The Language of Differential Forms

Home , Antisymmetric tensor, Cartan (crater), Contorsion tensor, Covariant derivative, Differential form, Exterior calculus

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 product and denoted by the wedge symbol, dxμ ∧ dxν , which is associative and antisymmetric, 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)). We also note, for later applications, that the dual of the identity operator is directly related to the scalar integration measure representing the hypervolume element of the given space–time manifold. From the deﬁnition (A.16) we have, in fact,

1 = μ1 ∧···∧ μD 1 ημ1···μD dx dx D! 0 1 D−1 = |g| ··· − dx ∧ dx ···∧dx 012D 1 = (−1)D−1 |g| d D x. (A.21)

Combining this result with the product rule

··· − μ1 μD = (− )D 1 ! , ημ1···μD η 1 D (A.22) we are led to the useful relation ··· ··· dxμ1 ∧···∧dxμD = |g| d D x ημ1 μD = d D x μ1 μD , (A.23) which will be frequently applied in our subsequent computations. The duality operation is necessarily required in order to deﬁne the scalar products appearing, for instance, in all action integrals. Consider in fact the exterior product between a p-form A and the dual of another p-form B. By using the deﬁnition (A.16) 308 Appendix A: The Language of Differential Forms and the relation (A.23) we obtain 1 ν ···ν μ μ A ∧ B = A ··· B 1 p η ··· ··· dx 1 ∧···∧dx D (D − p)! μ1 μp ν1 νp μp+1 μD − ··· μ ···μ = (− )D 1 D | | ν1 νp 1 p 1 d x g Aμ1···μp B δν1···νp − ··· = (− )D 1 ! D | | μ1 μp 1 p d x g Aμ1···μp B (A.24)

(in the second step we have applied the product rule (A.20)). The above result holds for forms of the same degree p (but p is arbitrary), and using Eq. (A.21) it can be rewritten as ··· ∧ = ∧ = ! μ1 μp . A B B A p 1 Aμ1···μp B (A.25)

Let us ﬁnally observe that—through the repeated application of the duality operation—we can express the divergence of a p form A by computing the exterior derivative of its dual, and by subsequently “dualizing” the obtained result. We obtain, in this way, the (p − 1)-form (d A) whose components exactly correspond to the divergence of the antisymmetric tensor A[μ1···μP ]. Consider, in fact, the exterior derivative of the dual form (A.16): 1 μ ···μ α μ + μ d A = ∂ |g|A 1 p ··· dx ∧ dx p 1 ∧···∧dx D . (A.26) (D − p)! α μ1 μD

Computing the dual we obtain 1 α μ ···μ d A = ∂ |g|A ··· 1 p ··· × (p − 1)!(D − p)! μ1 μp μp+1 μD 1 ··· ×√ μp+1 μD ν1 ∧···∧ νp−1 α ν ···ν − dx dx |g| 1 p 1 − +( − )( − ) = (− )D 1 p 1 D p ∇α ν1 ∧···∧ νp−1 , p 1 Aαν1···νp−1 dx dx (A.27) where 1 [αν1···νp−1] [αν1···νp−1] ∇α A = √ ∂α |g|A (A.28) |g| is the covariant divergence of a completely antisymmetric tensor, computed with a symmetric connection. By exploiting the above result we can also deﬁne a further differential operation acting on the exterior forms, represented by the so-called “co-differential” operator (or exterior co-derivative). The co-differential of a p-form is a mapping δ : Λp → Λp−1, deﬁning the (p − 1)-forma δ A such that

= ∇α μ1 ∧···∧ μp−1 . δ A p Aαμ1···μp−1 dx dx (A.29) Appendix A: The Language of Differential Forms 309

A comparison with Eq. (A.27) shows that exterior derivative d and co-derivative δ are related by δ = (−1)D−1+(p−1)(D−p)d. (A.30)

The notions of duality, exterior derivative and exterior product introduced above will be enough for the pedagogical purpose of this appendix, and will be applied to the geometric description of gravity illustrated in the following sections.

A.2 Basis and Connection One-Forms: Exterior Covariant Derivative

The language of exterior forms is particularly appropriate, in the context of differential geometry, to represent equations projected on the ﬂat tangent manifold. By using a the vierbeins Vμ (see Chap. 12), in fact, we can introduce in the tangent Minkowski space–time a set of basis 1-forms

a = a μ, V Vμ dx (A.31) and represent any given p-form A ∈ Λp on this basis as

= a1 ∧···∧ ap , A A[a1···ap ]V V (A.32)

= μ1 ··· μp where Aa1···ap Aμ1···μp Va1 Vap are the components of the form projected on the local tangent space. In this representation the formalism becomes completely independent of the particular coordinates chosen to parametrize the curved space–time manifold, at least until the equations are not explicitly rewritten in tensor components. On the other hand, in the absence of explicit curved indices (namely, of explicit representations of the diffeomorphism group), the full covariant derivative is reduced to a Lorentz-covariant derivative (see Sect.12.2). By introducing the connection 1-form, ab ab μ ω = ωμ dx , (A.33)

ab where ωμ is the Lorentz connection, we can then deﬁne the exterior, Lorentz- covariant derivative. Given a p-form ψ ∈ Λp, transforming as a representation of the Lorentz group with generators Jab in the local tangent space, the exterior covariant derivative is a mapping D : Λp → Λp+1, deﬁning the (p + 1)-form Dψ such that

i Dψ = dψ − ωab J ψ (A.34) 2 ab (see Eq. (12.22)). a Consider, for instance, a p-form A ∈ Λp vector-valued in the tangent space. The vector generators of the Lorentz group lead to the covariant derivative (12.30). The 310 Appendix A: The Language of Differential Forms corresponding exterior covariant derivative is given by

DAa = D Aa dxμ1 ∧···∧dxμp+1 = dAa + ωa ∧ Ab, (A.35) μ1 μ2···μp+1 b where dAa is the ordinary exterior derivative of Sect. A.1.2. Since the operator D is a 1-form and Aa is a p-form, the derivative DAa is a (p + 1)-form. We should note that DAa is transformed correctly as a vector under local Lorentz transformations, a a b DA → Λ b DA , (A.36) since the connection 1-form is transformed as a a c −1 k a −1 c ω b → Λ c ω k Λ b − (dΛ) c Λ b. (A.37)

This last condition is nothing more than the transformation law deduced in Exercise 12.1, Eq. (12.67), written, however, in the language of differential forms. The above deﬁnition can be easily applied to other representations of the local Lorentz group. If we have, for instance, a tensor-valued p-form of mixed type, a A b ∈ Λp, and we recall the deﬁnition (12.34) of the covariant derivative of a tensor object, we can immediately write down the exterior covariant derivative as

a a a c c a DA b = dA b + ω c ∧ A b − ω b ∧ A c. (A.38)

An so on for other representations of the local Lorentz group. It is important to stress that the differential symbol D operates on the p-form in a way which is independent on its degree p. Hence, the previous rules apply with no changes also to tensor-valued 0-forms. As an typical example we may quote here the metric ηab of the tangent Minkowski space–time: computing its exterior covariant derivative we ﬁnd

ab ab a cb b ac ab ba Dη = dη + ω cη + ω cη = ω + ω ≡ 0, (A.39)

(the result is vanishing thanks to the antisymmetry property of the Lorentz connection, ωab = ω[ab]). Another important tensor-valued 0-form in the tangent space is the fully antisymmetric symbol abcd. By applying the result of Exercise 12.3 we can easily compute the exterior covariant derivative Dabcd and check that, even in this case, this derivative is a vanishing 1-form. The properties of the covariant 1-form D, regarded as a mapping D : Λp → Λp+1, are the same as those of the exterior derivative d. Given, for instance, a p-form A and a q-form B, the covariant derivative of their exterior product obeys the rules

D(A ∧ B) = DA∧ B + (−1)p A ∧ DB, D(B ∧ A) = DB ∧ A + (−1)q B ∧ DA (A.40) Appendix A: The Language of Differential Forms 311

(see Eq. (A.15)). There is, however, an important difference concerning the second covariant derivative, which is in general non-vanishing being controlled by the space–time curvature. In fact, by applying the D operator to the generic (p + 1)-form Dψ of Eq. (A.34), and recalling the result of the commutator (14.69), we obtain

2 = ∧ = α ∧ β ∧ μ1 ∧···∧ μp D ψ D Dψ Dα Dβψμ1···μp dx dx dx dx

i ab α β μ μ =− R (ω)J ψ ··· dx ∧ dx ∧ dx 1 ∧···∧dx p 4 αβ ab μ1 μp i =− Rab J ∧ ψ, (A.41) 2 ab

ab where Rαβ is the Lorentz connection (12.54), and where we have deﬁned the curvature 2-form Rab as

ab 1 ab μ ν R = Rμν dx ∧ dx 2 a cb μ ν = ∂[μων] + ω[μ| cω|ν] dx ∧ dx ab a cb = dω + ω c ∧ ω . (A.42)

a If (in particular) ψ is a vector ﬁeld, ψ → A , and Jab correspond to the vector generators (12.29), then Eq. (A.41) becomes

2 a a b D A = R b ∧ A . (A.43)

This equation exactly reproduces, in the language of exterior forms, the result (12.51) concerning the commutator of two covariant derivatives applied to a Lorentz vector. We can ﬁnally check, as a simple exercise, that Eq. (A.43) can be directly obtained also by computing the exterior covariant derivative of Eq. (A.35). By applying the deﬁnition of D, and using the properties of the differential forms, we obtain, in fact:

2 a = ∧ a = ( a) + a ∧ c D A D DA d DA ω c DA = d2 Aa + dωa ∧ Ab − ωa ∧ dAb + ωa ∧ dAc + ωc ∧ Ab b b c b a a c b = dω b + ω c ∧ ω b ∧ A a b ≡ R b ∧ A , (A.44) where Rab is given by Eq. (A.42). 312 Appendix A: The Language of Differential Forms

A.3 Torsion and Curvature Two-Forms: Structure Equations

We have stressed in Chap. 12 that the Lorentz connection ω represents the non- Abelian “gauge potential” associated to the local Lorentz symmetry, and that the curvature R(ω) represents the corresponding “gauge field” (or Yang–Mills field). In the language of exterior forms the potential is represented by the connection 1-form, ωab, and the gauge field by the curvature 2-form, Rab, both defined in the previous section. In the previous section we have also introduced, besides the connection, another variable which is of fundamental importance for the formulation of a geometric model of the gravitational interactions: the 1-form V a, acting as a basis in the Minkowski tangent space. By recalling the vierbein metricity condition, Eq. (12.40), and considering its antisymmetric part

a ≡ a + a = Γ a ≡ a, D[μVν] ∂[μVν] ω[μ ν] [μν] Qμν (A.45) we can then associate to the 1-form V a the torsion 2-form Ra such that

a = a μ ∧ ν = a μ ∧ ν = a. R Qμν dx dx D[μVν]dx dx DV (A.46)

The Eqs. (A.42), (A.46) which deﬁne the curvature and torsion 2-forms in terms of the connection and basis 1-forms,

a a a a b R = DV = dV + ω b ∧ V , (A.47) ab ab a cb R = dω + ω c ∧ ω , (A.48) are called structure equations, as they control the geometric structure of the given manifold. We may expect that the curvature, being the Yang–Millsfield of the Lorentz group, satisfies a structure equation which is a direct consequence of the Lie algebra for that group, and which reflects the interpretation of the connection ω as the associated gauge potential. If also the torsion equation would be determined by the algebraic structure of some symmetry group, then also the 1-form V a could be interpreted as a gauge potential, and the torsion 2-form as the corresponding gauge field. In the following section it will be shown that the geometric structure described by Eqs. (A.47), (A.48) is indeed a direct consequence of the algebraic structure of the Poincaré group. More precisely, it will be shown that the torsion and the curvature defined by the above equations exactly represent the components of the Yang–Mills field for a non-Abelian gauge theory based on the local Poincaré symmetry. Appendix A: The Language of Differential Forms 313

A.3.1 Gauge Theory for the Poincaré Group

Consider a local symmetry group G, characterized by n generators X A, A = 1, 2,...,n, which satisfy the Lie algebra

C [X A, Xb]=ifAB XC , (A.49)

C C where f AB =−f BA are the structure constant of the given Lie group. In order to formulate the corresponding gauge theory (see Sect.12.1.1), let us A = A μ associate to each generator X A the potential 1-form h hμ dx , with values in the Lie algebra of the group, and deﬁne

≡ A μ. h hμ X Adx (A.50)

Let us then introduce the corresponding exterior covariant derivative,

i D = d − h, (A.51) 2 which we have written in units in which g = 1, where g is the dimensionless coupling constant. A The exterior product of two covariant derivatives deﬁnes the 2-form R = R X A, representing the gauge ﬁeld (or curvature):

i i D2ψ = D ∧ Dψ = d − h ∧ d − h ψ 2 2 i i i 1 =− dhψ + h ∧ dψ − h ∧ dψ − h ∧ hψ 2 2 2 4 i =− Rψ, (A.52) 2 where i R = R A X = dh − h ∧ h. (A.53) A 2

A Using the deﬁnition h = h X A, and the Lie algebra (A.49), we then obtain A A i B C R X A = dh X A − h ∧ h [X B , Xc] 4 1 = dhA + f Ah B ∧ hC X . (A.54) 4 BC A 314 Appendix A: The Language of Differential Forms

This clearly shows that the components of the gauge ﬁeld,

1 R A = dhA + f Ah B ∧ hC , (A.55) 4 BC are directly determined by the algebraic structure of the gauge group. Let us now consider the Poincaré group, namely the group with the maximum number of isometries in the ﬂat tangent space. It is characterized by ten generators,

X A ={Pa, Jab}, (A.56) where Jab =−Jba (in this case the group index A ranges over the 4 components of the translation generators, Pa, and the six components of the generators of Lorentz rotations, Jab). Let us associate to these generators an equal number of gauge potentials, represented by the 1-forms h A ={V a, ωab}, (A.57)

ab ba A where ω =−ω . The corresponding gauge (or Yang–Mills) ﬁeld R = R X A can then be decomposed into translation and Lorentz-rotation components,

A a ab R = R X A = R Pa + R Jab, (A.58) and the explicit form of the curvatures Ra and Rab in terms of the potential V a and ωab is ﬁxed by the Lie algebra of the group, according to Eq. (A.55). The Lie algebra of the Poincaré group is explicitly realized by the following commutation relations of generators:

[Pa, Pb] = 0,

[Pa, Jbc] = i (ηab Pc − ηac Pb) ,

[Jab, Jcd] = i (ηad Jbc − ηac Jbd − ηbd Jac + ηbc Jad) . (A.59)

A comparison with the general relation (A.49) then tell us that the nonvanishing structure constant are

d = d =− d fa,bc 2ηa[bδc] fbc,a ij = i j − i j , fab,cd 2ηd[aδb]δc 2ηc[aδb]δd (A.60) where the indices (or pairs of indices) corresponding to the generators Pa and Jab, respectively, have been separated by a comma. Inserting this result into the curvature (A.55) we then obtain the result that the gauge ﬁeld associated to the translations, Appendix A: The Language of Differential Forms 315

a a 1 a b cd 1 a cd b R = dV + f , V ∧ ω + f , ω ∧ V 4 b cd 4 cd b a 1 a cd b = dV + f , ω ∧ V 2 cd b = a + a cd ∧ b dV ηbdδc ω V a a b a = dV + ω b ∧ V ≡ DV , (A.61) exactly coincides with the torsion 2-form (A.47). Also, the gauge ﬁeld associated to the Lorentz rotations,

ab ab 1 ab ij cd R = dω + f , ω ∧ ω 4 ij cd 1 = dωab + η δaδb − η δaδb ωij ∧ ωcd 2 di j c ci j d 1 = dωab + ω a ∧ ωbd − ω a ∧ ωcb 2 d c ab a cb = dω + ω c ∧ ω , (A.62) exactly coincides with the Lorentz curvature (A.48). A gravitational theory based on a Riemann–Cartan geometric structure, characterized by curvature and torsion, can thus be interpreted as a gauge theory for the Poincaré group. The Einstein theory of general relativity corresponds to the limiting case Ra = DVa = 0 in which the torsion gauge field is vanishing, i.e. the potential associated to the translations is “pure gauge”. It is always possible, in principle, to formulate a model of space–time based on an arbitrary geometrical structure. In practice, however, the type of geometric structure which is more appropriate—and, sometimes, also necessarily required for the physical consistency of the model—turns out to be determined by the given gravitational sources. We have seen, for instance, that a symmetric (and metric compatible) connection may provide a satisfactory description of the gravitational interactions of macroscopic bodies; in the case of the gravitino field, instead, the presence of torsion is needed to guarantee a minimal and consistent (as well as locally supersymmetric) gravitational coupling to the geometry. In Sects.A.4.1 and A.4.2 it will be shown that, in the context of the so-called Einstein–Cartan theory of gravity, the torsion tensor is determined by the sources themselves—just like the curvature tensor—through the field equations of the adopted model of gravity. Hence, in that case, torsion cannot be arbitrarily prescribed any longer.

