Interference Pattern versus SR

[Mentz114]

I've just remembered that radar distance transforms exactly like wavelength between inertial frames, so obviously (radar distance)/wavelength is the same invariant as the one I showed earlier.

It is more intuitive because if the radar is moving towards the sender/receiver then the wavelength looks smaller, as does the distance, by the same factor. And both look longer if the radar is moving away.

[Edit] JDoolin, your post above is wrong, as demonstrated by the above.

Anyhow, I think this is hijacking the thread so we should keep this to PMs.

 

[Adel Makram]

Here is my calculation of Δt` in PDF

https://www.physicsforums.com/attachment.php?attachmentid=43252&stc=1&d=1327862756

 

[zonde]

I've just remembered that radar distance transforms exactly like wavelength between inertial frames, so obviously (radar distance)/wavelength is the same invariant as the one I showed earlier.

This seems very clear argument why number of wavelength between two events remains the same after LT.

 

[JDoolin]

I've just remembered that radar distance transforms exactly like wavelength between inertial frames

Who are you quoting here?

If you go back to your source for this, I'm sure you'll either find that they mean something different than you by radar distance, or wavelength, or inertial frames, or, they've made the exact same error you did.

Since "radar distance" and "wavelength" and "inertial frames" seem like pretty unambiguous terms to me, I suspect the most likely explanation is that they've made exactly the same error you did, which is using β in the LT and β in the Doppler shift equation without paying attention to the sign convention.

Specifically, they miscalculated the radar distance, not recognizing that it must go up as the observer accelerates toward the source.

This is VERY counterintuitive. This is such a common mistake among General Relativity Experts, I can't really blame you for it.

But please see

http://www.spoonfedrelativity.com/pages/Galilean-Transformation.php

and

http://www.spoonfedrelativity.com/pages/SR-Starter-Questions.php

and

http://www.spoonfedrelativity.com/pages/coordinate_concept_quiz.php

And TRY to understand!

 

[Mentz114]

I've just remembered that radar distance transforms exactly like wavelength between inertial frames

Who are you quoting here?

For example
http://www.phil-inst.hu/~szekely/PIRT_BP_2/papers/pierseaux_09_ft.pdf
Page 9 equation (26) and following. Especially,

... the transformation of length and wavelength are the same

This is such a common mistake among General Relativity Experts, I can't really blame you for it.

I'm not so you can. Here's me making some more mistakes,

http://www.blatword.co.uk/space-time/radarlc.pdf

 

[Mentz114]

This seems very clear argument why number of wavelength between two events remains the same after LT.

Yes, it is. It isn't intuitive that the spatial coordinate difference should transform the same way, but I can't see an error in the (trivial) calculation I did.

[Edit]In flat spacetime, radar distance is coordinate distance, so that solves that one. Dear me, is that the kind of mistake general relatvists typically make

 

[JDoolin]

For example
http://www.phil-inst.hu/~szekely/PIRT_BP_2/papers/pierseaux_09_ft.pdf
Page 9 equation (26) and following. Especially,

$$\sqrt{\frac{1+\beta}{1-\beta}}+\sqrt{\frac{1-\beta }{1+\beta}}=2 \gamma$$

assuming β is a real number between -1 and 1,

and assuming

$$\gamma = \frac{1}{\sqrt{1-\beta^2}}$$

Is that equation

(a) Always true?
(b) Sometimes true?
(c) never true


Let's make a small modification to that equation:

$$\sqrt{\frac{1+\beta}{1-\beta}}+\sqrt{\frac{1-\beta }{1+\beta}}=\pm 2 \gamma$$

Is that equation

(a) Always true?
(b) Sometimes true?
(c) never true

Now I don't mind looking stupid so long as I can learn something, so I'm going to ask the stupid questions. Where does radar time appear in the equation? Where does wavelength appear in the equation? What relevance does this equation have to our conversation up to this point?

 

[Mentz114]

$$\sqrt{\frac{1+\beta}{1-\beta}}+\sqrt{\frac{1-\beta }{1+\beta}}= 2 \gamma$$

The left hand side is symmetric in β. Changing its sign will not change the rhs which is always 2γ.

The sign convention for the LT - if two observers are separating then the relative velocity is +beta. If they are approaching the relative velocity is -beta.

This is true in both frames. Doing an LT can't make converging obervers begin to separate. My calculation is for positive beta ( separating) and gets the right result.

