- #1

albertrichardf

- 165

- 11

Anyway, I have tried "deriving" the space-time interval in the following way: consider that c is an invariant speed, which is the speed of an object in space-time. T, the proper time is also an invariant time interval, and distance = speed x time. To define an invariant interval, it would make sense to multiply the two, which gives:

[tex] s = cT [/tex]

To find out what this would be in space and time coordinates, square the equation, while keeping in mind that

t = T/y, where y is the gamma factor. After expanding y and some algebra, you obtain:

[tex] s^2 = t^2(c^2 - v^2) [/tex]

Now, suppose you have an object that is at position K in the proper time frame. It should stay at K because its velocity is zero. In another reference frame, moving at v relative to the proper time frame:

[tex] t' = yT [/tex]

[tex] x' = y(K - vT) = yK - yvT = yK - vt' [/tex]

If K = 0 the last equation becomes:

[tex] x' = -vt' [/tex]

If I replace for x' in my space-time interval, I obtain the following equation:

[tex] s^2 = c^2t^2 - x^2 [/tex]

which is how the interval is often shown. But the above only holds if K = 0, or if the space-time interval depends on the change in time and the change in position, which I'm thinking it does because it is an interval, but confirmation would be nice. Furthermore, if this is the case how could I proceed to define the 4-vector for position based on this? Would I have to use the Minkowski Pythagoras theorem or is there another way that I could do so?

Thank you for answering