2.3. Forms¶

We will principally examine two types of forms, namely alternating forms, and differential forms. As differential forms are a specialization of alternating forms, we will focus on defining and examining these only later. Instead, we begin by introducing alternating forms as \((0, p)\) tensors, along with operations on these objects.

Many of these ideas may be discussed fairly abstractly, and indeed, it may be argued it is easier to discuss the general actions of forms, to gain a familiarly with their behaviour, before providing a representation or decomposition. This is precisely what we will do here – to introduce a basis for alternating forms, we first define what a form is, operations on and with forms, and finally how the basis may be expressed in term of this language. This may feel a little abstract at first, but unfortunately we require some ground work to appreciate the physical significance of these objects.

This section will occupy itself with alternating forms, and introduce the differential form towards the end. This will set us up with a good foundation for discussing the exterior derivative and integration in the next section, before moving on to a detailed look at the metric.

2.3.1. Alternating real valued \(p\)-forms¶

Definition: Alternating real valued \(p\)-forms

Let \(V\) be a vector space, then the alternating real-valued forms of degree \(p\) are

\[\Lambda^p V := \left\{ \varphi : \underbrace{V \times V \times \ldots \times V}_{p} \rightarrow \mathbb{R}, \ \text{multilinear, alternating} \right\}.\]

Here, multilinear is equivalent to linear in each argument, and alternating implies for \(\pi \in S_n\)

\[\varphi \left( v_{\pi (1)}, v_{\pi (2)}, \ldots, v_{\pi (n) } \right) = \text{sgn}( \pi ) \varphi \left( v_{1}, v_{2}, \ldots, v_{n} \right).\]

There are a few noteworthy properties:

\(\Lambda^0 V = \mathbb{R}\),
\(\Lambda^1 V = V^\ast\), the space of \(1\)-forms on \(V\) is the dual space of \(V\),
\(\Lambda^p V = 0, \ \ \forall \ p > n = \text{dim}\, V\).

The last property can be reasoned given that the forms are alternating, and the degenerate choice of \(v \in V\) arguments for \(p > n\).

Hint

The space of \(p\)-forms \(\Lambda^pV\) is itself a vector space.

2.3.1.1. Vector space valued \(p\)-forms¶

Real valued alternating forms may be generalized to have vector space codomains.

Definition: Alternating vector space valued \(p\)-forms

Let \(V, W\) be vector spaces, then the alternating vector space valued forms of degree \(p\) are

\[\Lambda^p (V,W) := \left\{ \varphi : \underbrace{V \times V \times \ldots \times V}_{p} \rightarrow W, \ \text{multilinear, alternating} \right\}.\]

This permits the decomposition of any \(\varphi \in \Lambda^p (V, W)\) onto a basis \(a_i\) of \(W\)

\[\varphi = \varphi^a a_a,\]

with \(\varphi^a \in \Lambda^pV\).

2.3.1.2. Lie algebra valued \(p\)-forms¶

There is a special case where \(W\) may be a Lie algebra; then the form is decomposed onto the generators of the algebra \(T_a\), and the exterior product operator is replaced by commutators on the basis. Consider \(\varphi \in \Lambda^p (V, \mathfrak{g})\) and \(\psi \in \Lambda^q (V, \mathfrak{g})\), decomposed as

\[\begin{split}\varphi &= \varphi^a T_a, \\ \psi &= \psi^b T_b,\end{split}\]

with \(\varphi^a \in \Lambda^p V\), \(\psi^b \in \Lambda^q V\). We then redefine the exterior product by using the exterior product of the generators of the Lie algebra

\[\begin{split}\left[ \varphi, \psi \right] &= \varphi^a \wedge \psi^b \left[ T_a, T_b \right] \\ &= -(-1)^{pq} \left[ \varphi, \psi \right].\end{split}\]

2.3.2. Operations¶

2.3.2.1. Exterior product¶

Definition: Exterior product

Let \(\varphi \in \Lambda^p V\) and \(\psi \in \Lambda^q V\) be real-valued alternating forms of degree \(p\) and \(q\) respectively on a vector space \(V\). Their exterior product is defined

\[\begin{split}\wedge : \Lambda^p V \times \Lambda^q V \rightarrow & \, \Lambda^{p+q}V, \\ (\varphi, \psi) \mapsto & \, \varphi \wedge \psi.\end{split}\]

