Is this an incorrect treatment of SR? Oblique coordinates....

[Hiero]

To draw oblique coordinates with the coordinates measured perpendicular to each axis would be wrong, right?

I saw it done in a fairly popular book. It's usually the case that I'm the one who is wrong, but I think the book is incorrectly treating minkowski diagrams. Look at these images from the text:

(It says a bit earlier in the text that the S' frame moves along x axis.)
It says events a and b are "coincident" (simultaneous?) in the S' frame but differ in time in S. However the diagram seems to show b, the event which is spatially further (along the motion), occur sooner in time than a, which is not what SR predicts right?

This is a quite popular book... (though I have not read but this chapter)

 

[Orodruin]

Put that book away and never open it again.

 

[Ibix]

The axes are correct for something moving in the +x direction. The lines of constant x' and t' are wrong. The lines of constant t' should be parallel to the x' axis. Similarly the lines of constant x'. You can see this easily from the Lorentz transforms - set t' to some constant T' and work out what relationship this implies between x and t.

The point is that the axes are perpendicular in a Minkowski sense, not a Euclidean one.

What book is this?

 

[Hiero]

What book is this?

It is the last (14th) chapter of Mechanics by Kleppner-Kolenkow (2nd (latest?) edition). Only the last 3 chapters deal with SR, and upon skimming, only this final chapter mentions any Minkowski diagrams, so perhaps the majority of the book is good. (I have not read it.)

 

[Ibix]

I don't have it so I can't comment on the rest of it. The Minkowski diagrams are wrong.

People here seem to recommend Taylor and Wheeler's book for SR - there are some chapters online for free.

I wrote an interactive Minkowski diagram - it's at http://www.ibises.org.uk/Minkowski.html (Edit: link corrected) if you want to have a play around.

 

[Hiero]

People here seem to recommend Taylor and Wheeler's book for SR - there are some chapters online for free.

Spacetime Physics? This is indeed on my (perhaps optimistic) list of books to read soon.

Also I would like to see your Minkowski program but the link is invalid.

 

[Orodruin]

I am surprised. Kleppner-Kolenkow is usually considered a good textbook, but this error is just so blatant and fundamental that I would question it as a whole at face value. I have not read it myself.

 

[Orodruin]

Spacetime Physics? This is indeed on my (perhaps optimistic) list of books to read soon.

Also I would like to see your Minkowski program but the link is invalid.

He misspelled Minkowski in the link.
http://www.ibises.org.uk/Minkowski.html

 

[Ibix]

He misspelled Minkowski in the link.
http://www.ibises.org.uk/Minkowski.html

Now corrected above - thanks. I trusted my phone's auto-complete, but I must have typed it wrong when it learned Minkowski.html - I wonder how many times I've typed a broken link...

 

[Ibix]

I am surprised. Kleppner-Kolenkow is usually considered a good textbook, but this error is just so blatant and fundamental that I would question it as a whole at face value. I have not read it myself.

I had a quick Google for "Kleppner-Kolenkow errata" and found a few sheets. But at least the couple I looked at didn't mention this. Maybe no one uses it for the SR section?

 

[Orodruin]

Maybe no one uses it for the SR section?

I hope not ...

 

[Hiero]

I had a quick Google for "Kleppner-Kolenkow errata" and found a few sheets. But at least the couple I looked at didn't mention this. Maybe no one uses it for the SR section?

I also looked up the errata (perhaps lazily though) and could not find anything on this chapter. I am just thankful that I have PF to confirm things, because I was having a conceptual crisis trying to reconcile these two ways of measuring oblique coordinates. (I am relatively new to relativity so at first I assumed the book was okay and I was missing something.)

Your program is well made and fun to play with. I may even come back to it in doing certain problems; it could be enlightening to play with such a tool with a good problem in mind.

 

[Ibix]

Thanks - I had fun writing it.

One note: the coordinates aren't actually oblique. What's going on is much more closely related to the idea of a rotated set of axes - they remain orthogonal to one another, just pointing in a different direction. The reason the "rotated" (it's actually a hyperbolic rotation) Minkowski axes look non-orthogonal is that it isn't possible to represent Minkowski geometry on a Euclidean plane accurately. Both sets of axes are orthogonal to each other in the Minkowski sense. But we can only represent one pair as orthogonal in the Euclidean sense.

 

[Vanadium 50]

Kleppner-Kolenkow's section on relativity is terrible. It's needlessly complicated and oversimplifies all at the same time.

 

[robphy]

To draw oblique coordinates with the coordinates measured perpendicular to each axis would be wrong, right?

It seems this method is akin to the so-called "Brehme diagrams".
Unlike Loedel diagrams,
which appear to be Minkowski spacetime diagrams viewed in a special [center of velocity] frame.. then interpreted with Euclidean methods,
these Brehme diagrams seem not to be Minkowski spacetime diagrams reinterpreted.
I suspect that there is another transformation (not a Lorentz transformation) that goes from Loedel to Brehme, then interpreted with Euclidean methods.

Loedel and Brehme try to avoid the different scales along the two worldlines by viewing in a center-of-velocity frame,
then try to use Euclidean methods (as opposed to Minkowskian-geometry methods).

While the rest of Kleppner and Kolenkow is great [for mechanics],
I would (as others have suggested) find another place to learn relativity.

 

[Ibix]

Ok - a quick Google for Brehme diagrams suggests that they are drawn using one-forms instead of vectors. Am I understanding that right? The transformation law (derivation: http://www.farmingdale.edu/faculty/peter-nolan/pdf/relativity/Ch06Rel.pdf) certainly supports that.It would mean that the diagram in the OP is correct, it's just showing something unnecessarily complex for a three-chapter introduction to SR.

 

[Orodruin]

It would mean that the diagram in the OP is correct, it's just showing something unnecessarily complex for a three-chapter introduction to SR.

If you read the accompanying text it becomes very clear that it is wrong. In particular in the context of events lying on the same lines being simultaneous/at the same position in S'.

 

[Ibix]

If you read the accompanying text it becomes very clear that it is wrong. In particular in the context of events lying on the same lines being simultaneous/at the same position in S'.

But if I'm understanding the link I gave right, the primed frame drawn in the diagram has a velocity in the -x direction (contrary to what I said before when I thought this was a badly drawn Minkowski diagram). Shortcutting the algebra, it's because if you choose your sign convention so that $U_t$ and $U^t$ have the same sign then $U_x$ and $U^x$ have opposite signs in Minkowski space.

Not totally confident I understand this.

 

[Orodruin]

So it is a different type of space-time diagram that nobody today would use I suppose. I don't know if this is what was intended by the text cited by the OP, but outdated or wrong really leads to the same conclusion, don't use it to learn SR. (Indeed the same should be said of texts introducing relativistic mass.)

 

[vanhees71]

Kleppner-Kolenkow's section on relativity is terrible. It's needlessly complicated and oversimplifies all at the same time.

Hm, the Minkowski diagram in #1 is total nonsense (the right figure together with the explanation). I don't know what the lines they talk about are, for sure they are not what is claimed in the caption :-(.

 

[Dale]

I have had similar bad experiences with other textbooks that try to introduce relativity as a chapter or two at the end of a standard Newtonian physics text. I think that such introductions should be avoided. Study relativity with a dedicated source, or not at all.

Ben Crowells online text may be an exception.