Lorentz Transformation Derivation: Assumptions Req'd?

[Pencilvester]

In deriving the Lorentz transformation, is it required to assume that the transformation to get from coordinate system $\bf {x}$ to $\bf {x’}$ should be the same as that to get from $\bf {x’}$ to $\bf {x}$ (with the simple correction of flipping the velocity)? If no, could someone direct me to a derivation that does not assume this a priori? I’m having trouble deriving it myself without this assumption. If yes, what is the basis for this assumption?

 

[Nugatory]

Consider what happens when you transform the coordinates of an event from the unprimed frame to the primed frame and then back again... that will be enough to justify this requirement.

 

[Dale]

is it required to assume that the transformation to get from coordinate system xx\bf {x} to x′x′\bf {x’} should be the same as that to get from x′x′\bf {x’} to xx\bf {x} (with the simple correction of flipping the velocity)?

That is a consequence of the first postulate.

 

[Pencilvester]

Haha, duh. Thanks!

 

[Meir Achuz]

That is a consequence of the first postulate.

It is also required of the Galilean transformation.

 

[vanhees71]

Both the Poincare and the Galilei transformations can be derived from the 1st Newtonian postulate together with the assumption that time and space for any inertial observer are homogeneous (translation invariance in space and time) and that the space for any inertial observer is a 3D Euclidean affine space (implying that also rotations are a symmetry of space) and that the symmetry transformations of space and time together build a group. The "reciprocity property", i.e., that if an inertial frame $\Sigma'$ moves with velocity $\vec{v}$ wrt. to another inertial frame $\Sigma$ than $\Sigma$ moves with velocity $-\vec{v}$ relative to $\Sigma'$, can be derived from these symmetry assumptions and needs not to be postulated. The analysis reveals that the only possible space-time models obeying these assumptions are either the Galilei-Newton spacetime or the Einstein-Minkowski spacetime of special relativity.