Editing Spacetime (section)

==== Total momentum ====
[[File:Relativistic conservation of momentum.png|thumb|Figure 3–10. Relativistic conservation of momentum]]
To understand how the Newtonian view of conservation of momentum needs to be modified in a relativistic context, we examine the problem of two colliding bodies limited to a single dimension.

In Newtonian mechanics, two extreme cases of this problem may be distinguished yielding mathematics of minimum complexity:
: (1) The two bodies rebound from each other in a completely elastic collision.
: (2) The two bodies stick together and continue moving as a single particle. This second case is the case of completely inelastic collision.
For both cases (1) and (2), momentum, mass, and total energy are conserved. However, kinetic energy is not conserved in cases of inelastic collision. A certain fraction of the initial kinetic energy is converted to heat.

In case (2), two masses with momentums {{tmath|1=\boldsymbol{p}_{\boldsymbol{1} }=m_{1} \boldsymbol{v}_{\boldsymbol{1} } }}
and {{tmath|1=\boldsymbol{p}_{\boldsymbol{2} }=m_{2} \boldsymbol{v}_{\boldsymbol{2} } }} collide to produce a single particle of conserved mass {{tmath|1=m=m_{1}+m_{2} }} traveling at the [[center of mass]] velocity of the original system, <math>\boldsymbol{v_{c m}}=\left(m_{1} \boldsymbol{v_1}+m_{2} \boldsymbol{v_2}\right) /\left(m_{1}+m_{2}\right) </math>. The total momentum {{tmath|1=\boldsymbol{p=p_{1}+p_{2} } }} is conserved.

Fig.&nbsp;3-10 illustrates the inelastic collision of two particles from a relativistic perspective. The time components {{tmath|E_{1} / c}} and {{tmath|E_{2} / c}} add up to total ''E/c'' of the resultant vector, meaning that energy is conserved. Likewise, the space components {{tmath|1=\boldsymbol{p_{1} } }} and {{tmath|1=\boldsymbol{p_{2} } }} add up to form ''p'' of the resultant vector. The four-momentum is, as expected, a conserved quantity. However, the invariant mass of the fused particle, given by the point where the invariant hyperbola of the total momentum intersects the energy axis, is not equal to the sum of the invariant masses of the individual particles that collided. Indeed, it is larger than the sum of the individual masses: {{tmath|1=m>m_{1}+m_{2} }}.<ref name="Bais" />{{rp|94–97}}

Looking at the events of this scenario in reverse sequence, we see that non-conservation of mass is a common occurrence: when an unstable [[elementary particle]] spontaneously decays into two lighter particles, total energy is conserved, but the mass is not. Part of the mass is converted into kinetic energy.<ref name="Morin" />{{rp|134–138}}