Lorentz contraction and Spacetime diagram

[LCSphysicist]

Homework Statement

...

Relevant Equations

...

Hello, i can't understand how does the author found this expression relating $x_{c}$ and v. I already tried by a lot of geometrical ways, knowing that the tangent of the angle between the dotted line and the x-axis should be v, but the results are illogical. Could you help me? I am start to thinking it is a postulate ...

 

[PeroK]

It's from the Lorentz Transformation, which I assume is the "identical calculation from before".

 

[LCSphysicist]

It's from the Lorentz Transformation, which I assume is the "identical calculation from before".

But he derive the Lorentz transformation after this, before it there is just the time dilatation:

The figure he cites is another figure, but at least this expression for the time i could found using the interval invariance ds². For the length contraction, i can't see how he deduced it. And i don't want to use the interval invariance again, i am trying to gain some insights using just the graphic.

 

[PeroK]

But he derive the Lorentz transformation after this, before it there is just the time dilatation:
View attachment 278102
The figure he cites is another figure, but at least this expression for the time i could found using the interval invariance ds². For the length contraction, i can't see how he deduced it. And i don't want to use the interval invariance again, i am trying to gain some insights using just the graphic.

This is difficult without seeing the full derivation of everything in your book.

What book is this? Just out of interest.

 

[LCSphysicist]

This is difficult without seeing the full derivation of everything in your book.

What book is this? Just out of interest.

Bernard Schutz, a first course.

That's the image for the time dilatation, the curves in the graph are hyperbolae.

 

[PeroK]

A first course in GR?

 

[PeroK]

If so, a GR book is usually not the best place to learn SR. You generally get a concise review of SR aimed at refreshing what you already know and to familiarize yourself with the authors notation.

 

[Ibix]

I think what you are supposed to do is note that $\mathcal{C}$ lies on an invariant hyperbola (see the curve $\mathcal{EF}$ in Figure 1.11) that satisfies $-t^2+x^2=l^2$ (section 1.7). You also note that the $\bar{x}$ axis is the line $t=vx$ and solve simultaneously for $x$ and $t$.

I find Schutz helpful, but he does sometimes seem to skim over things which (IMO) need a bit more detail.

 

[anuttarasammyak]

Say angle between t axis and $\bar{t}$ axis $\theta$, law of sines on triangle ABC says
$$\frac{AC}{sin(\frac{\pi}{2}+\theta)}=\frac{AB}{sin(\frac{\pi}{2}-2\theta)}$$
$$AB=\frac{\cos 2\theta}{\cos\theta}AC=\frac{\cos^2\theta-sin^2\theta}{\cos\theta}AC$$

Magic : sin and cos are replaced by sinh and cosh,
where $\tanh \theta=v$ where c=1

$$AB=\sqrt{1-v^2}AC$$

We can do the similar to take $\theta$ pure imaginary angle.
I hope this magic is meaningful.

 

[guv]

Do you still need help? This seems like a trivial case of length contraction with $c$ replaced by 1. The formula is saying the proper length $l$ is contracted down to $\gamma l$.

Homework Statement:: ...
Relevant Equations:: ...

View attachment 278099
Hello, i can't understand how does the author found this expression relating $x_{c}$ and v. I already tried by a lot of geometrical ways, knowing that the tangent of the angle between the dotted line and the x-axis should be v, but the results are illogical. Could you help me? I am start to thinking it is a postulate ...