I Are equations for spacetime intervals correct?

Mike_bb
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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##

1.jpg


Are my equations for spacetime intervals correct?

Thanks!
 
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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.
 
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Mike_bb said:
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.
 
Dale said:
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.
 
SiennaTheGr8 said:
Nobody's perfect.
Nobody knows about it.
 
SiennaTheGr8 said:
Nobody's perfect.
My name is Nobody.
 
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