A.3.2 Bianchi Identities

Let us conclude Sect. A.3 by showing how the Bianchi identities, expressed in the language of exterior forms, can be easily deduced by computing the exterior covariant derivative of the two structure Eqs. (A.47), (A.48). 316 Appendix A: The Language of Differential Forms

The covariant derivative of the torsion gives the ﬁrst Bianchi identity, which reads

a a a b DR = dR + ω b ∧ R a b a b a b a c b = dω b ∧ V − ω b ∧ dV + ω b ∧ dV + ω c ∧ ω b ∧ V a b = R b ∧ V . (A.63)

The covariant derivative of the Lorentz curvature gives the second Bianchi identity, which reads:

ab = ab + a ∧ cb + b ∧ ac DR dR ω c R ω c R = dωa ∧ ωcb − ωa ∧ dωcb + ωa ∧ dωcb + ωc ∧ ωib c c c i b ac a ic +ω c ∧ dω + ω i ∧ ω ≡ 0. (A.64)

Note that the right-hand side of this equation is identically vanishing because, using the properties of the exterior forms introduced in Sects. A.1.1 and A.1.2,wehave

b ac a bc a cb ω c ∧ dω = dω c ∧ ω =−dω c ∧ ω , (A.65) so that the ﬁrst and the second-last term on the right-hand side exactly cancel each other. In addition,

b a ic a i bc a i cb ω c ∧ ω i ∧ ω = ω i ∧ ω c ∧ ω =−ω i ∧ ω c ∧ ω , (A.66) so that also the last and third to last term cancel each other. The Bianchi identities (A.63), (A.64) hold, in general, in a geometric structure satisfying the metricity condition ∇g = 0 (see Sect. 3.5), even in the case of nonvanishing torsion. In the absence of torsion we can easily check that the above identities are reduced to the known identities of the Riemann geometry, already presented in tensor form in Sect. 6.2. In fact, by setting Ra = 0, we ﬁnd that Eq. (A.63) becomes

A b R b ∧ V = 0, (A.67) and thus implies 1 a b μ ν α R[ | V| ]dx ∧ dx ∧ dx = 0, (A.68) 2 μν b α from which a a R[μν α] =−R[μνα] = 0, (A.69) which coincides with the ﬁrst Bianchi identity (6.14). From Eq. (A.64), on the other hand, Appendix A: The Language of Differential Forms 317

1 ab μ α β D[ R ] dx ∧ dx ∧ dx = 0, (A.70) 2 μ αβ from which ab D[μ Rαβ] = 0. (A.71)

In addition (see Chap.12),

ab ab ρ ab ρ ab ∇μ Rαβ = Dμ Rαβ − Γμα Rρβ − Γμβ Rαρ . (A.72)

By computing the totally antisymmetric part in μ, α, β, we ﬁnd that the Γ contribu- ρ tions disappear if the torsion is vanishing (Γ[μα] = 0). In that case Eq. (A.71) can be rewritten in the form ab ∇[μ Rαβ] = 0, (A.73) which coincides with the ﬁrst Bianchi identity (6.15).

A.4 The Palatini Variational Formalism

According to the variational method of Palatini, already introduced in Sect. 12.31, the connection and the vierbeins (or the metric) are to be treated as independent variables. In this section this method will be applied to the variation of the action written in the language of exterior forms: we will use, as fundamental independent variables, the basis 1-forms V a and the connection 1-form ωab. We will also restrict, for simplicity, to a space–time manifold with D = 4 dimensions (our computations, however, can be extended without difﬁculty to the generic D-dimensional case). Let us notice, ﬁrst of all, that the gravitational action (12.56)—which corresponds to the integral of the scalar curvature density over a four-dimensional space–time region—can be written as the integral of a 4-form as follows: 1 S = Rab ∧ (V ∧ V ) . (A.74) g 2χ a b

Using the deﬁnition of Lorentz curvature, Eq. (A.42), the deﬁnition of dual, Eq. (A.16), and the relation (A.23) we have, in fact:

1 1 Rab ∧ (V ∧ V ) = R ab V αV βη dxμ ∧ dxν ∧ dxρ ∧ dxσ a b 2 μν 2 a b αβρσ 1 √ = ab α β μνρσ 4 − Rμν Va Vb ηαβρση d x g 4 1 √ =− ab α β μ ν − ν μ 4 − Rμν Va Vb δαδβ δαδβ d x g 2 √ =−Rd4x −g (A.75) 318 Appendix A: The Language of Differential Forms

(in the second-last step we have used the product rule (A.20)inD = 4). The scalar curvature appearing here is defined as the following contraction of the Lorentz connection: = ab( ) μ ν , R Rμν ω Va Vb (A.76) in agreement with Eq. (12.55). The total action (for gravity plus matter sources) can then be written in the form 1 S = Rab ∧ (V ∧ V ) + S (ψ, V, ω), (A.77) g 2χ a b m where χ = 8πG/c4, ψ is the field representing the sources, and a possible appropriate boundary term is to be understood. In the following section this action will be varied with respect to V a and ωab, in order to obtain the field equations controlling the corresponding gravitational dynamics.

A.4.1 General Relativity and Einstein–Cartan Equations

In order to vary the action (A.77) with respect to V let us explicitly rewrite the dual operation referred to the basis 1-form of the local tangent space, according to Eq. (A.32). We obtain 1 (V ∧ V ) = V c ∧ V d . (A.78) a b 2 abcd The variation of the gravitational part of the action then gives 1 ab c d c d δV Sg = R ∧ δV ∧ V + V ∧ δV abcd 4χ 1 = Rab ∧ V c ∧ δV d , (A.79) 2χ abcd where we have used the anticommutation property of the exterior product of two 1-forms, δV c ∧ V d =−V d ∧ δV c, and the antisymmetry of the tensor in c and d. We should now consider the additional contribution arising from the variation of the matter action, which we can write, in general, as d δV Sm = θd ∧ δV . (A.80)

Here θd is a 3-form associated to the canonical energy-momentum density,

1 θ = θ i V a ∧ V b ∧ V c, (A.81) d 3! d iabc Appendix A: The Language of Differential Forms 319 whose explicit expression depends on the type of source we are including into our model (a few examples will be given below). By adding the two contributions (A.79), (A.80) we then obtain the ﬁeld equations

1 Rab ∧ V c =−χθ , (A.82) 2 abcd d reproducing the Einstein gravitational equations as an equality between 3-forms, vector-valued in the tangent Minkowski manifold. In order to switch to the standard tensor language let us extract the components of the forms using the deﬁnitions (A.42), (A.81), and multiply by the totally antisymmetric tensor μναβ. The left-hand side of Eq. (A.82) then gives

1 1 R abV a μναβ = R β − V β R, (A.83) 4 μν α abcd d 2 d where we have used the result of Exercise 12.4 (Eq. (12.75)). The right-hand side gives χ − θ i abcβ = χθ β. (A.84) 3! d iabc d The ﬁeld equation (A.82) thus provides the tensor equality

β β Gd = χθd , (A.85)

β where Gd is the Einstein tensor (A.83). The above equations are not completely determined, however, until we have not specified the connection to be used for the computation of the curvature, of the Einstein tensor, and of the energy-momentum tensor of the sources. To this aim we must consider the second field equation, obtained by varying the action (A.77) with respect to ω. We start with the variation of the curvature Rab(ω). From the definition (A.42) we have

ab ab a cb a cb δω R = dδω + δω c ∧ ω + ω c ∧ δω ab a cb b ac = dδω + ω c ∧ δω + ω c ∧ δω ≡ Dδωab. (A.86)

Let us now consider the gravitational action. Using the result (A.86), the deﬁnition of torsion (A.47), and the property Dabcd = 0 (see Sect. A.2), we obtain 1 ab c d δω Sg = Dδω ∧ V ∧ V abcd 4χ 1 = D δωab ∧ V c ∧ V d + 2δωab ∧ Rc ∧ V d (A.87) 4χ abcd 320 Appendix A: The Language of Differential Forms

(for the sign of the last term we have used Eq. (A.40)). The ﬁrst term of the above integral corresponds to a total divergence and can be expressed, thanks to the Gauss theorem, in the form of a boundary contribution. In fact, it is the four-volume integral of the exterior covariant derivative of a scalar- valued 3-form, i.e. it is an integral of the type μ ν α β DA = dA= ∂[μ Aναβ] dx ∧ dx ∧ dx ∧ dx Ω Ω Ω √ √ μναβ 4 μναβ = ∂μ Aναβη −g d x = dSμ −gη Aναβ Ω ∂Ω (A.88)

(we have used Eq. (A.23) and the Gauss theorem). In our case the 3-form A is given by ab c d A = δω ∧ V ∧ V abcd, (A.89) and since A is proportional to δω the above contribution is vanishing, because the variational principle requires δω = 0 on the boundary ∂Ω. We are thus left only with the second term of Eq. (A.87), which gives 1 δ S = δωab ∧ Rc ∧ V d . (A.90) ω g 2χ abcd

There is, however, a further possible contribution from the matter action Sm , whose variation with respect to ω can be expressed, in general, as ab δω Sm = δω ∧ Sab, (A.91)

where Sab =−Sba is an antisymmetric, tensor-valued 3-form related to the canonical density of intrinsic angular momentum. Its explicit from depends on the considered model of source (see the examples given below). Adding the two contributions (A.90) and (A.91) we ﬁnally obtain the relation

1 Rc ∧ V d =−χS , (A.92) 2 abcd ab which represents the field equation for the connection. Solving for ω, and inserting the result into Eq. (A.82), we have fully specified the geometry of the given model of gravity, and we can solve the equations to determine the corresponding dynamics. The two equations (A.82), (A.92) are also called Einstein–Cartan equations. In the particular case in which there are no contributions to Eq. (A.92)fromthe matter sources—or the contributions Sab are present, but are physically negligible— one obtains that the torsion is zero, and recovers the Einstein field equations of general relativity. In fact, if we rewrite Eq. (A.92) in tensor components, antisymmetrize, and Appendix A: The Language of Differential Forms 321 recall the rule (12.74), we arrive at the condition

1 c d μναβ 1 c μνβ Q[ V ] = Q V = 0, (A.93) 2 μν α abcd 2 μν abc namely at 1 Q c V β + Q c V β + Q c V β − Q c V β − Q c V β − Q c V β 2 ab c bc a ca b ac b ba c cb a = β + β − β = , Qab Qb Va Qa Vb 0 (A.94)

≡ c b where Qb Qbc . Multiplying by Vβ we ﬁnd that the trace must be vanishing, Qa = 0, and Eq. (A.94) reduces to:

c Qab ≡ 0. (A.95)

The condition of vanishing torsion, on the other hand, can also be written as Ra = DVa = 0, namely as

a ≡ a + a = . D[μVν] ∂[μVν] ω[μ ν] 0 (A.96)

This equation, solved for ω, leads to the Levi-Civita connection of general relativity (see Eqs. (12.41)–(12.48) with Q = 0). With such a connection Eq. (A.85) exactly reduces to the Einstein ﬁeld equations: to the left we recover the symmetric Einstein tensor, obtained from the usual Riemann tensor, and to the right we recover the symmetric (dynamical) energy-momentum tensor. For a torsionless geometry, and in the language of the exterior forms, the covariant conservation law of the energy-momentum tensor can be obtained by computing the exterior covariant derivative of Eq. (A.82). In fact, the derivative of the left-hand side is identically vanishing, 1 DRab ∧ V c = 0, (A.97) 2 abcd thanks to the second Bianchi identity (A.64). This immediately implies

Dθa = 0, (A.98) which reproduces to the conservation equation (7.35), when translated into the tensor language. For an explicit check of this result let us notice, ﬁrst of all, that Eq. (A.97) corresponds to the so-called “contracted Bianchi identity”, written in the language of exterior forms. Switching to the tensor formalism—i.e. considering the components of the forms, and antisymmetrizing— we obtain, in fact:

1 ∇ R abV c μναβ = 0. (A.99) 4 μ αβ ν abcd 322 Appendix A: The Language of Differential Forms

We have replaced Dμ with ∇μ because the difference between the two objects is represented by the contribution of the Christoffel symbols, which disappears after antisymmetrization in μ, α, β (see Eq. (A.72)). By using the result (12.75)forthe product of the antisymmetric tensors the above equation then reduces to:

1 ∇ R μ − V μ R = 0. (A.100) μ c 2 c

∇ = c By exploiting the metricity condition V 0 we can ﬁnally multiply by Vν , and rewrite our result as μ ∇μGν = 0, (A.101) which coincides indeed with the contracted Bianchi identity (6.26). Let us now consider the components of Eq. (A.98), use the deﬁnition (A.81), and antisymmetrize. By repeating the above procedure, and recalling that ∇μηρναβ = 0 (see Exercise 3.7), we get

1 1 ∇ θ ρη ημναβ =− ∇ θ μ = 0. (A.102) 6 μ a ρναβ 6 μ a a ∇ = Multiplying by Vν , and using V 0, we ﬁnally arrive at the condition

μ ∇μθν = 0, (A.103) which reproduces the covariant conservation of the energy-momentum tensor, in agreement with previous results (see Eq. (7.35)). Example: Free Scalar Field It is probably instructive to conclude our discussion of this generalized gravitational formalism with a simple example of matter ﬁeld which is not source of torsion: a massless scalar ﬁeld φ. Its action can be written (in units = c = 1): 1 S =− dφ ∧ dφ. (A.104) m 2

In fact, by applying the result (A.24) to the 1-form dφ, we obtain √ 4 μ dφ ∧ dφ =−d x −g∂μφ∂ φ, (A.105) so that the above action exactly coincides with the canonical action (7.37)ofafree scalar ﬁeld (with V (φ) = 0). The variation with respect to ω—which does not appear in Sm —is trivially zero: we thus recover the torsionless condition (A.95), and the connection reduces to the standard form used in the context of general relativity. The variation of the action (A.104) with respect to V represents a useful exercise for the calculus of exterior forms. Let us ﬁrst notice that δV dφ = 0, and that a nonzero Appendix A: The Language of Differential Forms 323

variational contribution is provided by the dual term only, δV ( dφ). By referring the dual to the tangent space basis we have, in particular:

1 dφ = V μ∂ φi V a ∧ V b ∧ V c. (A.106) 3! i μ abc Therefore: 1 δ dφ = ∂i φ δV a ∧ V b ∧ V c V 2 iabc 1 − δV j ∂ φV μi V a ∧ V b ∧ V c, (A.107) 3! μ j i abc where we have used the identity μ j =− j μ, δVi Vμ δVμ Vi (A.108)

j μ = j following from the relation Vμ Vi δi . Using again the deﬁnition of dual, we can rewrite Eq. (A.107) in compact form as follows: i a j δV dφ = ∂ φδV ∧ (Vi ∧ Va) − ∂ j φ δV . (A.109)

The variation of the scalar-ﬁeld action thus takes the form

1 a b a δV Sm ==− ∂ φ dφ ∧ δV ∧ (Va ∧ Vb) − ∂aφ dφ ∧ δV 2 1 =− ∂aφ dφ ∧ (V ∧ V ) ∧ δV b + ∂ φ dφ ∧ δV a 2 a b a (A.110)

(in the second step we have used, for the second term, the property A ∧ B = B ∧ A which holds if the two forms A and B are of the same degree, see Eq. (A.25)). The ﬁeld Eq. (A.82), in our case, becomes

1 χ Rab ∧ V c = ∂aφ dφ ∧ (V ∧ V ) + ∂ φ dφ . (A.111) 2 abcd 2 a d d The left-hand side, computed with a vanishing torsion, coincides with the usual symmetric Einstein tensor. Let us check that the right-hand side corresponds to the usual (symmetric) energy-momentum tensor of a massless scalar ﬁeld. By considering the components of the 3-form present on the right-hand side, and antisymmetrizing, we obtain 324 Appendix A: The Language of Differential Forms 1 1 1 ∂aφ∂ φ V i V j μναβ + ∂ φ∂ρφη ημναβ 2 2 μ adij ν α 6 d ρμνα 1 1 =− ∂aφ∂ φ V μV β − V β V μ + ∂ φ∂βφ 2 μ a d a d 2 d 1 = ∂ φ∂βφ − V β ∂ φ∂μφ = θ β, (A.112) d 2 d μ d which coincides indeed with the canonical tensor of Eq. (7.40) (for the free case with V = 0).

A.4.2 Spinning Sources and Riemann–Cartan Geometry

As a simple example of space–time geometry with nonvanishing torsion we will consider here a model in which the gravitational source is a massless Dirac ﬁeld, represented as a 0-form ψ, spinor-valued in the Minkowski tangent space. The matter action can then be written (in units = c = 1) as Sm =−i ψγ ∧ Dψ, (A.113)

a where γ = γa V is a 1-form, and Dψ is the 3-form obtained by dualizing the exterior covariant derivative of a spinor, defined according to Eq. (13.23). Using the result (A.24) we have, in fact, √ μ 4 − iψγ ∧ Dψ = iψγ Dμψ d x −g, (A.114) which leads to the covariant Dirac action (13.24) (with m = 0). By varying the spinor action with respect to V , and applying the definition (A.80), we obtain the 3-form θa = iψγa Dψ, (A.115) representing the gravitational source of the Einstein–Cartan gravitational equation (A.82). Note that this object is different from the dynamical energy-momentum tensor of the Dirac field computed in Exercise 13.3 (which is symmetric and acts as a source of the gravitational Einstein equations). In fact, by inserting θa in Eq. (A.82), extracting the components, antisymmetrizing, and finally projecting back to the curved space–time, we arrive at the following tensor equation:

Gαβ = iχψγα Dβψ, (A.116) with a right-hand side which is explicitly not symmetric in α and β. Such an asymmetry, which would be inconsistent in the context of the Riemann geometry, is appropriate instead to a Riemann–Cartan geometry with torsion. In that Appendix A: The Language of Differential Forms 325 case, in fact, the left-hand side of Eq. (A.116) is to be computed with a non-symmetric afﬁne connection (see Sect. 3.5), and turns out to be non-symmetric, unlike the usual Einstein tensor. In order to compute the torsion produced by the Dirac source, the action (A.113) has to be varied with respect to the connection ω. We recall, to this aim, that

1 ab Dψ = dψ + ω γ[ γ ]ψ (A.117) 4 a b (see Eq. (13.23)). We thus obtain i ab δω Sm =− ψγ ∧ δω γ[aγb] ψ 4 i ab =− δω ∧ ψ γγ[ γ ]ψ, (A.118) 4 a b

c where γ ≡ γc V , and where we have used the property γ ∧ δω = δω ∧ γ.By applying the deﬁnition (A.91) we ﬁnd that the Einstein–Cartan equation (A.92)for the connection becomes

1 c d i R ∧ V = χψ γγ[ γ ]ψ. (A.119) 2 abcd 4 a b The spinor current plays the role of source, and the torsion is no longer vanishing. To obtain the explicit expression of the torsion tensor we must rewrite the above equation in components, and antisymmetrize. For the left-hand side we already know the result, reported in Eq. (A.94). By repeating the same procedure for the right-hand side we obtain

i c 1 ρ μναβ i β ψγ γ[ γ ]ψ V η = ψγ γ[ γ ]ψ, (A.120) 4 a b 6 c ρμνα 4 a b and Eq. (A.119) becomes

β β β i β Q + Q V − Q V = χψγ γ[ γ ]ψ. (A.121) ab b a a b 4 a b

b The multiplication by Vβ now gives the torsion trace as

3 Q = i χψγ ψ, (A.122) a 8 a so that, moving all trace terms to the right-hand side:

i Q = χψ γ γ[ γ ] − 3η [ γ ] ψ. (A.123) abc 4 c a b c a b 326 Appendix A: The Language of Differential Forms

By recalling the relations (13.34), (13.36) among the γ matrices we can ﬁnally rewrite the torsion tensor by explicitly separating the vector and axial-vector contributions of the Dirac current: χ 5 d Q = ψγ γ ψ + iψγ[ η ] ψ . (A.124) abc 4 abcd a b c Once the torsion is determined, the corresponding Lorentz connection is obtained by solving the metricity conditions for the vierbeins, and is given (according to Eqs. (12.46)–(12.48)) by

ωcab = γcab + Kcab ≡ γcab − (Qcab − Qabc + Qbca) , (A.125) where γ is the Levi-Civita connection. With Q = 0, the Lorentz curvature determined by ω contain the contributions of the contortion K and defines a non-symmetric Einstein tensor, thus modifying the field equations with respect to the equations of general relativity. Another interesting consequence of the presence of torsion is the modification of the covariant form of the Dirac equation. In fact, the equation of motion following from the action (A.113), iγ ∧ Dψ = 0, can still be expressed in the standard form μ iγ Dμψ = 0, but the covariant derivative (A.117) is now referred to the generalized connection (A.125). The presence of torsion then introduces into the spinor equation non-linear “contact” corrections, also called “Heisenberg terms”. They can be easily determined by inserting into the Lorentz connection the explicit torsion tensor (A.124), and separating the torsion contributions by defining

1 ab 1 ab D = d + γ γ[ γ ] + K γ[ γ ] 4 a b 4 a b 1 ab = D + K γ[ γ ], (A.126) 4 a b

ab = ab μ where K Kμ dx is the one-form associated to the contorsion tensor of Eq. (12.48), and D is the spinor covariant derivative of general relativity (see Chap. 13), computed without torsion. We then obtain

μ μ i μ [a b] iγ Dμψ = iγ Dμψ + γ Kμabγ γ ψ 4 χ = iγμ D ψ + γcγ[aγb]ψ ψ (γ η − γ η ) ψ − i ψγ5γd ψ . μ 16 b ca a cb abcd (A.127)

Non-linear terms of this type are required, for instance, in the covariant equation of the Rarita–Schwinger ﬁeld to restore local supersymmetry, as already discussed in Sect. 14.3. Appendix A: The Language of Differential Forms 327

A.4.3 Example: A Simple Model of Supergravity

As a last application of the exterior calculus we will present here the action, and derive the corresponding ﬁeld equations, for the N = 1 supergravity model of Sect.14.3. μ Representing the gravitino ﬁeld as the 1-form ψ = ψμdx , spinor-valued in the tangent space, we can express the action for the Lagrangian (14.53) as follows, 1 i S = Rab ∧ V c ∧ V d + ψ ∧ γ γ ∧ Dψ, (A.128) 4χ abcd 2 5

a where γ = γa V , and where the operator D denotes the exterior, Lorentz-covariant derivative of Eq. (A.117). The reformulation of the gravitational part of this action into the usual tensor language has already be presented in Eq. (A.75). For the spinor part of the action we can use Eq. (A.23), which leads to the more explicit form

i i ψ γ γ D ψ dxμ ∧ dxν ∧ dxα ∧ dxβ = ψ γ γ D ψ μναβd4x, (A.129) 2 μ 5 ν α β 2 μ 5 ν α β in full agreement with the Lagrangian (14.53). The ﬁeld equations are obtained by varying with respect to V , ω and ψ. Starting with V we have i a δV S3/2 = ψ ∧ γ5γaδV ∧ Dψ 2 i = ψ ∧ γ γ Dψ ∧ δV a. (A.130) 2 5 a

By adding the variation of the gravitational part of the action, Eq. (A.79), we immediately obtain: 1 i Rab ∧ V c =− χψ ∧ γ γ Dψ. (A.131) 2 abcd 2 5 d Let us now translate this equation in the more convenient tensor language. The tensor version of the left-hand side has been reported in Eq. (A.83). By extracting the tensor components of the right-hand side we are led to the equation

i G β =− χψ γ γ D ψ μναβ d 2 μ 5 d ν α i = χψ γ γ D ψ μνβα 2 μ 5 d ν α β ≡ χθd , (A.132)

β where θd is the canonical tensor (14.65). Hence, we exactly recover the result previously given in Eq. (14.64). 328 Appendix A: The Language of Differential Forms

Let us now vary with respect to ω. By recalling the deﬁnition (A.117) of the spinor covariant derivative, and varying the gravitino action, we have: i ab δ S / = δω ∧ ψ ∧ γ γγ[ γ ] ∧ ψ. (A.133) ω 3 2 8 5 a b

By adding the variation of the gravitational action, Eq. (A.90), we arrive at the following ﬁeld equation for the connection:

1 c d i R ∧ V =− χψ ∧ γ γγ[ γ ] ∧ ψ. (A.134) 2 abcd 8 5 a b = c ν = ν Let us notice that γ γc Vν dx γν dx , so that we can exploit the relation (14.58) to express the product of Dirac matrices γ5γν γ[aγb]. By inserting the result into the above equation, and dropping terms which are vanishing for the anticommutation properties of the Majorana spinors (see Sect. 14.3.1), we are led to:

1 1 Rc ∧ V d =− χψ ∧ V cγd ∧ ψ 2 abcd 8 abcd 1 =− χψγc ∧ ψ ∧ V d . (A.135) 8 abcd Note that in the second line we have used the anticommutation property of the exterior product of two 1-forms, V c ∧ ψ =−ψ ∧ V c, and we have exchanged the d names of the indices c and d. From the above equation, factorizing V abcd, we can immediately deduce that the torsion 2-form is given by

1 Rc =− χψγc ∧ ψ (A.136) 4 (which corresponds, in the language of differential forms, to the tensor result (14.60)). Let us ﬁnally vary the action with respect to ψ. The result is the gravitino equation,

i γ γ ∧ Dψ = 0. (A.137) 2 5 By extracting the components, and antisymmetrizing, we arrive at the result

i γ γ D ψ μναβ = 0, (A.138) 2 5 ν α β which exactly reproduces the tensor equation (14.66). Appendix B Higher-Dimensional Gravity

As already shown in various parts of this book (Chap. 11, Appendix A), there are no difficulties in writing the gravitational equations in space–time manifolds with a total number of dimensions D > 4. The problem, if any, is to understand the possible relevance/pertinence of such models for a geometric description of gravity at the macroscopic level, and find the possible corrections to the four-dimensional gravitational interactions induced by the presence of the additional spatial dimensions (that we shall call, following the standard terminology, “extra” dimensions). Let us ask ourselves, first of all, why we should consider higher-dimensional models of gravity. The answer is simple: a higher-dimensional space–time is required by unified models of all fundamental interactions, such as supergravity and superstring models (see e.g. the books [5, 10, 18] of the bibliography). Ten-dimensional superstring theory, in particular, is at present the only unified theory able to include, besides gravity and the other interactions described by the fundamental (bosonic) gauge fields, also all the elementary (fermionic) components of matter. Such a theory also provides a model of quantum gravity valid (in principle) at all energy scales. If we accept the idea that a complete and theoretically consistent model of gravity needs to be formulated in a higher-dimensional space–time manifold, the question then becomes: how can we deduce, from such a model, the equations governing the gravitational interactions in D = 4? The answer is provided by the so-called mechanism of “dimensional reduction”, which basically tells us how our four-dimensional Universe is embedded into the higher-dimensional space. In this appendix we will briefly discuss two possibilities: the “old” Kaluza–Klein scenario, where the extra dimensions are compactified on a very small length scale, and the new “brane-world” scenario, where all fundamental interactions (but gravity) are confined on a four-dimensional “slice” of a higher- dimensional “bulk” manifold. As in the case of Appendix A, it should be clearly stressed that the aim of this appendix is that of providing only a first, pedagogical introduction to the above- mentioned problems. The interested reader is referred to other books for an exhaustive presentation of this subject, and for the discussion of its many aspects and problems (see e.g. the book [3] of the bibliography for the Kaluza–Klein scenario). © Springer International Publishing AG 2017 329 M. Gasperini, Theory of Gravitational Interactions, UNITEXT for Physics, DOI 10.1007/978-3-319-49682-5 330 Appendix B: Higher-Dimensional Gravity

Let us ﬁnally stress that, throughout this appendix, the capital Latin indices will be referred to tensor representations of a D-dimensional space–time manifold, and will thus assume the values A, B, C,...= 0, 1, 2, 3,...,D − 1.

