# A short relativistic story but what is its moral?

1. Aug 16, 2007

### bernhard.rothenstein

When I started to learn special relativity I knew that in a given inertial reference frame I the mass m, the speed u and the momentum p of a given particle are related by
p=mu (1)
Special relativity tought me that in an inertial reference frame I' which moves with speed V relative to I in the positive direction of the overlapped axes OX(O'X') (1) should read
p'=m'u'. (2)
Less or more complicated derivations lead to the following transformation equations for momentum and mass (g(V)=1/sqrt(1-V^2/c^2)
p=g(V)p'(1+V/u') (3)
m=g(V)m'(1+Vu'/c^2) (4)
"Old fashioned" physicists say that (4) relates the "relativistic mass" m and the "relativistic mass m') of the same particle measured by observers from I and I' respectively. If the particle is at rest in I' (u'=0) observers of that frame measure its "rest mass m(0) and (4) leads to
m=g(V)m(0) (5)
Taking into account that c has the same magnitude in all inertial reference frames in relative motion, all transformation equations remain "relativistically correct" if we multiply both their sides by a power of c. Doing so with (4) we obtain nothing interesting because mc and m'c have no physical meaning (no tardyon can move with speed c). Multiplying both sides 0f (4) with c^2 leads to E=mc^2 and E'=m'c^2 respectively which has the physical dimensions of energy (4) becoming
E=g(V)E'(1+Vu'/c^2)=g(V)(E'+Vp') (6)
(3) becoming
p=g(V)(p'+VE'/c^2). (7)