2.3. EINSTEIN’S THEORY OF GENERAL RELATIVITY 65
and by (2.3.15) we have
T ̃ijT ̃ij = tr[
(T ̃ij)(T ̃ij)T]
= tr[
(bkl)T(Tij)(bkl)(akl)(Tij)T(akl)T]
= tr[
(bkl)T(Tij)(Tij)T(akl)T]
= tr[(Tij)(Tij)T]
= TijTij.Here we have used the following property for matrices:
(2.3.16) tr[ABA−^1 ] =trB,
which can be shown by the fact that the eigenvaluesλ 1 ,···,λnofBare the same as those of
ABA−^1. Indeed we have
trB=λ 1 +···+λn=tr[ABA−^1 ].
Moreover, all general invariants are in contraction form:
(2.3.17) Akl^11 ······lksrBlk 11 ······lksr,
where the invariance holds true as well if the indices are exchanged. For example,AijBjiis
also invariant.
By the invariant form (2.3.17), the metric formds^2 of a Riemannian space is a general
invariant, i.e.
ds^2 =gijdxidxj=g ̃ijd ̃xidx ̃j.
- Covariant Derivatives. The covariance of differential equations requires that the
derivative operator∇be also covariant, i.e.∇is a general tensor operator.
Let us consider the usual derivatives
(2.3.18) ∂k=∂/∂xk for 1≤k≤n.
For a vector fieldA= (A^1 ,···,Ak), in the transformation (2.3.9) we have
(2.3.19) A ̃k=aklAl.
Differentiating both sides of (2.3.19), we have
∂A ̃k
∂x ̃j
=akl∂Al
∂xi∂xi
∂x ̃j+
∂akl
∂xi∂xi
∂ ̃xj
Al=aklbij∂Al
∂xi+
∂akl
∂xi
(2.3.20) bijAl,
wherebij=∂xi/∂ ̃xjare as in (2.3.11). In view of Definition2.29, we infer from (2.3.20)
that the usual derivative operators (2.3.18) are not tensor operators. Namely,
{
∂Ai
∂xj}
is not atensor, and the transformation formula (2.3.20) contains an extra term:
∂akl
∂xib
i
jAl.