B.1 Kaluza–Klein Gravity

The simplest example of higher-dimensional model gravity was provided almost one century ago by Kaluza and Klein,1 and was inspired by the wish of providing a geometric description not only of gravity but also of the other fundamental interaction known at that time, namely the electromagnetic interaction. The basic idea was that of interpreting the electromagnetic potential Aμ as a component of the metric of a five-dimensional space–time M5, and the U(1) gauge symmetry as an isometry of the five-dimensional geometry. This idea, as we shall see, can be extended (in principle) also to non-Abelian gauge fields, in the context of higher-dimensional manifolds with the appropriate geometric (and isometric) structure. But let us start with the simple case of pure D = 5 gravity, described by the action M3 S =− 5 dx5 |γ | R . (B.1) 2 5 5

Here γ5 is the determinant of the ﬁve-dimensional metric γAB, R5 is the Riemann 3 ≡ ( )−1 scalar curvature computed from γAB, and M5 8πG5 is the mass scale determining the effective gravitational coupling constant G5 in the ﬁve-dimensional space–time M5. Note that we are working in units = c = 1 and that, in these units, the gravitational coupling constant appearing in the action written for a D- D−2 2−D dimensional space–time has dimensions [G D]=L = M .InD = 4 the coupling is controlled by the usual Newton constant G, related to the Planck-length (or = 2 = −2 mass) scale by 8πG λP MP . A D-dimensional (symmetric) metric tensor has in general D(D + 1)/2 independent components, which become 15 in D = 5. It is thus always possible to parametrize γAB in terms of a 4-dimensional symmetric tensor gμν (with 10 independent components), a 4-dimensional vector Aμ (with four independent components) and a scalar φ (with one independent component). Including (for later convenience) a possible conformal rescaling of γAB we can thus set:

1T. Kaluza, Sitzungsber. Preuss. Akad. Wiss. Berlin 1921, 966 (1921); O. Klein, Z. Phys. 37, 895 (1926). Appendix B: Higher-Dimensional Gravity 331

= w( ) , γAB φ γ AB (B.2) where w(φ) is a positive (but arbitrary) scalar function of φ, and where

= − , = = , =− . γμν gμν φAμ Aν γμ4 γ4μ φAμ γ44 φ (B.3)

Let us recall that Greek indices run from 0 to 3, capital Latin indices from 0 to 4, and that the ﬁfth dimension corresponds to the index 4. Also, we are assuming that φ is positive. The inverse metric is then given by γ AB = w−1γ AB, where

μν μν μ4 4μ μ μα 44 −1 αβ γ = g , γ = γ = A = g Aα, γ =−φ + g Aα Aβ, (B.4)

μα = μ CB = B and where g gνα δν . It can be easily checked that the property γAC γ δA is identically satisfied. The parametrization of γAB in terms of the multiplet of dimensionless fields {gμν , Aμ, φ} is fully general, up to now, but useful in our context to discuss the transformation properties of the metric under particular coordinate transformations. In fact, let us consider the chart z A ={xμ, y} (we have called y the fifth coordinate z4), and the transformation to the new chart zA ={xμ, y} where

xμ = xμ, y = y + f (x). (B.5)

By applying the standard transformations rule of the metric tensor, Eq. (2.18), we ( ) readily obtain γAB z and ﬁnd that its new components are given by ( , ) = ( , ), ( , ) = ( , ) + ( ), gμν x y gμν x y Aμ x y Aμ x y ∂μ f x φ(x, y) = φ(x, y). (B.6)

The result for Aμ suggests that a geometric model which is isometric with respect to the transformation (B.5) should include an Abelian gauge symmetry, associated to the vector component Aμ of the metric tensor. That this is indeed the case is confirmed by the so-called procedure of “dimensional reduction”, needed to “extract” the geometry of our 4-dimensional space–time M4 from the given original M5 manifold. The Kaluza–Klein approach to this process is based on the topological assumption that M5 has the product structure M5 = M4 S1, where S1 is a compact one-dimensional space, topologically equivalent to a circle of radius Lc, and then parametrized by a coordinate y such that 0 ≤ y ≤ 2πLc. In that case any field defined on M5 (including gμν , Aμ and φ) is periodic in y, and can be expanded in Fourier series with respect to y. For the metric components, in particular, we have 332 Appendix B: Higher-Dimensional Gravity

∞ ( ) / ( ) = n ( ) iny Lc , gμν z gμν x e n=−∞ ∞ ( ) / ( ) = n ( ) iny Lc , Aμ z Aμ x e n=−∞ ∞ ( ) / φ(z) = φ n (x)einy Lc , (B.7) n=−∞

( (n))∗ = (−n) where all Fourier components satisfy the reality condition, i.e. gμν gμν , and so on. Once the y-dependence is fixed (according to the above Fourier expansion), dimensional reduction is achieved by inserting the metric components (B.2)–(B.4) into the action (B.1) and integrating over the fifth coordinate y. The result is an effective four-dimensional action2 involving the (complicated) mutual interactions (n) (n) of the infinite “towers” of four-dimensional fields (the Fourier modes gμν , Aμ , φ(n)) which—at least in the flat-space geometry and in the perturbative regime—are characterized by a mass which is growing with the Fourier index n, i.e. mn = n/Lc. This (low-energy) value of the mass can be easily obtained by expanding the full action (B.1) around the trivial Minkowski background, and putting γAB = ηAB + h AB +···. One then finds that the fluctuations h AB satisfy a linearized equation which reduces, in vacuum, to the five-dimensional d’Alembert equation, 2 −∇2 − 2 = , ∂0 ∂y h AB 0 (B.8) and that their Fourier components, taking into account the periodicity condition μ (B.7), are of the form h ∼ exp(−ikμx + iny/Lc). Hence they satisfy the dispersion relation n2 − ω2 + k2 + = 0, (B.9) 2 Lc

2 = 2/ 2 typical of massive modes with m n Lc . If we assume that the compactification scale Lc is very small (after all, as we shall see in a moment, the size of the fifth dimension has to be small enough to explain why it cannot be experimentally resolved at the present available energies), it follows that the massive modes with n = 0 must be very heavy. In the low-energy regime we can thus limit ourselves (at least in first approximation) to the zero modes only, assuming that all fields appearing in the Kaluza–Klein model are independent of the fifth coordinate y. In such a simplified case we can check explicitly that the given

2Such an action is also characterized by an infinite number of four-dimensional symmetries, as we may discover by Fourier expanding the parameters ξ A of the infinitesimal coordinate transformation z A → z A + ξ A(xμ, y). In fact, in order to respect the topological structure that we have assumed for M5, we have to restrict to coordinate transformations periodic in y, characterized by an infinitesimal A = A ( ) iny/Lc parameter which can be expanded as ξ n ξ(n) x e (see L. Dolan and M.J. Duff, Phys. Rev. Lett. 52, 14 (1984). Appendix B: Higher-Dimensional Gravity 333 model describes, in a four-dimensional space–time, the interactions of a gravitational (0) (0) (0) field gμν , a massless scalar φ and an Abelian gauge vector Aμ . In fact, let us compute explicitly the action (B.1) with the metric (B.2)–(B.4), assuming that g, A, φ depend only on x (and omitting the zero-mode index (0),for simplicity). For the metric determinant we immediately obtain √ 1/2 5/2 |γ5|= −gφ w (φ), (B.10) where g = det gμν . For the computation of the scalar curvature, and for a better illustration of the role played by the conformal factor w(φ), it is convenient to express the scalar curvature R5(γ), appearing in the action, in terms of the scalar curvature ( ) R5 γ computed for the conformally related metric γ AB (given in Eq. (B.2)). By recalling the general result for the conformal rescaling of the scalar curvature (see = w = e.g. the book [9] of the bibliography) we obtain, for γAB γ AB and in D 5, −1 A A R5(γ) = w R5 (γ) − 4∇ A∇ ln w − 3 ∇ A ln w ∇ ln w (B.11)

(the symbol ∇ A denotes the covariant derivative computed with the metric γ). The ﬁve-dimensional action (B.1) then becomes 3 2πLc M5 4 S =− dy d x |γ5|R5(γ) 2 0 M3 2πLc √ =− 5 dy d4x −gφ1/2w3/2(φ) R (γ) − 4∇ ∂ A ln w 2 5 A 0 A −3 (∂A ln w) ∂ ln w , (B.12)

∇ w w w where√ we√ have replaced A ln with ∂A ln , since is a scalar. By recalling that | |= − 1/2 γ5 gφ we have, also, √ A 1 A ∇ A ∂ ln w = √ √ ∂A −g φ∂ ln w −g φ 1 √ 1 = √ ∂ −g∂μ ln w + (∂μ ln w) ∂ ln φ , (B.13) −g μ 2 μ where we have replaced the index A with the index μ everywhere, since we are considering the limit in which all fields are independent of the fifth coordinate. It is now evident, from the action (B.12), that by choosing w(φ) = φ−1/3, i.e. ln w =−(1/3) ln φ, we can eliminate the non-minimal coupling to φ present in the four-dimensional part of the integration√ measure. With such a choice the measure reduces to the canonical form d4x −g, hence the first term in the second line of Eq. (B.13) contributes to the action as a total divergence (and can be dropped), while the second term becomes quadratic in the first derivatives of ln φ, and 334 Appendix B: Higher-Dimensional Gravity contributes to the kinetic part of the scalar action (together with the last term of Eq. (B.12)). The full action then becomes: 3 2πLc √ M5 4 1 μ S =− dy d x −g R5 (γ) + ∂μ ln φ (∂ ln φ) . (B.14) 2 0 3 Let us now evaluate the contribution directly arising from the scalar curvature of the five-dimensional metric γ AB. An explicit computation leads to √ √ 1 1 −g R (γ) = −g R(g) + φF F μν − (∂ ln φ)(∂μ ln φ) , (B.15) 5 4 μν 2 μ modulo a total divergence. Here R(g) is the scalar curvature associated to the = − four-dimensional metric gμν , and Fμν ∂μ Aν ∂ν Aμ.√ By inserting this result into Eq. (B.14), integrating over y, and defining σ =−(1/ 3) ln φ, we finally end up with the action √ M2 √ e− 3σ 1 S =− P d4x −g R + F F μν − ∂ σ∂μσ , (B.16) 2 4 μν 2 μ where we have identified the effective four-dimensional gravitational coupling with the usual Newton constant by setting:

2 ≡ ( )−1 = 3. MP 8πG 2πLc M5 (B.17)

Note that the ratio between the four- and five-dimensional coupling constants turns out to be controlled by the compactification scale Lc. In particular, if the coupling strength of D = 5 gravity is the same as in D = 4, i.e. M5 ∼ MP, then the size of ∼ −1 ∼ the compact five dimension must be in the Planck-length range, Lc MP λP. The effective action (B.16), obtained from the original Kaluza–Klein model through a dimensional reduction procedure, shows that the zero-mode content of a five-dimensional theory of pure gravity with one spatial dimension compactified on a circle can reproduce a canonical model of four-dimensional gravity, coupled to an Abelian gauge vector Aμ and to a scalar “dilaton” field σ. It should be noted, in this context, that we have the interesting appearance of a “non-minimal” scalar-vector coupling in front of the standard Maxwell Lagrangian.√ The vector field, however, has to be appropriately rescaled (Aμ → Aμ MP/ 2) in order to match the usual canonical normalization.

B.1.1 Dimensional Reduction from D = 4 + n Dimensions

The geometric description of gauge ﬁelds based on the Kaluza–Klein model of dimensional reduction can be extended to the case of non-Abelian symmetries, provided we consider space–time manifolds with a higher number of compact dimensions. Appendix B: Higher-Dimensional Gravity 335

The gauge group of the dimensionally reduced model corresponds, in that case, to the non-Abelian isometry group of the compact spatial dimensions. M = + Let us consider a space–time manifold D with D 4 n dimensions and with a topological structure Md = M4 KD−4, where KD−4 is a compact n- dimensional space characterized by an isometry group G generated by a set of { m } , = , ,..., N Killing vectors K(i) , where i j 1 2 N. Conventions: here and in the following sections we will split the D-dimensional coordinates as z A = (xμ, ym ), μ m where x , with μ, ν = 0, 1, 2, 3, will denote coordinates on M4, while y , with m, n = 4, 5,...D − 1, will denote coordinates on KD−4. In this section, the indices i, j, k, will run over the N generators of the isometry group. Suppose that the isometry group is non-Abelian, and that the generators satisfy a closed (non-trivial) Lie algebra of commutation relations. Let us onsider the differential representation (on the compact space KD−4) of the generators associated ≡ m to each Killing vector, Ki Ki ∂m (from now on we will omit, for simplicity, the round brackets on the group indices), and compute the commutation brackets , ≡ m n − m n . Ki K j Ki ∂m K j K j ∂m Ki ∂n (B.18)

m n It can be easily shown that, if Ki and K j are Killing vectors, then the right-hand side of the above equation is proportional to a Killing generator, too (recall the Killing properties illustrated in Sect. 3.3 and Exercise 3.4). We can thus write the commutation rules in the standard form

k Ki , K j = fij Kk , i, j, k = 1, 2,...,N, (B.19)

k k where fij =−f ji are the structure constant of the given isometry group G. Let us now generalize the previous parametrization of the higher-dimensional metric tensor γAB by introducing, in D dimensions, a symmetric 4 × 4 tensor gμν ,a ( − ) × ( − ) − m symmetric D 4 D 4 tensor φmn, and D 4 four-dimensional vectors Bμ (one vector Bμ for each of the D − 4 extra spatial dimensions). Note that the total number of components is again D(D + 1)/2, as appropriate to the metric γAB.More precisely, we shall use the following general ansatz:

− m n p = w gμν φmn Bμ Bν φmpBμ , γAB p − (B.20) φnp Bν φmn where we have included, again, the so-called “warp” factor w(φ) (a function of φ ≡ det φmn), possibly useful to restore the canonical normalization of the kinetic terms in the dimensionally reduced action. By computing γ = det γAB we obtain |γ| = wD/2 |φ|1/2 |g|, (B.21) 336 Appendix B: Higher-Dimensional Gravity and the inverse metric is given by:

μν m μα AB = w−1 g Bα g , γ n να − mn + αβ m n (B.22) Bαg φ g Bα Bβ

μα = μ mp = m where g gνα δν and φ φpn δn . We are now in the position of exploiting the isometries of the factorized geometry and showing that, after an appropriate dimensional reduction, to each one of the N isometries of the compact manifold KD−4 we can associate a vector transforming as a non-Abelian gauge potential of the reduced four-dimensional theory. Following (and extending to higher D) the Kaluza–Klein mechanism of the previous section, we shall implement the dimensional reduction by expanding the action around a sort of “ground state” configuration of the higher-dimensional geometry, representing an effective low-energy limit of our D-dimensional model of gravity. In particular, we will consider a configuration in which the metric gμν depends only on x; the tensor φmn is constant in four-dimensional space–time (but may depend on y); finally, the D − 4 four-vectors Bμ are parametrized in terms of N four-vectors i Aμ (one for each generator of the isometry group) which are only dependent on x (however, the y dependence of Bμ may appear through the Killing vectors needed to saturate the indices of the isometry group with the corresponding generators). Namely, we set

= ( ), = ( ), m ( , ) = i ( ) m ( ). gμν gμν x φμν φmn y Bμ x y Aμ x Ki y (B.23)

( ) i ( ) The metric gμν x and the N vector fields Aμ x (one for each Killing generators Ki ) play the role of the “zero-mode” fields gμν , Aμ of the D = 5 model of the previous i section. It is instructive to check explicitly that Aμ transforms as a non-Abelian gauge vector under the action of the isometry group G. To this aim we consider an infinitesimal isometry transformation zA = z A + ξ A, with generators

A = ( μ, m ), μ = , m ( , ) = i ( ) m( ). ξ ξ ξ ξ 0 ξ x y x Ki y (B.24)

We recall that, in general, the local inﬁnitesimal variation of the D-dimensional metric can be written, in general, as

M M M δγAB =−ξ ∂M γAB − γAM∂B ξ − γBM∂Aξ (B.25)

(see Eq. (3.53)). Let us concentrate on the variation of the “mixed” components, γμm, for the inﬁnitesimal transformation (B.24). We obtain

n n n δγμm =−γmn∂μξ − γμn∂mξ − ξ ∂nγμm. (B.26) Appendix B: Higher-Dimensional Gravity 337

The mixed components of the metric, on the other hand, are given, according to Eqs. (B.20) and (B.23), by = n = i ( ) ( ). γμm Bμφmn Aμ x Kim y (B.27)

Inserting the above expressions into Eq. (B.26)), and taking into account the x and y dependence of γ, , A and K (see Eqs. (B.23), (B.24), (B.27)), we have i = i − i n j − j n ( ) i . δ Aμ Kim Kim∂μ Aμ Kin ∂m K j K j ∂n Kim Aμ (B.28)

