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There's something about the lorentz transformations which is somewhat confusing to me, and that is how to treat the "x" coordinate. Supposing I have some spaceship which is moving from earth to some other planet located at a distance "D" (from earth) with a velocity v. Now, the spacetime coordinates of the events "1. leaving earth" and "2. reaching the planet" are (the spaceship frame is {S'} and that of earth is {S} ) :

[tex] (x_{1},t_{1})=(x'_{1},t'_{1})=(0,0) [/tex]

[tex] (x_{2},t_{2})=(D, \frac{D}{v} ) [/tex]

[tex] (x'_{2},t'_{2})=(0 , \gamma (t_{2} - (v/c^{2})x_{2})=(0 , \gamma (t_{2} - (v/c^{2})D) [/tex]

Now comes the confusing point which is how to treat [tex] x_{3} [/tex] which corresponds to the event of returning back to earth in the earth's frame. (in the spaceship frame it is [tex] x'_{3} = 0 [/tex] )

The Lorentz transformations relates coordinates and not distances so [tex] x_{3} = 0 [/tex] because the spaceship returns to the origin of earth and [tex] t_{3} = \frac{2D}{v} [/tex]. However, as I have seen in my notes:

[tex] x_{3} = 2D [/tex]

, that is, the distance that this spaceship travels is what is accounted for and not its coordinate.

Can anyone clear this point for me?

**Leaving earth:**[tex] (x_{1},t_{1})=(x'_{1},t'_{1})=(0,0) [/tex]

**Reaching the planet:**[tex] (x_{2},t_{2})=(D, \frac{D}{v} ) [/tex]

[tex] (x'_{2},t'_{2})=(0 , \gamma (t_{2} - (v/c^{2})x_{2})=(0 , \gamma (t_{2} - (v/c^{2})D) [/tex]

Now comes the confusing point which is how to treat [tex] x_{3} [/tex] which corresponds to the event of returning back to earth in the earth's frame. (in the spaceship frame it is [tex] x'_{3} = 0 [/tex] )

The Lorentz transformations relates coordinates and not distances so [tex] x_{3} = 0 [/tex] because the spaceship returns to the origin of earth and [tex] t_{3} = \frac{2D}{v} [/tex]. However, as I have seen in my notes:

[tex] x_{3} = 2D [/tex]

, that is, the distance that this spaceship travels is what is accounted for and not its coordinate.

Can anyone clear this point for me?

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