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By P. W Bridgman

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They will be in motion with respect to each other so that the S frame observer observes the other to moving at speed v an the +x direction and the S' frame observer will observe the other to be moving at the same speed in the -x' direction. Lets say we know the location of an event according to one observer's coordinate frame and wish to determine the location according to the other coordinate frame. We transform the coordinates of the event from the one to the other by doing a Lorentz coordinate transformation.

We transform the coordinates of the event from the one to the other by doing a Lorentz coordinate transformation. In this case the Lorentz coordinate transformation equations are ct = y(ct' + ~x') x = y(x' + ~ct') y = y' z =z' where we make the definitions ~ = vic and y = (l - ~2y1l2 A more compact form of the transformation that allows the boost to be in any direction rather then restricting it to a coordinate axis is ct = yct' + ~·r' r = ~ct' + r' + (y- Inverted equation becomes ct' = x' = y(ct - ~x) y(x - ~ct) y' = Y z' = z and in differential form equation becomes 1)(~·r'/~2)~ 32 Relativity Made Simple dct dx dy dz and the inverse differential form dct' dx' y' z' = y(dct' + ~dx') = y( dx' + ~dct') = dy' = dz' is = y(dct - ~dx) = y(dx - ~dct) =y =z c is called the Lorentz invariant speed and according to Einstein's second postulate it is the finite vacuum speed of light.

We call this quantity c2T 2 - X2 - y2 - Z2 the interval. We're going to see that in special relativity the interval takes the place of distance. But, there's a big difference - the interval is how far you went minus the time is took to get there. For a ray of light, the interval is always zero, because at light speed, 1 meter of distance takes 1 meter of time, and 1 - 1 = O. This fact, that all people see the same speed of light, will be elevated from a curiosity to a fundamental axiom. The equation c 2T2 - X2 - y2 - Z2 will similarly be elevated from a special equation about light to a fundamental equation about the distance between any two events.

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