Ordinary rotations preserve the RADIUS distance
of point A from the origin (refer to Fig.
).
By analogy, the Lorentz transformation preserves
the proper time 
of an event:
Again, the - sign in the latter equation makes a big difference!
In general, any quantity which we can define
(like )
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.
