18.1 Lorentz Transformation 373
Lik:=⎛
⎜
⎜
⎝
γ 00 −βγ
0100
0010
−βγ 00 γ⎞
⎟
⎟
⎠. (18.17)
18.1.5 General Lorentz Transformations
The product
Lik=Lin( 1 )Lnk( 2 )of two Lorentz transformationsLik( 1 )andLik( 2 )is also a Lorentz transformation. A
general Lorentz transformation can be expressed as a (multiple) product of special
Lorentz transformations. Notice that a rotation of the coordinate system, wherer^2 =
(r′)^2 andt=t′also obeys the invariance condition (18.1). A 4 by 4 matrix where
the first 3 by 3 elements are given by the matrix elements of the orthogonal matrix
Upertaining to the 3D rotation, with furthermore,L^44 =1 and the other elements in
the fourth row and fourth column put equal to zero, is also a Lorentz-transformation.
Thus the general Lorentz-transformation governs the interrelation between the
position and the time of two coordinate systems, one of which is rotated and moving
with a constant velocity with respect to the other coordinate system.
18.2 Lorentz-Vectors and Lorentz-Tensors
18.2.1 Lorentz-Tensors
The 3D scalars, vectors and tensors, as presented in Sect.2.5.2, are defined via their
transformation behavior under a rotation of the coordinate system. By analogy, the 4D
scalars, vectors and tensors needed for special relativity are defined via the behavior
of their components under a Lorentz transformation.
A quantityawith the 4 componentsa^1 ,a^2 ,a^3 ,a^4 is aLorentz vectorwhen its
components in the primed coordinate system are related to those in the original
system by the same transformation rule as obeyed by the 4-vector(r,ct),cf.(18.11),
i.e. when
(a′)i=Likak,(a′)i=Lkiak (18.18)
holds true. ALorentz scalaris a quantity which does not change under a Lorentz
transformation. ALorentz tensorof rankrequires a-fold product of Lorentz
matrices for the transformation of its 4 timescomponents. As an example, the
contra-variant components of a second rank tensorTare transformed according to