Interference Pattern versus SR

[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&#039;\\ c t&#039; \end{pmatrix}=\begin{pmatrix} \gamma &amp; -\gamma\beta_{obs} \\ -\gamma\beta_{obs} &amp; \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&#039;\\ c t&#039; \end{pmatrix}=\begin{pmatrix} \gamma &amp; \gamma\beta_{src} \\ \gamma\beta_{src} &amp; \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.