Lorentz Transforms Simplified: Understanding Ordinary Math and Physics

[JM]

Radar Length

Here are details. Draw a graph of 0<ct<9 vs 0<X<6. Inches gives a good size. The lines X1 = 1 + .4 ct and X2 = 2 + .4 ct represent the ends of an object of Length = 1 moving at speed v = .4 ct. The line cT = 1 + x represents a radar pulse emitted from cT = 1. The pulse meets X1 at point a (2.33, 3.33) and reflects back to X = 0 at cT1 = 5.67, then continues on to X2 at ( 4, 5) and reflects back to cT3 = 9. Subracting the X components leads to the apparent length La: Xb - Xa = ( cT3 - cTo)/2 -(cT1 - cTo)/2 =(cT3 - cT1)/2 = La
The next step is to subtract the distance,d, the object has moved during the time the light took to go from X1 to X2: d = v/c (( cT3 + cTo) - ( cT1 -cTo))/2. Subtraction gives the result for the length of the moving object Lm, as stated above.
OK so far?

 

[Chris Hillman]

Good and bad "explanations"

Hi, JM,

It's my impression that the derivation and use of the Lorentz transform can be explained clearly and completely using ordinary math and physics of the college level.

Of course; see for example http://physics.syr.edu/courses/modules/LIGHTCONE/ and http://casa.colorado.edu/~ajsh/sr/sr.shtml for two websites which offer "visual tutorials" into Minkowski's geometric interpretation of relativistic kinematics (str). (Rob, you are too modest!)

What then is the reason for the emphasis on the mysteries of slow clocks, shrinking rulers, and the twin paradox?

Twin paradox: good. Slowing clocks, shrinking rules: bad, bad, bad. Particularly if you want to understand the geometric interpretation which is in universal use in the physics literature.

Of course, str does not say that ideal clocks slow or that ideal rulers shrink; that wouldn't even make sense! Unfortunately, due to laziness (or sometimes genuine misconceptions), some authors do use this language. If you find it confusing, good for you! Fortunately you can find good books written by more careful authors.

 

[Doc Al]

The length of a moving object is given by Lm = (1-v/c)(cT3-cT1)/2.

OK.

The length of a stationary object is Ls = (cT2-CT1)/2.

OK.

But cT2 and cT3 are related by cT3 - cT2= (cT3 -cT1)v/c.

How did you conclude this?

Here are details. Draw a graph of 0<ct<9 vs 0<X<6. Inches gives a good size. The lines X1 = 1 + .4 ct and X2 = 2 + .4 ct represent the ends of an object of Length = 1 moving at speed v = .4 ct.

Seems to me that this represents a moving object of apparent length = 1 (what you called Lm earlier), since X2 - X1 = 1. (I had trouble following the rest of your post.)

 

[lalbatros]

It's my impression that the derivation and use of the Lorentz transform can be explained clearly and completely using ordinary math and physics of the college level. What then is the reason for the emphasis on the mysteries of slow clocks, shrinking rulers, and the twin paradox?

In other words, what makes the Lorentz transformation a physical law?
Maybe that's not so easy to answer, afer all!

My understanding is that the Lorentz transformation is about defining the coordinate transformations that leave the physical laws unchanged (invariant), and this is simple algebra indeed: a spherical wavefront needs to be invariant, something similar to the rotations studied on secondary school.

The surprise is in the physical consequences, these are not simple interpretations, these are very striking and were difficult to accept. The Lorentz invariance leads to strinking consequences: clock slowing down and specially the twin paradox. Fortunately these consequences are experimental facts now: particle lifetimes in accelerators, clocks drifts tested aboard planes.

That such important experimental facts are the consequence of the Maxwell's equations, and more specifically the Lorentz symmetry of the maxwell's equations is still very striking today.

However, considering "classical clocks" as mechanical devices, one realizes that imposing the Lorentz symmetry to the whole of physics -not only to electromagnetism-, was the real revolution. This revolution was necessary for the consistence of physics. And the consequences deserve all these discussions. Clocks are slowing down because the laws of mechanics are also relativistic.

Michel

 

[JM]

Radar Length

OK.

OK.

How did you conclude this?


Seems to me that this represents a moving object of apparent length = 1 (what you called Lm earlier), since X2 - X1 = 1. (I had trouble following the rest of your post.)

Thanks for your response. I tried to clarify but my reply was not accepted. I don't know why. I will try again later. For now note that a( 2.33, 3.33) indicates that point a is located at X = 2.33 cT = 3.33.

 

[JM]

Thanks, Chris. What is 'str'? I have read Minkowski's paper and find it to be opaque, is seems all math and no physics.
Your reply and the one following shows my problem; you say clocks and rods don't change, and the following person says they do. Until there is agreement between the two positions there will be confusion. Based on Einstiens analyses I think clocks all run at the same speed and objects don't shrink. Help!

 

[bernhard.rothenstein]

clocks

Thanks, Chris. What is 'str'? I have read Minkowski's paper and find it to be opaque, is seems all math and no physics.
Your reply and the one following shows my problem; you say clocks and rods don't change, and the following person says they do. Until there is agreement between the two positions there will be confusion. Based on Einstiens analyses I think clocks all run at the same speed and objects don't shrink. Help!

