Editing Special relativity (section)

=== 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>