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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