22   Energy and Momentum

 
The rest energE  of some object, the energy  E'  of the same object for a 'fast-moving observer' and momentum  p  of this object are connected in STR by the equation

This is the Pythagorean theorem applied to the appropriate Epstein diagram:

 

We have   sin(φ)  =  v / c   and   cos(φ)  =  √  . The kinetic fraction of  E'  is visualized by the difference  E'E .

The transformation of total energy is given by 

E' E / √

From the Epstein diagram we read

p · c  =  E' · sin(φ) =  E' · v / c  =  ( E / ) · v / c

and hence

p  =  E' · vc2  =  ( E / ) · vc2

This equation applies as well for the vectors  p  and  v

p  = ( E'c2 ) · v  =   E / ( c2 ·   ) · v

and it can be further extended to an equation of 4-vectors. The (contravariant) energy-momentum-4-vector is

p μ  =  ( E' / c , p )  =  ( E' / c , px , py , pz )  =  ( E / c2 ) · v μ  =  ( E / ( c2 ·   )) · ( c , )  =  ( E / ( c2 ·   )) · ( c , vx , vy , vz )

In the rest frame of the object this 4-vector reduces to  ( E / c , 0 , 0 , 0 )