Frame-Jumping & Incorrect Gamma Derivations Aplenty

[sponsoredwalk]

HI, I've been mulling over the derivation of the Lorentz factor gamma in my head for some time now & had worried about it a bit.

I've now discovered that what was bothering me was something called "frame-jumping".

This is where you use the Pythagorean Theorem to derive Gamma.

If you're unsure of what I'm talking about I'll point you to a video showing the derivation, it's only around a minute long, watch at time 12:45 to 13:30 or so and you'll see what I'm on about.

http://video.google.com/videoplay?docid=-6328514962912264988#

So
I'd just like to hear from an expert WHY every standard college level intro textbook
uses an incorrect derivation of the Lorentz Transforms...? If this is true.

Also, A VERY IMPORTANT THING I HOPE YOU'LL BE ABLE TO ANSWER.

In Kleppner & Kolenkow's Introduction to Mechanics will they use this derivation?

Thanks so much for any comments...

BTW: I found out about this thing called frame jumping from an online post,
here is the link if you're curious,

http://www.physforum.com/index.php?showtopic=21780

If you don't want to mull through the page I'll quote one important part of it,
the saw tooth thing is a reference to the hypotenuse line of motion that the
photon traveling perpendicular to the velocity.

One last thing: the "saw-tooth" path of the light beam in the observer frame is a CONSEQUENCE of the Lorentz transforms, explained later in the Einstein paper under the "aberration " (paragraph 7). Interestingly enough, I have never seen aberration mentioned as the reason for the saw-tooth path in any textbook. It is the correct reason, nevertheless. Since the equation of aberration is BASED on the Lorentz transforms and they are BASED on "gamma", you cannot use this type of reasoning in order to derive...gammma. So, your approach leads to a dead end. On the positive side, you now have the means of getting educated on the subject. And Einstein did a superb job, enjoy :-)

 

[bcrowell]

I don't think user Trout on physforum.com knows what he's talking about. There is nothing wrong with the derivation.

 

[Fredrik]

The derivation isn't wrong, but it contains a hidden assumption: It's assumed that all observers will agree about the length of a straight line that's perpendicular to the direction of motion.

My recommendation is that you don't worry too much about the mathematical rigor in the calculations that lead up to the definition of Minkowski spacetime. Think of it as a bunch of intelligent guesses rather than as a mathematical proof. This is OK because once we have seen that Minkowski spacetime appears to be a good candidate for a mathemathical model of space and time, we can use it to properly define a new theory of physics. The new theory is of course special relativity, and it's not defined by Einstein's postulates, but by a new set of axioms that tells us how to interpret the mathematics as predictions about results of experiments

Once the theory has a proper definition, so that it's perfectly clear what predictions it makes, we can compare those predictions to results of experiments, to see how accurate they are. I'm sure you already know (or have at least assumed) that the predictions of SR have been found to be extremely accurate. That's what really tells us that we've found a good theory, not the funny calculations we did to find the theory.

I don't remember what Kleppner-Kolenkow said about this, but I remember that I wasn't too fond of that book. I think the authors don't really care about subtle details like what you can derive and what you have to assume.

 

[yuiop]

The derivation isn't wrong, but it contains a hidden assumption: It's assumed that all observers will agree about the length of a straight line that's perpendicular to the direction of motion...

If the derivation starts with the two postulates of relativity, then the assumption that all observers agree on the length perpendicular to the direction of motion is a logical outcome of the two postulates.

For example let us imagine for a moment that one consequence of motion relative to the ether was for perdicular lengths to contract and that we had two rings, A and B that have the same radius when at rest with respect to each other. Now if A was at rest with the ether and B was moving relative to A, an observer at rest with A would see ring B pass inside ring A. An observer moving with ring B would also see ring B pass inside ring A, but this would mean that B would have to conclude that the consequence of A moving relative to his own frame is that the radius of A expands (or B could conclude that his rest frame has absolute motion). Therefore the second postulate of relativity, that the laws of physics are the same in all inertial reference frames, would be violated if transverse lengths altered in any way. Perhaps that point should be made clear, when presenting the Pythagorus derivation to students.

I sometimes think that the 2 postulates of relativity could be replaced by a single postulate along the lines of "Absolute motion can not be detected". I would be interested if there is a counterproof of that tentative assertion.

 

[espen180]

I sometimes think that the 2 postulates of relativity could be replaced by a single postulate along the lines of "Absolute motion can not be detected". I would be interested if there is a counterproof of that tentative assertion.

How does this contain the invariance of c?

 

[Ich]

I would be interested if there is a counterproof of that tentative assertion.

http://en.wikipedia.org/wiki/Emission_theory"

 

[Dale]

sponsoredwalk does bring up an interesting point. There is a transformation, known as the http://en.wikipedia.org/wiki/Woldemar_Voigt" , which has no length contraction in the direction of the boost, but instead has length expansion in the directions perpendicular to the direction of the boost. This transform also preserves the invariance of c, so it satisfies the second postulate.

However, if you look at the Voigt transform you will see that if you apply it with a boost of v and then again apply it with a boost of -v you do not get the identity transformation as you would expect. What that means mathematically is that the Voigt Transform does not define a group, and what that means physically is that the Voigt Transform does not satisfy the first postulate.

In addition, the Voigt Transform predicts a different time dilation than the Lorentz transform, so it is an easy matter for experiment to decide, which it has in favor of the Lorentz transform.

 

[yuiop]

How does this contain the invariance of c?

In the sense that if observers in two closed labs that have relative motion to each other, measure different velocities for the speed of light in their own labs, they would have a sense of absolute motion and this would violate the single postulate. Of course the single postulate might have to be worded slightly better. With a multitude of labs, all with different velocities they would notice that there is only one lab with a unique velocity where the speed of light is a minimum (or a maximum) and isotropic in all directions and this would be a preferred frame.