18.2 Lorentz-Vectors and Lorentz-Tensors 375
Thus the 4-velocity, divided byc, is a 4-dimensional unit vector. As a consequence,
the4-accelerationdefined by
bi=dui
dτ=
d^2 xi
dτ^2, (18.24)
obeys the relation
uibi=gikuibk= 0 , (18.25)
i.e. the 4-acceleration is orthogonal to the 4-velocity.
The 4-momentumpiof a particle with the restm,is
pi=mui. (18.26)The first three components ofpiare equal to
pμ=m(v)vμ,where the effective massm(v)is defined by
m(v):=γm=m
√
1 −v^2 /c^2. (18.27)
Here, the variablevoccurring inγis the magnitude of the velocityvof the moving
particle, as seen from the rest frame. The fourth component ofpiis equal to the
energyEof a free particle moving with speedv, divided byc,viz.
p^0 =m(v)c=E
c,
where
E:=m(v)c^2 =mc^2
√
1 −v^2 /c^2. (18.28)
The pertaining kinetic energy is
Ekin=E−mc^2 =mc^2(
1
√
1 −v^2 /c^2− 1
)
.
In the limitvc, this expression reduces to the non-relativistic limitEkin=^12 mv^2.