21.3.2 Covariant tensors
Every contravariant vector and tensor naturally has an associatedcovariant vectoror tensor of the
same rank, which is obtained bylowering all upper indicesusing the Minkowski metricημν. The
simplest case is for a contravariant vectorAμ, where we define the associated covariant vector by
Aμ≡ημνAν ⇔ Aμ=ημνAν (21.20)Under a Lorentz transformation Λ, the covariant vectorAμis mapped as follows,
Aμ→A′μ = ημνA′ν
= ημνΛνρAρ
= ημνΛνρηρσAσ (21.21)By our standard conventions of raising and lowering indices, we adopt the following notation,
ημνΛνρηρσ= Λμσ (21.22)Using the defining relations of Lorentz transformations,ημν=ηρσΛρμΛσν, we may reinterpret this
matrix as follows. contract the defining relation withημτgives,
δτν=ηρσΛρμημτΛσν= ΛστΛσν (21.23)Hence, Λμνis the inverse of the matrix Λμν. Thus, another way of expressing the transofmration
law for a covariant vector is in terms of the inverse of Λ,
Aμ→A′μ= ΛμνAν (21.24)Analogously, one refers to any objectAμ 1 ···μnwhich transforms under Λ by
Aμ 1 ···μn →A′μ 1 ···μn= (Λμ 1 ν^1 ···Λμnνn)Aν 1 ···νn (21.25)as a Lorentz tensor of rankn, or more precisely as a Lorentz tensor with lower indices or acovariant
tensor of rankn.
One very important example of a covariant vector is providedby the 4-derivative,∂μ≡∂
∂xμ(21.26)
In view of its defining relation, [∂μ,xν] =δμν, it is clear that∂μtransforms as a covariant vector,
∂′μ=∂
∂x′μ
= Λμν∂
∂xν
= Λμν∂ν (21.27)