6.1 Covariant Derivatives, Component View
85
Because of the Ricci Theorem these are the same, and there is in fact no ambiguity
in the notation. Another way to say the same thing is that the operations of raising
and lowering indices commutes with the operation of taking a covariant derivative.
It is also interesting to note a rather obvious converse of the Ricci Theorem: If
the covariant derivative of the metric tensor is zero then the connections are the
Christoffel connections. This follows because in the covariant derivative relation
(6.12) the first line is the same as (5.17), which leads to the Christoffel connections
in (5.19). Thus the Christoffel connections are dictated by the demands that they be
symmetric and the covariant derivative of the metric be zero.
6.2 Covariant Derivatives, Abstract View
As before in Sect. 5.6 we now consider vectors as invariant abstract objects that may
be expanded in a basis, conveniently taken to be a coordinate basis. Then the vector
and its change in moving to a nearby point are, as discussed in Sect. 5.6,
V = V
j
e j , d
V =
dV
i
+
i
k j V
j dx
k
e i , ,
i
k j = affine connections. (6.14)
For a field of vectors V
j
= V
j
(x
k ) we thus have the change
d
V =
∂ V
i
∂ x k dx
k
+
i
k j V
j dx
k
e i =
∂ V
i
∂ x k +
i
k j V
j
dx
k
e i
=
V
i ;k dx
k
e i .
(6.15)
This defines the coefficient array V
i ;k as determining the change in the vectot. Both
sides of (6.15) are invariant abstract vectors, so the last object in parentheses is the
ith component of the change in the vector. By the quotient theorem the array V
i ;k
then forms the components of a (1,1) tensor, as we have already discussed in terms
of components in Sect. 6.1. In terms of the basis vectors and forms we may write
that tensor as
∇
V = V
i ;k
e ⊗ ˜
dx
k
Covariant tensor derivative.
(6.16)
The various component arrays that we discussed in Sect. 6.1 now emerge as coefficients in tensor relations just as happened with tensors in general in Sect. 4.4.
The tensor character of the covariant derivative is made particularly clear in this
approach whereas in the component approach it required a bit of algebra to verify
its transformation as a tensor.
Consider next the derivative of a vector field along some given curve parametrized
as usual by the arc length s. From (6.15) we may define the derivative as
85
Because of the Ricci Theorem these are the same, and there is in fact no ambiguity
in the notation. Another way to say the same thing is that the operations of raising
and lowering indices commutes with the operation of taking a covariant derivative.
It is also interesting to note a rather obvious converse of the Ricci Theorem: If
the covariant derivative of the metric tensor is zero then the connections are the
Christoffel connections. This follows because in the covariant derivative relation
(6.12) the first line is the same as (5.17), which leads to the Christoffel connections
in (5.19). Thus the Christoffel connections are dictated by the demands that they be
symmetric and the covariant derivative of the metric be zero.
6.2 Covariant Derivatives, Abstract View
As before in Sect. 5.6 we now consider vectors as invariant abstract objects that may
be expanded in a basis, conveniently taken to be a coordinate basis. Then the vector
and its change in moving to a nearby point are, as discussed in Sect. 5.6,
V = V
j
e j , d
V =
dV
i
+
i
k j V
j dx
k
e i , ,
i
k j = affine connections. (6.14)
For a field of vectors V
j
= V
j
(x
k ) we thus have the change
d
V =
∂ V
i
∂ x k dx
k
+
i
k j V
j dx
k
e i =
∂ V
i
∂ x k +
i
k j V
j
dx
k
e i
=
V
i ;k dx
k
e i .
(6.15)
This defines the coefficient array V
i ;k as determining the change in the vectot. Both
sides of (6.15) are invariant abstract vectors, so the last object in parentheses is the
ith component of the change in the vector. By the quotient theorem the array V
i ;k
then forms the components of a (1,1) tensor, as we have already discussed in terms
of components in Sect. 6.1. In terms of the basis vectors and forms we may write
that tensor as
∇
V = V
i ;k
e ⊗ ˜
dx
k
Covariant tensor derivative.
(6.16)
The various component arrays that we discussed in Sect. 6.1 now emerge as coefficients in tensor relations just as happened with tensors in general in Sect. 4.4.
The tensor character of the covariant derivative is made particularly clear in this
approach whereas in the component approach it required a bit of algebra to verify
its transformation as a tensor.
Consider next the derivative of a vector field along some given curve parametrized
as usual by the arc length s. From (6.15) we may define the derivative as