Relativistic Momentum

Relativistic momentum: According to the first postulates of Einstein special theory of relativity, laws of physics are invariant in all inertial frames. We know that conservation of linear momentum in the absence of external force is the most fundamental and powerful law of physics. As per classical physics, the linear momentum of particle of mass moving with velocity is defined as . We will see that this form of definition of momentum fails to obey the principle of conservation under relativistic dynamics. Let us consider two spherical clays A & B of equal masses ‘ ' move with velocities and undergo inelastic collision with final velocities measured from the rest frame as shown in Fig. 1.        
                                      Fig 1: Inelastic collisions between two spherical clays A and B according to rest frame .

  Initial momentum,     ........................ ( 1 )
            Final momentum,        .......................   ( 2 )
Thus according to frame conservation of momentum is obeyed.

Let us view the same problem with respect to ' frame moving with velocity along direction as shown in Fig.2

                      
                       Fig.2 Inelastic collision between two spherical clays A & B as per inertial frame

As discussed in section (5) in module-1, the velocity transformation based on Lorentz transformation equations can be written as,

                                 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. ( 3 )

So,             
                                     . . . . . . . . . . . . . . .. ( 4 )
                           . . . . . . . . . . . . .. ( 5 )
                         . . . . . . . . . . .. ( 6 )
The total initial momentum in frame,             . . . . . . . . . . .. ( 7 )
And the final momentum,                    . . . . . . . . . . . . . . . . . . . . . . . . . .. ( 8 )
                                                                . . . . . . . . . . . . . . . . . . . . . . . . . .. ( 9 )