordinate in one frame of reference is a smaller value than the time coordinate in the other frame of reference. That is why they have a length contraction.

The Galilean transformation equations say t'=t, or from the other frame of reference, t2=t2'. The time coordinates in either frame of reference are the same in any set of Galilean transformation equations. That is why they do not have a length contraction.

[uncool]

In 1905 Albert Einstein changed perception of relativity by his idea of time dilation, which scientists subsequently claim to have proven by experiment. To quote Einstein, describing two frames of reference, K and K', "An event, wherever it may have taken place, would be fixed in space with respect to K by the three perpendiculars x,y, and z and with regard to time by a time value t. Relative to K', the same event would be fixed in respect of space and time by corresponding values of x',y',z',t'."

Einstein then substitutes the Lorentz equations in place of the Galilean transformation equations, which had been used with Isaac Newton's interpretation of absolute time before the Michelson-Morley experiment to describe relativity.

The problem is, however popular the idea may have become with scientists, it is mathematically incorrect. The correct way to describe relativity is with the Galilean transformation equations, but not with the absolute time interpretation that was formerly used. If a cesium isotope atom in K' has slower transitions than a cesium isotope atom in K, then a second in K' is not the same amount of time as a second in K. You cannot use them as equal amounts of time in equations for relativity. It is like using the rotation of Jupiter and the rotation of the earth as equal amounts of time because one rotation of Jupiter is called a day and one rotation of earth is called a day. The mathematically correct way to describe the different times in K and K' is by using the Galilean transformation equations twice, once for each frame of reference.

That defeats the idea of the Galilean transformation equations in the first place. The idea of the transformation equations is to be able to "translate" between the coordinates in each of the frames - you need one for every pair of frames, and they must be self-consistent in some way (namely, when you "translate" from frame A to frame B, and then from frame B to frame C, it must be the same as translating from frame A to frame C; also, transforming from frame A to itself must be trivial).

For frame of reference K we have

 

x'=x-vt

y'=y

z'=z

t'=t

What are x', y', z', and t'? In special relativity, those denote coordinates relative to frame K'; you've said that this is about frame of reference K alone without reference to K', so I have no idea what you mean by this.

=Uncool-

[swansont]

Well, it is fairly simple. If you have a length of time you call a second in K and a different length of time you call a second in K', if you use them in the same transformation equation you are going to have a length contraction in one frame of reference or the other.

The Lorentz equations say the time coordinate in one frame of reference is a smaller value than the time coordinate in the other frame of reference. That is why they have a length contraction.

The Galilean transformation equations say t'=t, or from the other frame of reference, t2=t2'. The time coordinates in either frame of reference are the same in any set of Galilean transformation equations. That is why they do not have a length contraction.

I think you are misunderstanding the equations. t' and t are not two different times in the same coordinate system. They are how the time is related in two coordinate systems. IOW, the Galilean transforms say exactly what you claim is wrong. You are claiming that t'≠ t, and the Galilean transforms say they are the same.

[rbwinn]

From K the time coordinates are t and t'. This is shown by the equation t'=t. What it means is that if t is the time in K, then t' is the same time in K'. The slower clock in K' is completely ignored because it does not show t'. A clock in K shows the time coordinates for both frames of reference.

But time is not absolute as scientists claim it has to be in the Galilean transformation equations. Anything that shows a rate of time can be used in the Galilean transformation equations. If we wanted to use the period of rotation for one of Jupiter's moons as time in the Galilean transformation equations, we could use that reference for time.

We want to use the time of the clock in K', which is a slower time than the time we used in K. We are not going to use t' for the time of this clock because t' was used for time in K' in our first set of equations and was defined to be t'=t, the time on a clock in K. So we say that time on a clock in K' is t2', and t2, the time coordinate in K is t2=t2', the time of a clock in K'. If there is another clock in another frame of reference that has a different rate of time than the rates of time in K and K', then we use a different set of Galilean transformation equations to show the time of that clock.

This proves that time is not absolute in the Galilean transformation equations and that they can be used to show any rate of time.

[John Cuthber]

I must have missed something.

"The problem is, however popular the idea may have become with scientists, it is mathematically incorrect. The correct way to describe relativity is with the Galilean transformation equations,"

yet

"In any event, if you are as good at mathematics as you claim to be, you should be able to see that every prediction of General Relativity you cite is also predicted by the Galilean transformation equations as I use them. "

 

Either they agree or they don't.

If they agree then it's redundant (and fails by comparison with GR which has a viable explanation).

On the other hand, as far as I know, there are no observations which do not agree with GR so, if this new version of physics doesn't agree with GR then it also doesn't agree with reality.

If your predictions don't agree with reality it isn't because reality has made a mistake.

[rbwinn]

Well, to me that is like saying that if Copernican astronomy does not agree with the epicycles of Ptolemaic astronomy, then it must be wrong.

