Differential Forms
Cartan’s calculus of differential forms is particularly useful in general relativity (and also in other fields of physics). We begin our discussion by repeating some algebraic preliminaries on exterior algebras. Then exterior differential forms and the as
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Differential Forms
Cartan’s calculus of differential forms is particularly useful in general relativity (but also in other fields of physics). We begin our discussion by repeating some algebraic preliminaries.
14.1 Exterior Algebra Let A be a commutative, associative, unitary R-algebra and let E be an A-module. In the following we have either A = R and E a finite dimensional real vector space or A = F(M) and E = X (M) for some differentiable manifold M. p We consider p the space T (E) of A-valued, p-multilinear forms on E and the subspace (E) of completely antisymmetric multilinear forms. Both spaces can naturally be turned into A-modules. In particular, 0 (E) := T 0 (E) := A and 1 (E) = T 1 (E) = E ∗ (dual module of E). The elements of p (E) are called (exterior) forms of degree p. Definition 14.1 We define the alternation operator on T p (E) by (AT )(v1 , . . . , vp ) :=
1 (sgn σ )T (vσ (1) , . . . , vσ (p) ) p!
for T ∈ T p (E), (14.1)
σ ∈ Sp
where the sum in (14.1) extends over the permutation group Sp of p objects; sgn σ denotes the signature of the permutation σ ∈ Sp . The following statements obviously hold: (a) A is an A-linear mapping from T p (E) onto (b) A ◦ A = A. In particular, Aω = ω for any ω ∈ p (E).
p
(E): A(T p (E)) =
p
(E);
N. Straumann, General Relativity, Graduate Texts in Physics, 607 DOI 10.1007/978-94-007-5410-2_14, © Springer Science+Business Media Dordrecht 2013
608
14
Definition 14.2 Let ω ∈
p
(E) and η ∈
ω ∧ η :=
q
Differential Forms
(E). We define the exterior product by
(p + q)! A(ω ⊗ η). p!q!
(14.2)
The exterior product has the following properties: (a) ∧ is A-bilinear; (b) ω ∧ η = (−1)p·q η ∧ ω; (c) ∧ is associative: (ω1 ∧ ω2 ) ∧ ω3 = ω1 ∧ (ω2 ∧ ω3 ).
(14.3)
In the finite dimensional case we have the Theorem 14.1 Let θ i , for i = 1, 2, . . . , n < ∞, be a basis for E ∗ . Then the set θ i1 ∧ θ i2 ∧ . . . ∧ θ ip , 1 ≤ i1 < i2 < . . . < ip ≤ n is a basis for the space p (E), p ≤ n, which has the dimension n n! , := p!(n − p)! p if p > n, p (E) = {0}. The Grassman algebra (or exterior algebra)
(14.4)
(14.5)
(E) is defined as the direct sum
p n (E) := (E), p=0
where the exterior product is extended in a bilinear manner to the entire (E) is a graded, associative, unitary A-algebra.
(E). Thus
Definition 14.3 For every p ∈ N0 we define the mapping E×
p p−1 (E) −→ (E),
(v, ω) −→ iv ω, where (iv ω)(v1 , . . . , vp−1 ) = ω(v, v1 , . . . , vp−1 ), 0 iv ω = 0 for ω ∈ (E).
The association (v, ω) −→ iv ω is called the interior product of v and ω.
(14.6)
14.2
Exterior Differential Forms
609
For every p, the interior product is an A-bilinear mapping and can thus be uniquely extended to an A-bilinear mapping E×
(E) −→
(E),
(v, ω) −→ iv ω. For a proof of the following theorem and other unproved statements in this section see, for example, [43], Chap. III. Theorem 14.2 For every fixed v ∈ E, the mapping iv : properties:
(E) −→ (E) has the
(i) iv is A-linear; (ii) iv ( p (E)) ⊆ p−1 (E); (iii) iv (α ∧ β) =
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