Classical Physics is coming back, RELOADED!!!

1100f,
I don't see any error in my analisis.
λ is the De Broglie "wave-lenght" which is a lenght, this means distance and according to Relativity Theory it must be transformed according to Lorentz Transform to give its value in another referential of observation.
The same reasoning is valid for the classical formulation and the four-vector formulation and the same results are obtained.

The analisis show that De Broglie Law is not invariant under a relativistic change of referentials.
It is a clear example that the first postulate of Relativity Theory fails: reality shows non-invariant laws in Physic exist, is a fact.

 

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The transformation of a distance according to Lorentz transformation is not just to take the distance and multiply it by the contraction factor. The Lorentz transformation acts on events. For a rigid rod, you have two events: one is that one end of the rod is at some position at some instant of time, the other is at the other end of the rod at the same instant. In another frame you must see to where the evnts are transformed. What is the position of one end of the rod at some instant (in the new frame) and where is the other end of the rod at the same instant in the new frame (I remind you that two events that are simultaneous in one reference frame are not necessarily simultaneous in another frame). Failing to understand this is a bad unerstanding of the theory of relativity.
The wavelength is not the distance between the two ends of some rigid body. You cannot use the Lorentz transformation the way you use it. As I told you, the wavelength is the distance between two crests of the wave.

According to you, even without the use of relativity, the relation λ = h/p does not hold since according to you the wavelength remains the same in all frames (and this is wrong).

The way to see if the relation is valid (again, for you to understand this, I am doing it non relativistically) is to find what should be the wavelength in the new reference frame and what should be the new momentum in the new reference frame. Sot that p--> p' and λ --> λ' and see that if in one frame p = h/λ to verify if p' = h/λ'.
To do this you must use the dispersion relations of the de-Broglie waves.

 

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martillo,

Special relativity says that the laws of physics are invariant when space time coordinates are transformed using the Lorentz transformation. The de Broglie wavelength is not an event in this sense.

Also, this vector:

isn't the one to use. You transform k.

Basically, the fact that an abstract mathematical quantity used in QM (which just happens to be called a length) doesn't transform by Lorentz doesn't in itself contradict relativity. A contradiction would be if you predicted a fringe pattern two different ways (eg. in a double-slit experiment) and got two different answers. This would be something physical and measurable.

 

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Maybe this will help you understand.
Forget for a moment SR and quantum mechanics.
Just look at sound waves. a loudspeaker gives a steady sound at a constant frequency. the wave produced has a wavelength. Now I am moving wrt this loudspeaker at some velocity v. Is the wavelength as measured by me the same as the wavelength measured by someone stationnary wrt loudspeaker?

 

The problem I see with Martillo is that simultaneity is so crucial to understanding relativity, but you simply can get what it means from a wikipedia page.

 

1100f,

I agree with martillo on this one. Always have. The wavelength of light, once emitted, does not change. What changes is the frequency at which the moving observers count those crests. A beam of light emitted from an object thousands of light years distant from the moving observers does not change in wavelength just because the observers change velocity relative to the beam, each observer counts the crests at different rates according to his velocity relative to the unchanging wavelength. Doppler shift is actually a change of frequency, not wavelength. The fundamental mistake in physics is to assume the emitter of the light is moving, and not the observer. The speed of light is a constant, a moving observer can determine his own change in speed wrt to emitter by the measuring the change in frequency of the emitted light.

 

Hi martillo,

Congratulations for actually using the Lorentz transform! I am honestly very impressed and pleased that you actually went to the effort to do so. Also, let me start out by saying that you did it correctly, and simply didn't notice that you already found the key to understanding the situation!

You used (0,λ,0,0) as the four-vector for a wavelength. What does this mean? Well, the wavelength is the spatial distance between two successive peaks at a given instant in time. So if the event (0,λ,0,0) is one peak, then there is another peak at (0,0,0,0). Now, the nice thing about using the full Lorentz transform is that you not only get length contraction, but also time dilation and the relativity of simultaneity.

How does that impact this scenario? Note that the transform of (0,λ,0,0) is (-βγλ,γλ,0,0) and the transform of (0,0,0,0) is (0,0,0,0). In the second frame the distance between the two events is length contracted as expected, but due to the relativity of simultaneity they do not occur at the same time! Since they do not occur at the same time they can not possibly represent the measurement of a wavelength. Thus, there is no contradiction in the fact that the wavelengths in the respective frames are not related to one another simply by length contraction. The key to understanding was hiding in the timelike component of the four-vector that you already (and correctly) calculated.

