18.1 Lorentz Transformation 371
xixi=−(r 12 +r^22 +r 32 )+c^2 t^2 =−r^2 +c^2 t^2. (18.4)The condition (18.1) for the Lorentz invariance is equivalent to
xixi=(x′)i(x′)i. (18.5)In this notation, the summation index always occurs as a pair of subscript and super-
script. Just as for the components of a position vector with respect to a non-orthogonal
basis, cf. Sect.2.2.2, the contra- and co-variant components of the 4-vector are linked
with each other by
xi=gikxk, xi=gikxk. (18.6)
In matrix notation, the metric tensor is given by
gik=gik:=⎛
⎜
⎜
⎝
−10 00
0 − 100
00 − 10
0001
⎞
⎟
⎟
⎠. (18.7)
Notice that
gigk=δki:=⎛
⎜
⎜
⎝
1000
0100
0010
0001
⎞
⎟
⎟
⎠, (18.8)
which is the 4-dimensional unit matrix. Furthermore, one has
xixi=gikxixk=gikxixk. (18.9)The parity operatorPreplacesrby−r, the time reversal operatorT replacestby
−t, cf. Sects.2.6.1and2.8. Clearly, the combined operationPTis needed for all
components of the 4-vector to reverse sign at once, viz.
PTxi=−xi. (18.10)Remarks on notation are in order. Sometimes,ctis treated as the first component
and the counting of the four components runs from 0 to 3, viz. the notationx^0 =ct,
xi =ri,i = 1 , 2 ,3 is used. Then the metric tensor has the diagonal elements
1 ,− 1 ,− 1 −1.
The notation due to Minkowski, where the fourth component of the vector isict,
with the imaginary uniti, avoids the use of a metric tensor. In this case, the square
of the pseudo-Euclidian norm of the vector isxkxk=r 12 +r 22 +r 32 −c^2 t^2.
The formulation of vectors and tensors in 4D-space with a metric tensor is pre-
ferred since it is more apt for the generalization from Special to General Relativity.