Lorentz Transforms Simplified: Understanding Ordinary Math and Physics

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

 

[robphy]

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?

While the mathematics is relatively simple, the physical interpretation by way of observations and thought-experiments is highly-nonintuitive [for us]. (Mathematics is probably too abstract for the general audience.)

 

[JM]

Thank you for your response, robphy. Part of ordinary physics is the interpretation of Lorentz transform results as describing the physical behavior of light waves. This interpretation seems to me to be sufficient. Is there some reason to follow a different interpretation? Could you elaborate on what observations and thought-experiments you have in mind?

 

[bernhard.rothenstein]

Thank you for your response, robphy. Part of ordinary physics is the interpretation of Lorentz transform results as describing the physical behavior of light waves. This interpretation seems to me to be sufficient. Is there some reason to follow a different interpretation? Could you elaborate on what observations and thought-experiments you have in mind?

I think that clock synchronization is the first step each learner should go through. Combined with the concept of space-time coordinates of the same event in two inertial reference frames in relative motion a combination of math and phys leads to the Lorentz-Einstein transformations.
sine ira et studio

 

[JM]

I think that clock synchronization is the first step each learner should go through. Combined with the concept of space-time coordinates of the same event in two inertial reference frames in relative motion a combination of math and phys leads to the Lorentz-Einstein transformations.
sine ira et studio

Do you find that you need to follow any interpretation other than the ordinary physics view that the results represent the behavior of the light waves? Can you direct me to your derivation?

 

[Daverz]

The results represent the behavior of all physical objects and fields. Do I misunderstand the question? That the math is simple doesn't mean that the results aren't weird.

 

[robphy]

Thank you for your response, robphy. Part of ordinary physics is the interpretation of Lorentz transform results as describing the physical behavior of light waves. This interpretation seems to me to be sufficient. Is there some reason to follow a different interpretation? Could you elaborate on what observations and thought-experiments you have in mind?

The behavior of light rays [which have finite speed] under the Lorentz transformation has implications for the non-intuitive notion of time in relativity. [Radar experiments are the probably the best ways to see them.] I think that various effects are emphasized in introductions to relativity in order to challenge the Newtonian common sense of students... which unfortunately [in many standard treatments] just seems to leave most of them confused.

 

[bernhard.rothenstein]

Let

Do you find that you need to follow any interpretation other than the ordinary physics view that the results represent the behavior of the light waves? Can you direct me to your derivation?

I think that it is not the behaviour of of the light wave but the behaviour of a light signal what is essential. That is why many instructors consider that the radar detection procedure of the space-time coordinates of the same event is a transparent way to derive the LET.
Please have a critical look at


arXiv.org > physics > physics/0602054
abstract

Starting with a thought experiment proposed by Kard, which derives the formula that accounts for the relativistic effect of length contraction, we present a "two line" transparent derivation of the Lorentz-Einstein transformations for the space-time coordinates of the same event. Our derivation make uses of Einstein's clock synchronization procedure.

Thanks for answering my thread.
the best things a physicist can offer to another one are information and constructive criticism

 

[JM]

The results represent the behavior of all physical objects and fields. Do I misunderstand the question? That the math is simple doesn't mean that the results aren't weird.

The Lorentz transforms are derived for the physics of light (electrodynamic) waves. By what reasoning are these equations applicable to solid bodies?

 

[JM]

The behavior of light rays [which have finite speed] under the Lorentz transformation has implications for the non-intuitive notion of time in relativity. [Radar experiments are the probably the best ways to see them.] I think that various effects are emphasized in introductions to relativity in order to challenge the Newtonian common sense of students... which unfortunately [in many standard treatments] just seems to leave most of them confused.

Examination of the physics of light implicit in (Einsteins) derivation of the LET reveals the mechanisms required to satisfy the Light Speed Postulate. Each term of the LET is related to these mechanisms. The procedure uses the same principles a student of physics already knows. Results calculated by LET for a spherical wave of light reduce to known results for an ordinary wave. I think this approach is less confusing than others.
Can you suggest a reference for the radar experiments?

 

[JM]

Prof. Rothenstein, Thank you for your information. I will look into your derivation.

 

[bernhard.rothenstein]

.
Can you suggest a reference for the radar experiments?[/QUOTE]
Please have a critical look at


arXiv.org > physics > physics/0409121

Physics, abstract
physics/0409121


Three levels of understanding physical relativity: Galileo's relativity, Up-to-date Galileo's relativity and Einstein's relativity: A historical survey


We present a way of teaching Einstein's special relativity. It starts with Galileo's relativity, the learners know from previous lectures. The lecture underlines that we can have three transformation equations for the space-time coordinates of the same event, which lead to absolute clock readings, time intervals and lengths (Galileo's relativity), to absolute clock readings but to relative time intervals and lengths (up-to-date Galileo transformations) and to relative clock readings time intervals and lengths.
a paradox is that in our era there are more Authors then readers

 

[JM]

I note some comments on radar methods. Checking references I didn't find what I wanted. Consider: Basic radar records the time of emission and return of the reflected signal to obtain position of the reflecting body. Pulses in succession determine velocity. If the body has reflective surfaces on its near and far sides a single pulse will produce two reflected pulses arriving at different times. It seems that these two reflections can be used to determine the length of the body. For a stationary body L=(ct3 - ct2)/2, where ct3 and ct2 are the second and first received reflected signals. For a body in motion the calculation is complicated, but if I didn't make a mistake, it shows that the length is the same, i.e. the measured length does not depend on the speed of motion. Has anyone done this already?

 

[bernhard.rothenstein]

radar detection

I note some comments on radar methods. Checking references I didn't find what I wanted. Consider: Basic radar records the time of emission and return of the reflected signal to obtain position of the reflecting body. Pulses in succession determine velocity. If the body has reflective surfaces on its near and far sides a single pulse will produce two reflected pulses arriving at different times. It seems that these two reflections can be used to determine the length of the body. For a stationary body L=(ct3 - ct2)/2, where ct3 and ct2 are the second and first received reflected signals. For a body in motion the calculation is complicated, but if I didn't make a mistake, it shows that the length is the same, i.e. the measured length does not depend on the speed of motion. Has anyone done this already?

As far as I know the radar detection procedure for the space-time coordinates of the same event leads to the conclusion that depending on the way in which the mirror moves (incoming or outgoing) time and space coordinates transform via the Bondi factor k. Distances and time intervals behave in the same way. Giving google to look for Bondi you will find the papers related to the subject. If I remember well Rosser in his Introduction to Special Relativity treats the problem.
sine ira et studio

 

[Daverz]

I don't know if this is helpful for JM, but it's possible to derive the Lorentz transformations without the second postulate but only assuming

* Principle of Relativity (first postulate)
* Isotropy of space
* Homogeneity of space and time
* Causality

When you do this, you find there are only two choices for transformations between inertial frames that are consistent with these assumptions

* Galilean transformations
* Lorentz transformations

No discussion of light rays or any other kind of electromagnetic interaction is needed. The Lorentz transformations thus derived will have a parameter with the units of velocity that must be bounded from above for causality to hold (let's call it "c"). You can then use experiment to choose between these transformations and fix the value of the parameter.

This is done in Doughty, Lagrangian Interaction (section 5.5). He refers to two papers from the 1970s in the American Journal of Physics (vol. 43, pages 434-437 and vol. 44, pages 271-277).

Rindler, in his book Essential Relativity in a section titled "Special Relativity without the Second Postulate", refers to a paper from 1910 and another from 1921, so it seems to be one of those bits of knowledge that is periodically forgotten and remembered. (This has also come up on the forum before.)

