Main Question or Discussion Point
Imagine there exists reference two frames, a frame O which is stationary and another, O' moving relative to O. If there is a four vector A--> (A0,A1,A2,A3). Then why is Aae->a as measured by the observer O equal to Aa'e->a' as measured by the observer O'?
The components are different, I know that the lengths of the vectors are the same, regardless of rotation or translation but assuming Galilean relativity, shouldn't velocity vectors not be equal in both frames? Does that mean velocity is not a four component vector? If not, why not? if so, why?
so if I applied the Lorentz transformation, Aa'e->a' = Λba'Abea', why is that important?
Are the vectors equal if and only if I applied the Lorentz transformation? If so why? if not? why not?