4.7 Volume Elements
55
Example 4.6 As the simplest example of a non-diagonal metric consider 2dimensional Euclidean space with Cartesian-like coordinates, but with a tilted
y axis as in Fig. 4.6.
From the figure the line element and the metric tensor are
ds
2
= dx
2
+ dy
2
+ 2 cos θ dxdy, g ik = =
g i · ·
g k =
1 cos θ
cos θ 1
. (4.86)
To get the metric in another way we express the coordinate basis in terms of
the orthonormal dyad,
g 1 = =
e 1 ,
g 2 = cos θ
e 1 + sin θ
e 1 with δ jk = =
e j · ·
e k
(4.87)
and find the same metric tensor (4.86).
Lastly, we work out the 2-volume element. The metric determinant from
(4.86) and the 2-volume or area are
|g| = sin
2
θ, dV 2 = sin θ dxdy.
(4.88)
This agrees with simple geometry and Fig. 4.6.
In this section we have chosen to develop the theory of invariant volume elements
and integrals in a simple way, depending on invariance arguments. However if
one pursues these ideas further he is lead naturally into the theory of p-forms
(Ohanian 1994; Misner 1973). When antisymmetric 2-tensors are considered such
forms become very useful. In this book we will develop the physics of gravity and
cosmology without the use of such p-forms but will discuss them very briefly in
Appendix 2 in Chap. 6.
Appendix 1: Differential Manifolds
The term manifold occurs in more mathematically oriented work. A manifold is an
open collection of points P with useful properties for physics applications; because
the collection is open there will be by definition a region around each point that
is also in the manifold. The points in a 1-dimensional manifold are in one to one
correspondence with an open set of the reals; an open set of the reals is defined as a
union of open intervals. Similarly the points in a 2-dimensional manifold are in one
to one correspondence with a pair of reals, and so forth for any dimension n. Thus
an n-dimensional manifold is an open set of points that can be labeled by an n-tuple
of reals in an open region, that is by coordinates.
55
Example 4.6 As the simplest example of a non-diagonal metric consider 2dimensional Euclidean space with Cartesian-like coordinates, but with a tilted
y axis as in Fig. 4.6.
From the figure the line element and the metric tensor are
ds
2
= dx
2
+ dy
2
+ 2 cos θ dxdy, g ik = =
g i · ·
g k =
1 cos θ
cos θ 1
. (4.86)
To get the metric in another way we express the coordinate basis in terms of
the orthonormal dyad,
g 1 = =
e 1 ,
g 2 = cos θ
e 1 + sin θ
e 1 with δ jk = =
e j · ·
e k
(4.87)
and find the same metric tensor (4.86).
Lastly, we work out the 2-volume element. The metric determinant from
(4.86) and the 2-volume or area are
|g| = sin
2
θ, dV 2 = sin θ dxdy.
(4.88)
This agrees with simple geometry and Fig. 4.6.
In this section we have chosen to develop the theory of invariant volume elements
and integrals in a simple way, depending on invariance arguments. However if
one pursues these ideas further he is lead naturally into the theory of p-forms
(Ohanian 1994; Misner 1973). When antisymmetric 2-tensors are considered such
forms become very useful. In this book we will develop the physics of gravity and
cosmology without the use of such p-forms but will discuss them very briefly in
Appendix 2 in Chap. 6.
Appendix 1: Differential Manifolds
The term manifold occurs in more mathematically oriented work. A manifold is an
open collection of points P with useful properties for physics applications; because
the collection is open there will be by definition a region around each point that
is also in the manifold. The points in a 1-dimensional manifold are in one to one
correspondence with an open set of the reals; an open set of the reals is defined as a
union of open intervals. Similarly the points in a 2-dimensional manifold are in one
to one correspondence with a pair of reals, and so forth for any dimension n. Thus
an n-dimensional manifold is an open set of points that can be labeled by an n-tuple
of reals in an open region, that is by coordinates.