6.2 Covariant Derivatives, Abstract View
87
This is zero according to the Ricci Theorem of Sect. 6.1.
In brief summary the equations and various component arrays in the preceding
Sect. 6.1 emerge in this section as coefficients in the abstract approach, just as
happened with tensors in general in Chap. 5.
6.3 The Divergence and Laplacian
In elementary vector calculus with Cartesian coordinates the divergence of a vector
field is defined as
div
B = ∇ ·
B =
dB x
dx
+
dB y
dy
+
dB z
dz
= B
i ,i , divergence.
(6.24)
The obvious covariant generalization of this is the contracted covariant derivative,
B
i ;i = B
i ,i +
k
kl B
l
.
(6.25)
This may be simplified into a form which contains no connections and is thus easy
to deal with. The contracted connection is
k
kl =
1
2
g
kn
g nk,l + g ln,k − g kl,n
=
1
2
g
kn g nk,l ,
(6.26)
where we have used the symmetry of the metric to cancel the second and third terms.
At this point we digress to recall some properties of matrices and determinants,
referred specifically to the metric tensor treated as a matrix. The inverse of the metric,
g
ik , may be calculated as g
ik
=
ik
/|g| where |g| is the determinant and
ki is the
cofactor matrix; the cofactor is found by crossing out the i row, and taking the
determinant with a sign (−1)
i+k . The determinant may be similarly expressed in
terms of the cofactor: choose a row, say i = 3, and the determinant is |g| = g 3k
3k .
This is often referred to as expansion in minors. From the above relations we see that
∂|g|
∂g jk
=
jk
= |g|g
jk
, so g
jk
=
1
|g|
∂|g|
∂g jk
.
(6.27)
Now we return to the expression for the contracted connection in (6.26) and
substitute the above to obtain several alternative ways to write it
k
kl =
1
2
g
kt g kt,l =
1
2
1
|g|
∂|g|
∂g kt
∂g kt
∂ x l =
1
2|g|
∂|g|
∂ x l =
1
2|g|
|g| ,l
=
1
2
(log|g|) ,l = (log
|g|) ,l =
√
|g|
,l
√
|g|
.
(6.28)
87
This is zero according to the Ricci Theorem of Sect. 6.1.
In brief summary the equations and various component arrays in the preceding
Sect. 6.1 emerge in this section as coefficients in the abstract approach, just as
happened with tensors in general in Chap. 5.
6.3 The Divergence and Laplacian
In elementary vector calculus with Cartesian coordinates the divergence of a vector
field is defined as
div
B = ∇ ·
B =
dB x
dx
+
dB y
dy
+
dB z
dz
= B
i ,i , divergence.
(6.24)
The obvious covariant generalization of this is the contracted covariant derivative,
B
i ;i = B
i ,i +
k
kl B
l
.
(6.25)
This may be simplified into a form which contains no connections and is thus easy
to deal with. The contracted connection is
k
kl =
1
2
g
kn
g nk,l + g ln,k − g kl,n
=
1
2
g
kn g nk,l ,
(6.26)
where we have used the symmetry of the metric to cancel the second and third terms.
At this point we digress to recall some properties of matrices and determinants,
referred specifically to the metric tensor treated as a matrix. The inverse of the metric,
g
ik , may be calculated as g
ik
=
ik
/|g| where |g| is the determinant and
ki is the
cofactor matrix; the cofactor is found by crossing out the i row, and taking the
determinant with a sign (−1)
i+k . The determinant may be similarly expressed in
terms of the cofactor: choose a row, say i = 3, and the determinant is |g| = g 3k
3k .
This is often referred to as expansion in minors. From the above relations we see that
∂|g|
∂g jk
=
jk
= |g|g
jk
, so g
jk
=
1
|g|
∂|g|
∂g jk
.
(6.27)
Now we return to the expression for the contracted connection in (6.26) and
substitute the above to obtain several alternative ways to write it
k
kl =
1
2
g
kt g kt,l =
1
2
1
|g|
∂|g|
∂g kt
∂g kt
∂ x l =
1
2|g|
∂|g|
∂ x l =
1
2|g|
|g| ,l
=
1
2
(log|g|) ,l = (log
|g|) ,l =
√
|g|
,l
√
|g|
.
(6.28)