Acting on vectors in \(V\)

\[\begin{split}(\varphi \wedge \psi) \left( v_1, \ldots, v_{p + q} \right) := & \\ \frac{1}{p! q!} \sum_{\pi \in S_{p+q} } \text{sgn} (\pi)\, \varphi \left( v_{\pi (1)}, \ldots, v_{\pi (p)} \right) & \psi \left( v_{\pi (p + 1)}, v_{\pi (p + q)} \right).\end{split}\]

If \(\varphi, \psi \in \Lambda^0V\), then \(\varphi \wedge \psi = \varphi \, \psi \in \mathbb{R}\).

Geometrically, we can consider the exterior product of \(\varphi, \psi \in \Lambda^1V\) as the determinant

\[\begin{split}(\varphi \wedge \psi) \left( v_1, v_2 \right) = & \ \text{det} \, \begin{bmatrix} \varphi(v_1) & \psi(v_1) \\ \varphi(v_2) & \psi(v_2) \end{bmatrix}, \\ \\ = & \ \varphi (v_1) \psi (v_2) - \varphi (v_2) \psi (v_1).\end{split}\]

The properties of the exterior product are

bilinearity,
associativity,
graded commutivity:

\[\varphi \wedge \psi = (-1)^{pq} \psi \wedge \varphi,\]

for \(\varphi \in \Lambda^pV\) and \(\psi \in \Lambda^qV\). This last property implies that \(\varphi \wedge \varphi = 0\) for the case of \(p\) odd.

The exterior product may also be derived from the extension of the direct product under the action of the alternating operator \(\mathcal{A}\) (see Tensor symmetry), that is

(2.7)¶\[\mathcal{A}(\varphi \otimes \psi) = \varphi \wedge \psi.\]

2.3.2.2. Pull back¶

The pullback is an active transformation between vector spaces of \(p\)-forms.

Definition: Pull back

Let \(F: W \rightarrow V\) be a linear transformation between two vector spaces \(W\) and \(V\). The pull back \(F^\ast\) is the induced linear mapping

\[\begin{split}F^\ast : \Lambda^p V & \rightarrow \Lambda^p W, \\ \varphi & \mapsto F^\ast \varphi,\end{split}\]

via

\[(F^\ast \varphi) \left( w_1, \dots, w_p \right) := \varphi \left( F w_1, \dots, F w_p \right).\]

The pull back is a homomorphism of the exterior algebras; that is to say

\(F^\ast\) is linear,
distributive:

\[F^\ast(\varphi \wedge \psi) = \left( F^\ast \varphi \right) \wedge \left( F^\ast \psi \right).\]

The additional property of the pull-back is

reverse composition,

(2.8)¶\[\left( F_1 \circ F_2 \right)^\ast = F^\ast_2 \circ F^\ast_1.\]

This property can be demonstrated manifestly.

We act \(\left( F_1 \circ F_2 \right)^\ast\) on a \(p\)-form \(\varphi \in \Lambda^pV\):

\[\left( F_1 \circ F_2 \right)^\ast \varphi \left( w_1, \dots, w_p \right) = \varphi \left( \left( F_1 \circ F_2 \right) w_1, \dots, \left( F_1 \circ F_2 \right) w_p \right),\]

which, by associativity of homomorphisms, is

\[\begin{split}&= F_2^\ast \varphi \left( F_1 w_1, \dots, F_1 w_p \right), \\ &= \left( F_2^\ast \circ F_1^\ast \right) \varphi \left( w_1, \dots, w_p \right), \\\end{split}\]

and therefore the property of eq. (2.8) holds.

2.3.2.3. Inner derivative¶

Sometimes called the interior product.

Definition: Inner derivative

Let \(\Lambda^p V\) be the space of alternating \(p\) forms on \(V\). The inner derivative is defined

\[\begin{split}\iota_v : \Lambda^p V \rightarrow & \, \Lambda^{p-1}V, \\ \varphi \mapsto & \, \iota_v \varphi,\end{split}\]

for some \(v \in V\). Acting on vectors \(v_i \in V\)

\[(\iota_v \varphi)(v_{1}, \ldots, v_{p-1}) := \varphi (v, v_q, \ldots, v_{p-1}).\]

The inner derivative is a contraction, which acts to fix a vector that \(\varphi\) acts on, by definition.

\(\iota_v\) is a derivation of the Grassman algebra \(\Lambda V\). The inner derivative has the following properties

linear map,
linear in \(v\), e.g. \(\iota_{v+u} = \iota_{v} + \iota_{u}\),
graded Leibniz rule, namely for some \(\varphi \in \Lambda^pV\) and \(\psi \in \Lambda^qV\)

\[\iota_v(\varphi \wedge \psi) = (\iota_v \varphi) \wedge \psi + (-1)^p \varphi \wedge (\iota_v \psi)\]

\(\iota_v \iota_u + \iota_u \iota_v = 0\), corollary \(\iota_v^2 = 0\)

The last point follows from the alternating property of forms.

Inner derivative

Is the classification of the inner derivative manifestly a derivation (i.e. obeys Leibniz)?

Without explicitly using the Leibniz property, we can derive the behaviour of the inner derivative when operating on the exterior product of forms.

This is best approached by first expanding the wedge product.

We will consider \(\varphi \in \Lambda^pV\) and \(\psi \in \Lambda^qV\). Let \(v_i \in V\), then

\[\iota_{v_1} (\varphi \wedge \psi) (v_2, \ldots, v_{p+q}) = (\varphi \wedge \psi) (v_1, v_2, \ldots, v_{p+q}), \]

by definition. Expanding the RHS

\[= \frac{1}{p! q!} \sum_{\pi \in S_{p+q}} \text{sgn}(\pi) \varphi (v_{\pi (1)}, \ldots, v_{\pi(p)}) \psi (v_{\pi (p+1)}, \ldots, v_{\pi(p+q)}),\]

Now we use that the inner derivative fixes a vector argument; we can rewrite the sum into two parts, namely those terms where \(\varphi\) acts on \(v_1\), and those where \(\psi\) does:

\[\begin{split}= \frac{p}{p! q!} & \sum_{\pi \in S_{p+q-1}} \text{sgn}(\pi) \varphi (v_{1}, v_{\pi(1)}, \ldots, v_{\pi(p-1)}) \psi (v_{\pi (p)}, \ldots, v_{\pi(p+q-1)}) \\ + \frac{q}{p! q!} (-1)^p & \sum_{\pi \in S_{p+q-1}} \text{sgn}(\pi) \varphi (v_{\pi (1)}, \ldots, v_{\pi(p)}) \psi (v_{1}, v_{\pi(p+1)}, \ldots, v_{\pi(p+q-1)}) ,\end{split}\]

where \((-1)^p\) is a continuity factor to account for the initial sign of \(\pi\) in the second summation term; effectively, there are \(p\) permutations of \(\pi\) before \(\psi\) acts on \(v_1\).

Note the factors of \(p\) and \(q\) on each term. This is from combinatorics; if \(D(n)\) denotes the number of permutations of \(n\), then an \(n\) permutation with a single fixed point is \(n D(n-1)\).

Now apply the definition of the inner derivative again

\[\begin{split}= \frac{p}{p! q!} & \sum_{\pi \in S_{p+q-1}} \text{sgn}(\pi) (\iota_{v_1} \varphi) (v_{\pi(1)}, \ldots, v_{\pi(p-1)}) \psi (v_{\pi (p)}, \ldots, v_{\pi(p+q-1)}) \\ + \frac{q}{p! q!} (-1)^p & \sum_{\pi \in S_{p+q-1}} \text{sgn}(\pi) \varphi (v_{\pi (1)}, \ldots, v_{\pi(p)}) (\iota_{v_1}\psi) (v_{\pi(p+1)}, \ldots, v_{\pi(p+q-1)}).\end{split}\]

Then, by noting that \(p/p! = 1/(p-1)!\), we can replace the expansions with wedge products again

\[\begin{split}=& \, \left((\iota_{v_1} \varphi) \wedge \psi \right) (v_2, \ldots, v_{p+q}) \\ & \, + (-1)^p \left(\varphi \wedge ( \iota_{v_1} \psi) \right)(v_2, \ldots, v_{p+q})\end{split}\]

which is precisely the graded Leibniz rule.

The inner derivative is manifestly a derivation, by virtue of the exterior product.

