Invariance of Space-Time Interval

In section 3.1.2 we found the invariance under Lorentz transformations of the squared space-time interval s 2 between two events connected by a light signal being solely based on the relativity principle of Einstein, i.e., the constant speed of light in all inertial frames, cf. Eq. (3.5), 

As a consequence of Eq. (C.l) and the homogeneity of space and time and the isotropy of space, we now formally prove the invariance of the space-time interval for any two events E and E2, cf. Eq. (3.6), 

Proof We thus consider these two arbitrary events with reference to two inertial frames IS and IS moving with velocity V relative to each other. We can always express the relationship between the four-dimensional distances S12 and s r, between these two events as 

4) must hold for any two events and therefore also for events connected by a light signal. From Eq. (C.1) we therefore find A v ) = 0. Furthermore, Eq. (C.4) must also hold for events in the infinitesimal neighborhood of each other where s 2 = ds is an infinitesimal, i.e., very small quantity. Eor those events we therefore arrive at 

Kelativistic Quantum Chemistry. Markus Reiher and Alexander Wolf 

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