 

[JDoolin]

Yes, it is. It isn't intuitive that the spatial coordinate difference should transform the same way, but I can't see an error in the (trivial) calculation I did.

[Edit]In flat spacetime, radar distance is coordinate distance, so that solves that one. Dear me, is that the kind of mistake general relatvists typically make

Yeah, that's pretty close! "the kind of mistake general relatvists typically make" is to first say "the meaning of distance is arbitrary" and then not to acknowledge the different definitions of distance.

So, obviously my list of "definitions of distance and length" is not complete since I have not included this "radar distance" in the list.

You have defined what you mean by radar distance here:

http://www.blatword.co.uk/space-time/radarlc.pdf

This looks pretty good to me. I haven't worked through it in close detail, but the way you've defined radar distance, L, here, you will be correct in saying the quantity

$$\frac{L}{\lambda }$$

is invariant IF you say L is the radar distance and λ is the wavelength of the reflected signal.However, in my understanding, the problem we've been working on is what happens to

$$\frac{L}{\lambda }$$

if L is the spatial distance to an event currently being observed, and λ is the wavelength of a NON-reflected signal.

This is an entirely different problem-set-up.

To clarify:

Spatial distance (to an event)
I look at my dresser, 10 feet away, and see an event that happened to it 10 nanoseconds ago, and say, "That event looks about 10 feet away."

More mathematically, if we have two events (x0,t0) and (x1,t1) then the spatial distance between those two events is |x1-x0|.

Radar distance (to an object)
I flash a light at an object, and see the return flash about 20 nanoseconds later, divide 20 nanoseconds by the speed of light and say, "I calculate it to be about 10 feet away."

So at least I understand now that we are definitely talking about two different situations; different contexts; different problems; different definitions of the same variables. With one definition of the variables, it is true to say L/λ is invariant. With another definition of variables, it is true to say L*λ is invariant.

 

[zonde]

I've been discussing this with Mentz114

The Number of Wavelengths is NOT an Invariant.

This is loosely based on Mentz114's proof, but with corrections and clarifications in the definitions of β, β_obs, and β_Away.

Say sender is sending short pulses of light toward receiver. Number of pulses in transit between two spacetime events (one on wordline of sender other on receiver's wordline) does not change with LT. I think this works as a model for number of wavelength as well.

And if we take your picture from post #50 and imagine performing LT with it I would say that number of light wordlines crossing horizontal line will change with LT. So that number of wavelength between wordlines of sender and receiver should change with LT.

 

[Mentz114]

So at least I understand now that we are definitely talking about two different situations; different contexts; different problems; different definitions of the same variables. With one definition of the variables, it is true to say L/λ is invariant. With another definition of variables, it is true to say L*λ is invariant.

In flat spacetime, radar distance is exactly the coordinate distance. I have demonstrated that both transform by the Bondi k-factor.

Did you see my remark about the sign convention in the LT ( post#148) ? I hope that clears up your previous misunderstanding.

 

[JDoolin]

In flat spacetime, radar distance is exactly the coordinate distance. I have demonstrated that both transform by the Bondi k-factor.

I don't know what you mean by "coordinate distance." Please compare it to my meaning of "spatial distance" between two events, in my last post.

Did you see my remark about the sign convention in the LT ( post#148) ? I hope that clears up your previous misunderstanding.

$$\sqrt{\frac{1+\beta}{1-\beta}}+\sqrt{\frac{1-\beta }{1+\beta}}= 2 \gamma$$

The left hand side is symmetric in β. Changing its sign will not change the rhs which is always 2γ.

Your math is correct. I was confused. γ will always be positive, and the left-hand-side will always be positive so long as -1<β<1, so my "smart" question was a red herring. However, my "stupid" questions are still pertinent. :) Namely, our conversation up to that point had been about spatial distance, and non-reflected signals. Suddenly we changed to radar-distance, and reflected signals.

Which is fine, of course, but that is why I was so confused. Why did we suddenly switch topics?

The sign convention for the LT - if two observers are separating then the relative velocity is +beta. If they are approaching the relative velocity is -beta.

I disagree with you. Here's the problem. If you have two sources, S1 in front of the observer, and S2 behind the observer, And your observer is going toward S1, then Obs and S1 are approaching, and O and S2 is separating. By your logic, you would have to do TWO DIFFERENT Lorentz Transformations.

