372 18 From 3D to 4D: Lorentz Transformation, Maxwell Equations
18.1.3 Lorentz Transformation Matrix
Components of the 4-vector in two coordinate systems which move with a constant
velocityvwithrespecttoeachotherarelinearlyrelatedviatheLorentztransformation
matrixL,viz.
(x′)i=Likxk,(x′)i=Lkixk. (18.11)
The condition (18.5) implies
LikLni=δnk. (18.12)
This relation is analogous to the unitarity condition (2.31) for the 3-dimensional
rotation matrix. In (18.12) the summation is over the fore indices. The corresponding
relation with a summation over the hind indices also holds true:
LkiLin=δkn. (18.13)As in the case of the orthogonal transformation discussed in Sect.2.41for a rotation in
3D, the reciprocal of the 4D Lorentz transformation matrixLis equal to its transposed
matrixL ̃, thusL−^1 =L ̃.
18.1.4 A Special Lorentz Transformation.
Consider a ‘primed’ coordinate system which moves with the constant velocityvin
the 1- orx-direction. With the abbreviations
β:=v
c,γ:=1
√
1 −β^2, (18.14)
the rule proposed by Lorentz for the interrelation of the components with respect to
these coordinate systems are
x′=γ(x−vt)=γ(x−βct), y′=y, z′=z, ct′=γ(ct−βx). (18.15)Clearly, forβ1 and consequentlyγ≈1, the Lorentz transformation rule (18.15)
reduces to the corresponding Galilei transformation wherex′=x−vtandt′=t.
A contra-variant Lorentz vectorais transformed according to
(a′)^1 =γ(a^1 −βa^4 ), (a′)^2 =a^2 ,(a′)^3 =a^3 ,(a′)^4 =γ(a^4 −βa^1 ).(18.16)The pertaining Lorentz transformation matrix, cf. (18.11), is