Editing Special relativity (section)

== Dynamics ==
Section ''{{slink|#Consequences derived from the Lorentz transformation}}'' dealt strictly with [[kinematics]], the study of the motion of points, bodies, and systems of bodies without considering the forces that caused the motion. This section discusses masses, forces, energy and so forth, and as such requires consideration of physical effects beyond those encompassed by the Lorentz transformation itself.

=== Equivalence of mass and energy ===
{{Main|Mass–energy equivalence}}

Mass–energy equivalence is a consequence of special relativity. The energy and momentum, which are separate in Newtonian mechanics, form a [[four-vector]] in relativity, and this relates the time component (the energy) to the space components (the momentum) in a non-trivial way. For an object at rest, the energy–momentum four-vector is {{nowrap|(''E''/''c'', 0, 0, 0)}}: it has a time component, which is the energy, and three space components, which are zero. By changing frames with a Lorentz transformation in the x direction with a small value of the velocity v, the energy momentum four-vector becomes {{nowrap|(''E''/''c'', ''Ev''/''c''<sup>2</sup>, 0, 0)}}. The momentum is equal to the energy multiplied by the velocity divided by ''c''<sup>2</sup>. As such, the Newtonian mass of an object, which is the ratio of the momentum to the velocity for slow velocities, is equal to ''E''/''c''<sup>2</sup>.

The energy and momentum are properties of matter and radiation, and it is impossible to deduce that they form a four-vector just from the two basic postulates of special relativity by themselves, because these do not talk about matter or radiation, they only talk about space and time. The derivation therefore requires some additional physical reasoning. In his 1905 paper, Einstein used the additional principles that Newtonian mechanics should hold for slow velocities, so that there is one energy scalar and one three-vector momentum at slow velocities, and that the conservation law for energy and momentum is exactly true in relativity. Furthermore, he assumed that the energy of light is transformed by the same Doppler-shift factor as its frequency, which he had previously shown to be true based on Maxwell's equations.<ref name=electro group=p/> The first of Einstein's papers on this subject was "Does the Inertia of a Body Depend upon its Energy Content?" in 1905.<ref name=inertia group=p>[http://www.fourmilab.ch/etexts/einstein/E_mc2/www/ Does the inertia of a body depend upon its energy content?] A. Einstein, ''Annalen der Physik''. '''18''':639, 1905 (English translation by W. Perrett and G.B. Jeffery)</ref> Although Einstein's argument in this paper is nearly universally accepted by physicists as correct, even self-evident, many authors over the years have suggested that it is wrong.<ref name=Jammer>{{cite book |title=Concepts of Mass in Classical and Modern Physics |author=[[Max Jammer]] |pages=177–178 |url=https://books.google.com/books?id=lYvz0_8aGsMC&pg=PA177 |isbn=978-0-486-29998-3 |publisher=Courier Dover Publications |date=1997 }}</ref> Other authors suggest that the argument was merely inconclusive because it relied on some implicit assumptions.<ref name= Stachel>{{cite book |title=Einstein from ''B'' to ''Z'' |page= 221 |author=John J. Stachel |url=https://books.google.com/books?id=OAsQ_hFjhrAC&pg=PA215 |isbn=978-0-8176-4143-6 |publisher=Springer |date=2002}}</ref>

Einstein acknowledged the controversy over his derivation in his 1907 survey paper on special relativity. There he notes that it is problematic to rely on Maxwell's equations for the heuristic mass–energy argument. The argument in his 1905 paper can be carried out with the emission of any massless particles, but the Maxwell equations are implicitly used to make it obvious that the emission of light in particular can be achieved only by doing work. To emit electromagnetic waves, all you have to do is shake a charged particle, and this is clearly doing work, so that the emission is of energy.<ref name=survey group=p>[https://archive.today/20240524161707/https://www.webcitation.org/5knwYbqwK?url=http://www.geocities.com/physics_world/abstracts/Einstein_1907A_abstract.htm ''On the Inertia of Energy Required by the Relativity Principle''], A. Einstein, Annalen der Physik 23 (1907): 371–384</ref><ref group=note>In a letter to Carl Seelig in 1955, Einstein wrote "I had already previously found that Maxwell's theory did not account for the micro-structure of radiation and could therefore have no general validity.", Einstein letter to Carl Seelig, 1955.</ref>

=== Einstein's 1905 demonstration of ''E'' = ''mc''<sup>2</sup> ===
In his fourth of his 1905 [[Annus mirabilis papers]],<ref name=inertia group=p/> Einstein presented a heuristic argument for the equivalence of mass and energy. Although, as discussed above, subsequent scholarship has established that his arguments fell short of a broadly definitive proof, the conclusions that he reached in this paper have stood the test of time.

Einstein took as starting assumptions his recently discovered formula for [[relativistic Doppler shift]], the laws of [[conservation of energy]] and [[conservation of momentum]], and the relationship between the frequency of light and its energy as implied by [[Maxwell's equations]].

{{multiple image
 | direction         = vertical
 | width             = 221
 | image1            = Einstein's derivation of E=mc2 Part 1.svg
 <!-- | caption1          = -->
 | image2            = Einstein's derivation of E=mc2.svg
 | caption2          = Figure 6-1. Einstein's 1905 derivation of ''E'' = ''mc''<sup>2</sup>
}}
Fig. 6-1 (top). Consider a system of plane waves of light having frequency <math>f</math> traveling in direction <math>\phi</math> relative to the x-axis of reference frame ''S''. The frequency (and hence energy) of the waves as measured in frame {{prime|''S''}} that is moving along the x-axis at velocity <math>v</math> is given by the relativistic Doppler shift formula that Einstein had developed in his 1905 paper on special relativity:<ref name="electro" group="p"/>
: <math> \frac{f'}{f} = \frac{1 - (v/c) \cos{\phi}}{\sqrt{1 - v^2/c^2}} </math>

Fig. 6-1 (bottom). Consider an arbitrary body that is stationary in reference frame ''S''. Let this body emit a pair of equal-energy light-pulses in opposite directions at angle <math>\phi</math> with respect to the x-axis. Each pulse has energy {{tmath|1= L/2 }}. Because of conservation of momentum, the body remains stationary in ''S'' after emission of the two pulses. Let <math>E_0</math> be the energy of the body before emission of the two pulses and <math>E_1</math> after their emission.

Next, consider the same system observed from frame {{prime|''S''}} that is moving along the x-axis at speed <math>v</math> relative to frame ''S''. In this frame, light from the forwards and reverse pulses will be relativistically Doppler-shifted. Let <math>H_0</math> be the energy of the body measured in reference frame {{prime|''S''}} before emission of the two pulses and <math>H_1</math> after their emission. We obtain the following relationships:<ref name=inertia group=p/>
: <math>\begin{align}
E_0 &= E_1 + \tfrac{1}{2}L + \tfrac{1}{2}L = E_1 + L \\[5mu]
H_0 &= H_1 + \tfrac12 L \frac{1 - (v/c) \cos{\phi}}{\sqrt{1 - v^2/c^2}} + \tfrac12 L \frac{1 + (v/c) \cos{\phi}}{\sqrt{1 - v^2/c^2}}   =  H_1 + \frac{L}{{\sqrt{1 - v^2/c^2}}}
\end{align}</math>

From the above equations, we obtain the following:
{{NumBlk2||<math>\quad\quad (H_0 - E_0) - (H_1 - E_1)  = L \left( \frac{1}{\sqrt{1 - v^2/c^2}}  - 1 \right) </math>  |6-1}}

The two differences of form <math> H - E </math> seen in the above equation have a straightforward physical interpretation. Since <math>H</math> and <math>E</math> are the energies of the arbitrary body in the moving and stationary frames, <math> H_0 - E_0 </math> and <math>H_1 - E_1</math> represents the kinetic energies of the bodies before and after the emission of light (except for an additive constant that fixes the zero point of energy and is conventionally set to zero). Hence,
{{NumBlk2||<math>\quad\quad K_0 - K_1  = L \left( \frac{1}{\sqrt{1 - v^2/c^2}}  - 1 \right) </math>  |6-2}}

Taking a Taylor series expansion and neglecting higher order terms, he obtained
{{NumBlk2||<math>\quad\quad K_0 - K_1 = \frac{1}{2} \frac{ L}{c^2} v^2 </math>  |6-3}}

Comparing the above expression with the classical expression for kinetic energy, ''K.E.''&nbsp;=&nbsp;{{sfrac|1|2}}''mv''<sup>2</sup>, Einstein then noted: "If a body gives off the energy ''L'' in the form of radiation, its mass diminishes by ''L''/''c''<sup>2</sup>."

