- #1

- 1,435

- 186

- TL;DR Summary
- Just a note, wonder where the rabbit hole leads

Suppose the time interval in the Lab frame is a multiple of the time interval in the Rocket frame ##\alpha \Delta t_L = \Delta t_R##, where ##0 < \alpha < 1## without loss of generality. Then the spacetime interval is

##\left( \Delta t_L\right) ^2-\left( \Delta x_L\right) ^2 = \left( \Delta t_R\right) ^2-\left( \Delta x_R\right) ^2\Rightarrow \left( \Delta t_L\right) ^2=\frac{\left( \Delta x_L\right) ^2 -\left( \Delta x_R\right) ^2}{1-\alpha^2}##

The thought ended abruptly there. I guess I just wanted to jot this down before I forgot, and hey, if you've got some good idea how to proceed from here: post it! Thanks!

Edit: something is off, like a minor error... but I can't put my finger on it. Should the bounds for be ##1< \alpha < \infty##?

##\left( \Delta t_L\right) ^2-\left( \Delta x_L\right) ^2 = \left( \Delta t_R\right) ^2-\left( \Delta x_R\right) ^2\Rightarrow \left( \Delta t_L\right) ^2=\frac{\left( \Delta x_L\right) ^2 -\left( \Delta x_R\right) ^2}{1-\alpha^2}##

The thought ended abruptly there. I guess I just wanted to jot this down before I forgot, and hey, if you've got some good idea how to proceed from here: post it! Thanks!

Edit: something is off, like a minor error... but I can't put my finger on it. Should the bounds for be ##1< \alpha < \infty##?

Last edited: