Appendix 2: p-Forms and Exterior Derivatives
91
index and thus have 4 components. A 4-form coefficient array must be a multiple of
the Levi-Cevita epsilon, which is discussed in Exercise 6.6, so the 4-form has only
1 independent component.
We will develop gravitational theory in this book without the use of p-forms so
we will not discuss them further. For the reader interested in the gravitational field
associated with the electromagnetic field they can be useful (Adler 1975; Misner
1973).
Exercises
6.1 Let us study a vector field in Euclidean 2-space. In Cartesian coordinates take
the field to be V
i
x
j
= (1, 1). This is a constant field represented by arrows
at 45° throughout space.
(a) Transform the vector field to polar coordinates and call it V
i . (We obtained
the transformation matrix in Example 4.1.) Note that it does not have
constant components. Lower the index and form the covariant field V k ,
which also does not have constant components.
(b) Sketch the field in terms of arrows in polar coordinates, and see that
the same picture results as with Cartesian coordinates. (Use the ideas
discussed in Example 4.4.)
(c) Calculate the covariant derivative V
i ;k in Cartesian coordinates, which of
course is trivial. What is it in polar coordinates.
6.2 Consider the vector field V
i
x
j
= (1, 0) in polar coordinates. Draw a picture
of it. Calculate its covariant derivative and its divergence.
6.3 For the covariant Laplacian in (6.32) …
(a) Calculate the Laplacian for a coordinate system in which the metric is
constant.
(b) Calculate it for Euclidean 3-space with cylindrical coordinates.
(c) Calculate it for Euclidean 3-space with spherical coordinates.
6.4 Calculate the Laplacian specifically for Minkowski space using Cartesian
coordinates. This is also called the d’Alembertian operator. Setting the
d’Alembertian of a function f equal to zero gives the scalar wave equation.
Show that one important type of solution is f (x − ct), where f is any twice
differentiable function. We will discuss this solution at length when we study
gravitational waves in Chap. 11.
6.5 In Exercise 5.3 we studied a line element of the form ds
2
= f (ρ)
2 dρ
2
+ρ
2 dϕ
2 .
(a) What is the square root of the metric determinant, and what is the invariant
volume element?
(b) Consider the special case f (ρ) = 1 − a/ρ, where a is a positive constant.
Note that the 1,1 metric component can be zero. What do you suppose
this means? Hint: calculate
√ |g| and think about the invariant volume
element. What peculiar features does this imply for the line element? A
similar peculiarity occurs for black holes, which we will discuss in Part
III.
91
index and thus have 4 components. A 4-form coefficient array must be a multiple of
the Levi-Cevita epsilon, which is discussed in Exercise 6.6, so the 4-form has only
1 independent component.
We will develop gravitational theory in this book without the use of p-forms so
we will not discuss them further. For the reader interested in the gravitational field
associated with the electromagnetic field they can be useful (Adler 1975; Misner
1973).
Exercises
6.1 Let us study a vector field in Euclidean 2-space. In Cartesian coordinates take
the field to be V
i
x
j
= (1, 1). This is a constant field represented by arrows
at 45° throughout space.
(a) Transform the vector field to polar coordinates and call it V
i . (We obtained
the transformation matrix in Example 4.1.) Note that it does not have
constant components. Lower the index and form the covariant field V k ,
which also does not have constant components.
(b) Sketch the field in terms of arrows in polar coordinates, and see that
the same picture results as with Cartesian coordinates. (Use the ideas
discussed in Example 4.4.)
(c) Calculate the covariant derivative V
i ;k in Cartesian coordinates, which of
course is trivial. What is it in polar coordinates.
6.2 Consider the vector field V
i
x
j
= (1, 0) in polar coordinates. Draw a picture
of it. Calculate its covariant derivative and its divergence.
6.3 For the covariant Laplacian in (6.32) …
(a) Calculate the Laplacian for a coordinate system in which the metric is
constant.
(b) Calculate it for Euclidean 3-space with cylindrical coordinates.
(c) Calculate it for Euclidean 3-space with spherical coordinates.
6.4 Calculate the Laplacian specifically for Minkowski space using Cartesian
coordinates. This is also called the d’Alembertian operator. Setting the
d’Alembertian of a function f equal to zero gives the scalar wave equation.
Show that one important type of solution is f (x − ct), where f is any twice
differentiable function. We will discuss this solution at length when we study
gravitational waves in Chap. 11.
6.5 In Exercise 5.3 we studied a line element of the form ds
2
= f (ρ)
2 dρ
2
+ρ
2 dϕ
2 .
(a) What is the square root of the metric determinant, and what is the invariant
volume element?
(b) Consider the special case f (ρ) = 1 − a/ρ, where a is a positive constant.
Note that the 1,1 metric component can be zero. What do you suppose
this means? Hint: calculate
√ |g| and think about the invariant volume
element. What peculiar features does this imply for the line element? A
similar peculiarity occurs for black holes, which we will discuss in Part
III.