Directional independence of motion in Lorentz Transformation

[Kanesan]

Hi All,

I am trying to understand the directional independence of LT. In contrast, we all know that Doppler Effect is dependent on the direction of motion. I have tried to find any reasoning or explanation and could not find one so far. May be I did not use correct terms in my searches. If anyone can explain or point to any existing explanation(s), that will be great.

I would like to clarify that V**2 is not an explanation; the "directional independence" is same as that Math term. I am looking to find any existing explanation(s) beyond that. If no other explanation exists, I have to take that as the answer!

If it has been discussed before and I missed, I apologize for that.

Thanks.

 

[Ibix]

The Lorentz transforms turn out to follow from the principle of relativity, that the laws of physics are the same in all inertial frames. If the laws cared whether they were approaching you or moving away, the universe would be a very different place.

I suppose you could say that what you are calling the directional independence of the transforms (symmetry might be a better way of putting it) encodes the fact that the laws of physics don't care who, what or where you are.

 

[Kanesan]

Thanks Ibix, I just wanted to know if I have missed something.

Sure, the universe/laws would not care. But, I just thought that the observer may find some difference like in Doppler (yes, the comparison has been my problem!). So, that's why I was trying to find more.

Looks like I was inventing a term called "directional independence", instead of using well known "symmetry"! lol.

 

[Dale]

I am trying to understand the directional independence of LT.

What do you mean by directional independence? A Lorentz transform to a frame moving at v in one direction is not the same transform as to a frame moving at v in another direction. The laws of physics are the same, but the coordinates are not.

 

[pervect]

If I'm understanding your question, you might try the keyword "isotropy" or "isotropic", which means having uniform physical properties in all directions.

This is one of the basic assumptions made in deriving the Lorentz transform. If you can explain what you want in more detail, I might be able to answer at more length, at the moment I think giving you the keyword is the best approach.

 

[Kanesan]

Sorry for the confusion. I will try to clarify. The time dilation and length contraction being independent of +v or -v of a moving object. That is what I tried to call "directional independence" of LT! I hope this explains the question better.

More precisely, I am trying to see any explanation for why Doppler frequency shift is direction dependent but time dilation is not. May seem like a weird comparison! But, that's what I am trying. (still I keep using the same term!)

I agree DaleSpam, the coordinates will change differently and that I find easier to understand.

Pervect, I thought "isotropy" is used for speed of light. No problem there. Not sure if it is used for the motion of objects, but it may fit there also?

 

[Dale]

Sorry for the confusion. I will try to clarify. The time dilation and length contraction being independent of +v or -v of a moving object. That is what I tried to call "directional independence" of LT! I hope this explains the question better.

Yes, that helps. This actually follows very easily from the Lorentz transform for differences in coordinates (in units where c=1):
$$\Delta t'=\frac{\Delta t-v\Delta x}{\sqrt{1-v^2}}$$
$$\Delta x'=\frac{\Delta x-v\Delta t}{\sqrt{1-v^2}}$$
Note, in the transform there are terms including $v$ (not directionally independent) and terms including $v^2$ (directionally independent).

Time dilation is given by computing $\Delta t'$ for a clock at rest in the unprimed frame. For such a clock $\Delta x = 0$ so:
$$\Delta t'=\frac{\Delta t-v\Delta x}{\sqrt{1-v^2}}=\frac{\Delta t}{\sqrt{1-v^2}}$$
Note that the directionally dependent term has dropped out, leaving only the directionally independent term. So even though the Lorentz transform as a whole has directional dependence, time dilation does not due to the specific requirement that $\Delta x =0$.

 

[Kanesan]

Thanks DaleSpam for going deeper to explain the Δt' equation. But that leaves one more question about the Δx. I thought the unprimed frame is assumed to be stationary always (as a convention of course). Then Δx will be zero always (due to that convention). Can you extend a bit on that point? May be I need to read a lot more!

Thanks for your help. I really appreciate it.

 

[Dale]

I thought the unprimed frame is assumed to be stationary always (as a convention of course). Then Δx will be zero always (due to that convention). Can you extend a bit on that point?

It is not always assumed as a convention. However, the standard time dilation formula only applies in cases where that assumption is met.

