GR September 29 Lecture
Bobby Bond
September 29, 2014
p-folds
Differential forms: A differential p-form is a (0,p) tensor which is completely antisymmetric.
Example:
Fµν = ∂µ Aν − ∂ν Aµ
(1)
Where 0 ≤ p ≤ n. A 0 form is fn and 1 form is a covector.
Wedge Product: p + q ≤ n. If A is a p-form and B is a q-form then the Wedge Product
is defined by
(A ∧ B)µ1 ...µp+q =
(p + q)!
A[µ1 ...µp Bµp+1 ...µp+q ]
p!q!
(2)
If A and B both 1-forms
(A ∧ B)µν = Aµ Bν − Aν Bµ
(3)
Exterior Derivative (d): If a A p-form then dA is defined by
(dA)µ1 ...µp+1 = (p + 1)∂[µ1 Aµ2 ...µp+1 ]
(4)
Examples of exterior derivatives on different forms:
φ is a 0-form
(dφ)µ = ∂µ φ
(5)
(dA)µν = 2(∂µ Aν − ∂ν Aµ ) ∝ Fµν
(6)
A is a 1-form
Acting on a Wedge Product
d(A ∧ B) = (dA) ∧ B + (−1)p A ∧ (dB)
1
(7)
The exterior derivative (d) is tensorial.
Since d is antisymmetric we have the following property
d(dA) = 0 ∀A
d2 = 0
(8)
We call B closed if dB = 0. We call B exact if B = dA. All exact forms are closed but
not all closed forms are exact (depends on the manifold).
Z p (M ) is the set of closed forms. B p (M ) is the set of exact forms.
p (M )
H p (M ) = B
Z p (M ) - elements are equivalent if they differ by an exact form Consider A,
B p-forms, both closed, not exact. Then A ∼ B if (A-B) is exact.
Hodge Dual (∗A): Let A be a p-form. ∗A is an n-p form.
(∗A)µ1 ...µn−p =
1 ν1 ...νp
µ ...µ Aν ...ν
p! 1 n−p 1 p
(9)
When p = n2 the spaces for A and ∗A are the same. Otherwise the spaces are not the
same.
∗ ∗ A = (−1)s+p(n−p) A
Where s is the number of - signs in the metric signature.
Maxwell: dF = 0 cf. F = dA implies F is exact.
F is a 2 form ∗F is also a 2 form.
d(∗F ) = ∗J where Jµ = (ρ, ii )
In 3D, if U,V are 1-forms then
∗(U ∧ V )i = jk
i Uj V k
(10)
is also a 1-form.
Integration:
Define
ε = µ1 ...µn dxµ1 ⊗ . . . ⊗ dxµn =
p
1p
|g|˜
µ1 ...µn dxµ1 ∧ . . . ∧ dxµn = |g|dn x
n!
(11)
This was simplified using the following relation
dn x = dx0 ∧. . .∧dxn =
1
1 ∂xµ
∂xµ
0
0
˜µ1 ...µn dxµ1 ∧. . .∧dxµn = | µ0 |˜
µ01 ...µ0n dxµ1 ∧. . .∧dxµn = | µ0 |·dn x0
n!
n! ∂x
∂x
(12)
2