I disagree, let me go through the derivation here, for brevity I am only going to consider a single spatial dimension and I will use the notation I showed above, but I will be explicit with all of my c terms etc.
p=(mγc,mγv) (definition of four momentum for a massive particle)
k=(f/c,1/λ) (definition of wave four-vector*)
p=hk (generalized deBroglie relationship)
L=((γ,-γv/c),(-γv/c,γ)) (Lorentz transform matrix from unprimed frame to particle's rest frame)

So, starting in the unprimed frame
k=(mγc/h,mγv/h)
λ=h/mγv
Lorentz transforming to the primed (rest) frame
k'=L.k=(mγ²c/h-mγ²v²/ch,0)=(mc/h,0)
λ'=infinity

Alternatively, starting in the primed frame
k'=(mc/h,0)
λ'=infinity
Lorentz transforming to the unprimed frame
k=L^-1.k'=(mγc/h,mγv/h)
λ=h/mγv

So you clearly see that the generalized de Broglie relationship is, in fact, invariant. Each frame can calculate their own results directly and each frame can transform their results to the other frame using the Lorentz transform as expected.

-Dale

*actually the wave four-vector is usually defined in units of radians/m but here I am using cycles/m because it makes it easier to express λ, that only requires a change from hbar to h in the deBroglie formula.

 

Dalespam,
What you have done is to find the Lorentz transformation of the vector k in both referentials in the both directions of transformations. But this is the transformation of the De Broglie formulation to the two referentials and of course they give the same formula, I agree with that.
The problem is that the formulation does not verify the second condition I have talked about: that to be really invariant a formula or even a four vector formulation MUST be consistent for its variables in the change of coordinates (change of referential). This means that if we make a change of coordinates every variable in the formula must give the same result if calculated with the formula in the second referential (after the change of coordinates) or if it is calculated directly with the coordinates transform (in this case Lorentz Transform). This last condition is not verified and so the formula and the four-vector formulation is NOT INVARIANT.

 

who was talking about the wavelength of light?

 

I usually are in agreement with 2inquisitive but here it seems the fact that doppler is not a linear function isn't being considered. That is the change in measured frequency is not the same as one would get if counting saw teeth by rubbing the blade at different speeds.

 

martillo, if you think that the generalized form of the deBroglie relationship, p=hk, is not invariant, then you are clearly wrong. This form, with only four-vectors and scalars, is called the manifestly invariant form because it is completely obvious simply by glancing at it that it is invariant! In addition, just to make things perfectly clear I went through the details in my previous post and showed explicitly that we get:
p/h=k
p'/h=k'
p'/h=L.k
p/h=L^-1.k'

I don't know what more I could do here to make the invariance obvious, and I really don't understand your continued objection. Your two "conditions" seem exactly the same to me and to me it seems that you are disagreeing with yourself! What is the difference between your two conditions? Perhaps it would help if you could express your requirements or objections mathematically, but again, if you think that a manifestly invariant equation is not invariant then you are obviously wrong.

-Dale

 

Hi martillo, something else to think about is what przyk said up above:

In the case of waves, what is the measurable quantity? One measurable quantity is the phase of the wave at any given event.
sin(phi(t, x)) = sin(2 pi (t f - x/λ)) = sin(2 pi s.k )
So phi(t,x) = s.k is a Lorentz scalar, and you see plainly that all frames will agree on the value of the phase of the wave at any event. Therefore all frames will necessarily agree on any experimental outcomes, such as those used to test the deBroglie relation. The inescapable logical conclusion is that the general deBroglie relationship is frame invariant and satisfies the first postulate.

-Dale

 

1100f,

Seems you were, 1100f. Its the frequency that changes due to the motion of the detector, not the wavelength. The wavelength should always be determined by the emitter's rest frame, IMO.

 

What are you speaking of when you say 'Doppler is not a linear function'? Are you speaking of second order Doppler effects, or the fact that 'classical' formulas were based on inaccurate Newtonian approximations of centuries ago?

 

In the formula p = h/λ nobody mentioned the it is light.

The wavelength is not determined by the frequency of the emitter. it is the distance between two consecutive crests of the wave

 

1100f, sound and light waves are both treated by the same formulas. Here is a cut & paste from mathpages to illustrate what I mean.

http://www.mathpages.com/rr/s2-04/2-04.htm
Of course the wavelength is the distance between crests, but what I was referring to was the particular wavelength is determined by the emitter, not the detector. The detector measures the frequency at which the crests are detected. To find the true wavelength, one must eliminate any Doppler shifts due to velocity, gravitational fields between the detector and emitter, etc. Ususally the spectrum of some specific element is found, such as iron, and then the Doppler shift calculated based on that line.

 

So you agree that the wavelength as measured in one reference frame is different from what is measured in another reference frame.

 

Not in Classical Physics where the "wave-lehght" remains the same but within Relativity lenght "contraction" exists and it must follow Lorentz Transform.

But the values obtained applying Lorentz Transform and that of applying De Broglie formula differ!

This cannot happen with invariant formulations.

De Broglie formulation is not invariant.

 

It is. The same equation is used and equally valid in all frames. No-one has ever needed to add some extra, frame-dependent term to correct for anything.

 

No it does not