Rindler has observed that Einstein's heuristic argument suggested merely that energy ''contributes'' to mass. In 1905, Einstein's cautious expression of the mass–energy relationship allowed for the possibility that "dormant" mass might exist that would remain behind after all the energy of a body was removed. By 1907, however, Einstein was ready to assert that ''all'' inertial mass represented a reserve of energy. "To equate ''all'' mass with energy required an act of aesthetic faith, very characteristic of Einstein."<ref name="Rindler0"/>{{rp|81–84}} Einstein's bold hypothesis has been amply confirmed in the years subsequent to his original proposal.

For a variety of reasons, Einstein's original derivation is currently seldom taught. Besides the vigorous debate that continues until this day as to the formal correctness of his original derivation, the recognition of special relativity as being what Einstein called a "principle theory" has led to a shift away from reliance on electromagnetic phenomena to purely dynamic methods of proof.<ref name="Fernflores_2018">{{cite book |last1=Fernflores |first1=Francisco |title=Einstein's Mass–Energy Equation, Volume I: Early History and Philosophical Foundations |date=2018 |publisher=Momentum Pres |location=New York |isbn=978-1-60650-857-2}}</ref>

=== How far can you travel from the Earth? ===
{{See also|Space travel under constant acceleration}}
Since nothing can travel faster than light, one might conclude that a human can never travel farther from Earth than ~&nbsp;100 light years. You would easily think that a traveler would never be able to reach more than the few solar systems that exist within the limit of 100 light years from Earth.  However, because of time dilation, a hypothetical spaceship can travel thousands of light years during a passenger's lifetime. If a spaceship could be built that accelerates at a constant [[Gravity of Earth|1''g'']], it will, after one year, be travelling at almost the speed of light as seen from Earth. This is described by:
<math display="block">v(t) = \frac{at}{\sqrt{1+ a^2t^2/c^2}} ,</math>
where ''v''(''t'') is the velocity at a time ''t'', ''a'' is the acceleration of the spaceship and ''t'' is the coordinate time as measured by people on Earth.<ref group=p>{{cite web|url = http://www.normalesup.org/~baglio/physique/acceleration.pdf|title = Acceleration in special relativity: What is the meaning of "uniformly accelerated movement" ?|date = 26 May 2007|access-date = 22 January 2016|publisher = Physics Department, ENS Cachan|last = Baglio|first = Julien}}</ref> Therefore, after one year of accelerating at 9.81&nbsp;m/s<sup>2</sup>, the spaceship will be travelling at {{nowrap|1=''v'' = 0.712 ''c''}} and {{nowrap|0.946 ''c''}} after three years, relative to Earth.  After three years of this acceleration, with the spaceship achieving a velocity of 94.6% of the speed of light relative to Earth, time dilation will result in each second experienced on the spaceship corresponding to 3.1 seconds back on Earth.  During their journey, people on Earth will experience more time than they do – since their clocks (all physical phenomena) would really be ticking 3.1 times faster than those of the spaceship. A 5-year round trip for the traveller will take 6.5 Earth years and cover a distance of over 6 light-years. A 20-year round trip for them (5 years accelerating, 5 decelerating, twice each) will land them back on Earth having travelled for 335 Earth years and a distance of 331 light years.<ref name=gibbskoks>{{cite web|author1=Philip Gibbs |author2=Don Koks |name-list-style=amp |title=The Relativistic Rocket |url=http://math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/rocket.html |access-date=30 August 2012}}</ref> A full 40-year trip at 1''g'' will appear on Earth to last 58,000 years and cover a distance of 55,000 light years. A 40-year trip at {{nowrap|1.1 ''g''}} will take {{val|148,000}} years and cover about {{val|140,000}} light years. A one-way 28 year (14 years accelerating, 14 decelerating as measured with the astronaut's clock) trip at 1''g'' acceleration could reach 2,000,000 light-years to the Andromeda Galaxy.<ref name=gibbskoks /> This same time dilation is why a muon travelling close to ''c'' is observed to travel much farther than ''c'' times its [[half-life]] (when at rest).<ref>[http://library.thinkquest.org/C0116043/specialtheorytext.htm The special theory of relativity shows that time and space are affected by motion] {{Webarchive|url=https://web.archive.org/web/20121021183616/http://library.thinkquest.org/C0116043/specialtheorytext.htm |date=2012-10-21 }}. Library.thinkquest.org. Retrieved on 2013-04-24.</ref>