In order to rewrite this transformation in a more transparent form we can now use the algebra of the isometry group given in Eqs. (B.18), (B.19), which implies:

n = n + k . K j ∂n Kim Ki ∂n K jm f ji Kkm (B.29)

Inserting this result into the last term of Eq. (B.28) we ﬁnd, after renaming indices: δ Ai K = K ∂ i − f i k Al μ im im μ kl μ − i j n + n . Aμ Ki ∂n K jm Kin∂m K j (B.30)

In the above equation, the contribution of the second line is identically vanishing thanks to the basic property of the Killing vectors ∇n Km + ∇m Kn = 0 (see Exercise 3.4), where ∇ denotes the covariant derivative computed with the metric φmn of the compact space KD−4. In fact, for any given (ﬁxed) pair of Killing vectors, of indices i and j,wehave

K n∂ K + K ∂ K n i n jm in m j = K n ∂ K + ∂ K − Γ p K − Γ p K i n jm m jn nm jp mn jp = n ∇ + ∇ ≡ . Ki n K jm m K jn 0 (B.31)

Γ = Γ( ) n = Here φ is the connection for the metric φmn;wehaveset∂m K j np mn ∂m(φ K jp), and we have eliminated the partial derivatives of φ by using the np metricity condition ∇mφ = 0. i Let us finally consider a local infinitesimal variation of the vector field Aμ at fixed value of Ki (namely, the field Aμ and the transformed field Aμ + δ Aμ are projected ( i ) = i on the same Killing vectors). In that case we have δ Aμ Kim Kimδ Aμ, and we can rewrite the result (B.30)as

i ( ) = i ( ) − i k ( ) l ( ). δ Aμ x ∂μ x fkl x Aμ x (B.32)

This is clearly the inﬁnitesimal transformation of the gauge potential of a non-Abelian i k symmetry group, with local parameter and structure constants fij . This can be explicitly checked by considering the gauge transformation for the non-Abelian vector potential Aμ already derived (in ﬁnite form) in Eq. (12.18), and 338 Appendix B: Higher-Dimensional Gravity expanding the group representation (12.10)as

i U = 1 + i Xi +··· , (B.33) where the group generators Xi satisfy the Lie algebra:

k Xi , X j = ifij Kk . (B.34)

In order to match the notation of this section we are denoting with i, j = 1, 2,...,N the indices with values in the group algebra. Also, we will use units in which the gauge coupling constant of Chap. 12 is ﬁxed to g = 1. By expanding Eq. (12.18)to ﬁrst order in we thus obtain i = i + i j − + i . Aμ Xi Aμ Xi i Aμ Xi X j X j Xi Xi ∂μ (B.35)

Hence, by using Eq. (B.34),

i ≡ i − i = i − i k l , δ Aμ Aμ Aμ ∂μ fkl Aμ (B.36) which exactly coincides with the variation (B.32) induced by an isometry of the compact manifold KD−4. The non-Abelian isometries of the compact space are thus associated to the presence of non-Abelian gauge fields in the effective geometric model in four space–time dimensions. It can be added that, by inserting the metric ansatz (B.20), (B.23) into the higher-dimensional Einstein action (and choosing the appropriate warp factor w(φ)) we end up, after dimensional reduction, with the canonical form of the four-dimensional Einstein–Yang–Mills action for the metric ( ) i gμν x and for the non-Abelian gauge potential Aμ. In such a context we also obtain an interesting generalization of Eq. (B.17), namely a relation between the size of the spatial (hyper)volume filled by the compact dimensions and the scale of the higher-dimensional gravitational coupling G D, defined by = 2−D 8πG D MD . In fact, let us consider the expansion of the D-dimensional Einstein action in the low-energy limit in which the geometry is described, in first approximation, by the ground state configuration (B.20), (B.23). We have M D−2 − D d D z |γ| R 2 D D−2 MD D−4 D/2 1/2 4 =− d y w |det φmn| d x |g| R(g) +··· . 2 KD−4 M4 (B.37)

Consider the pure gravity sector, and call VD−4 the proper (ﬁnite) hypervolume of the compact Kaluza–Klein extra-dimensional space (including a possible warp-factor contribution), Appendix B: Higher-Dimensional Gravity 339 D−4 D/2 1/2 VD−4 = d y w (y) |det φmn(y)| . (B.38) KD−4

Comparing Eq. (B.37) with the four-dimensional Einstein action, M2 − P d4x |g| R(g), (B.39) 2 we immediately obtain D−2 = 2. MD VD−4 MP (B.40)

−1 18 Since MP is known (MP = (8πG) 2.4 × 10 GeV), this is a constraint con- necting the strength of the higher-dimensional gravitational coupling to the size and the number of the compact extra dimensions. Let us consider, for instance, the simple isotropic case with a compactiﬁcation − ∼ D−4 scale of size Lc, the same for all D 4 extra dimensions. Then VD−4 Lc and Eq. (B.40) reduces to D−2 D−4 ∼ 2. MD Lc MP (B.41)

Again (as in D = 5) we obtain that a D-dimensional coupling of Newtonian strength, ∼ ∼ −1 ∼ −33 MD MP, implies a Planckian compactiﬁcation scale, Lc MP 10 cm. However, even larger compactiﬁcation scales are in principle allowed, provided the value of MD is smaller than Planckian. Solving Eq. (B.41)forLc we obtain, in general, (D−2)/(D−4) −17 1TeV 30/(D−4) Lc ∼ 10 cm 10 . (B.42) MD

We have referred MD to the TeV scale since this scale is, in a sense, preferred because of theoretical “prejudices” related to the solution of the so-called “hierarchy” problem (and possibly, also, of the cosmological constant problem3). Concerning the present observational results, we should mention the existence of gravitational experiments4 excluding the presence of extra dimensions down to −2 length scales Lc 10 cm. According to Eq. (B.42) this is compatible with MD ∼ 1 TeV provided the number of compact dimensions is n = D − 4 ≥ 2. In addition, high-energy experiments probing the standard model of strong and electroweak interactions have excluded (up to now) the presence of extra dimensions −15 down to scales Lc 10 cm. This seems to suggest MD 1TeV,orMD ∼ 1TeV but with an unexpectedly large number of extra dimensions, unless—as we shall see in Sect. B.2—there is some mechanism able to conﬁne gauge interactions inside three-dimensional space, making them insensitive to the extra dimensions.

3See for instance M. Gasperini, JHEP 06, 009 (2008). 4For a review see for instance E.G. Adelberg, B.R. Heckel and A.E. Nelson, Ann. Rev. Nucl. Part. Sci. 53, 77 (2003). 340 Appendix B: Higher-Dimensional Gravity

Before discussing this interesting possibility let us come back to the Kaluza– Klein scenario, with a compact extra-dimensional space and a topological structure Md = M4 KD−4. There is a problem, in D > 5, due to the fact that if we impose on the higher-dimensional metric γAB to satisfy the vacuum Einstein equations, and we look for low-energy solutions in which M4 coincides with the flat Minkowski space–time (gμν = ημν ), then we find, for consistency, that the manifold KD−4 has to be “Ricci flat”. This means, more precisely, that the Ricci tensor of the metric φmn must satisfy the condition Rmn(φ) = 0. This is possible, of course: the compact manifold, for instance, could be a torus, or a Kalabi–Yau manifold used in the compactification of superstring models. A Ricci-flat manifold, however, only admits Abelian isometries (see e.g. the book [3] k of the bibliography), hence all Killing vectors are commuting ( fij = 0) and the previous example reduces to a model with N Abelian gauge fields (an almost trivial generalization of the Kaluza–Klein model in D = 5). In order to solve this difficulty the model has to be generalized by dropping the original Kaluza–Klein idea that a physical four-dimensional model with gravity and matter fields can be derived from a pure gravity model in D > 4. We have to include non-geometric fields even in D > 4, possible representing non-Abelian gauge fields and/or sources of the extra-dimensional curvature contributing to Rmn = 0. There is an advantage with this procedure, as we shall see in the next section. In fact, the matter sources already present at a higher-dimensional level can automatically trigger the splitting of manifold MD into the product of two maximally symmetric manifolds—one of which is compact, while the other corresponds to our four-dimensional space–time—thus automatically implementing the so-called mechanism of “spontaneous compactification”.

B.1.2 Spontaneous Compactiﬁcation

Among the various mechanisms of spontaneous compactification (based on antisymmetric tensor fields, Yang–Mills fields, quantum fluctuations, monopoles, instantons, generalized higher-curvature actions, …), we will concentrate here on the case of the antisymmetric tensor fields, which has been inspired by the dimensional reduction of the supergravity theory formulated in D = 11 dimensions (and which also finds applications in the context of ten-dimensional superstring theory). Let us start by considering the general D-dimensional action for gravity with matter sources, 1 S =− d D x |γ| R(γ) + S , (B.43) 2 m where we have set to one the gravitational coupling, working in units where 8πG D = 2−D = MD 1. The corresponding gravitational equations are Appendix B: Higher-Dimensional Gravity 341

1 R − γ R = T , (B.44) AB 2 AB AB where TAB represents the contribution of Sm. Let us look for background solutions in which the geometry of the D-dimensional space–time manifold can be factorized as the product of two maximally symmetric spaces, MD = M4 MD−4, with metric

γμν = gμν (x), γmn = gmn(y), γμm = 0, (B.45) and with the corresponding Ricci tensors satisfying the conditions

Rμν =−Λx gμν , Rmn =−Λygmn, Rμm = 0, (B.46) where Λx and Λy are constant parameters (see e.g. Eq. (6.44)). This gives, for the D-dimensional scalar curvature,

μν mn R(γ) = g Rμν + g Rmn =−4Λx + (4 − D)Λy. (B.47)

Note that (like in the previous sections) we are splitting the D-dimensional coordinates x A into 4 coordinates xμ, with Greek indices running from 0 to 3, and D − 4 coordinates ym , with Latin indices running from 4 to D − 1. The above form of background geometry is clearly compatible with the Einstein equations (B.44) provided the sources satisfy the conditions

Tμν = Tx gμν , Tmn = Tygmn, Tμm = 0, (B.48) where Tx and Ty are constant parameters. Let us see that such conditions can be satisfied by the energy-momentum of an antisymmetric tensor field of appropriate rank. Consider the following action for the matter sources: ··· =− D | | M1 Mr , Sm k d x γ FM1···Mr F (B.49) where k is a model-dependent numerical coefficient (irrelevant for our discussion), and F is the field strength of a totally antisymmetric tensor A of rank r − 1, namely:

= . FM1···Mr ∂[M1 AM2···Mr ] (B.50)

The dynamical energy-momentum tensor associated to the action Sm , and deﬁned by the standard variational procedure (see Eq. (7.27)) referred to the metric γ AB,is then given by

M ···Mr 1 2 T =−2kr F ··· F 2 − γ F . (B.51) AB AM2 Mr B 2r AB 342 Appendix B: Higher-Dimensional Gravity

The variation of Sm with respect to A also provides the equation of motion of the tensor ﬁeld, NM2···Mr ∂N |γ|F = 0, (B.52) to be satisﬁed together with the Einstein equations (B.44). √ Let us now observe that, for our maximally symmetric background, |γ|= 1/2 1/2 | det gμν | | det gmn| . Wealso note that the constraints (B.48) imply, for the energy- momentum tensor (B.51), the following conditions:

··· − M2 Mr = , 2kr FμM2···Mr Fν Fx gμν ··· − M2 Mr = , 2kr FmM2···Mr Fn Fygmn (B.53) where Fx and Fy are constant parameters. This gives, in particular,

2 D − 4 Tx = 1 − Fx − Fy, r 2r 2 D − 4 T =− F + 1 − F . (B.54) y r x 2r y

As discovered5 in the context of D = 11 supergravity, there are two possibilities of obtaining a particular solution which simultaneously satisﬁes the conditions (B.53) and the equations of motion (B.52), and which is consistent with the assumed dimensionality split into 4 and D − 4 dimensions. • The ﬁrst possibility is to take r = 4 and set

μναβ μναβ cx μναβ F (x) = cx η = , (B.55) | det gμν |

(where cx is a constant), assuming that F = 0 for all the other components of the antisymmetric ﬁeld with one or more Latin indices. • The second possibility is to take r = D − 4 and set c m4···m D−1 m4···m D−1 y m4···m D−1 F (y) = cy η = √ , (B.56) | det gmn|

(where cy is a constant), assuming that F = 0 for all the other components with one or more Greek indices. We have denoted with η the totally antisymmetric tensors of the two maximally symmetric spaces with 4 dimensions and D − 4 dimensions (see Sect.3.2 for the deﬁnition of such tensor and a discussion of its basic properties). Thanks to the presence of antisymmetric tensors of appropriate rank it is thus possible to ﬁnd solutions with the required structure MD = M4 MD−4.Butlet

5P.G.O. Freund and M.A. Rubin, Phys. Lett. B97, 233 (1980). Appendix B: Higher-Dimensional Gravity 343 us see now if we can also obtain, in this “spontaneous” way, a configuration in which the extra-dimensional manifold MD−4 is compact and characterized by Λy > 0, in such a way to have a finite volume and to admit non-Abelian isometries. We can consider, to this aim, both possibilities (B.55), (B.56). Considering in particular Eq. (B.53) we find that the first case is characterized by r = 4, Fy = 0, while the second case by r = D − 4, Fx = 0. In both cases we obtain, from Eq. (B.54), the condition Tx + Ty = 0, and this immediately gives an important relation between the curvature scales Λx , Λy of the two spaces. In fact, by inserting the explicit configurations for the metric and the matter fields, Eqs. (B.46), (B.48), into the Einstein equations (B.44), and taking into account the constraint (B.47) for the scalar curvature, we obtain the relations:

D − 4 D − 6 Λ + Λ = T , 2Λ + Λ = T . (B.57) x 2 y x x 2 y y

Hence, by imposing Tx + Ty = 0, we immediately obtain

D − 5 Λ =− Λ . (B.58) x 3 y

This shows that, if we want a model with D > 5 and Λy > 0 (which admits the possibility of a compact extra-dimensional space with a non-Abelian isometry group), we must necessarily accept a four-dimensional maximally symmetric space with a negative cosmological constant, Λx < 0, namely with an anti-de Sitter (AdS) geometry. A background configuration AdS4 MD−4 does not look very realistic, because of the huge cosmological constant (|Λx |∼Λy) and also because of other phenomenological problems (such as the absence of four-dimensional “chiral” fermions, namely of fermions states of different helicity transforming as different representations of the gauge group). All the phenomenological problems are basically related to the nonvanishing (and negative) value of the cosmological constant of M4, which forbids a four-dimensional Minkowski geometry. In order to recover the Minkowski solution even for D > 5 the simplest possibility is probably that of accepting a Ricci-flat extra-dimensional space, setting Λy = 0 and giving up non-Abelian isometries. In that case the Yang–Mills gauge fields must be already present in the higher-dimensional action, where indeed they can themselves trigger the mechanism of spontaneous compactification (on a Ricci-flat manifold). This is what happens, for instance, in the so-called “heterotic” string model (see e.g. the books [10, 18] of the bibliography), where the chiral fermion problem is indeed solved in this way. Another possibility is that of adding a suitable cosmological constant ΛD to the D-dimensional action (B.43), in such a way as to exactly cancel the contribution of Λx (hence allowing D = 4 Minkowski solutions), while keeping a positive constant in the compact space MD−4 (to guarantee the presence of non-Abelian isometries). This, however, would require a high degree of “fine tuning” to exactly match the 344 Appendix B: Higher-Dimensional Gravity various contributions. In addition, the ad hoc introduction of ΛD would explicitly break the supersymmetry of the higher-dimensional supergravity action. An alternative mechanism, which relaxes the need for fine tuning—still providing a Ricci-flat four-dimensional geometry, Rμν = 0, together with a non Ricci-flat compact space, Rmn = 0—is based on the presence of a non-minimally coupled scalar field φ in the higher-dimensional action. Such a configuration is typical of the bosonic sector of superstring models, and we will present here a simple example based on the following D-dimensional action: −φ D e M M ···M S =− d x |γ| R(γ) + ∂ φ∂ φ + V (φ) + kF ··· F 1 r , 2 M M1 Mr (B.59) where φ is the so-called “dilaton” field. By varying the action with respect to γ and φ we obtain, respectively, the gravitational equation

