Editing Lorentz transformation (section)

== Derivation of the group of Lorentz transformations ==
{{Main|Derivations of the Lorentz transformations|Lorentz group}}

An ''[[Event (relativity)|event]]'' is something that happens at a certain point in spacetime, or more generally, the point in spacetime itself. In any inertial frame an event is specified by a time coordinate {{math|''ct''}} and a set of [[Cartesian coordinate]]s {{mvar|x}}, {{mvar|y}}, {{mvar|z}} to specify position in space in that frame. Subscripts label individual events.

From Einstein's [[Postulates of special relativity|second postulate of relativity]] (invariance of [[Speed of light|{{mvar|c}}]]) it follows that:
{{NumBlk||<math display="block">c^2(t_2 - t_1)^2 - (x_2 - x_1)^2 - (y_2 - y_1)^2 - (z_2 - z_1)^2 = 0 \quad \text{(lightlike separated events 1, 2)}</math>|{{EquationRef|D1}}}}
in all inertial frames for events connected by ''light signals''. The quantity on the left is called the ''spacetime interval'' between events {{math|1=''a''{{sub|1}} = (''t''{{sub|1}}, ''x''{{sub|1}}, ''y''{{sub|1}}, ''z''{{sub|1}})}} and {{math|1=''a''{{sub|2}} = (''t''{{sub|2}}, ''x''{{sub|2}}, ''y''{{sub|2}}, ''z''{{sub|2}})}}. The interval between ''any two'' events, not necessarily separated by light signals, is in fact invariant, i.e., independent of the state of relative motion of observers in different inertial frames, as is [[Derivations of the Lorentz transformations#Invariance of interval|shown using homogeneity and isotropy of space]]. The transformation sought after thus must possess the property that:
{{NumBlk||<math display="block"> \begin{align}
& c^2(t_2 - t_1)^2 - (x_2 - x_1)^2 - (y_2 - y_1)^2 - (z_2 - z_1)^2 \\[6pt]
= {} & c^2(t_2' - t_1')^2 - (x_2' - x_1')^2 - (y_2' - y_1')^2 - (z_2' - z_1')^2  \quad \text{(all events 1, 2)}.
\end{align}
</math>|{{EquationRef|D2}}}}
where {{math|(''t'', ''x'', ''y'', ''z'')}} are the spacetime coordinates used to define events in one frame, and {{math|(''t′'', ''x′'', ''y′'', ''z′'')}} are the coordinates in another frame. First one observes that ({{EquationNote|D2}}) is satisfied if an arbitrary {{math|4}}-tuple {{mvar|b}} of numbers are added to events {{math|''a''{{sub|1}}}} and {{math|''a''{{sub|2}}}}. Such transformations are called ''spacetime translations'' and are not dealt with further here. Then one observes that a ''linear'' solution preserving the origin of the simpler problem solves the general problem too:
{{NumBlk||<math display="block">\begin{align}
& c^2t^2 - x^2 - y^2 - z^2 = c^2t'^2 - x'^2 - y'^2 - z'^2 \\[6pt]
\text{or} \quad & c^2t_1t_2 - x_1x_2 - y_1y_2 - z_1z_2 = c^2t'_1t'_2 - x'_1x'_2 - y'_1y'_2 - z'_1z'_2
\end{align}</math>|{{EquationRef|D3}}}}
(a solution satisfying the first formula automatically satisfies the second one as well; see [[polarization identity]]). Finding the solution to the simpler problem is just a matter of look-up in the theory of [[classical group]]s that preserve [[bilinear form]]s of various signature.<ref group=nb>The separate requirements of the three equations lead to three different groups. The second equation is satisfied for spacetime translations in addition to Lorentz transformations leading to the [[Poincaré group]] or the ''inhomogeneous Lorentz group''. The first equation (or the second restricted to lightlike separation) leads to a yet larger group, the [[conformal group]] of spacetime.</ref> First equation in ({{EquationNote|D3}}) can be written more compactly as:
{{NumBlk||<math display="block">(a, a) = (a', a') \quad \text{or} \quad a \cdot a = a' \cdot a',</math>|{{EquationRef|D4}}}}
where {{math|(·, ·)}} refers to the bilinear form of [[Signature (quadratic form)|signature]] {{math|(1, 3)}} on {{math|'''R'''{{sup|4}}}} exposed by the right hand side formula in ({{EquationNote|D3}}). The alternative notation defined on the right is referred to as the ''relativistic dot product''. Spacetime mathematically viewed as {{math|'''R'''{{sup|4}}}} endowed with this bilinear form is known as [[Minkowski space]] {{mvar|M}}. The Lorentz transformation is thus an element of the group {{math|O(1, 3)}}, the [[Lorentz group]] or, for those that prefer the other [[metric signature]], {{math|O(3, 1)}} (also called the Lorentz group).<ref group=nb>The groups {{math|O(3, 1)}} and {{math|O(1, 3)}} are isomorphic. It is widely believed that the choice between the two metric signatures has no physical relevance, even though some objects related to {{math|O(3, 1)}} and {{math|O(1, 3)}} respectively, e.g., the [[Clifford algebra]]s corresponding to the different signatures of the bilinear form associated to the two groups, are non-isomorphic.</ref> One has:
{{NumBlk||<math display="block">(a, a) = (\Lambda a,\Lambda a) = (a', a'), \quad \Lambda \in \mathrm{O}(1, 3), \quad a, a' \in M,</math>|{{EquationRef|D5}}}}
which is precisely preservation of the bilinear form ({{EquationNote|D3}}) which implies (by linearity of {{math|Λ}} and bilinearity of the form) that ({{EquationNote|D2}}) is satisfied. The elements of the Lorentz group are [[Rotation group SO(3)|rotations]] and ''boosts'' and mixes thereof. If the spacetime translations are included, then one obtains the ''inhomogeneous Lorentz group'' or the [[Poincaré group]].