That means that there are occasions where the time dilation formula simply does not apply. For example, for two events on a pulse of light there is no reference frame where $\Delta x=0$ so the time dilation formula cannot be used to transform the $\Delta t$ between those two events.

 

[Kanesan]

Thanks DaleSpam for pointing out there are exceptions to the convention. That is going to take some time for me to get a grasp!

When I read back the derivation you showed, though we had the directional element (v) during the derivation from coordinates, in the end the time dilation equation lost it. So, we are back to where we were, though clearly we started well. We have the same result whichever way we approach. So, we have the result showing "directional independence" but no explanation of why it is so or what it means. May be in some way I am thinking that the equation is the "result" we get and we need an "explanation" for it. I am not sure whether that thinking is philosophically correct! Sorry if it looks dragging.

 

[Nugatory]

So, we have the result showing "directional independence" but no explanation of why it is so or what it means. May be in some way I am thinking that the equation is the "result" we get and we need an "explanation" for it. I am not sure whether that thinking is philosophically correct!

Directional independence is something that we observe - it is a fact that every imaginable experiment that we have ever done has shown that we live in a universe with directional independence (more formally, spatial "isotropy" as pervect said). Physics won't tell us why; instead the universe is telling us that our physics has to be that way.

 

[Kanesan]

Thanks Nugatory. It is easy and reasonable. I am fine with that. Same way, if I see Doppler effect alone, I am fine with that also.

But when I see them together (Time Dilation and Doppler Effect), I get the problem. One is direction dependent while other is not!
This has been a nagging problem for me almost a decade!

There is no question about spatial isotropy from the light perspective. I am not sure if we can call the motion of object relative to observer as isotropic, so I keep using "direction independent" kind of terms. To the observer, the symmetry is reduced due to the presence of the object. The object may not care about the observer, but observer's perceptions may not be isotropic, especially when that object is the subject of observation... so goes the thought there. I admit that I may not be using most appropriate term there. Still, we could leave that for now as it is only a minor point as we all think.

 

[Dale]

When I read back the derivation you showed, though we had the directional element (v) during the derivation from coordinates, in the end the time dilation equation lost it. So, we are back to where we were, though clearly we started well.

I am not sure what you are asking about now. If you start with the Lorentz transformation and a clock at rest in unprimed frame then you derive the directional independence of time dilation.

Are you looking for a derivation of the Lorentz transform? If not then I am at a loss for what you think is still missing.

 

[PAllen]

Maybe this will help?

The Lorentz transform is direction dependent, period (as is the Galilean transform, and anything based on the principle of relativity - which preceded SR by centuries). Theories and models from them, that are Lorentz invariant are also direction dependent. Everything related to the question of "which way is it going" are affected by direction (Doppler and a zillion other things are in this category). A very few special observables, are not direction dependent. These are the exceptions, and it is only for these you might inquire, if you insist, "why?". In the case of time dilation, you can 'explain' it by saying time dilation is the ratio of the time component of a timelike 4-vector to its norm. Neither of these quantities depends on spatial direction, so you derive the so called "clock hypothesis" - that speed is the only thing affecting clock rates in an inertial frame in SR.

For length contraction, there is probably some cute statement you can take as 'explanation' in terms of the expansion tensor, bur I have no interest in working it out. It is easy to derive from the Lorentz transform that the only contraction is along the line of motion, and not affected by a reversal of direction on the same line of motion. This is actually a much less general statement than the clock hypothesis. There are exactly two, and only two, velocities (speed + direction) that produce the same contraction. It is really a symmetry of direction reversal (only).

 

[Chestermiller]

Let me understand correctly. You want to know why the relativistic Doppler effect is direction dependent, while time dilation and length contraction are not, correct?

Chet

 

[Kanesan]

Sorry DaleSpam if my comment was confusing. There was no new question except that I retraced that whichever way we arrive at the time dilation, we lost the direction dependence in the end of derivation. So, the derivations show the result but not any reason as to why so different.

Chet, I was only comparing normal Doppler effect with time dilation. Did not think about relativistic Doppler yet.

PAllen, similar to time dilation being a ratio of times, Doppler is also a ratio of times (frequency ratio is in effect time ratio) and both use relative velocity. So, mostly I am trying to compare the Doppler effect and time dilation. Still they have very different forms of result, in terms of direction of motion being important or not.

