Are equations for spacetime intervals correct?

[Mike_bb]

Hello!

I considered connections between spacetime intervals in 2D (see pic below) and their coordinates/times. I derived following equations ($A_1$ and $A_2$ are coefficients that depend on coordinates and times):

$A_1(x_1,x_2,t_1,t_2)(c^2\Delta t^2-\Delta x^2) = A_2(x_1',x_2',t_1',t_2')(c^2\Delta t'^2-\Delta x'^2)$
$A_1(x_1,x_2,t_1,t_2)(c^2\Delta t'^2-\Delta x'^2) = A_2(x_1',x_2',t_1',t_2')(c^2\Delta t^2-\Delta x^2)$

It leads to well-known equation:

$c^2\Delta t^2-\Delta x^2 = c^2\Delta t'^2-\Delta x'^2$

I also derived this equation in a different way. I used coordinates and time of events A and B ($K$ is coefficient that depends on coordinates and times of events):

$c^2\Delta t^2-\Delta x^2 = K(x_A=0,t_A=0,x_B=v\Delta t,t_B=\Delta t)(c^2\Delta t'^2-\Delta x'^2)$

$c^2\Delta t'^2-\Delta x'^2 = K(x_A=0,t_A=0,x_B=v\Delta t,t_B=\Delta t)(c^2\Delta t^2-\Delta x^2)$

$c^2\Delta t^2-\Delta x^2 = K(c^2\Delta t'^2-\Delta x'^2)$

$c^2\Delta t'^2-\Delta x'^2 = K(c^2\Delta t^2-\Delta x^2)$

It leads to $c^2\Delta t^2-\Delta x^2 = c^2\Delta t'^2-\Delta x'^2$

Are my equations for spacetime intervals correct?

Thanks!

 

[Dale]

The final equations are correct. But I prefer to use a positive for the space coordinates and a negative for the time coordinate. It is just a matter of taste.

 

[Nugatory]

Are my equations for spacetime intervals correct?

Yes, although you can avoid a bit of clutter if you use seconds for time and light-seconds for distance so that all the $c^2$ factors disappear.

 

[Mike_bb]

Dale, Nugatory

Big thanks!

 

[SiennaTheGr8]

The final equations are correct. But I prefer to use a positive for the space coordinates and a negative for the time coordinate. It is just a matter of taste.

Nobody's perfect.

 

[Mike_bb]

Nobody's perfect.

Nobody knows about it.

 

[Sagittarius A-Star]

Nobody's perfect.

My name is Nobody.