 

[bernhard.rothenstein]

radar detection

I don't know if this is helpful for JM, but it's possible to derive the Lorentz transformations without the second postulate but only assuming

* Principle of Relativity (first postulate)
* Isotropy of space
* Homogeneity of space and time
* Causality

When you do this, you find there are only two choices for transformations between inertial frames that are consistent with these assumptions

* Galilean transformations
* Lorentz transformations

No discussion of light rays or any other kind of electromagnetic interaction is needed. The Lorentz transformations thus derived will have a parameter with the units of velocity that must be bounded from above for causality to hold (let's call it "c"). You can then use experiment to choose between these transformations and fix the value of the parameter.

This is done in Doughty, Lagrangian Interaction (section 5.5). He refers to two papers from the 1970s in the American Journal of Physics (vol. 43, pages 434-437 and vol. 44, pages 271-277).

Rindler, in his book Essential Relativity in a section titled "Special Relativity without the Second Postulate", refers to a paper from 1910 and another from 1921, so it seems to be one of those bits of knowledge that is periodically forgotten and remembered. (This has also come up on the forum before.)

it is up to jm to decide. have you ever tried to teach SR that way?

 

[Daverz]

it is up to jm to decide. have you ever tried to teach SR that way?

I think the derivation is too lacking in concreteness for a first introduction, but
it does point up the fact that the LT does not depend on properties of electromagnetic radiation or interaction of any kind. I also find it fascinating that if you make the assumptions above and crank through the logic and math that the Galilean and Lorentz transformations pop out.

Worth looking up if you have access to a library that carries Am. J. Phys. or the books mentioned above.

 

[bernhard.rothenstein]

I think the derivation is too lacking in concreteness for a first introduction, but
it does point up the fact that the LT does not depend on properties of electromagnetic radiation or interaction of any kind. I also find it fascinating that if you make the assumptions above and crank through the logic and math that the Galilean and Lorentz transformations pop out.

Worth looking up if you have access to a library that carries Am. J. Phys. or the books mentioned above.

that is my opinion as well and I am fond of simple and special relativity with human face. I know the papers and I think that it is hard to teach them without mnemonic helps i.e. without using notices during the lecture a condition I consider making part from the deontology of teaching. Thanks for your answer and help.

 

[Doc Al]

It seems that these two reflections can be used to determine the length of the body.

Sure.

For a stationary body L=(ct3 - ct2)/2, where ct3 and ct2 are the second and first received reflected signals.

OK.

For a body in motion the calculation is complicated, but if I didn't make a mistake, it shows that the length is the same, i.e. the measured length does not depend on the speed of motion.

Why don't you show what you did that led you to that conclusion.

 

[JM]

As far as I know the radar detection procedure for the space-time coordinates of the same event leads to the conclusion that depending on the way in which the mirror moves (incoming or outgoing) time and space coordinates transform via the Bondi factor k. Distances and time intervals behave in the same way. Giving google to look for Bondi you will find the papers related to the subject. If I remember well Rosser in his Introduction to Special Relativity treats the problem.
sine ira et studio

The focus of this discussion is whether moving objects actually contract or whether the calculated shortening is caused by the method of measurement.Einstein said that the moving object 'appeared to contract'. French says that the contraction is not in the body,but in the measurements. I note in one of your papers is a figure that shows both contraction and dilation. Is there general agreement that physical bodies do not change length?
The radar methods that I have seen don't appear to use both projected and reflected rays, but use only the projected ray. The quantity of my interest is the length of the moving body, not coordinates. My proposed radar method uses both rays and a single observer to eliminate the moving observer.
Does this clarify my question posed in my radar comment.

 

[JM]

Daverz: In re your comment #15. I have seen many derivations of the LET, but not the one you mentioned. In trying to find my way through the maze of different analyses and claims, I am trying to find something that can be generally agreed on. Einsteins method has the advantage that it relates to physics and uses methods of the college level. I note that his derivation of the LET does not use advanced math, relativity, or ideas about slow clocks, shrinking objects or twins. His work on the Maxwell equations enforces relativity, but also does not involve slow clocks, etc. If these ideas are correct they could form a basis for examination of other derivations and their relation to physics.

 

[bernhard.rothenstein]

contraction, dilatation or nothing?

