Editing Michelson–Morley experiment (section)

=== Length contraction and Lorentz transformation ===
{{Further|History of special relativity|History of Lorentz transformations}}
A first step to explaining the Michelson and Morley experiment's null result was found in the [[Length contraction|FitzGerald–Lorentz contraction hypothesis]], now simply called length contraction or Lorentz contraction, first proposed by [[George Francis FitzGerald|George FitzGerald]] (1889) in a letter to same journal that published the Michelson-Morley paper, as "almost the only hypothesis that can reconcile" the apparent contradictions. It was independently also proposed by [[Hendrik Lorentz]] (1892).<ref group=A name=lorentz95 /> According to this law all objects physically contract by <math display="inline">L/\gamma</math> along the line of motion (originally thought to be relative to the aether), <math display="inline">\gamma=1/\sqrt{1-v^2/c^2}</math> being the [[Lorentz factor]]. This hypothesis was partly motivated by [[Oliver Heaviside]]'s discovery in 1888 that electrostatic fields are contracting in the line of motion. But since there was no reason at that time to assume that binding forces in matter are of electric origin, length contraction of matter in motion with respect to the aether was considered an [[ad hoc hypothesis]].<ref group=A name=AIMiller />

If length contraction of <math display="inline">L</math> is inserted into the above formula for <math display="inline">T_\ell</math>, then the light propagation time in the longitudinal direction becomes equal to that in the transverse direction:

:<math>T_\ell=\frac{2L\sqrt{1-\frac{v^2}{c^2}}}{c}\frac{1}{1-\frac{v^2}{c^2}}=\frac{2L}{c} \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}=T_t</math>

However, length contraction is only a special case of the more general relation, according to which the transverse length is larger than the longitudinal length by the ratio <math display="inline">\gamma</math>. This can be achieved in many ways. If <math display="inline">L_1</math> is the moving longitudinal length and <math display="inline">L_2</math> the moving transverse length, <math display="inline">L'_1=L'_2</math> being the rest lengths, then it is given:<ref group=A name=lorentz04 />

:<math>\frac{L_2}{L_1}=\frac{L'_2}{\varphi}\left/\frac{L'_1}{\gamma\varphi}\right.=\gamma.</math>

<math display="inline">\varphi</math> can be arbitrarily chosen, so there are infinitely many combinations to explain the Michelson–Morley null result. For instance, if <math display="inline">\varphi=1</math> the relativistic value of length contraction of <math display="inline">L_1</math> occurs, but if <math display="inline">\varphi=1/\gamma</math> then no length contraction but an elongation of <math display="inline">L_2</math> occurs. This hypothesis was later extended by [[Joseph Larmor]] (1897), Lorentz (1904) and [[Henri Poincaré]] (1905), who developed the complete [[Lorentz transformation]] including [[time dilation]] in order to explain the [[Trouton–Noble experiment]], the [[Experiments of Rayleigh and Brace]], and [[Kaufmann–Bucherer–Neumann experiments|Kaufmann's experiments]]. It has the form

:<math>x'=\gamma\varphi(x-vt),\ y'=\varphi y,\ z'=\varphi z,\ t'=\gamma\varphi\left(t-\frac{vx}{c^2}\right)</math>

It remained to define the value of <math display="inline">\varphi</math>, which was shown by Lorentz (1904) to be unity.<ref group=A name=lorentz04 /> In general, Poincaré (1905)<ref group=A name=poincare05 /> demonstrated that only <math display="inline">\varphi=1</math> allows this transformation to form a [[Lorentz group|group]], so it is the only choice compatible with the [[principle of relativity]], ''i.e.,'' making the stationary aether undetectable. Given this, length contraction and time dilation obtain their exact relativistic values.