88
6 Tensor Analysis
Note that if the determinant g is negative, as it generally is in relativity theory, the
√
|g| in (6.18) must be replaced by
√
−|g|; equivalently we may interpret
√
|g| as
being the root of the absolute value of the determinant.
These expressions are often of use. With the final form in (6.28) we may return
to the expression (6.25) for the divergence and write it as
B
k ;k = B
k ,k +
k
kl B
l
= B
k ,k +
√ |g|
,k
√ |g|
B
k
=
1
√ |g|
|g|B
j
, j
generalized divergence.
(6.29)
Thus we have expressed the divergence in an elegant form that contains no connection
but. only the metric determinant and an ordinary derivative; one need not calculate
the connections.
The form for the divergence (6.29) and the invariant volume element discussed
in Sect. 4.7 combine beautifully in giving a covariant version of Gauss’s law for
integrals; see Exercise 6.9.
In elementary vector calculus the Laplacian is defined as the divergence of the
gradient of a scalar,
div grad φ = ∇ · ∇φ = ∇
2
φ =
∂
2
φ
∂ x 2 +
∂
2
φ
∂ y 2 +
∂
2
φ
∂z
Laplacian.
(6.30)
The natural generalization of this is to use the above definition of divergence on the
gradient of a scalar, or
∇
2
φ =
g
i j
φ , j
;i
, generalized Laplacian.
(6.31)
As we have shown for the divergence this may be written without connections as
g
i j
φ , j
;i
=
1
√
|g|
|g|g
i j
φ j
,i
.
(6.32)
This form is quite useful for doing vector analysis in a curvilinear coordinate
system, and gives the familiar textbook expressions for the Laplacian in spherical
and cylindrical coordinates with ease. It is important in tensor analysis in general
relativity.
Example 6.1 Let us work out the simple but nontrivial example of the Laplacian in polar coordinates. Call the scalar function f . From Example 4.1 we
have the metric and its inverse,
6 Tensor Analysis
Note that if the determinant g is negative, as it generally is in relativity theory, the
√
|g| in (6.18) must be replaced by
√
−|g|; equivalently we may interpret
√
|g| as
being the root of the absolute value of the determinant.
These expressions are often of use. With the final form in (6.28) we may return
to the expression (6.25) for the divergence and write it as
B
k ;k = B
k ,k +
k
kl B
l
= B
k ,k +
√ |g|
,k
√ |g|
B
k
=
1
√ |g|
|g|B
j
, j
generalized divergence.
(6.29)
Thus we have expressed the divergence in an elegant form that contains no connection
but. only the metric determinant and an ordinary derivative; one need not calculate
the connections.
The form for the divergence (6.29) and the invariant volume element discussed
in Sect. 4.7 combine beautifully in giving a covariant version of Gauss’s law for
integrals; see Exercise 6.9.
In elementary vector calculus the Laplacian is defined as the divergence of the
gradient of a scalar,
div grad φ = ∇ · ∇φ = ∇
2
φ =
∂
2
φ
∂ x 2 +
∂
2
φ
∂ y 2 +
∂
2
φ
∂z
Laplacian.
(6.30)
The natural generalization of this is to use the above definition of divergence on the
gradient of a scalar, or
∇
2
φ =
g
i j
φ , j
;i
, generalized Laplacian.
(6.31)
As we have shown for the divergence this may be written without connections as
g
i j
φ , j
;i
=
1
√
|g|
|g|g
i j
φ j
,i
.
(6.32)
This form is quite useful for doing vector analysis in a curvilinear coordinate
system, and gives the familiar textbook expressions for the Laplacian in spherical
and cylindrical coordinates with ease. It is important in tensor analysis in general
relativity.
Example 6.1 Let us work out the simple but nontrivial example of the Laplacian in polar coordinates. Call the scalar function f . From Example 4.1 we
have the metric and its inverse,