Lorentz transform gone wrong

[Physicsperson123]

Here is the space-time diagram of an observer:

Here is the diagram as seen by an observer travelling from left to right:

I have attempted to represent the axis system of the moving observer on the axis system of the stationary observer in the following diagram:

Event D seems to lie in accordance with the book diagram on the x' and y' axis. Events B and C, however, lie below the x'-axis, contrary to the book diagram.
What am I doing wrong, and what is the transformation shown in the book?
Help would be greatly appreciated!

 

[Ibix]

I don't think there's anything wrong with your diagram. I'm not sure what the book is doing - it looks a bit like they did a Euclidean rotation instead of a Lorentz boost. Which book is it?

 

[Dale]

The diagram in the book does not look correct to me. It looks like they just drew dots without actually applying the Lorentz transform.

For one thing, the Lorentz transform is linear. So since A, B, and C are in a line in one frame then they will be in a line in all frames. The book doesn’t show that

 

[Physicsperson123]

I don't think there's anything wrong with your diagram. I'm not sure what the book is doing - it looks a bit like they did a Euclidean rotation instead of a Lorentz boost. Which book is it?

Hi, thanks for your feedback!
I'm quite surprised that the book is inaccurate: Black Holes by Brian Cox and Jeff Forshaw

 

[Ibix]

Hi, thanks for your feedback!
I'm quite surprised that the book is inaccurate: Black Holes by Brian Cox and Jeff Forshaw

There are mistakes in almost all textbooks. I haven't read that one, and I'm not sure of its quality nor where it lies on the popsci fluff/proper textbook axis.

In figure 2.3 are A, B and C colinear? They don't look like it (which is bad), but that could be just the page being curved when you photographed it.

 

[Physicsperson123]

There are mistakes in almost all textbooks. I haven't read that one, and I'm not sure of its quality nor where it lies on the popsci fluff/proper textbook axis.

In figure 2.3 are A, B and C colinear? They don't look like it (which is bad), but that could be just the page being curved when you photographed it.

Hi, A, B and C aren't colinear
I'm just starting out the book, so I think I'll just stick it out and see how it goes I guess

 

[PeroK]

Here is the space-time diagram of an observer:
View attachment 334003
Here is the diagram as seen by an observer travelling from left to right:
View attachment 334004
I have attempted to represent the axis system of the moving observer on the axis system of the stationary observer in the following diagram:
View attachment 334002
Event D seems to lie in accordance with the book diagram on the x' and y' axis. Events B and C, however, lie below the x'-axis, contrary to the book diagram.
What am I doing wrong, and what is the transformation shown in the book?
Help would be greatly appreciated!

This is the problem with diagrams. Best to stick with algebra. It's more reliable!

 

[DrGreg]

I just searched for the book title, and it's a "popular" book aimed at the general public rather than an academic text book, although some of the reviews suggest that it's more technical than most popular books.

Both authors are professors, so they ought to have got this right. Maybe the diagram drawing was delegated to someone other than the authors, and the authors never noticed the error.

ISBN 9780008390648, only just published in paperback a few weeks ago, but previously released in hardback.

 

[PeterDonis]

A, B and C aren't colinear

They sure look that way on the first diagram; they all lie on the horizontal "space" axis. Which means they should all lie on a line going up and to the right with a slope of less than 45 degrees in the second diagram.

Also, point D is placed incorrectly on the second diagram; it should be to the right of the vertical axis, not to the left. At least, that's the case if we take points B and C as both being above the horizontal axis in the second diagram.

Finally, given the above, the second diagram is labeled incorrectly. It is actually a transform of what things would look like to an observer moving right to left in the first diagram, not left to right. In other words, the observer who is at rest in the first diagram is moving left to right in the second.

(Alternatively, if the labeling of the second diagram is correct, then point D is placed correctly, but points B and C are not; they should be below the horizontal axis, on a line going down and to the right with a slope of less than 45 degrees.)

Even though this is a pop science book, I am surprised and disappointed to find errors this simple in it, given that the authors are professors.

 

[Ibix]

They sure look that way on the first diagram;

Agreed, but they aren't in figure 2.3 which was what I asked about. Which means that this isn't somebody using Minkowski diagram software like mine that can do Euclidean or Minkowski geometry and accidentally leaving it in the wrong mode. They are separately drawn diagrams, drawn wrong.

 

[vanhees71]

Here is the space-time diagram of an observer:
View attachment 334003
Here is the diagram as seen by an observer travelling from left to right:
View attachment 334004
I have attempted to represent the axis system of the moving observer on the axis system of the stationary observer in the following diagram:
View attachment 334002
Event D seems to lie in accordance with the book diagram on the x' and y' axis. Events B and C, however, lie below the x'-axis, contrary to the book diagram.
What am I doing wrong, and what is the transformation shown in the book?
Help would be greatly appreciated!

I think the book has the opposite relative velocity than what you've drawn in your diagram, but with this qualification both are correct (modulo the fact that in the book A, B, and C are collinear in one frame, so they should be collinear in any frame).

I think in this respect your diagram is better since it depicts the original and new reference frame in one diagram and leaves the events, where they are, i.e., it clearly shows that the physics of course doesn't change at all by just changing the description between inertial frames.

The only thing that's missing is the construction of the "unit tics" on the new axes, which have to be constructed with the corresponding time- and space-like unit hyperbolae.

 

[Ibix]

I think the book has the opposite relative velocity than what you've drawn in your diagram, but with this qualification both are correct

That was my first thought too, but no, because in that case D is the wrong place. The angle DAC is roughly a right angle in both diagrams, where it should be "scissored" outwards or inwards by the boost.

 

[vanhees71]

Hm, then it's just very sloppy and confusing :-(. I'd rather construct the new axes as usual in the same Minkowski diagram as done by the OP. The only step missing is to construct the unit tic marks on the new axes by the unit hyperbolae.

 

[Vanadium 50]

Maybe the diagram drawing was delegated to someone other than the authors

Almost certainly.

The authors provide a sketch, which then goes to the "art department". They create the actual diagram. Usually the authors get it back as artwork, not as it will appear on the page (that's when it comes back from the "compositor") so any context has to be from memory or other drafts the authors have around.

Did the authors goof? Perhaps. But the surprising thing isn't that it happened. It's that it doesn't happen more often.

 

[vanhees71]

Hm, nowadays "the authors" prepare their plots, diagrams, and other "artwork" themselves on their computers. At least my university physics department has no more an "art department"...

 

[PeterDonis]

I think the book has the opposite relative velocity than what you've drawn in your diagram

It depends on whether you think point D in the second diagram is placed wrong (left of the vertical axis instead of right), or whether you think points B and C in the second diagram are placed wrong (above the horizontal axis instead of below). See post #9.

 

[Sagittarius A-Star]

(Alternatively, if the labeling of the second diagram is correct, then point D is placed correctly, but points B and C are not; they should be below the horizontal axis, on a line going down and to the right with a slope of less than 45 degrees.)

I tried to create a LT, which leaves point D approximately in place in the 2nd diagram. Then I get the red dots.