In passing from the second to the third line, we have relabeled the summation indices so that as to
expose the combinationxμxνin both cases. This relation has to hold for allxμ, so that we must
have
ημν=ηρσΛρμΛσν (21.14)This relation defines all Lorentz transformations. It is instructive to count their number of in-
dependent parameters. The matrix Λ has 16 real components, but obeys 16 relations expressed
through the equation of two matrices which are both automatically symmetric. A real symmetric
4 ×4 matrix has 10 real independent components, so 16−10 = 6 independent parameters, which
precisely accounts for 3 rotations and 3 boosts.
21.3 General Lorentz vectors and tensors
The starting point for introducing 4-vector notation in thepreceding section was the quantityxμ,
which, under a Lorentz transformation Λ behaves linearly inxμ,
xμ→x′μ= Λμνxν (21.15)One refers to any objectVμ= (V^0 ,V^1 ,V^2 ,V^3 ) which transforms under Λ by
Aμ→A′μ= ΛμνAν (21.16)as aLorentz vector. To be more precise, one sometimes refers toAμas aLorentz vector with upper
index, or acontravariant vector.
21.3.1 Contravariant tensors
The transformation law of the product of nvectorsxμii withi = 1, 2 ,···,n follows from the
transformation law of each vector,xμi →x′iμ= Λμνxνi, and we have,
xμ 11 ···xμnn→x′ 1 μ^1 ···x′nμn= (Λμ^1 ν 1 ···Λμnνn)xν 11 ···xνnn (21.17)The productxμ 11 ···xμnnis atensor of rankn. 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 (21.18)as a Lorentz tensor of rankn, or more precisely as a Lorentz tensor with upper indices or a
contravariant tensor of rankn. A special case is whenn= 0, where we obtain,
A→A′=A (21.19)namely a tensor of rank 0, more commonly referred to as aLorentz scalar.