2.3.3. Basis¶

A basis of \(\Lambda^pV\) is the exterior product of the dual basis

\[\beta^{i_1} \wedge \beta^{i_2} \wedge \ldots \wedge \beta^{i_p}\]

with \(1 \leq i_1 < i_2 < \ldots < i_p \leq n = \text{dim}\, V \).

Consequently, since each \(p\)-form may be described over an \(n\)-dimensional basis, the dimensionality of \(\Lambda^pV\) is the permutations of \(p\) of the \(n\) possible basis vectors \(\beta^i\), without repetition, due to the alternating property. Therefore, we conclude

(2.9)¶\[\text{dim}\, \Lambda^p V = {n \choose p}.\]

It is always the case that \(\text{dim}\, \Lambda^n V = 1\).

For example, consider \(n = 3\), the possible bases for different degrees of forms are

\[\begin{split}\Lambda^1V : & \ \left\{ \beta^1, \ \beta^2, \ \beta^3 \right\}, \\ \Lambda^2V : & \ \left\{ \beta^1 \wedge \beta^2, \ \beta^1 \wedge \beta^3, \ \beta^2 \wedge \beta^3 \right\}, \\ \Lambda^3V : & \ \left\{ \beta^1 \wedge \beta^2 \wedge \beta^3 \right\}.\end{split}\]

The basis may also be obtained through use of eq. (2.7), writing

\[\mathcal{A} \left( \beta^{i_1} \otimes \dots \otimes \beta^{i_p} \right) = \beta^{i_1} \wedge \dots \wedge \beta^{i_p}.\]

Note that a set of such a basis is referred to as a frame, using the same terminology as in Frames.

2.3.3.1. Orientation¶

Definition: Orientation

An orientation of vector space \(V\) with \(n = \text{dim} \, V\) is the choice of one of the two parts of \(\Lambda^nV - \{0\}\).

An orientation is the choice of a positive or negative slice of \(\omega \in \Lambda^nV\). It may be defined by the choice of basis \(b_i\) on \(V\)

\[\omega = \beta^1 \wedge \ldots \wedge \beta^n.\]

Given an orientation \(\omega\), then a basis \(b_i\) is oriented if

\[\omega \left( b_1, \ldots, b_n \right) > 0.\]

Orientations become very useful when we discuss integration in the next chapter.

2.3.4. Differential forms¶

Definition: Differential \(p\)-form

For a point \(x\) in some open subset \(\mathscr{U} \subset \mathbb{R}^n\), and its corresponding tangent space \(T_x \mathscr{U}\), the differential \(p\)-forms are the mappings

\[\begin{split}\varphi : \ \mathscr{U} &\rightarrow \Lambda^p T_x \mathscr{U}, \\ x &\mapsto \varphi_x.\end{split}\]

The set of these mappings is denoted \(\Lambda^p \mathscr{U}\).

Differential forms inherit the properties of alternating forms, and similarly use the notation

\[\Lambda \mathscr{U} := \bigoplus_{p=0}^n \Lambda^p \mathscr{U},\]

to define the direct sum.

A noteworthy set of differential forms are

those mapping to \(\Lambda^p T_x \mathscr{U} = \mathbb{R}\) , so that \(\Lambda^p \mathscr{U}\) is the set of smooth functions on \(\mathscr{U}\),
the cotangent space denoted \(\left( T_x \mathscr{U} \right)^\ast = \Lambda^1 T_x \mathscr{U}\).

The primary difference between alternating forms and differential forms is that in the latter, everything is defined pointwise on \(x \in \mathscr{U}\) ; as such, given a frame \(b_i\), the basis of \(T_x \mathscr{U}\) induced is \(b_i (x)\), and consequently the basis of the cotangent space \(\left( T_x \mathscr{U} \right)^\ast\) is \(\beta^i (x)\), implying the transformation eq. (2.5) extends as

\[\beta^{\prime i}(x) = \gamma (x)^i_{\phantom{i}j} \beta^j(x).\]

In the case of a frame \(\frac{\partial}{\partial x^i}\), the dual frame is written \(\text{d}x^i\), which omits the argument \(x\). We then find the analogue of eq. (2.4) as

\[\text{d}x^i \left( \frac{\partial}{\partial x^j} \right) = \delta^i_{\phantom{i}j},\]

with \(\delta^i_{\phantom{i}j}\) being a constant function for all \(x\).

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