Sunday 13 December 2015

special relativity - Invariance of spacetime interval directly from postulate


In Special Relativity, the spacetime interval $$\mbox{d}s^2 = \mbox{d}t^2 - \mbox{d}x^2 - \mbox{d}y^2 - \mbox{d}z^2 \tag{$\star$}$$ between two events is well known to be invariant under Lorentz transformations, i.e. identical for inertial observers.



Once one assumes the speed of light to be constant for all inertial observers, it is easy to see that $(\star)$ is indeed invariant if the events are lightlike separated. If I recall correctly, it was possible (assuming homogeneity and isotropy of space) to then also derive that $(\star)$ must be invariant for arbitrary events (i.e. also ones which are timelike or spacelike separated) but I don't recall the details.


Can anyone help me out with this?




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