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##
Are my equations for spacetime intervals correct?
Thanks!
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!