I need your help. I've just begun learning about space time diagrams. Below is my doubt:

An event is shown as a point on a spacetime diagram.

enter image description here

Another frame has coordinates of space and time as:

$$\begin{aligned} c t^{\prime} &=(\cosh \theta)(c t)-(\sinh \theta) x \\ x^{\prime} &=(-\sinh \theta)(c t)+(\cosh \theta) x \\ y^{\prime} &=y \\ z^{\prime} &=z \end{aligned}$$ It's said that the new spacetime diagram could be drawn on the previous spacetime diagram as shown below:

enter image description here

Then is that the time coordinate of the event in the new frame can be found as shown below (red line is parallel to the $x'$ line, $(a, b)$ is the intersection point of red line with $ct'$ axis)

enter image description here

Hence the new time coordinate is $c t^{\prime}=\sqrt{a^2+b^2}$ ? Is that correct?

  • $\begingroup$ Not it is wrong, you can not use Pythagoras in this diagram. imagine a=1 what would b be since the x' system moves with v. $\endgroup$
    – trula
    Sep 23 at 13:07
  • 1
    $\begingroup$ Related : Your identical closed post here Angle in a spacetime diagram. $\endgroup$
    – Frobenius
    Sep 24 at 2:53