Relativistic Momentum Conservation

Given that energy and momentum are clearly very closely related concepts in relativistic dynamics, we conjecture that conservation of energy also implies conservation of momentum.

Figure 3.12: Relativistic inelastic collision.

Consider the situation illustrated in Figure 3.12. In the laboratory frame, a particle of rest mass $m_0$, moving with speed $v$, collides with another particle of rest mass $m_0$ that is stationary. After the collision, the two particles stick together, and the composite particle, whose relativistic mass is $M$, moves off in the same direction as the originally moving particle at speed $V$. Now, the relativistic mass of the originally moving particle is

$$m = \frac{m_0}{\sqrt{1-v^2/c^2}}.$$ (3.188)

Thus, if mass/energy is conserved in the collision then the net relativistic mass before the collision must match that after the collision. In other words,

$$m+ m_0 = M,$$ (3.189)

which implies that

$$M = \left(1+\frac{1}{\sqrt{1-v^2/c^2}}\right)m_0.$$ (3.190)

On the other hand, if momentum is also conserved in the collision then the net relativistic momentum before the collision must match that after the collision. In other words,

$$m\,v = M\,V,$$ (3.191)

which yields

$$\frac{m_0}{\sqrt{1-v^2/c^2}}= \left(1+\frac{1}{\sqrt{1-v^2/c^2}}\right)m_0\,V,$$ (3.192)

where use has been made of Equations (3.188) and (3.190). The previous equation implies that

$$V = \frac{v}{1+\sqrt{1-v^2/c^2}},$$ (3.193)

which can be rearranged to give

$$V = \frac{v-V}{1-v\,V/c^2}.$$ (3.194)

However, there is another way of obtaining the previous equation. Let us transform to a frame of reference that moves, with respect to the laboratory frame, with speed $V$ parallel to the motion of the original moving particle. The composite particle appears stationary in this reference frame. Now, if momentum is conserved, then the new reference frame is the center-of-mass frame. (See Section 1.6.1.) Consequently, our two particles must approach one another with equal and opposite velocities, $V$, before the collision, as shown in the figure. However, when the transformation of velocity, (3.122), is applied to the originally moving particle, we obtain

$$V= \frac{v-V}{1-v\,V/c^2},$$ (3.195)

which is identical to Equation (3.194). Note that we obtained Equation (3.194) from considerations of energy and momentum conservation, whereas we obtained the previous equation from a consideration of momentum conservation alone. Hence, we deduce that momentum conservation in relativistic dynamics implies energy conservation, and vice versa.