21.3.3 Contraction and trace
Two vectorsAμ,Bμmay becontractedto form their Minkowski inner product,
A·B≡ημνAμBν=AμBμ=AμBμ (21.28)We have already encountered this inner product of two 4-vectors, one of positionxμ= (ct,x), and
one of momentumkμ= (ω/c,k), which is given explicitly by,
k·x=−ωt+k·x (21.29)This inner product is invariant under Lorentz transformations. More generally, two tensorsAμ^1 ···μn, Bμ^1 ···μn
of the same ranknmay be contracted to form a scalar,
A·B=Aμ 1 ···μnBμ^1 ···μn (21.30)One may also contract two tensorsA, andB, of ranksm+pandn+prespectively overpindices
to yield a tensor of rankm+n,
Aμ 1 ···μmρ 1 ···ρpBν^1 ···νnρ^1 ···ρp=Cμ 1 ···μmν^1 ···νn (21.31)A particularly important contraction of this type consistsin taking a trace by contracting a tensor
Aof rankm+2 with the Minkowski metric tensorη(note that the pair of indices has to be specified)
to yield a tensor of rankm,
Aμ 1 ···μi···μj···μmημiμj=Bμ 1 ···μ̂i···μ̂j···μm (21.32)Later on, we shall describe further linear operations on tensors, namely symmetrization and anti-
symmetrizations, with the help of which general tensors maybe decomposed into their irreducible
components.
21.4 Classical relativistic kinematics and dynamics
In non-relativistic mechanics, the relation between energyE, massm, velocityvand momentum
pis given by
E=
1
2
mv^2 =
p^2
2 m(21.33)
To obtain the relativistic invariant modification of these relations, we begin by identifying the
relevant Lorentz 4-vectors of the problem. It turns out thatthe correct 4-vector which generalizes
the 3-dimensional momentum vectorpis
pμ= (E/c,p) (21.34)Under a Lorentz transformation Λ,pμindeed transforms as a contravariant vector,
pμ→p′μ= Λμνpν (21.35)