I know for miself the folowing equations relating clock readings and time intervals
t=gt0 (1)
t0=t'0D (2)
t=g(t'+Vx'c^-2)= gt'(1+Vu'c^-2). (3)
where g is the Lorentz factor and D the Doppler factor.
Equation (1) is best illustrated by the light clock approach and relates the proper time interval t0measured as a difference between the readings of the same moving clock and the non-proper time interval t measured as a difference between the readings of two distant clocks synchronized in the stationary reference frame.
Equation (2) establishes a relationship between two proper time intervals whereas equation (3) establishes a relationship between two non-proper time intervals, measured as a difference between the readings of two distant clocks at rest in the involved inertial reference frames respectively and synchronized in each of them in accordance with Einstein's clock synchronization procedure.
Is it necessary to mention that one of the clocks of I goes slower then the clocks of I'? I consider no! and that is the reason why I am still interested in srt.

 

[JM]

Radar Length

DocAl, I tried to upload the entire graph but my file size is too big. Can we proceed a little at a time? Have you made the graph of cT vs X, and drawn the three lines indicated? Point a is where the radar pulse meets line 1, the near end of the moving object, and point b is where it meets line 2 , the far end. The reflections of the pulse from these points arrive at X = 0 at cT1 and cT3. The difference leads to the formula for the apparent length La, apparent because the effect of motion has not been accounted for.
Let me know how you are doing.

 

[JM]

Prof. Rothenstein, Thanks for your comments. I have read the analysis of the light clock, and I agree with the math. It's the conclusion, stated as " moving clocks run slow' that I don't see. Some writers say that all motions slow down, even though the moving people can't detect the slowing. Wouldn't it be just as correct to say that stationary clocks run fast, or that none of the clocks change tempo and it's just the comparison that's different? Its hard to express my objections, I just think that 'slow clocks' hasn't been proven.

 

[Doc Al]

DocAl, I tried to upload the entire graph but my file size is too big. Can we proceed a little at a time? Have you made the graph of cT vs X, and drawn the three lines indicated? Point a is where the radar pulse meets line 1, the near end of the moving object, and point b is where it meets line 2 , the far end. The reflections of the pulse from these points arrive at X = 0 at cT1 and cT3. The difference leads to the formula for the apparent length La, apparent because the effect of motion has not been accounted for.
Let me know how you are doing.

I'm doing just fine. What is it that you are trying to graph? I couldn't understand what you were trying to do in your previous posts. You seem to be under the impression that if only you "account for the effect of motion" then relativistic effects will disappear. No idea what you mean by that.

Seems like the graphing thing is slowing you down. Instead, tell me how you arrived at this (incorrect) result:

But cT2 and cT3 are related by cT3 - cT2= (cT3 -cT1)v/c.

 

[JM]

Radar Length

Thanks for staying with me, Doc. To see this relation add the points c( 3.33,3.33), d( 4, 3.33), e( 3,33.4.33), and f(4, 3.67). The object is at a-c, it has moved from c to d. e is where the pulse would reflect from the far end if the object were stationary, this reflection would arrive at X= 0 at cT2. The point f is where the extension of this reflection meets the vertical line b-d. The triangle b-f-e is a 45-45-90 triangle with base b-f, which =cT3-cT2, and altitude c-d, which = the distance moved. Since the base =2 x the altitude, the relation for cT3 - cT2 results.
This operation involves only the observer at X=0, not anyone on the object so how would relativity be involved?

 

[Doc Al]

To see this relation add the points c( 3.33,3.33), d( 4, 3.33), e( 3,33.4.33), and f(4, 3.67).

What do these coordinates represent? Rather than give a string of numbers, just tell me what you are trying to do.

The object is at a-c, it has moved from c to d. e is where the pulse would reflect from the far end if the object were stationary, this reflection would arrive at X= 0 at cT2. The point f is where the extension of this reflection meets the vertical line b-d. The triangle b-f-e is a 45-45-90 triangle with base b-f, which =cT3-cT2, and altitude c-d, which = the distance moved. Since the base =2 x the altitude, the relation for cT3 - cT2 results.

Don't really have a clue what you are doing here. I assume you are trying to describe how the pulses reflect from the rear and front of the object. If so, do it systematically. Describe: The pulse reflecting from the rear of the object; the receipt of that pulse by the stationary observer; the pulse reflecting from the front of the object (where and when does this happen, according to the stationary observer's calculation?); the receipt of this pulse by the stationary observer.

This operation involves only the observer at X=0, not anyone on the object so how would relativity be involved?

Don't really know what you mean here. As stated before, you can use the time difference between the reflected pulses to measure the length of the moving object. As you stated yourself:

The length of a moving object is given by Lm = (1-v/c)(cT3-cT1)/2.

Now to predict what that answer will be is another matter altogether. For that you need relativity! If you ignore relativity, of course you'll get the bogus result that the lengths are the same.

 

[JM]

DocAl, I have only one observer, I don't care what other observers are looking at. Why do I need relativity? ( You're right, I do get the moving length = the stationary length.)

 

[Doc Al]

DocAl, I have only one observer, I don't care what other observers are looking at. Why do I need relativity?

You only need it if you would like to correctly predict the results of any measurements you might make on moving things. Up to you!

You don't seem to realize that using that radar formula for the length of the moving object automatically includes relativistic effects. Measure T3 and T1, do the calculation, and you'll find that Lm < Ls. (Of course, since we're not really doing any measurements, we have to use our knowledge of physics to predict what that answer would be.)

( You're right, I do get the moving length = the stationary length.)

That's because you are ignoring relativity. Please show how you get it and we can point out what you are missing.