1 1 R − γ R +∇ (∂ φ) + γ ∂ φ∂ M φ − γ ∇ ∂ M φ AB 2 AB A B 2 AB M AB M φ = e (TAB + γABV ) , (B.60) and the dilaton equation M M φ R(γ) +∇M ∂ φ − ∂M φ∂ φ = 2e V (B.61)

(see e.g. the book [9] of the bibliography). Here V = ∂V/∂φ, and TAB is the energy- momentum tensor of Eq. (B.51). The variation with respect to A leads then to the equation of motion (B.52) for the antisymmetric tensor, exactly as before. Let us look again for factorized solutions with the structure MD = M4 MD−4, where the metric satisﬁes the conditions (B.45), (B.46), the antisymmetric tensor the condition (B.48), and, in addition, the scalar ﬁeld is a constant, φ = φ0. Inserting this ansatz into the gravitational equations we obtain

R(γ) −Λ − = eφ0 (T + V ) , x 2 x 0 R(γ) −Λ − = eφ0 T + V , (B.62) y 2 y 0 while the dilaton equation (B.61)gives

( ) = φ0 , R γ 2e V0 (B.63) = ( ) = ( / ) where V0 V φ0 and V0 ∂V ∂φ φ=φ0 . We now use for the antisymmetric tensor ﬁeld the Freund–Rubin solutions (B.55), (B.56), both characterized by the condition Tx + Ty = 0, which implies (combined with Eq. (B.62)): φ0 Λx + Λy + R(γ) =−2e V0. (B.64) Appendix B: Higher-Dimensional Gravity 345

We are interested, in particular, in a Ricci-ﬂat solution for the four-dimensional space–time M4. This means—using Eq. (B.47) which expresses the scalar curvature R(γ) in terms of Λx and Λy—that we a re looking for solutions characterized by:

R(γ) Λ = 0,Λ=− (B.65) x y D − 4

This choice can simultaneously satisfy all equations of our model (and, in particular, the dilaton condition (B.63) and the condition (B.64) for the antisymmetric ﬁeld) provided V D − 4 =− . (B.66) V D − 5 φ0

We can thus obtain the sought geometrical structure without fine adjustment of free dimensional parameters, at the price of imposing a simple differential condition on the functional form of the potential. In this particular case, for instance, the condition is satisfied by an exponential potential with V ∼ exp[−φ(D − 4)/(D − 5)]). This model of spontaneous compactification can be easily generalized to (more realistic) cases in which the dilaton coupling to the Einstein action is described by an arbitrary function f (φ) replacing exp(−φ). In that case6 the previous Eq. (B.66) is to be replaced by a condition relating (V /V )0 to ( f /f )0.

B.2 Brane-World Gravity

Another approach to the problem of the dimensional reduction, not necessarily alternative to the Kaluza–Klein scenario, is based on the assumption (suggested by superstring models of unified interactions) that the charges sourcing the gauge interactions are confined on 3-dimensional hypersurfaces called “Dirichlet branes” (or D3-branes), and that the associated gauge fields can propagate only on the “world- volume” swept by the time evolution of such branes. It follows that the gauge interactions are insensitive to the spatial dimensions orthogonal to the brane, even in the limiting case in which such dimensions are infinitely extended. According to such a scenario—also called “brane-world” scenario—we are thus living on a four- dimensional “slice” of a D-dimensional space–time (the so-called “bulk” manifold). In these models, however, gravity behaves differently from the other fundamental interactions, and can propagate along all existing spatial directions. Hence the gravitational theory must be formulated, in general, in D dimensions, and its equations determine the metric and the curvature not only of the brane but also of the whole D-dimensional bulk space–time. We have thus to face, even in this context, the problem already met in the context of the Kaluza–Klein scenario: how to obtain (at least as a ground state solution, valid

6See e.g. M. Gasperini, Phys. Rev. D31, 2708 (1985). 346 Appendix B: Higher-Dimensional Gravity in the low-energy limit) a flat Minkowski geometry in the four-dimensional space– time of the brane in which we live? Also: how to explain why we have not found (so far) any gravitational evidence of the extra dimensions? are they compactified on very small distance scales like in the Kaluza–Klein scenario? In the following sections it will be shown that the compactification of the dimensions external to the brane is a possibility,butnot a necessity as in the Kaluza–Klein context. Here will first introduce a very simple model of gravity in D dimensions to show that it is possible, in general, to obtain exact solutions describing a flat four-dimensional space–time associated to a brane embedded in a curved (higher- dimensional) bulk manifold. Let us start with the following action for a D-dimensional space–time manifold M D, M D−2 S = d D x |g | − D R + Lbulk + Sbrane, (B.67) D 2 D D p

Lbulk where we have included the Lagrangian density D , generically representing all gravitational sources possibly present in the bulk manifold and contributing to the geometry described by the D-dimensional metric gAB. We have put into explicit brane evidence the term Sp , representing the action of a p-dimensional brane (p-brane, for short) embedded in MD, with p + 1 < D. It also contributes to the bulk geometry, in two ways: with its own energy-momentum density, and with the energy-momentum density of all gravitational sources possibly living on it—namely, all matter fields and their quantum fluctuations confined on the (p + 1)-dimensional hypersurface Σp+1 swept by the brane evolution. brane To obtain an explicit form of Sp let us now recall that a p-brane is an elementary object extended along p spatial dimensions: for instance, a 0-brane is a point-particle, a 1-brane is a string, a 2-brane is a two-dimensional membrane, and so on. The action which controls the free dynamics of these objects is proportional to the integral determining the (p + 1)-dimensional “world-volume” of the hypersurface Σp+1 described by their evolution in time. For a 0-brane, for instance, the action is given by the well known line integral along the “world-line” Σ1 described by the particle (see Eq. (5.1)). For a 1-brane we have the surface integral over the string “world-sheet” Σ2. For a 2-brane the action is proportional to the volume integral over a three-dimensional world-volume Σ3 (see Fig. B.1). And so on for higher values of p. μ Let us concentrate on a p-brane embedded in MD, and let us call ξ = (ξ0, ξ1,...,ξ p) the coordinates of the intrinsic chart used to describe the geometry of A 0 1 D−1 Σp+1 (the world hypersurface of the brane), while we call x = (x , x ,...,x ) the coordinates on MD. The parametric equations, describing the embedding of Σp+1 into MD (see Sect. 2.1), can then be expressed in the form

x A = X A(ξμ), A = 0, 1,...,D − 1, μ = 0, 1,...,p, (B.68) and the so-called “induced metric” on the hypersurface ΣP+1 is given by Appendix B: Higher-Dimensional Gravity 347

Fig. B.1 Examples of “world-volumes”, and corresponding action integrals, for a particle (p = 0), a string (p = 1), and a two-dimensional membrane (p = 2). The four-dimensional space–time in which we live could be the one described by the time evolution of a 3-brane embedded in a higher-dimensional bulk manifold

∂ X A ∂ X B h = g . (B.69) μν ∂ξμ ∂ξν AB

The covariant action for an “empty” p-brane (written in the so-called Nambu–Goto form, which directly generalizes the point-particle action) thus becomes brane = p+1 | |. Sp Tp d ξ h (B.70) Σp+1

Here h = det hμν , and Tp—the so-called “tension”—is a constant representing the vacuum energy density of the brane, i.e. the vacuum energy per unit of proper 348 Appendix B: Higher-Dimensional Gravity p-dimensional volume of the brane. It should be noted that, if the brane also contains matter ﬁelds in addition to the vacuum energy, then the “cosmological” constant Tp must be replaced by the Lagrangian density Lp including (besides Tp) the contribution of all gravitational sources living on the brane. Such a Lagrangian is in general ξ-dependent, and, unlike Tp, has to be kept under the integral. The above brane action can also be rewritten in an equivalent form which avoids the explicit presence of the square root—and is thus more convenient for variational computations—at the price of introducing an auxiliary tensor ﬁeld γμν , acting as a Lagrange multiplier, and representing the “intrinsic” Riemannian metric of the manifold Σp+1. Such an equivalent form is the so-called Polyakov action, T ∂ X A ∂ X B Sbrane = p d p+1ξ |γ| γμν g − (p − ) , p μ ν AB 1 (B.71) 2 Σp+1 ∂ξ ∂ξ

μν where γ = det γμν . Its variation with respect to γ gives the constraint

1 1 h − γ γαβh − γ (p − 1) = 0, (B.72) μν 2 μν αβ 2 μν which is identically solved by γμν = hμν , where hμν is defined by Eq. (B.69). Using μν μν = μ = + this result to eliminate γ , and using the identity h hμν δμ p 1, one then finds that the above Polyakov action exactly reduces to the Nambu–Goto form of Eq. (B.70). It is finally convenient, for our purpose, to take into account that the brane contribution to the total action (B.67) is localized exactly at the brane position specified by the embedding equations (B.68), and it is vanishing for x A = X A(ξ). We can brane thus express Sp in the same way as the other terms of the action, i.e. as an integral over the D-dimensional bulk volume, provided we integrate over an appropriate delta-function distribution. We can write, in particular, brane = D | | Lbrane, Sp d x gD D (B.73) where

Lbrane = D T ∂ X A ∂ X B = √ p d p+1ξ |γ| γμν g − (p − 1) δ D(x − X(ξ)). | | μ ν AB 2 gD Σp+1 ∂ξ ∂ξ (B.74)

In that case the total action (B.67) becomes M D−2 S = d D x |g | − D R + Lbulk + Lbrane , (B.75) D 2 D D D Appendix B: Higher-Dimensional Gravity 349 and can be easily varied with respect to the three independent ﬁelds of our model, A μν namely gAB, X and γ . The variation with respect to gAB gives the bulk Einstein equations,

1 R − g R = M2−D T bulk + T brane , (B.76) AB 2 AB D AB AB where the energy-momentum tensor of the sources is provided by the standard vari- AB Lbulk ational deﬁnition (7.26), (7.27), performed with respect to g and applied to D Lbrane and D . For the brane, in particular, we have T T brane = √ p d p+1ξ |γ|γμν ∂ X ∂ X δ D(x − X(ξ)), (B.77) AB | | μ A ν B gD Σp+1

A = A/ μ Lbrane A where ∂μ X ∂ X ∂ξ . The variation of D with respect to X gives the brane equation of motion, μν B ∂μ |γ|γ ∂ν X gAB(x) = x=X(ξ) 1 μν M N = |γ|γ ∂μ X ∂ν X ∂AgMN(x) . (B.78) 2 x=X(ξ)

Finally, the variation with respect to γμν gives the constraint (B.72), which leads to identify γμν with the induced metric hμν . Let us now consider the particular case p = 3, where the brane space–time Σ4 has the appropriate number of dimensions to represent a possible model of our macroscopic space–time. Let us also assume, for simplicity, that the bulk space–time has only one additional dimension, so that D = 5 (like in the original Kaluza–Klein proposal). Finally, let us concentrate on a very simple example where the only bulk contribution to gravity from the space external to the brane comes from the vacuun energy density, hence has the form of a cosmological constant Λ. We set, more precisely, Lbulk =−M D−2Λ, so that:

2−D bulk = Λ . M TAB gAB (B.79)

In this context we will look for particular solutions of Eqs. (B.76), (B.78) describing a ﬂat (Minkowski) hypersurface Σ4 embedded in a (generally curved) ﬁve-dimensional bulk manifold M5. Let us call x A = (xμ, y) the bulk coordinates, and suppose that the hypersurface Σ4 is rigidly localized at y = 0, described by the following trivial embedding equations:

A = A( ) = A μ, 4 ≡ = , x X ξ δμ ξ x y 0 A = 0, 1, 2, 3, 4 μ = 0, 1, 2, 3 . (B.80) 350 Appendix B: Higher-Dimensional Gravity

Also, suppose that Σ4 has a globally flat geometry described by the Minkowski metric 2 ημν , and that the bulk metric is conformally flat, gAB = f (y)ηAB, with a conformal factor f 2 which depends only on the y coordinate parametrizing the spatial direction normal to the brane. Since our configuration is symmetric under y →−y reflections we thus look for a “warped” five-dimensional geometric structure described by the following line-element: 2 2 μ ν 2 ds = f (|y|) ημν dx dx − dy . (B.81)

We can easily check that, for this type of background, the induced metric (B.69) 2 reduces to hμν = f ημν = γμν , and that the brane equation (B.78) is identically sat- isﬁed thanks to the reﬂection symmetry, which implies (∂ f/∂y)y=0 = 0 (see below). Let us then consider the Einstein equations (B.76). For the energy-momentum of the sources we easily get, from Eq. (B.79), −3 B bulk = Λ B , M5 TA δA (B.82) and, from Eq. (B.77), 4 brane T4 = 0 ν brane = −1 ν ( ). Tμ f T3δμδ y (B.83)

The ﬁve-dimensional Christoffel connection associated to the metric (B.81), on the other hand, has the following nonvanishing components:

f f f Γ 4 = ,Γ4 = η ,Γν = δν , (B.84) 44 f μν f μν 4μ f μ

(a prime denotes differentiation with respect to y). Deﬁning F = f /f we then obtain, from the components of the Einstein tensor,

1 G 4 = R 4 − R =−6 f −2 F 2, 4 4 2 1 G ν = R ν − δν R =−3 f −2 F + F 2 δν . (B.85) μ μ 2 μ μ The Einstein equations (B.76), decomposed into the directions normal and tangential to the brane space–time Σ4, thus reduce, respectively to:

6F 2 =−Λf 2 (B.86) + 2 =−Λ 2 − −3 ( ). 3F 3F f M5 T3 f δ y (B.87)

Note that f depends on the modulus of y, so that the second derivative of f (present into F ) contains the derivative of the sign function, which generates a delta-function Appendix B: Higher-Dimensional Gravity 351 contribution to the left-hand side of Eq. (B.87). We have thus to match separately the ﬁnite parts of this equation and the coefﬁcients of the singular contributions at y = 0. In order to solve the above system of equations it is convenient to adopt the explicit representation |y|=y(y), (y) = θ(y) − θ(−y), (B.88) where θ(y) is the Heaviside step function and (y) the sign function, satisfying the properties: 2 = 1, = 2δ(y). (B.89)

We can thus set ∂ f f = (y), (B.90) ∂|y| and Eq. (B.86) becomes ∂ f 2 Λ =− f 4, (B.91) ∂|y| 6 which admits real solutions provided Λ<0. Assuming that the bulk cosmological constant is negative, and integrating, we then obtain the particular exact solution

Λ 1/2 f (|y|) = (1 + k|y|)−1 , k = − (B.92) 6

Inserting this solution into the metric (B.81) we exactly obtain for the bulk space–time an anti-de Sitter (AdS) geometry, written in the conformally ﬂat parametrization. We have still to solve the second Einstein equation (B.87), which contains the explicit contribution of the brane. Using Eqs. (B.88)–(B.90) we can recast our equa- tionintheform:

3 ∂2 f 6 ∂ f + δ(y) =−Λf 2 − M−3T f δ(y). (B.93) f ∂|y|2 f ∂|y| 5 3

The finite part of this equation is identically satisfied by the solution given in Eq. (B.92). By equating the coefficients of the delta-function terms we are led to a condition between the tension of the brane and the curvature scale of the AdS bulk geometry: = 3 = 3 (− Λ)1/2 . T3 6kM5 M5 6 (B.94)

If this condition is satisﬁed we obtain the so-called Randall–Sundrum model,7 in which the positive contribution of the vacuum energy density of the brane (represented by its tension T3) is exactly canceled by an opposite contribution generated

7L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 4960 (1999). 352 Appendix B: Higher-Dimensional Gravity by the external bulk sources, and the geometry of the brane-world Σ4 is allowed to be of the ﬂat Minkowski type, as required.

B.2.1 De Sitter Gravity on the Brane

The model we have considered, with an empty 3-brane and the vacuum bulk sources of Eq. (B.79), is also compatible with more general solutions than the Minkowski one illustrated before. In particular, there are solutions where the brane-world manifold Σ4 is a maximally symmetric manifold with constant positive curvature (namely, it is characterized by an intrinsic de Sitter geometry). In this section we will briefly discuss this possibility, which may have important cosmological applications in the context of the inflationary phase typical of our primordial Universe (see e.g. the books [12, 14] of the bibliography). Let us start with the embedding configuration parametrized by the equations (B.80), assuming however that the bulk geometry is described by the following (less trivial) line-element: 2 2 μ ν 2 ds = f (|y|) gμν dx dx − dy , (B.95) where gμν is the de Sitter metric given by (see also Exercise 2.2, Eq. (2.42))

2Ht g00 = 1, gij =−e δij, i, j = 1, 2, 3 (B.96)

(we have set x0 = t, and H is a constant). As before, the warp factor f (y) is invariant 2 under y →−y reflections. For this geometric configuration we obtain γμν = f gμν , and we find that the brane equation of motion (B.78) is identically satisfied. Let us then consider the Einstein equations (B.76). The nonvanishing components of the Christoffel connection, for the metric (B.95), are

Γ j = j ,Γ0 = 2Ht , 0i Hδi ij He δij Γ B = B ,Γ4 = , 4A FδA μν Fgμν (B.97) where, as before, F = f /f and the prime denotes differentiation with respect to y. The corresponding Einstein tensor takes the form 4 = −2 2 − 2 , ν = −2 2 − 2 − 2 ν . G4 6 f H F Gμ 3 f H F F δμ (B.98)

By inserting the gravitational sources (B.82), (B.83), we are ﬁnally lead to the Einstein equations Appendix B: Higher-Dimensional Gravity 353 6 H 2 − F 2 = Λf 2, (B.99) 2 − 2 − 2 = Λ 2 + −3 ( ), 3 H F F f M5 T3 f δ y (B.100) which obviously reduce to the previous equation (B.86), (B.87) for the Minkowski brane-world solution with H = 0. To solve these equations we follow the same procedure as before, using the relations (B.88)–(B.90). The ﬁrst Einstein equation (B.99) thus reduces to

∂ f 2 Λ = f 2 H 2 + k2 f 2 , k2 =− . (B.101) ∂|y| 6

For Λ<0 the parameter k is real, and the above equation has the particular exact solution H 1 f (|y|) = , (B.102) k H ( + | |) sinh k 1 k y which determines the curved geometry of the bulk space–time external to the brane, according to Eq. (B.95). This warp factor describes a rather complicated geometric structure; note however that in the limit H → 0 (in which Σ4 reduces to a ﬂat Minkowski manifold) we easily recover the solution (B.92) corresponding to the well known anti-de Sitter geometry, with a curvature scale controlled by the parameter k. To ﬁx the allowed value of H we now consider the second Einstein equation (B.100) which, by using the relations (B.88)–(B.90), can be recast in the form

3 ∂2 f 6 ∂ f 3H 2 − − δ(y) = Λf 2 + M−3T f δ(y). (B.103) f ∂|y|2 f ∂|y| 5 3

The ﬁnite part of this equation (y = 0) is identically satisﬁed by the solution (B.102). By imposing the equality of the delta-function terms we obtain the following condition on f (y), evaluated at y = 0:

∂ f T (0) =− 3 f 2(0). (B.104) | | 3 ∂ y 6M5

This implies, using the explicit form of Eq. (B.102),

Λ −1/2 H = T3 − , cosh 3 (B.105) k 6M5 6 which fixes the de Sitter scale H of the brane-world manifold Σ4 in terms of the five- dimensional gravitational coupling M5, of the energy density of the bulk vacuum, Λ, and of the brane, T3. Note that in the limit H → 0 we exactly recover the condition (B.94), needed to obtain a brane-manifold with a flat Minkowski geometry. 354 Appendix B: Higher-Dimensional Gravity

B.2.2 Dirichlet Branes and Gauge Fields Conﬁnement

As stressed at the beginning of Sect. B.2, the model of four-dimensional space–time as a brane-world manifold embedded in a higher-dimensional background suggests an alternative approach (and solution) to the problem of dimensional reduction. According to string theory, in fact, there are branes—called Dirichlet branes— able to confine the fundamental gauge fields and their sources on the world-manifold spanned by the evolution of such branes. In that case the gauge interactions cannot propagate along the spatial directions orthogonal to the brane, hence they become fully insensitive to the possible presence of extra dimensions. The only exception to this rule is gravity, even if, in such a context, we may expect a partial (low-energy) confinement also for the gravitational interactions. This last effect, typical of gravity on a brane-world manifold, will be discussed in the next sections. This section will be devoted—in view of their importance—to a brief introduction to the Dirichlet branes and their properties. We need to recall, to this aim, only a few basic notions concerning the theory of a free bosonic string propagating in a higher-dimensional (flat) space–time. Let us start with the Polyakov action (B.71), written for a free string (p = 1), and expressed in the so-called “conformal gauge” in which the√ intrinsic metric of the world-sheet Σ2 is flat, γμν = ημν = diag(1, −1), so that |γ|=1. Note that for a one-dimensional object like a string we can always choose this gauge, thanks to the invariance of the Polyakov action under local transformations of the type ω(ξ) μν −ω(ξ) μν ω(ξ) γμν → e γμν , γ → e γ , |γ|→e |γ| (B.106)

(also called Weyl, or conformal, transformations) The world-sheet metric γμν has indeed only three independent components, which can be completely fixed by imposing three arbitrary conditions following from the invariance under general coordinate transformations, ξμ → ξμ, with μ = 0, 1, and from the invariance under the conformal transformations (B.106). Let us call ξ0 = τ and ξ1 = σ the world-sheet coordinates in the conformal gauge, and denote with a dot the derivative with respect to τ and with a prime the derivative with respect to σ. The Polyakov action (B.71) for a string thus takes the form τ2 π ˙ S1 = dτ dσ L(X, X ), (B.107) τ1 0 where T T L(X˙ , X ) = 1 ημν ∂ X A∂ X = 1 X˙ A X˙ − X A X , (B.108) 2 μ ν A 2 A A and where we have assumed, following the usual conventions, that the spatial coordinate along the string ranges from 0 to π. For the purpose of this section we can also assume, for simplicity, that the string is embedded in a D-dimensional flat space– Appendix B: Higher-Dimensional Gravity 355 time, with metric gAB = ηAB. The contraction of the capital Latin indices in the action is thus performed with the Minkowski metric, and there are no contributions to the string action due to a non-trivial external geometry. The string equations of motion can now be deduced by varying the above action with respect to X A, by imposing the standard conditions that the variation is zero at A A the initial and final times of the motion, δX (τ1) = 0 = δX (τ2), and by imposing also suitable conditions needed to eliminate the variational contributions at the spatial boundaries σ = 0eσ = π. In fact, after the variation and an integration by parts, we obtain τ2 π ∂L δS = dτ dσ δ ∂ X A 1 ∂(∂ X A) μ τ1 0 μ τ2 π ∂L τ2 π ∂L = dτ dσ∂ δX A − dτ dσ ∂ δX A. μ ( A) μ ( A) τ1 0 ∂ ∂μ X τ1 0 ∂ ∂μ X (B.109)

If the boundary contributions—represented by the ﬁrst term in the second line of the above equation—are all vanishing, then we immediately arrive at the Euler– Lagrangian equations of motion,

∂L ∂L ∂L 0 = ∂ = ∂ + ∂ = X¨ A − X A. (B.110) μ A τ ˙ A σ A ∂(∂μ X ) ∂ X ∂ X which in our case describe the motion of a free string, embedded in a D-dimensional Minkowski manifold and parametrized by the equations X A = X A(τ, σ). Let us now consider the boundary contributions to Eq. (B.109), which can be explicitly written as τ2 π ∂L I ≡ dτ dσ∂ δX A b μ ∂(∂ X A) τ1 0 μ τ2 π ∂L A ∂L A = dτ dσ ∂τ δX + ∂σ δX ∂ X˙ A ∂ X A τ1 0 τ π π ∂L 2 τ2 ∂L = dσ δX A + dτ δX A . (B.111) ˙ A ∂ X A 0 ∂ X τ1 τ1 0

The ﬁrst integral in the last line does not contribute because of the usual condition A A to be imposed at the time boundaries, δX (τ1) = 0 = δX (τ2). The second integral is identically vanishing provided we impose the following boundary condition: π ∂L π π 0 = δX A = X δX A = X X˙ Aδτ + X Aδσ ∂ X A A 0 A 0 0 ≡ ˙ A π . X A X δτ 0 (B.112) 356 Appendix B: Higher-Dimensional Gravity

For a closed string (with no free ends), satisfying the periodicity condition X A(τ, σ) = X A(τ, σ + π), the above condition is always automatically satisfied. For an open string, with two non-coincident ends corresponding to the values σ = 0 and σ = π of the spatial coordinate, the above boundary condition has to be imposed at each end of the string, and can be satisfied in two ways. The first possibility is the so-called Neumann boundary condition, = = , X A σ=0 0 X A σ=π (B.113) which allows the ends of the strings to move, in such a way as no momentum is flowing off the string through its ends. There is, however, a second possibility, called Dirichlet boundary condition and specified by ˙ A = = ˙ A , X σ=0 0 X σ=π (B.114) which imposes on the ends of the string to keep fixed. If the string is moving through a higher-dimensional space–time, with X A = (X 0, X 1, X 2,...,X D−1), it is always possible to impose Neumann boundary conditions on the first p + 1 coordinates {X 0, X 1, X 2,...,X p} (including time), and Dirichlet boundary conditions on the remaining D − 1 − p spatial coordinates {X p+1, X p+2,...,X D−1}. In such a case the ends of the open strings are fixed along the Dirichlet directions, but are free to move on the p-dimensional (space-like) hypersurface Dp given by the equations

X i = const, p + 1 ≤ i ≤ D − 1, (B.115) as well as on the associated hypervolume Σp+1 spanned by the evolution in time of Dp. The hypersurface defined by Eq. (B.115)iscalled“p-dimensional Dirichlet brane”, or (for short) Dp-brane. In Fig. B.2 we have illustrated a simple example of D2-brane embedded in a four-dimensional Minkowski space–time. On the other hand, in the unified models all fundamental interactions suggested by superstring theory, the (Abelian and non-Abelian) charges which are sources of the gauge fields are localized just at the ends of the open strings. By using adapted boundary conditions, in order to appropriately localize the position of the string ends, it is thus possible to confine the charges on a Dp-brane, and formulate models where the gauge fields only propagate through the (p + 1)-dimensional space–time 8 manifold associated to the time evolution of the Dp-brane.

8See for instance P. Horawa and E. Witten, Nucl. Phys. B460, 506 (1996); Nucl. Phys. B475,94 (1996). Appendix B: Higher-Dimensional Gravity 357

(b)

(a)

Fig. B.2 The ﬁgure shows two possible open string states, case (a) and case (b), both characterized by Neumann boundary conditions along the directions (x0, x1, x2), and Dirichlet boundary condi- 3 3 tions along x . The condition x = constant deﬁnes the Dirichlet D2-brane, which in this case corresponds to the plane {x1, x2}. The two ends of an open string can belong to the same plane (case (a)) if x3(σ = 0) = x3(σ = π), or to different but parallel planes (case (b)) if x3(σ = 0) = x3(σ = π)

B.2.3 Gravity Conﬁnement in Four Dimensions

If we take seriously the possibility that the world explored by fundamental (strong and electroweak) interactions is the four-dimensional space–time of a D3-brane, embedded in a higher-dimensional manifold, we still have to face the problem of why we have not yet detected the extra dimensions by means of gravitational experiments. Indeed gravity, unlike the other gauge interactions, is expected to propagate along all spatial directions. A possibility is that the dimensions external to Σ4 have a very small, compact size, not accessible to presently available experimental sensitivities (as also assumed in the context of the Kaluza–Klein scenario). In the brane-world scenario, however, there is a second possibility based on an effect of “gravity confinement”: an appropriate curvature of the bulk geometry can force the long-range component of tensor interactions to be strictly localized on Σ4, just like the vector gauge interactions. In that case only a residual, short-range tail of the gravitational interaction (mediated by massive tensor particles) may propagate in the directions orthogonal to Σ4, and make the extra dimensions (in principle) detectable by experiments probing small enough corrections to long-range gravitational forces. This interesting possibility can be illustrated considering the simple, five-dimensional Randall–Sundrum model introduced at the beginning of Sect.B.2, and by expanding to first order the fluctuations of the bulk metric tensor, gAB → A gAB + δgAB,atfixed brane position, δX = 0, around the background metric gAB defined by the solution (B.81). Let us call the fluctuations δgAB = h AB, and let us compute the perturbed action up to terms quadratic in h AB. We are interested, in particular, in the transverse and traceless part of the fluctuations of the four-dimensional geometry, δgμν = hμν , which describes the propagation of gravitational waves (see Chap. 9) in the brane space–time Σ4. In the linear approx- 358 Appendix B: Higher-Dimensional Gravity imation they are decoupled from other (scalar and extra-dimensional) components of δgAB. We shall thus assume that our perturbed geometric configuration is characterized by the following metric fluctuations:

α μν ν hμ4 = 0, hμν = hμν (x , y), g hμν = 0 = ∂ hμν . (B.116)

For the computation of the perturbed, quadratic action we will follow the straightforward procedure introduced in Sect.9.2 (which leads to the result (9.48)), taking into account, however, that we are now expanding around the non-trivial ﬁve- dimensional geometry (B.81). After using the unperturbed background equations we obtain M3 δS =− 5 d5x |g | h ν ∇ ∇ Ah μ 8 5 μ A ν M3 =− 5 d5xf3 h ν h μ − h ν h μ − 3Fh ν h μ , 8 μ ν μ ν μ ν (B.117) where the covariant derivative ∇A is referred to the unperturbed metric gAB, and where = 2 − 2 ∂t ∂i is the usual d’Alembert operator in four-dimensional Minkowski space. ν Integrating by parts to eliminate h , decomposing hμ into the two independent polarization modes (see Eq. (9.15), and tracing over the polarization tensors, the action for each polarization mode h = h(t, xi , y) can then be written as: M3 δS = 5 dy f 3 d4x h˙2 + h∇h − h2 . (B.118) 4

