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Proper Time and Lorentz Invariants

There is one important difference between ordinary ROTATIONS and the LORENTZ TRANSFORMATIONS: the former preserve the RADIUS distance

$$r \; = \; \sqrt{x^2 + y^2} \; = \; \sqrt{x'^2 + y'^2}$$ (23.6)

 of point  A  from the origin, whereas the latter preserve the PROPER TIME  $\tau$  of an event:

$$\tau \; = \; \sqrt{c^2t^2 - x^2} \; = \; \sqrt{c^2t'^2 - x'^2}$$ (23.7)

The - sign in the latter is important!

In general, any quantity which we can define (like $\tau$) that will have the same value in every inertial reference frame, regardless of relative motion, may be expected to become very precious to our bruised sensibilities. The has dismantled most of our common sense about which physical observables are reliable, universal constants and which depend upon the reference frame of the observer; if we can specifically identify those properties of a quantity that will guarantee its invariance under LORENTZ TRANSFORMATIONS, then we can at least count on such quantities to remain reliably and directly comparable for different observers. Such quantities are known as LORENTZ INVARIANTS.

The criterion for LORENTZ INVARIANCE is that the quantity in question be the scalar product of two 4-vectors, or any combination of such scalar products. What do we mean by 4-vectors? {Space and time} is the classic example, but I think I will defer the formal definition until we have seen a few more..