=== Elastic collisions ===
Examination of the collision products generated by particle accelerators around the world provides scientists evidence of the structure of the subatomic world and the natural laws governing it. Analysis of the collision products, the sum of whose masses may vastly exceed the masses of the incident particles, requires special relativity.<ref name="Idema_2022">{{cite web |last1=Idema |first1=Timon |title=Mechanics and Relativity. Chapter 14: Relativistic Collisions |url=https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_Mechanics_and_Relativity_(Idema)/14%3A_Relativistic_Collisions |website=LibreTexts Physics |date=17 April 2019 |publisher=California State University Affordable Learning Solutions Program |access-date=2 January 2023}}</ref>

In Newtonian mechanics, analysis of collisions involves use of the [[conservation of mass|conservation laws for mass]], [[conservation of momentum|momentum]] and [[conservation of energy|energy]]. In relativistic mechanics, mass is not independently conserved, because it has been subsumed into the total relativistic energy. We illustrate the differences that arise between the Newtonian and relativistic treatments of particle collisions by examining the simple case of two perfectly elastic colliding particles of equal mass. (''Inelastic'' collisions are discussed in [[Spacetime#Conservation laws]]. Radioactive decay may be considered a sort of time-reversed inelastic collision.<ref name="Idema_2022"/>)

Elastic scattering of charged elementary particles deviates from ideality due to the production of [[Bremsstrahlung]] radiation.<ref name="Nakel_1994">{{cite journal |last1=Nakel |first1=Werner |title=The elementary process of bremsstrahlung |journal=Physics Reports |date=1994 |volume=243 |issue=6 |pages=317–353 |doi=10.1016/0370-1573(94)00068-9|bibcode=1994PhR...243..317N }}</ref><ref>{{cite book |last1=Halbert |first1=M.L. |editor1-last=Austin |editor1-first=S.M. |editor2-last=Crawley |editor2-first=G.M. |title=The Two-Body Force in Nuclei |date=1972 |publisher=Springer |location=Boston, MA. |chapter=Review of Experiments on Nucleon-Nucleon Bremsstrahlung}}</ref>

==== Newtonian analysis ====
[[File:Elastic collision of moving particle with equal mass stationary particle.svg|thumb|Figure 6–2. Newtonian analysis of the elastic collision of a moving particle with an equal mass stationary particle]]
Fig. 6-2 provides a demonstration of the result, familiar to billiard players, that if a stationary ball is struck elastically by another one of the same mass (assuming no sidespin, or "English"), then after collision, the diverging paths of the two balls  will subtend a right angle. (a) In the stationary frame, an incident sphere traveling at 2'''v''' strikes a stationary sphere. (b) In the center of momentum frame, the two spheres approach each other symmetrically at ±'''v'''. After elastic collision, the two spheres rebound from each other with equal and opposite velocities ±'''u'''. Energy conservation requires that {{abs|'''u'''}} = {{abs|'''v'''}}. (c) Reverting to the stationary frame, the rebound velocities are {{nowrap|'''v''' ± '''u'''}}. The dot product {{nowrap|1=('''v''' + '''u''') ⋅ ('''v''' − '''u''') = '''v'''<sup>2</sup> − '''u'''<sup>2</sup> = 0}}, indicating that the vectors are orthogonal.<ref name="Rindler0"/>{{rp|26–27}}