The dot denotes differentiation with respect to t = x0, the prime with respect to y, 2 ij and ∇ = δ ∂i ∂ j is the Laplace operator of 3-dimensional Euclidean space. The variation with respect to h ﬁnally gives the vacuum propagation equation for the linear ﬂuctuations of the geometry on the manifold Σ4:

h − h − 3Fh = 0. (B.119)

It differs from the usual d’Alembert wave equation because the fluctuations are coupled to the gradients of the bulk geometry, through their intrinsic dependence on the fifth coordinate y. In order to solve the above equation it is now convenient to separate the dependence of h on the bulk and brane coordinates by setting μ h(x , y) = vm(x)ψm (y), (B.120) m Appendix B: Higher-Dimensional Gravity 359 and we find, in this way, that the new variables v,ψ satisfy the following (decoupled) eigenvalue equations:

v =− 2v , m m m + ≡ −3 3 =− 2 . ψm 3Fψm f f ψn m ψm (B.121)

If the spectrum is continuous, the sum of Eq. (B.120) is clearly replaced by integration over m. It is also convenient to rewrite the equation for ψ in canonical (Schrodinger- like) form, by introducing the rescaled variable ψm, such that: 3 −1/2 ψm = f M5 ψm (B.122)

−1/2 (the dimensional factor M5 has been inserted for later convenience). The equation for ψ then becomes + 2 − ( ) = , ψm m V y ψm 0 (B.123) where 3 f 3 f 2 V (y) = + , (B.124) 2 f 4 f or, using the explicit background solution (B.92),

15 k2 3kδ(y) V (y) = − . (B.125) 4 (1 + k|y|)2 1 + k|y|

This is a so-called “volcano-like” potential, as the first term of V (y) is symmetric and peaked at y = 0, but the peak is in correspondence of a negative delta-function singularity, which looks like the crater of a volcano. It is well known, from one-dimensional quantum mechanics, that the Schrodinger equation with an attractive delta-function potential admits one bound state only, associated with a square-integrable wave function which is localized around the position of the potential. In our case such a configuration corresponds to the eigenvalue m = 0, and to the reflection-symmetric solution of Eq. (B.123) given by

3/2 ψ0 = c0 f , (B.126) where c0 is a constant to be determined by the normalization condition. In this respect, it is important to stress that ψ0, defined as in Eq. (B.122) (with ψ0 dimensionless), has the correct canonical normalization to belong to the L2 space of square integrable functions with measure dy (as in conventional quantum mechanics). Also, ψ0 turns out to be normalizable even for an infinite extension of the dimension normal to the brane. In that case, by imposing the standard normalization to one, we obtain the condition 360 Appendix B: Higher-Dimensional Gravity +∞ 2 2 2 c0 c0 1 = dy ψ0 = dy = (B.127) −∞ (1 + k|y|)3 k which fixes c0 as a function of k (i.e. of Λ.seeEq.(B.92)). We can also express the same result in terms of the non-canonical variable ψ but, in that case, we must use 3 inner products with (non-canonical) dimensionless measure dy = dyM5 f . The example of the case m = 0 clearly show how the massless components of the metric fluctuations (corresponding to long-range gravitational interactions) can be localized on the brane at y = 0: such a localization occurs not because the fifth dimension is compactified on a very small length scale, but because the massless modes are “trapped” in a bound state generated by the five-dimensional bulk curvature. In this case, in particular, it is the AdS geometry which forces massless fluctuations to be peaked around the brane position. Let us now take into account the massive part of the fluctuation spectrum, considering the Schrodinger equation (B.123) with m = 0. Even in that case there are exact solutions, with a continuous spectrum of positive values of m which extends up to infinity. However, as we shall see, these solutions are not bound states of the potential, and are not localized on the brane space–time Σ4. To obtain such solutions we can follow the standard quantum-mechanical treat- ment of a delta-function potential. Looking for reflection-symmetric functions ψm(|y|) we first rewrite Eq. (B.123)as

d2ψ dψ m + 2δ(y) m + (m2 − V ψ = 0, (B.128) d|y|2 d|y| m where V is given by Eq. (B.125). Outside the origin (y = 0) this reduces to a Bessel equation, whose general solution can be written as a combination of Bessel functions Jν and Yν of index ν = 2 and argument α = m/(kf):

−1/2 ψm = f [Am J2(α) + Bm Y2(α)] . (B.129)

Imposing on this expression to satisfy Eq. (B.128)alsoaty = 0, and equating the coefﬁcients of the delta-function terms, we obtain an additional condition which relates the two integration constants Am and Bm:

J1(m/k) Bm =−Am . (B.130) Y1(m/k)

The general solution can thus be rewritten as m m ψ = c f −1/2 Y J (α) − J Y (α) , (B.131) m m 1 k 2 1 k 2 where cm is an overall constant factor, to be determined by the normalization condition Appendix B: Higher-Dimensional Gravity 361 ∗ ≡ 3 ∗ = ( , ). dyψmψn dy M5 f ψmψn δ m n (B.132)

Here δ(m, n) corresponds to the Kronecker symbol for a discrete spectrum, and to the Dirac delta function for a continuous spectrum of values of m and n. In particular, for values different from zero the spectrum is continuum, and the normalization condition gives m 1/2 m m −1/2 c = J 2 + Y 2 , (B.133) m 2k 1 k 1 k which completely ﬁxes the amplitude of the massive modes of the tensor metric ﬂuctuations. Using the asymptotic behavior of the Bessel functions J2(α), Y2(α), with α = m/(kf) = m(1 + k|y|)/k, we see that the above solutions, instead of being damped, are asymptotically oscillating for y →±∞: hence, they cannot be localized on the brane. We may thus expect from these massive modes new (and genuinely higher- dimensional) effects: in particular, short-range corrections which are sensitive to the presence (and to the number) of the extra dimensions, and which bear the direct imprint of the bulk geometry. The effect of the massive modes will be discussed in the next section.

B.2.4 Gravitational Short-Range Corrections

For a quantitative estimate of the gravitational corrections induced by the massive fluctuations of the brane-world geometry we need to compute, first of all, the effective coupling strengths of the massive modes. Such couplings can be obtained from the canonical form of the effective action (B.118), after its dimensional reduction obtained by integrating out the y dependence appearing in the ψm components of the fluctuations. We insert, for this purpose, the expansion (B.120) into the action (B.118), and note that the term h2 is proportional (modulo a total derivative) to the mass term of the mode ψm. In fact: 3 2 = v v 3 dy f h m n dy f ψm ψn m.n d = v v dy f 3ψ ψ − ψ f 3ψ m n dy m n m n m.n 3 2 = vmvn dy f m ψmψn. (B.134) m.n

In the last step we have neglected a total derivative and used Eq. (B.121)forψm. Integrating over y, and taking into account the orthonormality condition (B.132), we 362 Appendix B: Higher-Dimensional Gravity arrive at a dimensionally reduced action which contains only the components vm(x) of the metric ﬂuctuations: M2 δS ≡ δS = 5 d4x v˙2 + v ∇2v − m2v2 . (B.135) m 4 m m m m m m

The summation symbol used here synthetically denotes that the contribution of the massless mode m = 0 has to be summed to the integral performed over the continuous spectrum of all massive modes (i.e. over all positive values of m up to +∞). Let us now introduce the variable hm representing the effective ﬂuctuations of the four-dimensional Minkowski metric evaluated on the hypersurface Σ4, namely:

( ) = ( , ) ≡ v ( ) ( ). hm x [hm x y ]y=0 m x ψm 0 (B.136)

In terms of this variable, the action (B.135) becomes 2 M ˙ 2 2 δS = 5 d4x h + h ∇2h − m2h . (B.137) 4ψ (0) m m m m m m

A comparison with the canonical form of the action for the tensor ﬂuctuations of the Minkowski geometry (see Eq. (9.48), traced over the two polarization modes) immediately lead us to conclude that the effective coupling constant for the mode hm is given by ( ) ≡ ( ) = −2 2 ( ). 8πG m MP m M5 ψm 0 (B.138)

Note that this effective coupling depends not only on the scale M5 typical of bulk gravitational interactions, but also on the position of the brane on the bulk manifold (since the bulk is curved, and its geometry is not translational invariant). In the case of massless ﬂuctuations, using Eqs. (B.122), (B.126), (B.127), we ﬁnd 1/2 ψ0 = (k/M5) ; the corresponding coupling parameter, that we may identity with the usual Newton constant G, is then given by

( ) ≡ = k . 8πG 0 8πG 3 (B.139) M5

For the massive ﬂuctuations, instead, the coupling is mass dependent: using the ( ) = −1/2 ( ) deﬁnitions ψm 0 M5 ψm 0 and the solutions (B.131), (B.133), we obtain

α [Y (α )J (α ) − J (α )Y (α )]2 8πG(m) = 0 1 0 2 0 1 0 2 0 , (B.140) 3 2( ) + 2( ) 2M5 J1 α0 Y1 α0 where α0 = m/k. Note that G(m) is referred to a continuous spectrum of values of m, hence it represents the effective coupling in the inﬁnitesimal mass interval between m and m + dm. Appendix B: Higher-Dimensional Gravity 363

We are now in the position of estimating the effective gravitational interactions on the four-dimensional brane-manifold Σ4, including the contribution of all (massless and massive) modes. Let us consider, as a simple but instructive example, the static gravitational ﬁeld produced by a point-like source of mass M localized on the brane. The linearized propagation equation for tensor metric ﬂuctuations on the Minkowski space–time of the brane, including the sources, is given by Eq. (8.10). Including a possible mass μν term, and using the effective coupling (B.140), we obtain for a generic mode hm :

μν 1 + m2 h =−16πG(m) τ μν − ημν τ . (B.141) m 2

→−∇2 ij → = μν → 0 = 00 → In the static limit we have , τ 0, τ η τμν τ0 ρ, and hm 2φm , where φm is the effective gravitational potential associated to the ﬂuctuations of mass m.Fromthe(0, 0) component of the above equation we then obtain −∇2 + 2 ( ) =− ( ) ( ), m φm x 4πG m ρ x (B.142) which represents a generalized Poisson equation controlling the massive mode contributions to the total static potential. The general solution for φm can be expressed using the standard method of the Green function, i.e. by setting 1 φ (x) =− d3x G (x, x)4πG(m)ρ(x), (B.143) m 4π m

where Gm (x, x ) satisﬁes 2 2 −∇ + m Gm (x, x ) = 4πδ(x − x ). (B.144)

Hence, by Fourier transforming, we obtain the following Green function,

d3 p ei p·(x−x ) G (x, x) = 4π , (B.145) m (2π)3 p2 + m2 valid for modes of arbitrary mass m. For the massless mode, in particular, the Green function is given by ∞ ( | − |) G ( , ) = 2 sin p x x = 1 . 0 x x dp (B.146) π 0 p|x − x | |x − x |

Inserting this result in Eq. (B.143), and considering a point-like source with ρ(x) = Mδ3(x), we obtain the well-known Newtonian solution 364 Appendix B: Higher-Dimensional Gravity

GM φ (0) =− , (B.147) m r where r =|x| (we have used the effective gravitational coupling of Eq. (B.139)). For a massive mode the Green function is given by

2 ∞ p2 sin(p|x − x|) e−m|x−x | G (x, x) = dp = , (B.148) m 2 2 π 0 p + m p|x − x | |x − x | and we obtain the potential

G(m)M φ (0) =− e−mr , (B.149) m r where the effective G(m) is deﬁned by Eq. (B.140). The total static potential produced by the point-like source is ﬁnally given by the sum of all massless and massive contributions, namely by

∞ = = + φ φm φ0 dm φm 0 m GM 1 ∞ =− 1 + dm G(m)e−mr . (B.150) r G 0

In the limit of weak fields, at large distances from the source, we see that the contribution of the massive fluctuations is exponentially suppressed, so that the dominant contribution to the above integral comes from the small-mass regime. For weak fields, we can then obtain an approximate estimate of the short-range corrections by using the small argument limit (m → 0) of the Bessel function appearing in the definition (B.140)ofG(m). In this limit we obtain

m m 8πG(m) −→ = 8πG (B.151) kM3 2k2 m → 0 2 5

(we have used Eq. (B.139)). The effective potential thus becomes, in the weak ﬁeld limit, GM 1 ∞ φ =− 1 + dm me−mr 2 r 2k 0 GM 1 =− 1 + . (B.152) r 2k2r 2

It follows that, in such a low-energy regime, the higher-dimensional corrections become important only at distance scales which are sufficiently small with respect to the curvature scale of the higher-dimensional manifold in which the brane is Appendix B: Higher-Dimensional Gravity 365 embedded. This means, in the particular case we are considering, that the corrections are important at distances r k−1, where k−1 is the curvature radius of the five- dimensional AdS bulk geometry (see Eq. (B.92)). At larger scales of distance the gravitational interaction experienced on the brane becomes effectively four-dimensional, quite irrespectively of the compactification and size of the extra dimensions. This result can be extended to space–times where the brane geometry is described by Ricci-flat metrics different from the Minkowski metric, and where the total number of dimensions is D > 5. References

1. Aharoni, J.: The Special Theory of Relativity. Oxford University Press, Oxford (1959) 2. Anderson, J.L.: Principles of Relativity Physics. Academic Press, New York (1967) 3. Appelquist, T., Chodos, A., Freund, P.G.O.: Modern Kaluza–Klein Theories. Ben- jamin/Cummings, Menlo Park (1985) 4. Becker, F., Becker, M., Schwarz, J.H.: String Theory and M Theory. Cambridge University Press, Cambridge (2007) 5. Castellani, L., D’Auria, R., Frè, P.: Supergravity and Superstrings: A Geometric Perspective. World Scientific, Singapore (1991) 6. Ciufolini, E., Gorini, V., Moschella, U., Frè, P. (eds.): Gravitational Waves. Institute of Physics Publishing, Bristol (2001) 7. Dodelson, S.: Modern Cosmology. Academic Press, San Diego (2003) 8. Durrer, R.: The Cosmic Microwave Background. Cambridge University Press, Cambridge (2008) 9. Gasperini, M.: Elements of String Cosmology. Cambridge University Press, Cambridge (2007) 10. Green, M.B., Schwartz, J., Witten, E.: Superstring Theory. Cambridge University Press, Cam- bridge (1987) 11. Hawking, S.W., Ellis, G.R.F.: The Large Scale Structure of Spacetime. University Press, Cam- bridge (1973) 12. Kolb, E.W., Turner, M.S.: The Early Universe. Addison Wesley, Redwood City (1990) 13. Landau, L.D., Lifshitz, E.M.: The Classical Theory of Fields. Pergamon Press, Oxford (1971) 14. Liddle, A.R., Lyth, D.H.: Cosmological Inflation and Large-Scale Structure. Cambridge Uni- versity Press, Cambridge (2000) 15. Maggiore, M.: Gravitational Waves. Oxford University Press, Oxford (2007) 16. Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation. Freeman, San Francisco (1973) 17. Ohanian, H.C., Ruffini, R.: Gravitation and Spacetime. W. W. Norton and Co., New York (1994) 18. Polchinski, J.: String Theory. Cambridge University Press, Cambridge (1998) 19. Rindler, W.: Essential Relativity. Springer, Berlin (1977) 20. Rindler, W.: Introduction to Special Relativity. Oxford University Press, Oxford (1991) 21. Ryan, M.P.,Shepley, L.C.: Homogeneous Relativistic Cosmologies. Princeton University Press, Princeton (1975) 22. Straumann, N.: General Relativity and Relativistic Astrophysics. Springer, Berlin (1991) 23. Wald, R.: General Relativity. The University of Chicago Press, Chicago (1984) 24. Weinberg, S.: Gravitation and Cosmology. