Proof of the Fundamental Theorem of Calculus

[digi99]

It is proved in this topic (last answer):

https://www.physicsforums.com/showthread.php?t=536987

 

[Simon Bridge]

No proofs there.

 

[digi99]

That topic is so long that most people don't read it anymore (misundertanding, than not, not easy readable anymore) understandable you have to spread your time. But I was right there in some parts (not the absolute points of course).

But short:

- I was calculating there a time dilation in frame A for B in frame B compared to a light wave, formula time dilation (lesser time) = V/C (or t' = t. (1 - V/C) or x' = x . (1 - V/C)) (I considered your own movement V. t is subtracted from the same light wave in frame A and B, the end effect is it will be smaller for B and distances too of course).

- after Lorentz is t' = γ . t . (1 - V/C) or x' = γ . x . (1 - V/C) (also described as x' = γ . ( x - V. t) or t' = γ . (t - V.x/C2)

- the time dilation calculated by A is a factor γ bigger after Lorentz as part of the calculation, as well t as V/C are corrected

- V/C seen in time is exactly the part V.t of the light wave in distance, so it looks you skipped time (V/C or V.t in distance of the total passsing length of the light wave)

- after Lorentz both parts are multiplied with γ but the effect is t' = t. 1/γ for B itselves (time dilation already invloved)

- for every other point in frame B is the same (V.t is subtracted from the light wave)

- thoughts are exactly the same as for a passing train (in 1 direction only, the driving direction)

- in thoughts is it now easier to see time dilation in mind, a light wave presents relative time and can be seen as a clock between two events, in our normal calculations of distances/time we don't consider the skipping part in relative time during the movement itselves (absolute time does not exists)

So your own movement compared with the light wave is V.t in frame A (or V/C in time dilation), after Lorentz in frame B γ . V.t (total length light wave smaller) and γ . V/C time dilation.

So the skipped part of a passing light wave gives the time dilation.

Now is the question what is the meaning of the time dilation calculated in frame A, it seems not the really time dilation (but the responceable part is proofed), therefore you need Lorentz.

Or maybe you don't need Lorentz and this is the time dilation (V/C), in that case V > C is possible. Maybe there are experiments where the time dilation is exactly meassured (so not just a time dilation) and confirmed Lorentz is absolutely right (CERN do your best).

First I thought now I can calculate it in another form (γ included without Lorentz, but after many many hours I gave up, impossible, too few facts).

 

[Dale]

digi99, I am sorry, I have tried multiple times now, but I cannot decipher your "proof" nor this explanation of your proof. I don't know what you are trying to prove, what your initial assumptions or starting formulas are, nor can I follow your derivation.

It is possible that you never had a class that required you to write formal proofs. I would recommend that you spend some time at this site http://zimmer.csufresno.edu/~larryc/proofs/proofs.html to get some good tips on how to organize your thoughts so that other people can follow your proof and be convinced by your logic.

Also, it is considered spam to post the same post multiple times. You should pick the one place where you think it fits best, not spam the forum. Linking to it from other places is OK if you want.

 

[Simon Bridge]

I was calculating there a time dilation in frame A for B in frame B compared to a light wave, formula time dilation (lesser time) = V/C (or t' = t. (1 - V/C) or x' = x . (1 - V/C)) (I considered your own movement V. t is subtracted from the same light wave in frame A and B, the end effect is it will be smaller for B and distances too of course).

I'm guessing that part of the problem we have understanding you is that English is not your first language.

I don't know what you mean by "lesser time".

I don't know what V/C is supposed to be - perhaps the relative speed divided by the speed of light?

t'=t(1-V/C) looks like it is supposed to be the time-dilation formula - in which case it should look more like this: t'=t \sqrt{1-v^2/c^2} - is that what you meant? (If you quote me you'll see how I did that - it will help you a lot to learn to use LaTeX.)

The bit about light waves is a mystery because you have not explained which light wave and what it is doing there. The traditional derivation for time dilation uses a light-beam - is this what you are referring to?

the time dilation calculated by A is a factor γ bigger

None of the observers need to do any calculations to get the time dilation - all they have to do is look at each other's clocks.

... and so it goes.

You see - to make a "proof" you have to be very clear about everything you say. Otherwise it just looks like you have become confused.

 

[digi99]

Thanks DaleSpam and Simon, I shall do my best not to spam anymore (unless the name DaleSpam). In fact I considered later that I came to the same things as in the very beginning of my first topic (in this forum) about light waves. It is for me also difficult to understand that many of you don't see what I mean in my topic. I had the same problems already in another forum, so there I stopped already (in my own language). I will do it now in a slow unspammed way, no hurry anymore. The spam is, I wanted to complete the topics, because many topics and all half answered/solved does not help either. But maybe can it be solved what is wrong. If no, no problem at all and let it rest. My answers are only to give views to physicists, maybe it leads to something (not for me because I have now to less knowledge). I must learn a lot in a short time, because I don't need to know all, my interest is only relativity and light for this moment.

I am a starter in relativity (mathematics background, a long time ago, but I have of course not learn the language of physicists and English is not my first language) , but I will take gass back because it takes too much time of my own work (independent). I read some books about Brian Green later. That was the reason that I wanted first to understand the basics of relativity because many books tells only a part of it (e.g. they don't start with the behaviour of light, so that was an immediately problem for me with my exact thinking, so did I come to my first topic in a physicist forum to get a view of light).

I think totally I gave enough information, but I know I got very few feedback (only from Ghwellsjr but ended in a kind of doppler effect what I did not mean).

It's for me (mathematics thinking) a big question you don't see the relation t'=t(1-V/C) (before Lorentz) and after Lorentz t'=γ.t(1-V/C). I tried to explain how I come to t'=t(1-V/C) by thinking in the Newton way.

I think Simon has not seen my drawing in answer #11 of the related topic as first in this topic meant (was a topic with a bad start, yes my first). If you have seen that (Simon), I think you understand how I came to that formula (what it means).

My thinkings were let's show a light wave (the same light wave in fact) in as well frame A (object A in rest, moving object B) and frame B (object B in rest). Because light has lead to time dilation and interesting subjects in relativity, whithout the secrets of light there was not been a very well known Einstein I guess.

(object A and B are persons now) So a light wave starts in frame A (x = 0) and a person B is moving at the same time. Person A may meassure what the speed is of person B in it's frame, and that is speed V. But person A thought I use the light wave as a clock to meassure the speed of person B (consider it now as thought experiment otherwise you will see Dopller effect, and better is to see person B on the y-axes and the light wave in the middle on that y-axes between person A and B). So A meassured the total length of the moving light wave (speed C) during time t (normally meassured with a normal clock) and recalculated the time used by dividing that length / C. That time is t too of course, but as in Newton you have to subtract your own movement (V.t) from that total meassured length of the passing light wave, so recalculated t_seen_from_a_for_B = t(1-V/C), t is here the conventional method with a normal clock or from the light wave (without subtracting B's own movement in frame A).

So A thought already, B must undergo a time dilation by calculation (A meassured t), but for B it must be t(1-V/C) (see drawing in #14).

After Lorentz for B, it's t' = γ . t . (1-V/C).

What is the relation. I conclude you may see a passing light wave as relative time (it is passing A), if you move you skip a part of that light wave (seen by B) and so B's time will be lesser at the same time A's it sees (logically because B skips a part of the light clock used, that's the light wave).

You must understand now what I mean ... there is only a factor γ more after the Lorentz transformation. The consequences are that B sees a smaller lightwave (compared to A) where it's total passing length (compared to A) is γ . V . t smaller or γ . V/C in time lesser (of course near γ . t). So the part of the skipped piece of passing light wave is responceable for the time dilation in B (at the same time compared with A time goes slower for B).

But I still don't know (unclear books) if it affects B's age ... (without twin paradox conditions) ...

 

[digi99]

Maybe is my explanation not clear enough (for Simon / DaleSpam), I turned my computer on.

A and B are comparing both times in a specific period by meassuring a passing light wave while both in rest (and uses the same light wave).

That's t for A and t' for B, but A tried to predict the time for B and saw already in it's prediction a time dilation V/C (he did not know that later a factor γ would be involved more because of Lorentz).

In fact what I want to say for a passing light wave is the same as for a normal clock, if you move a normal clock time goes slower but that is more difficult to see.

That's why I take a light wave in this example, generally I may say, a passing light wave presents (relative) time, (absolute time does not exist) like a clock (light is also used as a clock), if you move (compare with a train) you see lesser light passing. The final effect is you will see a smaller light wave (while you are moving at the same time the light wave is going smaller, but if you compare it to the size A it sees, in that size sees A already a time dilation for B).

Confused, I am not, but it is slowly a very difficult and heavy task for me to explain ... so difficult I would never expect before I started it with university people (I see it as very simple) ... next time with something else I let to read it first to others before placing it in a forum ...

 

[digi99]

This was in first instance a thought of me, but because I "proofed as a starter" you see the same time dilation (without factor γ) in the Lorentz formula, I think partly it can be true (but without the factor γ). So the question is, what meaning has that time dilation that A calculated in it's own frame ?

 

[ghwellsjr]

I think totally I gave enough information, but I know I got very few feedback (only from Ghwellsjr but ended in a kind of doppler effect what I did not mean).

Digi99 is trying to provide a simple way to illustrate time dilation which simply means a clock running slower the faster an observer moves. If you look at posts #1 and #6 on his link on the first post of this thread you will see the clearest explanation. He starts with a monochrome light source and two observers who have "special clocks" that can count the wave cycles of the light coming from the light source during a period of one second. The first observer's "special clock" counts out the same number of cycles as the light source is emitting in one second which he calls t. The second observer is moving at speed v away from the light source. He will count t(1-v/c) cycles coming from the light source. Thus, a very simple way to show that a moving observer's "special clock" runs slower than a stationary observer's "special clock". Note that at v=0 the "special clock" runs at the regular rate. At v=c the "special clock" comes to a standstill.

 

[digi99]

Digi99 is trying to provide a simple way to illustrate time dilation which simply means a clock running slower the faster an observer moves. If you look at posts #1 and #6 on his link on the first post of this thread you will see the clearest explanation. He starts with a monochrome light source and two observers who have "special clocks" that can count the wave cycles of the light coming from the light source during a period of one second. The first observer's "special clock" counts out the same number of cycles as the light source is emitting in one second which he calls t. The second observer is moving at speed v away from the light source. He will count t(1-v/c) cycles coming from the light source. Thus, a very simple way to show that a moving observer's "special clock" runs slower than a stationary observer's "special clock". Note that at v=0 the "special clock" runs at the regular rate. At v=c the "special clock" comes to a standstill.

Fantastic Ghwellsjr (thank you, a big relief), it is exactly what I meant. But it is only a partially explanation because the factor γ has to be found too in this way.

In fact I was trying to make Lorentz visible in a simple way with my topic. I had the thought (for to explain it simple to others) see passing light waves as the time, if you move you see lesser light waves passing so time goes slower. And that's a fact now (in fact simple because light is a exact clock for relative time, t(1-V/C) is visible as explained and in Lorentz). The same is valid for another type of clock but more difficult to understand/make visible.

I bought the books Brian Green and Sander Bias and was starting with professor Sander Bias, but I stopped because my problems with light.

So I did come to my topic while thinking about it. Now I was looking again in his book how he found the Lorentz formula in his time space diagrams (learned from that book in the start chapters) because I could not. And he is doing it in the exact way I was thinking, also in his book he works with the relative speed V/C and finds γ in that way. A very good book to understand relativity, all explained in detailed space time diagrams.

So this is a very good day for me. I can stop now with spamming and read the books slowly in the coming weeks, I am fully prepared now and shall read them more easily.

 

[Dale]

It's for me (mathematics thinking) a big question you don't see the relation t'=t(1-V/C) (before Lorentz) and after Lorentz t'=γ.t(1-V/C). I tried to explain how I come to t'=t(1-V/C) by thinking in the Newton way.

OK, this formula is incorrect. The correct formula is t'=\gamma(t-vx/c^2). See here.

Where did this formula come from come from?

For clarity, let's introduce the following notation. Let all primed quantities refer to quantities measured in B's frame and let all unprimed quantities refer to quantities in A's frame. Let's use subscripts a, b, and c to refer to the coordinates of A, B, and the light pulse. Finally, let's denote the relative velocity of the frames by an unsubscripted v. So v_a'=-v_b=v and v_b'=v_a=0 and v_c=v_c'=c.

 

[digi99]

OK, this formula is incorrect. The correct formula is t'=\gamma(t-vx/c^2). See here.

I am happy you asked this because I was on the last moment so confused with the derivations in my first topic (some are right, some are not right) I was not sure anymore and you forced me now to clarify the last things. Gladly it still fits. It took a few hours (a lot a papers).

The expression t(1-V/C) and x(1-V/C) are for the meassured light wave length (passing light wave) in frame A (how A it sees/calculated for B). So x is positioned on the light line and time t.

In frame B you must take the coordinates of the transformed light line and so you must take t' and x'. Lorentz : x' = γ . (x - v.t) = γ . (x - v/c . t . c) = γ . (x - v/c . x) = γ . x . (1 - v/c). And t' = γ . t . (1 - v/c). Pfff...

 

[Dale]

The expression t(1-V/C) and x(1-V/C) are for the meassured light wave length (passing light wave) in frame A (how A it sees/calculated for B). So x is positioned on the light line and time t.

OK, so since you are describing the light then using the notation I described above this should be written: t'_c=\gamma t_c(1-v/c) which can be derived from the Lorentz transform as follows
By the second postulate t_c c=x_c
By the Lorentz transform for the light wave t_c'=\gamma(t_c-vx_c/c^2)
So by substitution t_c'=\gamma(t_c-v(t_c c)/c^2) = \gamma t_c(1-v/c)

In frame B you must take the coordinates of the transformed light line and so you must take t' and x'. Lorentz : x' = γ . (x - v.t) = γ . (x - v/c . t . c) = γ . (x - v/c . x) = γ . x . (1 - v/c). And t' = γ . t . (1 - v/c). Pfff...

Huh? Please use the notation I suggested, or propose your own clear notation and I will use it. But I cannot tell if you intend these to be general coordinate transformations or if they are the coordinates of some specific worldline such as the worldline of the light pulse.

 

[digi99]

OK, so since you are describing the light then using the notation I described above this should be written: t'_c=\gamma t_c(1-v/c) which can be derived from the Lorentz transform as follows
By the second postulate t_c c=x_c
By the Lorentz transform for the light wave t_c'=\gamma(t_c-vx_c/c^2)
So by substitution t_c'=\gamma(t_c-v(t_c c)/c^2) = \gamma t_c(1-v/c)

Hi DaleSpam, I take only conclusions by analysing so I learn from you and others. So what you wrote here is what I suggested (you did it in the right way). Tc is the same time for the moving object.

So from now on you can explain time dilation in a very simple way everybody in the world could understand, no magic anymore with complex drawings.

If you see a passing light wave as (relative) time, if you move, the light wave is slower passing you so time is going slower (time dilation). You have to compare it to the original size of the light wave when standing still, the finally effect will be when moving that the light wave you see wil being smaller, just as time do. It relates to the counting cycles of a light wave (the total length of the passing light wave).

I hope that I have added something extra to physics, I am sure this helps by analysing further by thinking lesser complex concerning time dilation.

Is it now allowed to place links to this topic to complete my other topics (last time), maybe there could be a new option in the future in this forum that you can update a topic without to place it as first in the queu of answers ?

 

[Simon Bridge]

OK - so walk me through the process of getting the time dilation from a light wave?
Do I need a special light source or can I pick any of the normal environmental ones (Sun, moon stars)?

if you move, the light wave is slower passing you

Um - no: light waves pass me just as fast when I move as when I don't.
I see fewer waves per second if I head away from the light source (doppler shift) but is it me that is moving or the light source? How do I tell?

 

[Dale]

So what you wrote here is what I suggested (you did it in the right way). Tc is the same time for the moving object.

In my notation t_c is the time of an arbitrary event on the worldline of the light pulse in A's reference frame.

So from now on you can explain time dilation in a very simple way everybody in the world could understand, no magic anymore with complex drawings.

Sure, if you are willing to accept the Lorentz transform then time dilation is very simple and doesn't require any drawings to explain.

If you see a passing light wave as (relative) time, if you move, the light wave is slower passing you so time is going slower (time dilation).

No, v_c=v_c'=c. The light wave passes at the same speed in every frame

You have to compare it to the original size of the light wave when standing still, the finally effect will be when moving that the light wave you see wil being smaller, just as time do. It relates to the counting cycles of a light wave (the total length of the passing light wave).

This is the first mention here about counting cycles. I thought we were describing a brief pulse of light. If you are talking about a continuous source of coherent light, then is this source at rest in A's frame or B's frame?

We will need to modify our notation. I suggest that we replace the subscript c with a subscript number indicating which cycle of the light wave is referenced.

 

[ghwellsjr]

In my notation t_c is the time of an arbitrary event on the worldline of the light pulse in A's reference frame.

Sure, if you are willing to accept the Lorentz transform then time dilation is very simple and doesn't require any drawings to explain.

No, v_c=v_c'=c. The light wave passes at the same speed in every frame

This is the first mention here about counting cycles. I thought we were describing a brief pulse of light. If you are talking about a continuous source of coherent light, then is this source at rest in A's frame or B's frame?

We will need to modify our notation. I suggest that we replace the subscript c with a subscript number indicating which cycle of the light wave is referenced.

DaleSpam, Please reread my previous post:

Digi99 is trying to provide a simple way to illustrate time dilation which simply means a clock running slower the faster an observer moves. If you look at posts #1 and #6 on his link on the first post of this thread you will see the clearest explanation. He starts with a monochrome light source and two observers who have "special clocks" that can count the wave cycles of the light coming from the light source during a period of one second. The first observer's "special clock" counts out the same number of cycles as the light source is emitting in one second which he calls t. The second observer is moving at speed v away from the light source. He will count t(1-v/c) cycles coming from the light source. Thus, a very simple way to show that a moving observer's "special clock" runs slower than a stationary observer's "special clock". Note that at v=0 the "special clock" runs at the regular rate. At v=c the "special clock" comes to a standstill.

 

[Dale]

If he is counting wave cycles then it sounds like Doppler shift, not time dilation.

 

[ghwellsjr]

That's what I told him and that's what Simon Bridge told him, although it's not normal Doppler because he's basing the time duration on the stationary frame instead of the moving observer's frame.

I even pointed out that if the moving observer changes direction and returns to the stationary observer, both their "special clocks" will end up with the same "time" on them instead of what should be happening according to the Twin Paradox.

But he still thinks its a better way to illustrate time dilation even though he realizes that it only "works" in one direction and even though it only "works" correctly at v=0 and v=c.

 

[digi99]

I am very surprised we still are not agreed.

I am looking just to the found formulas, there is no doubt I guess they are right. Pure Lorentz but expressed in total length of the passing light waves and related time you did not seen before I guess, the same time as the time for the moving object but calculated in another way but with the same result (in fact I am not counting periods, that's a practice problem maybe, but I considered only the length of the passed light signal, do it in mind please, people you explain don't think in frequencies, cycles etc. only students at universities). But at the same time, that passing light signal is getting smaller because of the limitation C. So yes, the speed is always C, so you have to see in mind that light is going slower for a little moment and immediately because of that is going smaller (everything is going smaller, the total length and the periods if you like, the difference is because of the time dilation. Formulas don't lie otherwise I could not say this.

You see this all I think too difficult maybe you studied physics. But you have to think like average people. And the formulas are right, so I don't tell nonsence and it has nothing to do with Doppler (see the light waves on a distance). What I tell you is not important, look only to the formulas.

For me is time the same term like distance. Nature says distance / time = lightspeed C all times.

So take a ruler and consider that as time (exactly the same as for distance).

Let's move that ruler by another person (with eg. seconds drawned on it) in front of your eyes. Than you see time passing IN MIND (like with light). If you move (with light direction is not important, in both directions you should experience same effects because of the limitation C) that ruler is a little bit going slower really, because it cannot be going slower (C) it's by nature immediately corrected and the ruler is going shorter (IN MIND). The same for light waves, what is the problem ? You can make a machine for students with sensors in the ground, when they walk with your simulated light waves they will go smaller.

This you see in formulas at both sides. Before Lorentz you see a light wave which length expresses time (same time A sees in his frame for B). After Lorentz that piece of light wave is going smaller and so does time with factor 1/γ, this time experiences B (but not aware of). DaleSpam, this is not the example of the pulse problem just light, that I discuss in that topic. You see before Lorentz you can think in the Newton way, see light as an object and subtract your own movement as you normally do. That is translated in the formulas (but a factor γ more). The speed of the light waves before and after Lorentz are still C at any moment (before Lorentz the total length of light waves is shorter because of B's movement, but it's time too, so still speed C).

If I am wrong (that's possible), than the formulas are wrong (but I don't believe)!

What is nice to this view, you can see light as an object and at the same time it is a clock.

By the way I thought today, if V could not be greater C, than there is nothing special about light, maybe because of his mass 0, it can have the highest speed possible detemined by nature. That's not a secret of light but from nature (there is probably one object with mass 0). If V could be grreater C, than the secret lies in light (time). So as long is not proofed V > C, I don't think anymore about that secret (not that I would found anything in other dimensions etc. but is not worth the time).

 

[digi99]

I never gave up/failed in a project how much time it will cost (pure passion) but maybe this would be the first time in life (frustrated feeling, but soon forgotten I guess).

I have think about it deeply what I am explaining difficult, I am sure it will now be clear (without confusions).

We do it now in the reverse order (but I found this all in the other order).

The formule I found : x' = γ . x . (1 + V/C) and t' = γ . t . (1 + V/C) for the passing light wave in frame B during the time B moves in frame A. It expresses time for B in frame B (included time dilation).

Now let's analyse the part t . (1 + V/C) or x . (1 + V/C) = x - v.t

Than we get my drawing (see answers above). What you see is that the originally light wave (representing t in frame A, has cut by v.t, so this light wave is going shorter and the time it represents too (original size belonging to frame A). In the end formula this is a little bit too much and corrected by γ. And we know the real end result is that all will be smaller (length and amplitude).

Ok now in steps.

So I could say see the time as a passing light wave (the light wave in frame A), if you move (as B in A) you see lesser light wave passing (yes light wave has cut with B's movement = v.t) so the time is going slower (yes total length light wave until now = x . (1 - V/C) = x - v.t and time also t = (1- V/C), speed still C).

That is it really going smaller, I left because that makes it difficult but can be said for who is interested (yes that fits because the light wave was indeed cut by B's movement, this was a little bit too much and corrected by γ how B it really sees in frame B, but totally shorter (compared to A), in reality also the amplitude, so there is a time dilation). Two times are different at the same moment so a time dilation included.

Question : do I lie now and why ?

When I later have read the book from professor Sander Bais, I can also specify the factor γ in my drawing (I think and quickly seen in his book because he works too with V/C).

 

[ghwellsjr]

You are describing a situation in which B moves away from A, correct? After awhile, if B turns around and moves toward A, does your idea still work?

 

[digi99]

You are describing a situation in which B moves away from A, correct? After awhile, if B turns around and moves toward A, does your idea still work?

Hi George, I read your other answer later about time dilation/age.

It's bad that I made errors again in the last answer for other readers confusing, it must be 1 - V/C in all cases (sometimes I said 1 + V/C).

Yes time dilation would be the same, so I guess the Lorentz formula would be the same, still 1- V/C as component (would be very strange otherwise, I used Lorentz, but coordinates for B starts not in 0,0 so another derivation with a same result, coordinates changes and direction changes, so that gives the same result).

I have the idea, because you point me on the direction of the light wave, that direction does not matter (already think about it in parts of my first topic). But it is not the direction that fits in an easy explanation for others. But I am sure that your movement is still v.t and still x . (1 -V/C) length of light wave will be passing (while you are going in a different direction than the light wave, light can never passing you with a speed > C).

But maybe because you have studied physics, you tell me now something different so my thinking is not right anymore ?

 

[Dale]

Pure Lorentz but expressed in total length of the passing light waves

There is nothing in there about the length of the light. If you want to talk about the length of a pulse of light then you will need two separate worldlines, one for the front of the pulse and one for the back. I would recommend using a subscript 0 to indicate the front of the pulse and a subscript 1 to indicate the back of the pulse. Also, without loss of generality you can set your unit of time and distance such that the duration and length of the pulse is 1.

(in fact I am not counting periods, that's a practice problem maybe, but I considered only the length of the passed light signal

That doesn't really matter. The wavelength and the frequency are inevitably linked, so if you change the wavelength then you must change the frequency. It is Doppler whether you count periods or measure wavelengths.

everything is going smaller, the total length and the periods if you like, the difference is because of the time dilation.

The difference in both length and frequency is due to Doppler shift, time dilation is a small part of the Doppler shift.

And the formulas are right, so I don't tell nonsence and it has nothing to do with Doppler (see the light waves on a distance). What I tell you is not important, look only to the formulas.

So far, the formulas seem to have little to do with what you are saying. Can you express your formulas in the notation I have suggested for clarity? Where is the length of the light pulse?

 

[digi99]

There is nothing in there about the length of the light. If you want to talk about the length of a pulse of light then you will need two separate worldlines, one for the front of the pulse and one for the back. I would recommend using a subscript 0 to indicate the front of the pulse and a subscript 1 to indicate the back of the pulse. Also, without loss of generality you can set your unit of time and distance such that the duration and length of the pulse is 1.

That doesn't really matter. The wavelength and the frequency are inevitably linked, so if you change the wavelength then you must change the frequency. It is Doppler whether you count periods or measure wavelengths.

The difference in both length and frequency is due to Doppler shift, time dilation is a small part of the Doppler shift.

So far, the formulas seem to have little to do with what you are saying. Can you express your formulas in the notation I have suggested for clarity? Where is the length of the light pulse?

Hi DaleSpam,

I am not talking about pulse light, I am only using just 1 long light wave (= light signal) with speed C. Suppose that light wave is already left hours ago before, and see that light wave from coordinates 0,0 as it arrives on the moment B starts moving. I don't know the name for such light wave ... maybe a beam ..

The coordinate x and x' represent the total length of the passing light wave (maybe you think the wave length, as related with some cycles, no I mean the length of the traveled path of the passing light signal), so what have I to do with cycles and frequency in this case ?

Would this be a communication problem, that when I say total length of the light wave, you think to the wave length (some cycles) ? (I mean the total length of the traveled path of the passing light wave between 0,0 and 0,x)

 

[Dale]

I am not talking about pulse light, I am only using just 1 long light wave (= light signal) with speed C. Suppose that light wave is already left hours ago before, and see that light wave from coordinates 0,0 as it arrives on the moment B starts moving. I don't know the name for such light wave ... maybe a beam ..

Whether you call it a pulse, beam, wave, or signal is not really important. What is important is the math. Since it has a length it must have a front and a back which means you need two worldlines to describe it, not just one.

The coordinate x and x' represent the total length of the passing light wave

X and x' are coordinates, not lengths. A length is the difference between two coordinates, i.e. in my suggested notation something like L_c=x_1-x_0

(maybe you think the wave length, as related with some cycles, no I mean the length of the traveled path of the passing light signal), so what have I to do with cycles and frequency in this case ?

Any signal may be decomposed into a sum of sine waves using the Fourier transform. Even if your signal is not repetitive you still have frequencies and wavelengths and Doppler shift.

Would this be a communication problem, that when I say total length of the light wave, you think to the wave length (some cycles) ? (I mean the total length of the traveled path of the passing light wave between 0,0 and 0,x)

Yes, there have been a number of communication problems and now some math problems, which is why I have requested that you be clear in your notation. I don't understand why you are so reluctant to do so.

 

[Simon Bridge]

@digi99: in order to communicate effectively it is required that there is a common language. This is a subject which has been under active study now for a long time, so there is a language that has already been developed for it. If you refuse to use that language you will be unable to communicate your ideas. The others have been trying to get you to use that language with the special notation: I urge you to adopt it.

Please bear in mind that this is such a well studied field that it is very unlikely that you have come up with anything not thought of before. Try to listen to what people here are trying to tell you because they are genuinely trying to help you avoid some quite common pitfalls that you seem determined to jump down into. (Specifically, but not restricted to, the idea that a particular reference frame is "really stationary" and everyone else just thinks they are stationary when they are "in fact" moving.)

The closest I can figure is that you believe you have come up with a method of teaching about time dilation that is simpler than the history-tested approaches commonly used to date. The trouble is that the simplifications involve ignoring quite a large chunk of relativity... which is what we want to teach. By concentrating on light pulses and paths like that you introduce ideas which will lead to worse confusions later.

So, instead, we require the students to do some hard work at the start, to make more important concepts easier to learn later. We also have to be careful not to leave too much hanging loose for pseudoscience and crackpots to take advantage of.

 

[digi99]

Whether you call it a pulse, beam, wave, or signal is not really important. What is important is the math. Since it has a length it must have a front and a back which means you need two worldlines to describe it, not just one.

X and x' are coordinates, not lengths. A length is the difference between two coordinates, i.e. L_c=x_1-x_0

Ok DaleSpam, I understand and shall give the situation again with the right notation.

I have think about it deeply what I am explaining difficult, I am sure it will now be clear (without confusions).

I have in frame A a light wave following the x-as from (0,0) to (X_c, T_c), I have B moving from (0,0) to (X_b,T_b) with speed V_b. T_c = T_b and lightspeed V_c = C. The length of the light wave L_c = X_c - 0 = X_c. That's all I have, I look only to the length of the light wave for B (we meassure nothing extra, what Ghwellsjr said for counting is from an old topic, so this is the situation in mind). Seen from frame B all variables have an accent '.

We do it now in the reverse order (but I found this all in the other order).

The formule I found : X'_c = γ . X_c . (1 - V_b/V_c) and T'_c = γ . T_c . (1 - V_b/V_c) for the passing light wave in frame B during the time (T_b) B moves in frame A. It expresses time (T'_c) for B in frame B (included time dilation). The length of the light wave seen from B is L'_c.

Now let's analyse the part T_c . (1 - V_b/V_c) or X_c . (1 - V_b/V_c) = X_c - V_b . T_b

Than you get my drawing (see answers above). What you see is that the originally light wave (representing T_c in frame A, length L_c has cut off by V_b . T_b, so this light wave is going shorter (L_c - V_b . T_b) and the time it represents too (L_c / V_c in the original size belonging to frame A). In the end formula (T'_c) this is a little bit too much and corrected by γ. And we know the real end result is that all will be smaller (length and
amplitude).

Ok now in steps.

So I could say see the time as a passing light wave (the light wave in frame A seen from position (0,0) to (X_c, T_c), if you move (as B in A) you see lesser light wave passing (yes light wave has cut off by B's movement = V_b . T_b) so the time is going slower (yes total length light wave until now L'_c = X_c . (1 - V_b/V_c) = X_c - V_b .

T_b from (X_b, T_b) to (X_c, T_c) and related time is T_c . (1- V_b/V_c), speed still V_c).

That it is really going smaller, I don't tell because that makes it difficult but can be said for who is interested (yes that fits because the light wave was indeed cut off by B's movement L'_c, this was a little bit too much and corrected by γ how B it really sees in frame B, but totally shorter (compared to A, L_c), in reality also the amplitude, so there is a time dilation). Two times T_c (L_c / V_c) and T'_c (= L'_c / V'_c) are different
at the same moment so a time dilation included.

Question : do I lie now and why ?

When I later have read the book from professor Sander Bais, I can also specify the factor γ in my drawing (I think and quickly seen in his book because he works too with V_b/V_c).

 

[digi99]

@digi99: in order to communicate effectively it is required that there is a common language. This is a subject which has been under active study now for a long time, so there is a language that has already been developed for it. If you refuse to use that language you will be unable to communicate your ideas. The others have been trying to get you to use that language with the special notation: I urge you to adopt it.

Please bear in mind that this is such a well studied field that it is very unlikely that you have come up with anything not thought of before. Try to listen to what people here are trying to tell you because they are genuinely trying to help you avoid some quite common pitfalls that you seem determined to jump down into. (Specifically, but not restricted to, the idea that a particular reference frame is "really stationary" and everyone else just thinks they are stationary when they are "in fact" moving.)

The closest I can figure is that you believe you have come up with a method of teaching about time dilation that is simpler than the history-tested approaches commonly used to date. The trouble is that the simplifications involve ignoring quite a large chunk of relativity... which is what we want to teach. By concentrating on light pulses and paths like that you introduce ideas which will lead to worse confusions later.

So, instead, we require the students to do some hard work at the start, to make more important concepts easier to learn later. We also have to be careful not to leave too much hanging loose for pseudoscience and crackpots to take advantage of.

Hi Simon,

I understand what you are telling me, but I was only looking for a method for non-students, just working people in other professions, and I was asking you what you find from my method. In fact is only for my website, my idea was, ok learn what relativity is and learn others in a more simple way (what is the meaning that I learned it for myself and not giving through). And if that simple way is not the exact way but close to it, that's no problem I guess otherwise a lot of people just don't know.

But it took more and more time than I thought before ... because I am on a level with experts and I am not ... so after this subject I return to normal life and read my books, meanwhile I can ask a question ... and if you all don't agree I don't teach it on my website in this way ..

Of course I have not the illusion to find something new but I was surprised on many forums that nobody was knowing the expression (1 - V/C) in this relation, so I was digging further and came to the formula (but just Lorentz, nothing special but maybe never used before looking to light only, that's nice for a starter like me) ...

 

[Dale]

Ok DaleSpam, I understand and shall give the situation again with the right notation.

Thanks, this is helpful. I will walk through it step-by-step with you. Again, for simplicity I am using units where c=1.

I have in frame A a light wave following the x-as from (0,0) to (X_c, T_c),

Are you familiar with the parametric equation of a line? We can write this worldline as: (x_c,t_c)=(t,t). Do you see how this equation contains both the event (0,0) and (x_c,t_c) and it constrains it so that it has the correct speed to be light dx_c/dt=1=c?

I have B moving from (0,0) to (X_b,T_b) with speed V_b.

Similarly, for B we have the worldline (x_b,t_b)=(v_b t,t).

The length of the light wave L_c = X_c - 0 = X_c.

No, it isn't. When we measure the length of something we measure the distance between the front and back at the same time. For instance, my car is about 4 m long. When I am driving at 100 kph if I took your approach and measured the the length of the car at t=0 and t=T_b=1h then I would get that my car is 100 km long. This is clearly not correct. I need to measure the position of the front and the back at the same time in order to get the length.

So, here is my recommendation, instead of using one worldline for the light use two, one for the front (subscript 0) and one for the back (subscript 1). Set your unit of distance so that the length of the light is 1.
(x_0,t_0)=(t,t)
(x_1,t_1)=(t-1,t)
L_c=|x_1-x_0|=|(t-1)-t|=1

Do you follow this so far?

 

[digi99]

Do you follow this so far?

Yes DaleSpam, I can follow it and I am very curious to the result ...

 

[Dale]

OK, so you want to find
L_c'=|x_1'-x_0'|

From the Lorentz transform:
t'=\gamma(t-vx) and x'=\gamma(x-vt)
So in B's frame we have
(x_0',t_0')=\gamma(x_0-v t_0, t_0-v x_0)=\gamma(t-v t,t-v t) =
\frac{1}{\sqrt{1-v^2}}(t(1-v),t(1-v)) = \sqrt{\frac{1-v}{1+v}}(t,t)
Which is essentially just the relativistic Doppler formula.

Similarly we have
(x_1',t_1')=\gamma(x_1-v t_1, t_1-v x_1)=\gamma((t-1)-v t,t-v (t-1))

So now
L_c'=|x_1'-x_0'|=|\gamma((t-1-vt)-(t-vt))|=\gamma

 

[digi99]

Today/tomorrow I can finish my topic, because I wanted to finish it in the good way and so spent my time to it. So I was reading everything again in my notes, and I found many mistakes (that I have learned now, order your notes otherwise it will be a mess, I did it all in a hurry between my work which asks a lot of time from me too, and besides I am a starter with mathematics knowledge from a long time ago and not much physics, but I learn quick).

In my old topic (see link to that old topic in first answer in this topic) I wrote once that I was wrong and came with another formula (something different but right) and that was the end. But my statement was/is still there (see later).

Than I came in discussion with DaleSpam.

You can use the Lorentz transform to change coordinates for any object or path. It is not limited to light nor to paths of objects which are stationary in one of the frames.

And that I misinterpret probably. Lorentz is meant to calculate time and distance how the moving object B it experiences (like a person) in rest (but moving in frame A). So you may use that formula only to calculate time and distance when you are in rest. That found factor 1/γ (in case of constant speed) you may use than for corrections in time and distances in the objects rest frame. What I did was now to use the transformed coordinates also for other movements in frame A for frame B. In fact you may only transform coordinates for other moving objects with the factor 1/γ. So I came again in the wrong way on the wrong formulas as first found in my old topic.

So I thought now knowing what is right, maybe I can find a formula now which relates 1/γ and (1 - V/C). Yes I found, so you can see the relation between the prediction from A of the time dilation in frame A and the really time dilation in frame B. That's is just as I thought a linear relation.

The formula is generally in itex (yes I am willing to learn Simon, it's not the whole effect what I wanted, later maybe) :

t'=t . 1/γ . 1/\sqrt{1+v^2/c^2} . (1-v/c)

And of course for length's/distances too.

So new times and distances are expressed in a way I thought in frame A (prediction time dilation). And in fact in diagrams like prof. Sander Bais is working like that too in thinking (seeing a light wave as just something else).

Generally you may I think always say when you move compared to somebody in rest, times goes slower compared to a passing light wave. When you move, you move a specified distance in frame A, at the same time the light wave travels a specified distance (started together with you, mover). So in that time you traveled also a distance, when you arrive has the light wave lesser distance traveled seen by you, so lesser time than in frame A at the same time. This can only give a time dilation (light wave was a clock too). A light wave cannot travel the same distance seen from A and seen from B because B has moved.

If you change the direction of the light wave does not matter in this thinking, only the positive distance you moved is important (not the direction for calculating time).

I wrote (digi99): The length of the light wave L_c = X_c - 0 = X_c.

No, it isn't.

This I don't understand, the traveled path of a light wave is c.t in the derivation for Lorentz too. In my eyes is the traveled path also c.t, so L_c = X_c (started at origin (0,0)). The traveled path is equal to the length of the passing light wave (was a beam). Or must we not talk about length but just over the length of a traveled path (maybe is meassuring length for an object something else as for light itself) ?

DaleSpam, I wait first for an answer on this new situation. Finally you will see a shorter light wave (and so related time), this must be visible finally in a diagram too ...

 

[digi99]

Stupid, formula is wrong, rest story is correct, maybe found a new one.

 

[Dale]

This I don't understand, the traveled path of a light wave is c.t in the derivation for Lorentz too. In my eyes is the traveled path also c.t, so L_c = X_c (started at origin (0,0)). The traveled path is equal to the length of the passing light wave (was a beam).

The length of something is not at all the same as the distance traveled. If I drive my car for an hour the distance traveled by my car is 100 km, but the length of my car is still just 4 m. The two concepts are completely different.

 

[digi99]

So later maybe I find one after reading Sander Bais.

For now the most closes formula's are from my old topic.

ΔXb_light = ΔXa_light . (1 - V/C).
ΔTb_light = ΔTa_light . (1 - V/C)

After Lorentz:

ΔXb_light = γ . (ΔXa_light - V/C . ΔXb_seen_from_a) (ΔXb_seen_from_a = V . ΔTa_light = V . T)

ΔTb_light = γ . ΔTa_light . (1 - V2/C2)

Expressed in X' and T' for the moving object B :

T' = γ . (T - V/C . X/C)

 

[digi99]

The length of something is not at all the same as the distance traveled. If I drive my car for an hour the distance traveled by my car is 100 km, but the length of my car is still just 4 m. The two concepts are completely different.

Ok understand, but I (we) have to see the distance of the traveled light wave in a diagram.
That's my statement. And you know that will be shorter always (in some way related to your movement, in fact I say nothing special or not understandable I think).

 

[Dale]

Ok understand, but I (we) have to see the distance of the traveled light wave in a diagram.
That's my statement. And you know that will be shorter always (in some way related to your movement, in fact I say nothing special or not understandable I think).

Yes, as long as you do not confuse length and distance traveled you can calculate either, in fact, I have done both above. The distance traveled is given by the Doppler formula:
(x_0',t_0')=\sqrt{\frac{1-v}{1+v}}(t,t)

 

[digi99]

Yes, as long as you do not confuse length and distance traveled you can calculate either, in fact, I have done both above. The distance traveled is given by the Doppler formula:
(x_0',t_0')=\sqrt{\frac{1-v}{1+v}}(t,t)

I forgot to thank you for your answer (math). I think there is an error in your formula (the same as you thought some time ago, you forgot to divide by c^2 for the time transformation, I am never sure so I say it only).

But the situation has changed now, my formula is not valid anymore compared with after Lorentz. It can now be having another meaning.

The distance of light is c.t in frame A and c.t' in frame B used in the derivation of Lorentz, you can't say after that derivation, that way of thinking is not valid anymore. So the distance traveled by the light wave is c.t in frame A and c.t' in frame B (just a fact). You willl find the same result in a diagram. And don't think how do I meassure that, maybe you get the doppler effect in that way.

Compared to a train, A sees x meters train passing (a special device counts the meters), B is moving y from A into the driving direction of the train, and he counts x-y meters train (with that special device in front of his eyes). The same is valid for a passing light wave presenting time by distance (his movement from B gives a time dilation). As I think time is a equal term like distance, distance / time = always the lightspeed.

The meaning of (1 - V/C) can be different explained, maybe is it valid in each little piece during the movement of B.

1/y = \sqrt{(1-v/c)(1+v/c)} and you see a relation with (1 - v/c), so a lineair relation is not to expect.

But if you say in each little piece of movement Δb_light = Δa_light . (1 - V/C), maybe you find by the math integration 1/γ. My math knowledge is from 35 years ago (as I studied it), sometimes I use formulas for my work but the more difficult things are gone (an integration, no it's gone, maybe if I try hard but now I don't see the point anymore).

 

[Dale]

I forgot to thank you for your answer (math). I think there is an error in your formula (the same as you thought some time ago, you forgot to divide by c^2 for the time transformation, I am never sure so I say it only).

You are correct, there are many factors of c or c² missing, but it is not a mistake. I was explicitly using units where c=1 specifically so that I could neglect those factors, that is why I made such a point of my units so many times. If you want to use units where c is not 1 then you can analyze the units of each expression to determine where to put the factors of c. For example, in the expression 1-v, 1 is unitless and v has units of speed, so you need to divide v by c to get a unitless number that you can subtract from 1.

The distance of light is c.t in frame A and c.t' in frame B used in the derivation of Lorentz, you can't say after that derivation, that way of thinking is not valid anymore. So the distance traveled by the light wave is c.t in frame A and c.t' in frame B (just a fact).

Yes, this is the second postulate, the invariance of c.

As I think time is a equal term like distance, distance / time = always the lightspeed.

Yes, in all frames.

The meaning of (1 - V/C) can be different explained, maybe is it valid in each little piece during the movement of B.

1/y = \sqrt{(1-v/c)(1+v/c)} and you see a relation with (1 - v/c), so a lineair relation is not to expect.

All fine.

But if you say in each little piece of movement Δb_light = Δa_light . (1 - V/C)

This equation is incorrect. I derived the correct one above.

 

[digi99]

OK, so you want to find
L_c'=|x_1'-x_0'|

From the Lorentz transform:
t'=\gamma(t-vx) and x'=\gamma(x-vt)
So in B's frame we have
(x_0',t_0')=\gamma(x_0-v t_0, t_0-v x_0)=\gamma(t-v t,t-v t) =
\frac{1}{\sqrt{1-v^2}}(t(1-v),t(1-v)) = \sqrt{\frac{1-v}{1+v}}(t,t)
Which is essentially just the relativistic Doppler formula.

Similarly we have
(x_1',t_1')=\gamma(x_1-v t_1, t_1-v x_1)=\gamma((t-1)-v t,t-v (t-1))

So now
L_c'=|x_1'-x_0'|=|\gamma((t-1-vt)-(t-vt))|=\gamma

Thanks DaleSpam, I understand now (I had not look enough). You just transformed the light line from frame A to frame B and indeed that formula of my is involved now (very good). One thing I don't understand is, you find γ as length while the length contraction is 1/γ. Can you explain that ?

Later today I come back on my statement, my subject after many weeks (to close it and going to use for 100% on my website, I will formulate again), but you have explained the formula ... but is also logic that it is valid in my thinking, only in a very little piece of movement and never as one total piece ... because nature is correcting immediately because of the constant speed C ...

 

[Dale]

One thing I don't understand is, you find ³ as length while the length contraction is 1/³. Can you explain that ?

The length contraction formula does not apply. Length contraction only applies when the object is at rest in one of the frames. A pulse of light is not at rest in any frame.

This is a very common mistake and is the reason I recommend against using the length contraction and time dilation formulas. They automatically fall out of the Lorentz transforms when appropriate, and using the Lorentz transforms prevents accidentally using the length contraction and time dilation formulas when you shouldn't.

 

[digi99]

The length contraction formula does not apply. Length contraction only applies when the object is at rest in one of the frames. A pulse of light is not at rest in any frame.

This is a very common mistake and is the reason I recommend against using the length contraction and time dilation formulas. They automatically fall out of the Lorentz transforms when appropriate, and using the Lorentz transforms prevents accidentally using the length contraction and time dilation formulas when you shouldn't.

Very interesting, what you say sounds logically, light is never at rest. Ok in length gives it another value than expected, possible it's light and speed c, the important factor, but it's path is c.t' = c. 1/γ . t. When you consider this as length e.g. on level t, so c.t is the length in frame A, is than the length γ . c . t or 1/γ . c . t, that's confusing, something I don't see right ? Light waves can not be bigger than other objects while moving ?

What I said about little pieces is not true too, because the speed is constant V, but maybe when you consider little pieces ΔV.t in (1 - V/C) you will find 1/γ or γ with math integration. Is this ever tried (I think so but without results) ?

 

[digi99]

That's what I told him and that's what Simon Bridge told him, although it's not normal Doppler because he's basing the time duration on the stationary frame instead of the moving observer's frame.

I even pointed out that if the moving observer changes direction and returns to the stationary observer, both their "special clocks" will end up with the same "time" on them instead of what should be happening according to the Twin Paradox.

But he still thinks its a better way to illustrate time dilation even though he realizes that it only "works" in one direction and even though it only "works" correctly at v=0 and v=c.

You were right from the beginning with the Doppler effect, I don't get it but I go to read again (besides I had wrong transformations), because it will always there I guess in the rest frame of a moving object. But I come with a conclusion I will place on my website, later I will make a new formulation. But maybe I understand because if something moves, you are in the "middle" of a light wave, is the reason probably. But if you consider a light wave on distance from both frames, also Doppler ?

 

[digi99]

OK, so you want to find
L_c'=|x_1'-x_0'|

From the Lorentz transform:
t'=\gamma(t-vx) and x'=\gamma(x-vt)
So in B's frame we have
(x_0',t_0')=\gamma(x_0-v t_0, t_0-v x_0)=\gamma(t-v t,t-v t) =
\frac{1}{\sqrt{1-v^2}}(t(1-v),t(1-v)) = \sqrt{\frac{1-v}{1+v}}(t,t)
Which is essentially just the relativistic Doppler formula.

Similarly we have
(x_1',t_1')=\gamma(x_1-v t_1, t_1-v x_1)=\gamma((t-1)-v t,t-v (t-1))

So now
L_c'=|x_1'-x_0'|=|\gamma((t-1-vt)-(t-vt))|=\gamma

Hi DaleSpam,

I want it let it go, while reading my books I could judge it again with more knowledge, but it confuses me how you use Lorentz, I still not get the picture how to use it, that was all wrong how I did it, but now I see, you use Lorentz how I did ?

I was/am thinking now (see an answer from me) that you may use Lorentz for transforming the coordinates for the moving object only, so you could see how B sees time (and distances of course) but all his coordinates are transformed to x' = 0 for any time (t' = 1/γ.t).

But I understand while looking to your solution, you may transform all coordinates from frame A to frame B with the Lorentz formula, so you see every point how B it sees at rest. So other speedlines you can transform in that way, so you see that speedline how B it sees. But transforming other points does not give t' = 1/γ . t anymore, this is my confusion now ?
After many weeks I have at least learn how to use Lorentz (cannot be that difficult, so I don't make such errors a next time).

Besides 1-V/C is a wrong thought of me because even in little pieces is never t' = (1 - V/C) . t at any moment. It's always the very know triangle relation between c.t, c.t' and v.t, but difficult to see mathematically if as well B and a light wave are all on the x-axes in the same direction. But maybe after reading my books.

Besides I thought today, looking to the length contraction, near the light speed is all very small comparing to standing still, so a foton with light speed is never visible for us (mass 0).

Depending on your answers concerning Lorentz I have to check what I was doing wrong in the Lorentz transformations where I found expressions with (1 - V/C) because I did it in the same way as you did.

 

[digi99]

That's what I told him and that's what Simon Bridge told him, although it's not normal Doppler because he's basing the time duration on the stationary frame instead of the moving observer's frame.

I even pointed out that if the moving observer changes direction and returns to the stationary observer, both their "special clocks" will end up with the same "time" on them instead of what should be happening according to the Twin Paradox.

But he still thinks its a better way to illustrate time dilation even though he realizes that it only "works" in one direction and even though it only "works" correctly at v=0 and v=c.

You were right from the beginning with the Doppler effect, I don't get it but I go to read again (besides I had wrong transformations), because it will always there I guess in the rest frame of a moving object. But I come with a conclusion I will place on my website, later I will make a new formulation. But maybe I understand because if something moves, you are in the "middle" of a light wave, is the reason probably. But if you consider a light wave on distance from both frames, also Doppler ?

I am a starter so understanding goes a little bit slower for me. Now I understand that my formula (question was has it a meaning) is the Doppler effect, in case it would be there, so not always is in another frame.

 

[digi99]

OK, so you want to find
L_c'=|x_1'-x_0'|

From the Lorentz transform:
t'=\gamma(t-vx) and x'=\gamma(x-vt)
So in B's frame we have
(x_0',t_0')=\gamma(x_0-v t_0, t_0-v x_0)=\gamma(t-v t,t-v t) =
\frac{1}{\sqrt{1-v^2}}(t(1-v),t(1-v)) = \sqrt{\frac{1-v}{1+v}}(t,t)
Which is essentially just the relativistic Doppler formula.

Similarly we have
(x_1',t_1')=\gamma(x_1-v t_1, t_1-v x_1)=\gamma((t-1)-v t,t-v (t-1))

So now
L_c'=|x_1'-x_0'|=|\gamma((t-1-vt)-(t-vt))|=\gamma

I learned from your Star Track series, never give up because I am still not satisfied with this topic (who is right or not is not important, but I don't get it).

See answer #18 in this topic. I agreed already with Ghwellsjr that I came to the Doppler effect when I was combining some formulas in math. Exact the same formulas you did found now and with formulas I thought valid.

But in my old topic I came with a kind of basic formulas too in the very beginning so I thought the relation is there between 1 - V/C before and after Lorentz. But I thought later the way I found them is not right, so I came to the last formulas close to it (#36 in this topic, a summarization of my original topic, the formulas where I found the Doppler effect too, not in frequencies but just as you in transformed coordinates of a light wave).

In the wrong formulas I thought they are not right because the relation 1/γ was not found, like you proved with the length Lc. So it is possible, that Lorentz is not valid for a light wave itself (speed c) but you used the Lorentz formulas also for transformations of a light wave. Very very confusing all.

Additional: Ok I understand the Doppler effect now, it's of course everywhere in the rest frame of the moving object, because everything moves in that frame with the same speed. Clear. But so there is a relation between 1 - V/C before and after Lorentz (I suppose now it may be used for transform all the linespeeds included light itselves, maybe all moving objects in the rest frame of the moving object has a length factor γ, only objects in rest 1/γ).

BUT I am talking in my topic about the total length of the path of a light wave (not the wave length, language confusion) and that must be before Lorentz c.t and after c.t' (and t' = 1/γ . t and not γ . t) and that I don't see back in the diagram. Funny but my statement is still there what I said (except time dilation calculated before Lorentz, that's be seen as the Doppler effect but you may explain that how you like, the path of the light wave is going shorter to explain the constant speed, the path (distance) presents time, so time will be lesser, or time dilation explained) ...

 

[digi99]

Last answer #47 changed ...

 

[Dale]

Light waves can not be bigger than other objects while moving ?

Why not? The thing to do here is not to simply apply the length contraction formula, but to derive a general formula that would work for v=0 and v=c. Some objects will contract some will expand, all depending on the various velocities. It would be a tedious exercise.

But maybe I understand because if something moves, you are in the "middle" of a light wave, is the reason probably. But if you consider a light wave on distance from both frames, also Doppler ?

Light from a source that is coming towards you is blueshifted and light from a source that is going away from you is redshifted.

But I understand while looking to your solution, you may transform all coordinates from frame A to frame B with the Lorentz formula, so you see every point how B it sees at rest. So other speedlines you can transform in that way, so you see that speedline how B it sees. But transforming other points does not give t' = 1/γ . t anymore, this is my confusion now ?

This is correct. You may transform all coordinates from A to B using the Lorentz formula. You are correct that not all points will give t'=t/γ, which is why it is important to use the whole formula, and not just part.

So it is possible, that Lorentz is not valid for a light wave itself (speed c) but you used the Lorentz formulas also for transformations of a light wave. Very very confusing all.

You can certainly use the Lorentz transform for a light wave. Light waves are governed by Maxwell's equations, and Maxwell's equations are invariant under the Lorentz transform.

What you cannot do is use a Lorentz transform to get a frame where a light wave is at rest. In all frames, the light wave will travel at c.

BUT I am talking in my topic about the total length of the path of a light wave (not the wave length, language confusion) and that must be before Lorentz c.t and after c.t' (and t' = 1/γ . t and not γ . t) and that I don't see back in the diagram.

In post 32 I did both the total length of the path (Doppler formula) and the length of the light wave (γ). Is that clear? I am not sure if you are confused or not.

 

[digi99]

In post 32 I did both the total length of the path (Doppler formula) and the length of the light wave (γ). Is that clear? I am not sure if you are confused or not.

Yes indeed DaleSpam it is clear now, if you fill in c.t there you will find t' = 1/γ .t so the light wave has traveled path c.t'. Of course is length from something that is moving different than in rest.

Thanks very much, thanks to you I understand Lorentz now fully. Also good for others to read I guess because without study you don't find this easily in popular books or internet.

Later I come with a reformulation again in a try to explain time dilation in a few sentences without complex drawings most people will only confuse (most are no Beta people).

So the meaning of the formula 1 - V/C is the Doppler effect, while you move (v.t) the frequency of the light wave will change, but the traveled path is still t. Only after Lorentz it will be t'. So I may not talk about a time dilation seen in frame A, it's just the Doppler effect.

So I go to explain it something different but in somehow equal words with the path of a light wave (the most exact clock) as main role ..