Appendix A The Language of Differential Forms This appendix—with the only exception of Sect.A.4.2—does not contain any new physical notions with respect to the previous chapters, but has the purpose of deriving and rewriting some of the previous results using a different language: the language of the so-called differential (or exterior) forms. Thanks to this language we can rewrite all equations in a more compact form, where all tensor indices referred to the diffeomorphisms of the curved space–time are “hidden” inside the variables, with great formal simplifications and benefits (especially in the context of the variational computations). The matter of this appendix is not intended to provide a complete nor a rigorous introduction to this formalism: it should be regarded only as a first, intuitive and oper- ational approach to the calculus of differential forms (also called exterior calculus, or “Cartan calculus”). The main purpose is to quickly put the reader in the position of understanding, and also independently performing, various computations typical of a geometric model of gravity. The readers interested in a more rigorous discussion of differential forms are referred, for instance, to the book [22] of the bibliography. Let us finally notice that in this appendix we will follow the conventions introduced in Chap. 12, Sect. 12.1: latin letters a, b, c,...will denote Lorentz indices in the flat tangent space, Greek letters μ, ν, α,... tensor indices in the curved manifold. For the matter fields we will always use natural units = c = 1. Also, unless otherwise stated, in the first three Sects. A.1–A.3 we will assume that the space–time manifold has an arbitrary number D of dimensions, with signature (+, −, −, −,...). A.1 Elements of Exterior Calculus Let us start with the observation that the infinitesimal (oriented) surface-element dx1dx2 of a two-dimensional differentiable manifold is antisymmetric with respect → = → to the transformation which exchanges the two coordinates, x1 x1 x2 and x2 = | / |=− x2 x1, since the Jacobian determinant of such a transformation is ∂x ∂x 1. © Springer International Publishing AG 2017 303 M. Gasperini, Theory of Gravitational Interactions, UNITEXT for Physics, DOI 10.1007/978-3-319-49682-5 304 Appendix A: The Language of Differential Forms Hence: dx1dx2 =− dx2dx1. (A.1) With reference to a generic volume element dx1dx2 ···dxD of a higher-dimensional manifold let us then introduce the composition of differentials called exterior prod- uct and denoted by the wedge symbol, dxμ ∧ dxν , which is associative and anti- symmetric, dxμ ∧ dxν =−dxν ∧ dxμ. Let us define, in this context, an “exterior” differential form of degree p—or, more synthetically, a p-form—as an element of the linear vector space Λp spanned by the external composition of p differentials. Any p-form can thus be represented as a homogeneous polynomial with a degree of p in the exterior product of differentials, ∈ Λp =⇒ = μ1 ∧···∧ μp , A A A[μ1···μp ]dx dx (A.2) μi ∧ μ j =− μ j ∧ μi where dx dx dx dx for any pair of indices, and where A[μ1···μp ] (the so-called “components” of the p form) correspond to the components of a totally antisymmetric tensor of rank p. A scalar φ, for instance, can be represented as a μ 0-form, a covariant vector Aμ as a 1-form A, with A = Aμdx , an antisymmetric μ ν tensor Fμν as a 2-form F, with F = Fμν dx ∧ dx , and so on. In a D-dimensional manifold, the direct sum of the vector spaces Λp from 0 to D defines the so-called Cartan algebra Λ, D Λ = Λp. (A.3) p=0 In a given linear vector space Λ the exterior product is a map Λ × Λ → Λ, whose properties can be represented in the coordinate differential base dxμ1 ∧ dxμ2 ··· by a composition law which satisfies the properties of (1) bilinearity: (α dxμ1 ∧···dxμp + β dxμ1 ∧···dxμp ) ∧ dxμp+1 ∧···∧dxμp+q = (α + β)dxμ1 ∧···∧dxμp ∧ dxμp+1 ∧···∧dxμp+q (A.4) (α and β are arbitrary numerical coefficients); (2) associativity: (dxμ1 ∧···dxμp ) ∧ (dxμp+1 ∧···dxμp+q ) = dxμ1 ∧···∧dxμp+q ; (A.5) (3) skewness: [ ] dxμ1 ∧···∧dxμp = dx μ1 ∧···∧dxμp . (A.6) This last property implies that the exterior product of a number of differentials μp > D is identically vanishing. Appendix A: The Language of Differential Forms 305 Starting with the above definitions, we can now introduce some important operations concerning the exterior forms. A.1.1 Exterior Product The exterior product between a p-form A ∈ Λp and a q-form B ∈ Λq is a bilinear and associative mapping ∧:Λp × Λq → Λp+q , which defines the (p + q)-form C such that = ∧ = μ1 ∧···∧ μp+q . C A B Aμ1···μp Bμp+1···μp+q dx dx (A.7) The commutation properties of this product depend on the degrees of the forms we are considering (i.e. on the number of the components we have to switch), and in general we have the rule: A ∧ B = (−1)pq B ∧ A. (A.8) A.1.2 Exterior Derivative The exterior derivative of a p form A ∈ Λp can be interpreted (for what concerns μ the product rules) as the exterior product between the gradient 1-form dx ∂μ and the p-form A. It is thus represented by the mapping d : Λp → Λp+1, which defines the (p + 1)-form dAsuch that = μ1 ∧···∧ μp+1 . dA ∂[μ1 Aμ2···μp+1]dx dx (A.9) For a scalar φ, for instance, the exterior derivative is represented by the 1-form μ dφ = ∂μφdx . (A.10) The exterior derivative of the 1-form A is represented by the 2-form μ ν dA= ∂[μ Aν]dx ∧ dx , (A.11) and so on for higher degrees. An immediate consequence of the definition (A.9) is that the second exterior derivative is always vanishing, d2 A = d ∧ dA≡ 0, (A.12) regardless of the degree of the form A. We can also recall that a p-form A is called closed if dA= 0, and exact if it satisfies the property A = dφ, where φ is a (p − 1)- form. If a form is exact then it is (obviously) closed. However, if a form is closed then 306 Appendix A: The Language of Differential Forms it is not necessarily exact (it depends on the topological properties of the manifold where the form is defined). Another consequence of the definition (A.9) is that, in a space–time with a α α symmetric connection (Γμν = Γνμ ), the gradient ∂μ appearing in the exterior- derivative operator can be always replaced by the covariant gradient ∇μ. In fact, ∇ = − Γ α − Γ α −··· , μ1 Aμ2μ3... ∂μ1 Aμ2μ3... μ1μ2 Aαμ3... μ1μ3 Aμ2α... (A.13) so that all connection terms disappear after antisymmetrization, and =∇ ≡∇ μ1 ∧···∧ μp+1 . dA A [μ1 Aμ2···μp+1]dx dx (A.14) Finally, again from the definition (A.9) and from the commutation rule (A.8), we can obtain a generalized Leibnitz rule for the exterior derivative of a product. Consider, for instance, the exterior product of a p-form A and a q-form B.By recalling that d is a 1-form operator we have d(A ∧ B) = dA∧ B + (−1)p A ∧ dB, d(B ∧ A) = dB ∧ A + (−1)q B ∧ dA. (A.15) And so on for multiple products. A.1.3 Duality Conjugation and Co-differential Operator Another crucial ingredient for the application of this formalism to physical models is the so-called Hodge-duality operation, which associates to each p-form its (D − p)- dimensional “complement”. The dual of a p-form A ∈ Λp is a mapping : Λp → ΛD−p, defining the (D − p)-form A such that 1 μ ···μ μ + μ A = A 1 p η ··· ··· dx p 1 ∧···∧dx D . (A.16) (D − p)! μ1 μpμp+1 μD We should recall that the fully antisymmetric tensor η is related to the Levi-Civita antisymmetric density by the relation = | | ημ1···μD g μ1···μD (A.17) √ √ (see Sect. 3.2, Eq. (3.34)). We should also note that the use of |g| instead of −g is due to the fact that the sign of det gμν , in an arbitrary number of D space–time dimensions and with the signature (+, −, −, −,...), depends on the number (even or odd) of the D − 1 spacelike components. Appendix A: The Language of Differential Forms 307 It may be useful to point out that the square of the duality operator does not coincides with the identity, in general. By applying the definition (A.16), in fact, we obtain 1 μ ···μ ν ν A = A ··· η 1 D η ··· ··· dx 1 ∧···∧dx p p!(D − p)! μ1 μp μp+1 μD ν1 νp ··· p(D−p) D−1 1 μ1 μp ν ν = (−1) (−1) δ ··· A ··· dx 1 ∧···∧dx p p! ν1 νp μ1 μp = (−1)p(D−p)+D−1 A. (A.18) The factor (−1)D−1 comes from the product rules of the totally antisymmetric tensors since, in D − 1 spatial dimensions (and with our conventions), we have D−1 012···D−1 D−1 012···D−1 = (−1) = (−1) . (A.19) The product rules thus become, in general, ··· − μ ···μ μ1 μD = (− )D 1( − )! 1 p , ην1···νp μp+1···μD η 1 D p δν1···νp (A.20) μ1···μp where δν1···νp is the determinant defined in Eq. (3.35). The additional factor (−1)p(D−p), appearing in Eq. (A.18), comes from the switching of the p indices of A with the D − p indices of its dual (such a switching is needed to arrange the indices of η in a way to match the sequence of the product rule (A.20)).