So what is the viable explanation of General Relativity? As far as I can tell it is based on a length contraction that only exists in the minds of scientists. They found a way to explain any experimental results no matter what they may be.

But there is one experiment General Relativity cannot explain. Explain how an astronaut with a clock in his satellite gets the same speed for the satellite as a scientist on the ground with a clock that shows a different rate of time.

[John Cuthber]

"So what is the viable explanation of General Relativity? "

The speed of light is constant, because you can calculate it from the permittivity and permeability of a vacuum, and you can't have a speed measured WRT a vacuum.

If you say the speed of light in vacuo is constant then you get various consequences.

You can make predictions from those consequences- such as the precession of Mercury.

If you check, you find that Mercury does indeed precess in the way predicted.

 

 

As for this "Well, to me that is like saying that if Copernican astronomy does not agree with the epicycles of Ptolemaic astronomy, then it must be wrong. " you clearly need to think it through some more.

 

Does your idea predict anything different from GR?

If so, then as far as we know, it's wrong because no experimental evidence disagrees with GR.

 

On the other hand, if your idea does agree with GR then what advantage does it offer?

It gives the same answers and it doesn't have a clearly established basis (unlike GR); why bother with it?

 

"But there is one experiment General Relativity cannot explain. Explain how an astronaut with a clock in his satellite gets the same speed for the satellite as a scientist on the ground with a clock that shows a different rate of time. "

I think you will find that GR does explain that perfectly well, but you haven't understood it.

Edited October 13, 2013 by John Cuthber

[Sensei]

Does your idea predict anything different from GR?

If so, then as far as we know, it's wrong because no experimental evidence disagrees with GR.

 

On the other hand, if your idea does agree with GR then what advantage does it offer?

It gives the same answers and it doesn't have a clearly established basis (unlike GR); why bother with it?

 

GR predicts existence of gravitational waves. That were never observed and detected, even though there was pumped hundred millions dollars in building devices for detecting them for 53 years since 1960.. Just a thought.

[John Cuthber]

 

GR predicts existence of gravitational waves. That were never observed and detected, even though there was pumped hundred millions dollars in building devices for detecting them for 53 years since 1960.. Just a thought.

Probably a good thing since many of those detectors could only detect a local cataclysmic event.

 

Not observing something weak and rare doesn't mean it doesn't exist- it just means it's weak and rare.

[Bignose]

Well, to me that is like saying that if Copernican astronomy does not agree with the epicycles of Ptolemaic astronomy, then it must be wrong.

I know you wrote this facetiously, but in a way, the answer is yes. The history on this is a good case... as better and more detailed observations of the movements of the celestial bodies were gathered, the epicycle model was expanded and expanded to try to match the observations. They were making models of cyles within cycles within cycles to match it.

 

On the other hand, they had the Copernican model that was making predictions that were coming true time and time again. And it was easier to boot. To a very significant extent, it was the continual agreement with prediction and measurement that convinced people.

 

And, yes, the data gathered when while in an 'epicycle' paradigm is still valuable as data gathered. What happened is the Copernican model could accurate match that old data as well as more recently gathered data.

 

This actually is precisely what I was asking you to do above. The data gathered to compare with SR and GR is still valued as data. Unless you have a compelling reason to know it is wrong, you can't just ignore it. That's why I want you to make a comparison with the known data & measurements, the predictions made by GR, and then your predictions.

 

Once again, if you can show your predictions are more accurate, you will make your case far, far stronger. And frankly, your refusal to do these calculations really makes your case weaker. All I am asking you to do is compare apples to apples here. If you truly think your idea is right, I don't understand why you wouldn't want to jump at this opportunity. What is the worst that could happen? That you find our you're wrong? Join the club. Everyone makes mistakes. Finding that mistake early means you can go back and correct it and try again sooner. That good! And the best that can happen is you are shown completely right. This is a no-brainer to me. Either way, you are helped. So, I don't get this reluctance...

[michel123456]

If i understand clearly what is being discussed here:

rbwinn states that if one uses a different time in a different FOR, there is no length contraction and no time dilation. Which is correct. Each FOR observes his own time as "regular" and observes no own length contraction and no own time dilation.

The Lorentz transformation is not about that.

As stated by Swansont, Lorentz transformation is used to describe what one FOR observes "happening" in another FOR.

[rbwinn]

Well, General Relativity is based on the same false premise that Special Relativity was, that two clocks with different rates would show the same velocity. Working from reality, I explained relativity to show that an abserver with a slower clock will see a faster speed. For instance, we put two passengers on a train, one with a slow clock and one with a faster clock and tell them, Use your clock and the mile markers by the track to determine the speed of the train.

According to scientists, they could get the same speed, because if the faster clock showed proper time, then it shows the same time as a clock in the frame of reference of the track, and a clock in the frame of reference of the track shows the same speed for the train as the slower clock the other passenger has. Reality would indicate that the passenger with the slower clock would get a faster speed for the train.

[uncool]

From K the time coordinates are t and t'.

The time coordinates of what? Are they both the time coordinate for the same event relative to frame K?

This is shown by the equation t'=t. What it means is that if t is the time in K, then t' is the same time in K'.

So t' is the time coordinate for the event relative to frame K', not to frame K?

The slower clock in K' is completely ignored because it does not show t'. A clock in K shows the time coordinates for both frames of reference.

But time is not absolute as scientists claim it has to be in the Galilean transformation equations. Anything that shows a rate of time can be used in the Galilean transformation equations. If we wanted to use the period of rotation for one of Jupiter's moons as time in the Galilean transformation equations, we could use that reference for time.

I have no clue what you are trying to say here. The idea of the Galilean transformation is that it tells you how to "translate" from coordinates relative to one frame to coordinates relative to the other frame. There must be a unique way to do so. But you seem to be describing two different transformations from K to K', if I've read your post right.

=Uncool-

Well, General Relativity is based on the same false premise that Special Relativity was, that two clocks with different rates would show the same velocity.

This way of phrasing it could easily hide many, many misunderstandings. That doesn't mean that it definitively does, but I'd be very careful with it. The premise for SR is that anyone will measure the speed of light to be a constant - approximately 3.0*10^8 m/s.

Working from reality, I explained relativity to show that an abserver with a slower clock will see a faster speed. For instance, we put two passengers on a train, one with a slow clock and one with a faster clock and tell them, Use your clock and the mile markers by the track to determine the speed of the train.

Relativity does presume that clocks are "synchronized" if they are comoving, which excludes "slower and faster" clocks that are comoving, as in this example.

According to scientists, they could get the same speed, because if the faster clock showed proper time,

"Proper time"? What do you mean by that? There is a meaning for "proper time" in relativity, but it doesn't have the properties that you are ascribing to it.

then it shows the same time as a clock in the frame of reference of the track, and a clock in the frame of reference of the track shows the same speed for the train as the slower clock the other passenger has. Reality would indicate that the passenger with the slower clock would get a faster speed for the train.

You are setting up a strawman here. This is not what special relativity says.

=Uncool-

[rbwinn]

Well, it is not really as difficult as you are trying to make it. If the coordinates in K are (x,y,z,t) and the coordinates in K' are (x',y',z',t'), all you do is provide the coordinates for the event. Suppose that the event takes place at (5 light sec,0,0,1 sec) in K. K' is traveling at ,25 light sec/sec relative to K. Then according to the Galilean transformation equations

 

x'=(5 light sec-.25 light sec/sec(1 sec))=4.75 light sec

y'=0

z'=0

t'= 4 sec

 

t' is not shown by a clock in K'. It is shown by a clock in K. The clock in K' is irrelevant to these equations.

But you want to know what a clock in K' would read. Well, that depends on what is being used as the standard for time in K'. It could be related to t in K. For example, scientists say t2' would equal (t-vx/c^2)/sqrt(1-v^2/c^2) and figure the value for t2' from the Lorentz equations. It does not matter how t2' relates to t. The Galilean transformation equations work for any standard of time. For instance if the clock in K' was faster than the one in K and was twice as fast, we would just compute the value of t2' from t2'=2t. Scientists claim they can determine what t2' would be at the coordinates given if t= 1 sec from General Relativity or the Lorentz equations. That is fine with me. I don't care what a clock in K' reads when a clock in K reads 1 sec. It reads something. Let scientists determine it by experiment. My own estimate would be about .75 sec using Galilean transformation equations to estimate the time. Maybe it is more. Maybe it is less. Let scientists determine it by experiment, and then they cannot complain about it. But since we do not have any scientists, we will just use .75 sec for purposes of showing how the equations work. So using the Galilean transformation equations as seen by an observer in K' using a clock in K',

 

x=x'+v'(t2')

5 light sec=4.75 light sec + v'(.75sec)

v'=.33333 light sec/sec

0=y'

0=z'

t2=.75 sec

 

So the observer in K' sees K' traveling with a speed of .333333 light sec/sec instead of the .25 light sec/sec seen by an observer in K, and his clock is slower, which is what reality would also indicate.

As I said before, in order to find fault with these equations, you are going to have to do more than take times from the second set of equations and try to get them to work in the first equations. The standard of time in the first equations is faster than the standard of time in the second set of equations.

Now I understand that scientists have all kinds of shortcuts which require this kind of improper mathematics dating back to the time when James Clerk Maxwell and other scientists were deriving equations using the Galilean equations and absolute time, which had the speed the same measured from either frame of reference, but whenever you do it with these equations, I am just going to say, There are two different speeds. You cannot do it.