Twin Paradox (3 objects version)

[PeterDonis]

Equation (1) would also be valid, if I had chosen the same event as origin for both frames.

Yes. This is in agreement with what I said. Just remove the $\Delta$s from Equation (1) and you have the Lorentz transformation bewteen the two frames when they both have the same event as the origin.

 

[Sagittarius A-Star]

Just remove the $\Delta$s from Equation (1) and you have the Lorentz transformation bewteen the two frames when they both have the same event as the origin.

Yes. If $\Delta x$ gets replaced by $x$ (and the same for y, z, t coordinates) , this means, that $x_2 - x_1$ gets replaced by $x - 0$. This would be equivalent to defining the origin as one of the two events.

Source for Lorentz transformation of deltas and differentials (equations 7 and 8):
http://www.scholarpedia.org/article...nematics#Galilean_and_Lorentz_transformations

 

[vanhees71]

Of course, Minkowski space is not only a Lorentzian vector space but an affine manifold with the translations as additional symmetry, i.e., the complete continuous symmetry group is the proper orthochronous Poincare group, including space-time translations.