Explaining Pythagoras' Theorem & Its Impact on Acceleration

[D.A.Peel]

Can anyone explain to me why Pythagoras' Theorem governs the rate of change, of mass, length and time within accelertated bodies?
It's a simple theorem learned by most children by the age of eleven, so one would expect the answer to this question to be quite simple as well.

 

[Ich]

It's actually not the well-known theorem of Euclidean space, but a different version belonging to Minkowski space, where the length of a vector (e.g. a time interval) is calculated from its components using
s² = t² - x² - y² - z². (or -t² + x² + y² + z² as a matter of convention)
In one spatial dimension, this becomes
s² ("true" elapsed time) = t² - x² = t²*(1-v²) (less than elapsed coordinate time).
The same logic gives relativistic mass: it is the "time component" of a vector (Energy-Momentum vector) which has a length equal to the rest mass. Modern usage is to call the time component energy, not relativistic mass.
It's a different situation for length contraction: what we define as "length" is actually not a component of a vector, but a one-dimensional slice of a two-dimensional entity, the measuring rod, which extends both in space and in time. Therefore the different result.