Contrast this with what I've said.

(From http://www.spoonfedrelativity.com/pages/Number-of-Wavelengths-Is-Not-Invariant.php

In both of the equations below, (x,t) represent the coordinates of events in some <i>initial</i>, or source's reference frame, and (x',t') represent the coordinates of events in some <i>final</i>, or observer's reference frame.
$$\begin{pmatrix} x'\\ c t' \end{pmatrix}=\begin{pmatrix} \gamma & -\gamma\beta_{obs} \\ -\gamma\beta_{obs} & \gamma \end{pmatrix} \begin{pmatrix} x\\ ct \end{pmatrix}$$
In this equation &beta;_obs is the velocity of the observer in the "current" reference frame. You can also write this equation as
$$\begin{pmatrix} x'\\ c t' \end{pmatrix}=\begin{pmatrix} \gamma & \gamma\beta_{src} \\ \gamma\beta_{src} & \gamma \end{pmatrix} \begin{pmatrix} x\\ ct \end{pmatrix}$$
In this equation &beta;_src is the velocity of the source in the observer's reference frame.

This is true in both frames. Doing an LT can't make converging obervers begin to separate. My calculation is for positive beta ( separating) and gets the right result.

I'm not saying converging observers separate. I'm saying a particular pair of events along the world-line of the source, and the world-line of the observer separate. The intersection of the past-light-cone of the observer, with the world-line of the object the observer is moving toward. i.e. the image of the object in the observer's reference frame.

If two observers are located at the same point at the same time, and both are observing the same event, the one moving toward the event will see the image farther away, and the one moving away from the event will see the image closer.

Again, if you have doubt of this, I refer you to http://www.spoonfedrelativity.com/pages/SR-Starter-Questions.php,

 

[Mentz114]

JD, I hardly know how to address the inaccuracies and misunderstandings in your last post. I'll give a summary of what is going on.

1. We've defined phase to be the ratio of the distance between the receiver and emitter and the wavelength of the light, in some lab frame. The emitter and receiver are stationary wrt each other and the lab.

2. It has been shown that for 2 distance measures, coordinate distance and radar distance that they transform like wavelength, so both, divided by the wavelength give a Lorentz invariant. This is to be expected because in flat spacetime, coordinate distance and radar distance are the same.

This remark

Namely, our conversation up to that point had been about spatial distance, and non-reflected signals. Suddenly we changed to radar-distance, and reflected signals.

The radar pulses have no connection to the light in the 'lab' frame whose phase we are discussing. They are used for the radar distance measurement between the receiver and emitter from a moving frame.

And we didn't 'change to radar distance', it was added to the scenario to try to convince you of the invariance. Now you accuse me of trickery.

You are awesomely missing the point and I haven't got the energy to convince you, and this thread is not the place to do it either.

 

[JDoolin]

Say sender is sending short pulses of light toward receiver. Number of pulses in transit between two spacetime events (one on wordline of sender other on receiver's wordline) does not change with LT. I think this works as a model for number of wavelength as well.

And if we take your picture from post #50 and imagine performing LT with it I would say that number of light wordlines crossing horizontal line will change with LT. So that number of wavelength between wordlines of sender and receiver should change with LT.

What you're saying sounds basically right to me. But let me try to say the same thing in my own words.

If you've got two specific events, and draw a line between them, and then count the number of peaks between them, the number you counted won't change. That's because you're counting intersections of space-like world-lines, and null world-lines.

However, if you draw a line of simultaneity, it's a different story, because a line of simultaneity is not drawn between two specific events. The events selected are observer dependent. Hence, you get a different number of wavelengths.

The events themselves are invariant. The intersections of worldlines are invariant. But the number of wavelengths perceived simultaneously varies.

 

[JDoolin]

Here is a spacetime diagram from the reference frame of the observer:

As you can see, the segment EA is greater than the segment FC. That means as the observer passes the first object, the approaching object looks relatively far away. But as the observer passes the second object, the receding object looks relatively nearby.

On the other hand, the length of segments AG + GB is smaller than segment CH + HD. This means that if the observer establishes the radar distance from event A toward the approaching object, it appears relatively smaller, and if you establish the radar distance from event C, toward the receding object, the radar distance is relatively longer.

So the image distance from A is longer than the image distance from C.
and the radar distance from A is shorter than the radar distance from C.