378 18 From 3D to 4D: Lorentz Transformation, Maxwell Equations
to the simpler assumption made by Lorentz and Einstein: they- andz-components
are not affected when the motion is inx-direction.
The timet′shown by a clock moving with the primed system, as seen from the
originalsystematthepositionx=vt,ist′=γt( 1 −v^2 /c^2 )=γ−^1 .Thetimedelay,
expressed byt′/tisγ−^1 and consequently equal to
√
1 −v^2 /c^2 for the Lorentz-
transformation (18.15) and 1−v^2 /c^2 , for the Voigt-transformation. Experiments on
time delay confirm the validity of the Poincaré-Einstein choice=1 and thus the
Lorentz-transformation (18.15).
18.1 Exercise: Doppler Effect
Letω 0 be the circular frequency of the electromagnetic radiation in a system which
moves with velocityv=vexwith respect to the observer, who records the frequency
ω. Determine the Doppler-shiftδω=ω 0 −ωfor the two cases, where the wave
vector of the radiation is parallel (longitudinal effect) and perpendicular (transverse
effect) to the velocity, respectively.
Hint: use the Lorentz transformation rule (18.16) for the components of the 4-wave
vectorKi,cf.(18.31). Furthermore, identifyω′withω 0 and usek 1 ≡kx.
18.3 The 4D-Epsilon Tensor
18.3.1 Levi-Civita Tensor
In 4D, the totally antisymmetric isotropic tensor, which is analogous to the 3D epsilon
tensor, is a tensor of rank 4. Here ‘isotropic’ means, the tensor is form invariant
under a Lorentz transformation, just as the unit tensor and the metric tensor. The
antisymmetric 4D-epsilon tensor is also calledLevi-Civita tensor. By analogy to
(4.1), it is defined according to
εkmn=−εkmn:=∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
δ1kδ 1 δ1mδ1n
δ2kδ 2 δ2mδ2n
δ3kδ 3 δ3mδ3n
δ4kδ 4 δ4mδ4n∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
. (18.36)
This implies
εimn=1 , k,,m,n=even permutation of 1234
− 1 , k,,m,n=odd permutation of 1234
0 , k,,m,n=else,(18.37)
e.g. one hasε^1234 =1 andε^2134 =−1.