4.4 Tensors, Component View
47
which is the equation in the barred frame. Now since the vector V τ is arbitrary its
coefficients in the last expression must be equal, so we have
∂ x
ω
∂ x α
∂ x
τ
∂ x β T
αβ
= T
ωτ .
(4.48)
This is the transformation law of a tensor so the theorem is proven for this special
case. The general case is when the array is any rank and the vector is arbitrary, and
the proof goes through as above.
Theorem 11 Our next theorem is very simple but often useful. If a tensoris zero in
one coordinate system then it must be zero in all coordinate systems. This is obvious
from the definition (4.38).
The existence of a metric tensor allows us to associate a covariant vector with
any contravariant vector. As we discussed in Sect. 4.3 we lower an index according
to
V α = g αβ V
β
.
(4.49)
By the above Theorems 3 and 4 this is indeed a covariant vector. As discussed in
Sect. 4.3 we define the inverse metric array g
λτ by
g
λτ g τ ν = δ
λ
ν ,
(4.50)
and use it to raise an index as follows
g
μρ V ρ = V
μ
.
(4.51)
A simple but important theorem tells us that the raising and lowering operations are
consistent inverses of each other.
Theorem 12 The Kronecker delta transforms as a (1, 1) 2nd rank tensor; it is thus
peculiar in that its components are the same in all systems. Proof is straightforward
with the use of Theorem 1.
δ
α
β =
∂ x
α
∂ x ρ
∂ x
ω
∂ x
β
δ
ρ
ω =
∂ x
α
∂ x ρ
∂ x
ρ
∂ x
β
= δ
α
β .
(4.52)
Because of this the relation defining the inverse metric assures us that it is a second
rank contravariant tensor according to the Quotient Theorem. As is clear from above,
the operations of raising and lowering are consistent: if we first lower and then raise
an index we regain the original vector.
It is evident that we may raise or lower an index in any tensor in exactly the same
way as with vectors. For example we may form
T
αβ g βκ = T
α
κ .
(4.53)
47
which is the equation in the barred frame. Now since the vector V τ is arbitrary its
coefficients in the last expression must be equal, so we have
∂ x
ω
∂ x α
∂ x
τ
∂ x β T
αβ
= T
ωτ .
(4.48)
This is the transformation law of a tensor so the theorem is proven for this special
case. The general case is when the array is any rank and the vector is arbitrary, and
the proof goes through as above.
Theorem 11 Our next theorem is very simple but often useful. If a tensoris zero in
one coordinate system then it must be zero in all coordinate systems. This is obvious
from the definition (4.38).
The existence of a metric tensor allows us to associate a covariant vector with
any contravariant vector. As we discussed in Sect. 4.3 we lower an index according
to
V α = g αβ V
β
.
(4.49)
By the above Theorems 3 and 4 this is indeed a covariant vector. As discussed in
Sect. 4.3 we define the inverse metric array g
λτ by
g
λτ g τ ν = δ
λ
ν ,
(4.50)
and use it to raise an index as follows
g
μρ V ρ = V
μ
.
(4.51)
A simple but important theorem tells us that the raising and lowering operations are
consistent inverses of each other.
Theorem 12 The Kronecker delta transforms as a (1, 1) 2nd rank tensor; it is thus
peculiar in that its components are the same in all systems. Proof is straightforward
with the use of Theorem 1.
δ
α
β =
∂ x
α
∂ x ρ
∂ x
ω
∂ x
β
δ
ρ
ω =
∂ x
α
∂ x ρ
∂ x
ρ
∂ x
β
= δ
α
β .
(4.52)
Because of this the relation defining the inverse metric assures us that it is a second
rank contravariant tensor according to the Quotient Theorem. As is clear from above,
the operations of raising and lowering are consistent: if we first lower and then raise
an index we regain the original vector.
It is evident that we may raise or lower an index in any tensor in exactly the same
way as with vectors. For example we may form
T
αβ g βκ = T
α
κ .
(4.53)