Editing Spacetime (section)

==== Energy and momentum conservation ====
In a Newtonian analysis of interacting particles, transformation between frames is simple because all that is necessary is to apply the Galilean transformation to all velocities. Since {{tmath|1=v' = v - u}}, the momentum {{tmath|1=p' = p - mu}}. If the total momentum of an interacting system of particles is observed to be conserved in one frame, it will likewise be observed to be conserved in any other frame.<ref name="Morin" />{{rp|241–245}}

Conservation of momentum in the COM frame amounts to the requirement that {{math|1=''p''&nbsp;=&nbsp;0}} both before and after collision. In the Newtonian analysis, conservation of mass dictates that {{tmath|1=m=m_{1}+m_{2} }}. In the simplified, one-dimensional scenarios that we have been considering, only one additional constraint is necessary before the outgoing momenta of the particles can be determined—an energy condition. In the one-dimensional case of a completely elastic collision with no loss of kinetic energy, the outgoing velocities of the rebounding particles in the COM frame will be precisely equal and opposite to their incoming velocities. In the case of a completely inelastic collision with total loss of kinetic energy, the outgoing velocities of the rebounding particles will be zero.<ref name="Morin" />{{rp|241–245}}

Newtonian momenta, calculated as {{tmath|1=p = mv}}, fail to behave properly under Lorentzian transformation. The linear transformation of velocities {{tmath|1=v' = v - u}} is replaced by the highly nonlinear
{{tmath|1= v^{\prime} = (v-u) / (1- {v u}/{ c^{2} } )}} so that a calculation demonstrating conservation of momentum in one frame will be invalid in other frames. Einstein was faced with either having to give up conservation of momentum, or to change the definition of momentum. This second option was what he chose.<ref name="Bais" />{{rp|104}}

{{multiple image
<!-- Layout parameters -->
 | align             = right
 | direction         = vertical
 | width             = 250
<!--image 1-->
 | image1            = Energy-momentum diagram for pion decay (A).png
 | width1            = <!-- displayed width of image; overridden by "width" above -->
 | alt1              = 
 | caption1          = Figure 3-12a. Energy–momentum diagram for decay of a charged pion.
<!--image 2-->
 | image2            = Energy-momentum diagram for pion decay (B).png
 | width2            = <!-- displayed width of image; overridden by "width" above -->
 | alt2              = 
 | caption2          = Figure 3-12b. Graphing calculator analysis of charged pion decay.
}}
The relativistic conservation law for energy and momentum replaces the three classical conservation laws for energy, momentum and mass. Mass is no longer conserved independently, because it has been subsumed into the total relativistic energy. This makes the relativistic conservation of energy a simpler concept than in nonrelativistic mechanics, because the total energy is conserved without any qualifications. Kinetic energy converted into heat or internal potential energy shows up as an increase in mass.<ref name="Morin" />{{rp|127}}

{{smalldiv|1=
'''Example:''' Because of the equivalence of mass and energy, elementary particle masses are customarily stated in energy units, where {{nowrap|1=1 MeV = 10<sup>6</sup>}} electron volts. A charged pion is a particle of mass 139.57&nbsp;MeV (approx. 273 times the electron mass). It is unstable, and decays into a muon of mass 105.66&nbsp;MeV (approx. 207 times the electron mass) and an antineutrino, which has an almost negligible mass. The difference between the pion mass and the muon mass is 33.91&nbsp;MeV.
: {{SubatomicParticle|Pion-}} → {{SubatomicParticle|link=yes|Muon-}} + {{SubatomicParticle|link=yes|Muon antineutrino}}

Fig.&nbsp;3-12a illustrates the energy–momentum diagram for this decay reaction in the rest frame of the pion. Because of its negligible mass, a neutrino travels at very nearly the speed of light. The relativistic expression for its energy, like that of the photon, is {{tmath|1=E_{v}=p c,}} which is also the value of the space component of its momentum. To conserve momentum, the muon has the same value of the space component of the neutrino's momentum, but in the opposite direction.

Algebraic analyses of the energetics of this decay reaction are available online,<ref>{{cite web|last1=Nave|first1=R.|title=Energetics of Charged Pion Decay|url=http://hyperphysics.phy-astr.gsu.edu/hbase/Particles/piondec.html|website=Hyperphysics|publisher=Department of Physics and Astronomy, Georgia State University|access-date=27 May 2017|archive-date=21 May 2017|archive-url=https://web.archive.org/web/20170521075304/http://hyperphysics.phy-astr.gsu.edu/hbase/Particles/piondec.html|url-status=live}}</ref> so Fig.&nbsp;3-12b presents instead a graphing calculator solution. The energy of the neutrino is 29.79&nbsp;MeV, and the energy of the muon is {{nowrap|1=33.91  MeV − 29.79  MeV = 4.12 MeV}}. Most of the energy is carried off by the near-zero-mass neutrino.
}}