==== Relativistic analysis ====
[[File:Relativistic elastic collision of equal mass particles.svg|thumb|Figure 6–3. Relativistic elastic collision between a moving particle incident upon an equal mass stationary particle]]
Consider the elastic collision scenario in Fig.&nbsp;6-3 between a moving particle colliding with an equal mass stationary particle. Unlike the Newtonian case, the angle between the two particles after collision is less than 90°, is dependent on the angle of scattering, and becomes smaller and smaller as the velocity of the incident particle approaches the speed of light:

The relativistic momentum and total relativistic energy of a particle are given by
{{NumBlk2||<math>\quad\quad \vec{p} = \gamma m \vec{v} \quad \text{and} \quad E = \gamma m c^2 </math>  |6-4}}

Conservation of momentum dictates that the sum of the momenta of the incoming particle and the stationary particle (which initially has momentum = 0) equals the sum of the momenta of the emergent particles:
{{NumBlk2||<math>\quad\quad \gamma_1 m \vec{v_1} + 0 = \gamma_2 m \vec{v_2}  +  \gamma_3 m \vec{v_3} </math>  |6-5}}

Likewise, the sum of the total relativistic energies of the incoming particle and the stationary particle (which initially has total energy mc<sup>2</sup>) equals the sum of the total energies of the emergent particles:
{{NumBlk2||<math>\quad\quad \gamma_1 m c^2 + m c^2 = \gamma_2 m c^2 + \gamma_3 m c^2 </math>  |6-6}}

Breaking down ({{EquationNote|6-5}}) into its components, replacing <math>v</math> with the dimensionless {{tmath|1= \beta }}, and factoring out common terms from ({{EquationNote|6-5}}) and ({{EquationNote|6-6}}) yields the following:<ref name="Champion_1932" group="p"/>
{{NumBlk2||<math>\quad\quad \beta_1 \gamma_1 = \beta_2 \gamma_2 \cos{\theta} +  \beta_3 \gamma_3 \cos{\phi} </math>  |6-7}}
{{NumBlk2||<math>\quad\quad \beta_2 \gamma_2 \sin{\theta} = \beta_3 \gamma_3 \sin{\phi} </math>  |6-8}}
{{NumBlk2||<math>\quad\quad \gamma_1 + 1 = \gamma_2 + \gamma_3 </math>  |6-9}}

From these we obtain the following relationships:<ref name="Champion_1932" group="p">{{cite journal |last1=Champion |first1=Frank Clive |title=On some close collisions of fast β-particles with electrons, photographed by the expansion method. |journal=Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character |year=1932 |volume=136 |issue=830 |pages=630–637 |publisher=The Royal Society Publishing |doi=10.1098/rspa.1932.0108 |bibcode=1932RSPSA.136..630C |s2cid=123018629 |doi-access=free }}</ref>
{{NumBlk2||<math>\quad\quad \beta_2 = \frac{\beta_1 \sin{\phi}}{ \{ \beta_1^2 \sin^2{\phi} +  \sin^2(\phi + \theta )/\gamma_1^2 \}^{1/2} }    </math>  |6-10}}
{{NumBlk2||<math>\quad\quad \beta_3 = \frac{\beta_1 \sin{\theta}}{ \{ \beta_1^2 \sin^2{\theta} +  \sin^2(\phi + \theta )/\gamma_1^2 \}^{1/2} }    </math>  |6-11}}
{{NumBlk2||<math>\quad\quad \cos{(\phi + \theta)} = \frac{ (\gamma_1 - 1) \sin{\theta} \cos{\theta} }{ \{ (\gamma_1 + 1)^2 \sin^2 \theta + 4 \cos^2 \theta  \}^{1/2} }    </math>  |6-12}}
For the symmetrical case in which <math> \phi = \theta</math> and {{tmath|1= \beta_2 = \beta_3 }}, ({{EquationNote|6-12}}) takes on the simpler form:<ref name="Champion_1932" group="p"/>
{{NumBlk2||<math>\quad\quad \cos{\theta} = \frac{\beta_1}{ \{ 2/\gamma_1 + 3 \beta_1^2 - 2 \}^{1/2} }    </math>  |6-13}}