Tensors for Physics

374 18 From 3D to 4D: Lorentz Transformation, Maxwell Equations

(T′)ik=LinLkmTnm. (18.19)

Equations of physics, properly formulated in terms of Lorentz tensors of ranks
= 0 , 1 , 2 ,...areLorentz invariantand consequently are in accord withSpecial
Relativity.

18.2.2 Proper Time, 4-Velocity and 4-Acceleration

Letτbe the time in the co-moving coordinate system, i.e. in a system moving with
a particle. It is calledproper time(Eigenzeit). Time differences dtin a space-fixed
coordinate system are related to the proper time differences dτby

dt=γdτ=

dτ

√

1 −β^2

. (18.20)

Thistime-dilationexplainstheprolongedlivetimeoffastmovingπ-mesonsobserved
in high altitude radiation.
The proper time
dτ=

√

1 −β^2 dt (18.21)

is a Lorentz scalar, i.e. it is invariant under Lorentz transformations. This is inferred
from

dτ^2 =

(

1 −

v^2

c^2

)

dt^2 =

1

c^2

(dt^2 −dr·dr)=

1

c^2

gikdxidxk.

Herev^2 =dv·dvand dr=vdtwere used.
The4-velocity, defined by
ui=

dxi

dτ

, (18.22)

is a Lorentz vector. Its components are

u^1 =γv 1 , u^2 =γv 2 , u^3 =γv 3 , u^4 =γc.

Notice that the 3D velocityvis the derivative of the position vector with respect to
twhereas the proper timeτoccurs in (18.22).
The norm of the 4-velocity is constant, its square is given by

uiui=gikuiuk=c^2. (18.23)