Editing Spacetime (section)

=== Conservation laws ===
{{Main|Conservation law}}

In physics, conservation laws state that certain particular measurable properties of an isolated physical system do not change as the system evolves over time. In 1915, [[Emmy Noether]] discovered that underlying each conservation law is a fundamental symmetry of nature.<ref>{{cite arXiv|last1=Byers|first1=Nina|title=E. Noether's Discovery of the Deep Connection Between Symmetries and Conservation Laws|eprint=physics/9807044|year=1998}}</ref> The fact that physical processes do not care ''where'' in space they take place ([[space translation symmetry]]) yields [[conservation of momentum]], the fact that such processes do not care ''when'' they take place ([[time translation symmetry]]) yields [[conservation of energy]], and so on. In this section, we examine the Newtonian views of conservation of mass, momentum and energy from a relativistic perspective.

==== 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}}

==== Choice of reference frames ====
{{multiple image|align=right|image1=2-body Particle Decay-Lab.svg|width1=115|image2=2-body Particle Decay-CoM.svg|width2=105|caption1=Figure 3-11. <br />(above) '''Lab Frame'''.<br />(right) '''Center of Momentum Frame'''.|&nbsp;}}
The freedom to choose any frame in which to perform an analysis allows us to pick one which may be particularly convenient. For analysis of momentum and energy problems, the most convenient frame is usually the "[[center-of-momentum frame]]" (also called the zero-momentum frame, or COM frame). This is the frame in which the space component of the system's total momentum is zero. Fig.&nbsp;3-11 illustrates the breakup of a high speed particle into two daughter particles. In the lab frame, the daughter particles are preferentially emitted in a direction oriented along the original particle's trajectory. In the COM frame, however, the two daughter particles are emitted in opposite directions, although their masses and the magnitude of their velocities are generally not the same.<ref name="Idema_2022">{{cite web |last1=Idema |first1=Timon |title=Mechanics and Relativity. Chapter 4.3: Reference Frames |url=https://phys.libretexts.org/Bookshelves/University_Physics/Mechanics_and_Relativity_(Idema)/04%3A_Momentum/4.03%3A_Reference_Frames |website=LibreTexts Physics  |date=17 April 2019 |publisher=California State University Affordable Learning Solutions Program |access-date=6 July 2024}}</ref>

==== 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.
}}