Value of this 'Science' Channel as quick intro

[PAllen]

No, it doesn't tell you that. Different types of diagrams have to be interpreted differently. In a space-propertime diagrams a meeting is not a crossing of the paths, but the arrival at the same space coordinates after traveling along the same path length (cooridante time).

Ok, so you rely on the formula $dt^2=dx^2+d\tau^2$, plus the convention that paths passing through origin are actually coinciding at an event, but that no where else does an apparent intersection have any meaning? Strictly speaking, any world line in a space proper time diagram can be arbitrarily shifted vertically by any amount without changing the physics for that world line. Thus, you apparently must insist on only considering world lines that coincide at some event, and requiring that that event be the origin.

 

[PAllen]

@PeterDonis is asking about use for predictions in professional research literature, if I understand him correctly.

The literature is filled with quantitative use of spacetime diagrams. There is also professionally published literature using @robphy ’s tilted graph paper to make exact predictions (though perhaps all such publications are authored by @robphy ).

 

[PeterDonis]

Thread closed for moderation.

 

[berkeman]

An off-topic discussion about videos by @ScienceClic has been deleted and he will not be able to post again in this thread. After a Mentor discussion, this thread is now reopened.

 

[robphy]

The literature is filled with quantitative use of spacetime diagrams. There is also professionally published literature using @robphy ’s tilted graph paper to make exact predictions (though perhaps all such publications are authored by @robphy ).

I think I'm the only one who has published the rotated graph paper diagrams.

I do know of two presentations by others who have used it in their slides:

• https://web.archive.org/web/20131107065620/http://www.physics.pomona.edu/sixideas/VisTalk.pdf (see pg 24 - 50 - 66.)
• https://www.einsteinianphysics.com/iit-docs/Time%20in%20SRT%20and%20GRT.pdf (pg 15, 19)
• [forthcoming?] appendix in a revised edition of short relativity text

 

[MikeGomez]

An off-topic discussion about videos by @ScienceClic has been deleted and he will not be able to post again in this thread.

I thought this is thread is a discussion about a video by @ScienceClic. Could you please clarify what exactly is off topic here?

Specifically I was interested to hear what @D.S.Beyer finds to be dubious in another video that he mentions which visualizes 4D space-time .

 

[berkeman]

I thought this is thread is a discussion about a video by @ScienceClic. Could you please clarify what exactly is off topic here?

This is still under discussion by the Mentors. More in a bit...

 

[vanhees71]

I think I'm the only one who has published the rotated graph paper diagrams.

I do know of two presentations by others who have used it in their slides:

• https://web.archive.org/web/20131107065620/http://www.physics.pomona.edu/sixideas/VisTalk.pdf (see pg 24 - 50 - 66.)
• https://www.einsteinianphysics.com/iit-docs/Time%20in%20SRT%20and%20GRT.pdf (pg 15, 19)
• [forthcoming?] appendix in a revised edition of short relativity text

Isn't this just Bondi's k-formalism visualized in a pretty clever didactical way of "rotated graph paper".

Behind this is of course the representation of vectors $x \in \mathbb{R}^{1,1}$ in terms of "light-cone coordinates", $(\xi,\eta)=(x^0-x^1,x^0+x^1)$. The lines $\xi=\text{const}$ are the light cones for light moving in positive 1-direction, and $\xi=\text{const}$ those for light moving in negative 1 direction.

Since $x_{\mu} x^{\mu}=(x^0)^2-(x^1)^2=\xi \eta$ an Lorentz transformation, which leaves this Minkowski quadratic form invariant, is given by
$$\xi'=A \xi, \quad \eta'=\frac{1}{A} \eta.$$
This means
$$x^{\prime 0}-x^{\prime 1}=A(x^0-x^1), \quad x^{\prime 0}+x^{\prime 1}=\frac{1}{A}(x^0+x^1).$$
From this you get
$$x^{\prime 0}=\frac{1}{2} [(A+1/A) x^0+ (1/A-A) x^1], \quad x^{\prime 1}=\frac{1}{2}[(1/A-A) x^0+(1/A+A) x^1].$$
The origin of the primed system moves with velocity $v$ given by
$$\beta\frac{v}{c}=\frac{A-1/A}{A+1/A}=\frac{A^2-1}{A^2+1}.$$
i.e.,
$$A=\sqrt{\frac{1+\beta}{1-\beta}},$$
Since $A \in \mathbb{R}$ you must have $|\beta|<1$ and the transform reads, as it should
$$x^{\prime 0}=\gamma(x^0-\beta x^1), \quad x^{\prime 1}=\gamma(-\beta x^0+x^1), \quad \gamma=\frac{1}{\sqrt{1-\beta^2}}.$$
The nice thing with this representation and the "rotated graph paper" is that the area of the "diamonds" spanned by the coordinates lines $\xi=\text{const}$ and $\eta=\text{const}$ stay the same under Lorentz transformations. The consequences for the geometry of the Minkowski space in this rotated-graph-paper representation is given in detail in @robphy 's nice paper

https://arxiv.org/abs/1111.7254

 

[D.S.Beyer]

I thought this is thread is a discussion about a video by @ScienceClic. Could you please clarify what exactly is off topic here?

Specifically I was interested to hear what @D.S.Beyer finds to be dubious in another video that he mentions which visualizes 4D space-time .

I'm going to save that for another thread since this thread is ...uh... in 'flux'.
I'll make sure to remember to @ you, cause it's going to be fun.