 

[Chestermiller]

With the Doppler effect, there is the superimposed effect of transmission of a signal from a source to a receptor. This is not isotropic, and depends on whether the source and receptor are moving towards each other or moving away from each other. Time dilation has no such signal transmission effect, and all the measurements for time dilation are assumed to be made at the spot.

Chet

 

[PAllen]

Sorry DaleSpam if my comment was confusing. There was no new question except that I retraced that whichever way we arrive at the time dilation, we lost the direction dependence in the end of derivation. So, the derivations show the result but not any reason as to why so different.

Chet, I was only comparing normal Doppler effect with time dilation. Did not think about relativistic Doppler yet.

PAllen, similar to time dilation being a ratio of times, Doppler is also a ratio of times (frequency ratio is in effect time ratio) and both use relative velocity. So, mostly I am trying to compare the Doppler effect and time dilation. Still they have very different forms of result, in terms of direction of motion being important or not.

1) Forget relativity, and imagine non-relativistic Doppler. Does it depend on direction? Of course. How can it be different relativistically - Newtonian mechanics and SR both the the same principle of relativity, and are both theories with homgeneity and isotropy. The difference between Newtonian theory and SR, is that in Newtonian theory, after factoring out Doppler, you conclude ideal clocks all tick the same, while in SR you find (direction independent) time dilation after factoring out classical (direction dependent) Doppler. Note, the raw observation of moving clocks shows strong direction dependence. After factoring out this direction dependence, in SR, you are left with a direction independent time dilation (that depends only on speed).

2) Doppler is not the the time component of a vector, nor the norm of a vector. In 4-vector notation, it is the dot product of a null vector and a timelike vector, with the null vector reflecting the direction of propagation of light. So, of course it depends on the direction of propagation of light. You have one timelike vector for the detector. You cannot expect the dot product of this vector with two different null vectors to be the same! This gets at the point many have made, but you seem resistant to: the Lorentz transform (just like the Galilean transform) is direction dependent. You don't seek reasons for why things change in a direction dependent way under such a transform. Instead, you may seek to understand the special exceptions that do not.

 

[PAllen]

I can't resist a somewhat sarcastic comment. Why does it take so much longer to get to Vienna from Paris going West compared to going East? This is very mysterious ...

 

[Dale]

So, the derivations show the result but not any reason as to why so different.

The derivation IS the reason. That is why we do derivations. They show how the conclusions logically from the assumptions.

In this case the derivation shows why the directional independence follows from the Lorentz transform and a clock which is stationary in the unprimed frame. So I am not sure what could qualify as a "reason as to why" if not a derivation. What could possibly be a better reason?

 

[Kanesan]

Thank you all for the contributions. It will take some time for me to digest everything (put together all the inputs) to get a better picture. Without that first, it will be simply dragging. Yes, I am slow indeed! lol. [I have similar problem going from old Fortran to object oriented programing!]
I really appreciate all the patient answers.

 

[Dale]

No problem. While you are thinking, one other thing that may help is to show how to derive the "directional dependence" of the Doppler shift:

As before, we start with the Lorentz transform for differences in coordinates (in units where c=1):
$$\Delta t'=\frac{\Delta t-v\Delta x}{\sqrt{1-v^2}}$$
$$\Delta x'=\frac{\Delta x-v\Delta t}{\sqrt{1-v^2}}$$
Again, note that in the transform there are terms including $v$ (not directionally independent) and terms including $v^2$ (directionally independent).

The Doppler shift is given by computing $\Delta t'$ for a pulse of light in the unprimed frame. For such a pulse of light $\Delta x = \Delta t$ so:
$$\Delta t'=\frac{\Delta t-v\Delta x}{\sqrt{1-v^2}}=\frac{\Delta t(1 - v)}{\sqrt{1-v^2}}=\Delta t \sqrt{\frac{1 - v}{1 + v}}$$
Note that the directionally dependent term has not dropped out, so the Doppler shift is directionally dependent.

 

[Kanesan]

Thanks DaleSpam for adding the Doppler side derivation (similar to time dilation derivation) to be included in the full analysis/thought process. It will surely help to consider both equations and derivations fully/properly.