The focus of this discussion is whether moving objects actually contract or whether the calculated shortening is caused by the method of measurement.Einstein said that the moving object 'appeared to contract'. French says that the contraction is not in the body,but in the measurements. I note in one of your papers is a figure that shows both contraction and dilation. Is there general agreement that physical bodies do not change length?
The radar methods that I have seen don't appear to use both projected and reflected rays, but use only the projected ray. The quantity of my interest is the length of the moving body, not coordinates. My proposed radar method uses both rays and a single observer to eliminate the moving observer.
Does this clarify my question posed in my radar comment.

from the paper

arXiv.org > physics > physics/0507016
physics/0507016


Length measurement of a moving rod by a single observer without asssumptions concerning its magnitude
Subj-class: Physics Education

We extend the results presented by Weinstein concerning the measurement of the length of a moving rod by a single observer, without making assumptions concerning the distance between the moving rod and the observer who measures its length.
Full-text: PDF only

and from the references therein, you can see that the "photographic" detection of the length of a moving rod can lead to contraction dilatation or to no change in the length. the procedure involves the light rays which start at different times from the different points of the moving rod but arrive at the camera simultaneously. please have a critical look at it

 

[JM]

Prof. Rothenstein; In regard to your response #22: I am now looking at the conclusions of the paper you mentioned, and may I quote "...they are the result of measurement...". Just so I may understand clearly, does your conclusion mean that you conclude that actual physical bodies do not actually change length, but that the calculated changes are only a property of the measurement method?
I am pressing on this point because I think it is very important to understanding Einsteins special relativity. Some comments that I have seen seem to be saying that bodies actually contract.

 

[jtbell]

The answer to the question, "Do moving objects in SR actually contract?" depends on exactly what you mean by "actually contract."

 

[bernhard.rothenstein]

physicists and taylors have something in common

Prof. Rothenstein; In regard to your response #22: I am now looking at the conclusions of the paper you mentioned, and may I quote "...they are the result of measurement...". Just so I may understand clearly, does your conclusion mean that you conclude that actual physical bodies do not actually change length, but that the calculated changes are only a property of the measurement method?
I am pressing on this point because I think it is very important to understanding Einsteins special relativity. Some comments that I have seen seem to be saying that bodies actually contract.

I personally think that physicists and taylors have in common the fact that they clothe, the first mother Nature the second humans. At the proof of the product they contract there and dilate in other parts. What mother Nature does is that She prevents us from finding out if we are at rest or in a state of uniform motion. Einstein and before him others tought us how to avoid attempts to find out methods for contradicting Galileo. The problem has much in common with the perpetuum mobile and conservation of energy.
In short length contraction and time dilation are the result of the measurement procedure. If I try to measure the length of a moving rod I do not contract or dilate its proper length, measuring the period of a moving clock I do not change its proper period. Probably Heisenberg has something to add. That is my humble oppinion and that is why I am still interested in special relativity. Otherwise...
Thank you for your pleasant way of conversation. That is the best way for learning from each other. You noticed that English is not my first language but I think you can understand the essence of my message.

 

[Daverz]

While it's useful to define a "proper" length, the length of an object measured in the rest frame, but I see no reason that the measurement of the length of the same object made in a moving frame should be considered second class in any way.

 

[bernhard.rothenstein]

did I mention the concept of second class length? please be more specific. we all learn from each other.

 

[Daverz]

I was just thinking out loud.

 

[JM]

Radar Length

The length of a moving object is given by Lm = (1-v/c)(cT3-cT1)/2. The length of a stationary object is Ls = (cT2-CT1)/2. But cT2 and cT3 are related by cT3 - cT2= (cT3 -cT1)v/c. Substitution shows that they are the same. The 'cT's are the times of the radar pulses returned from the object.

 

[JM]

The answer to the question, "Do moving objects in SR actually contract?" depends on exactly what you mean by "actually contract."

I read a discussion about two objects connected by a string. One idea presented was that as the objects accelerated in unison the string would break due to the Lorentz contraction. That sounds like 'actually contracting'.

 

[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.