Tensors for Physics

370 18 From 3D to 4D: Lorentz Transformation, Maxwell Equations

is invariant, for two coordinate systems moving with a constant velocity with respect
to each other. More specifically, the linear relation between the coordinates and the
time

r→r′, t→t′

has to be such thats^2 =(s′)^2 , i.e.

c^2 t^2 −r^2 =c^2 (t′)^2 −r′^2 , (18.1)

or, withx,y,zinstead ofr 1 ,r 2 ,r 3 ,

c^2 t^2 −(x^2 +y^2 +z^2 )=c^2 (t′)^2 −(x′^2 +y′^2 +z′^2 ). (18.2)

When the coordinate systems are chosen such that thex- and also thex′-direction is
parallel to the constant velocityv, one hasy′=y,z′=zand (18.2) reduces to

c^2 t^2 −x^2 =c^2 (t′)^2 −x′^2.

The same relation applies for differencesdtanddxbetween timestand positions
x.From

c^2 dt^2 −dx^2 =c^2 (dt′)^2 −dx′^2 = 0

follows

dx

dt

=

dx′

dt′

=c,

i.e. the speed of light is the same in both coordinate systems.
Four-dimensional vectors, endowed with the appropriate metric, allow to express
c^2 t^2 −r^2 as a 4D scalar product. Then (18.1) becomes analogous to the condition
that the 3D scalar product is invariant under a rotation of the coordinate system.

18.1.2 4-Vectors.

Contra- and co-variant 4-vectorsxiandxi, withi= 1 , 2 , 3 ,4, are introduced by

xi=(r 1 ,r 2 ,r 3 ,ct), xi=(−r 1 ,−r 2 ,−r 3 ,ct). (18.3)

With the Einstein summation convention for the four Roman indices, the scalar
product of the 4-vectors is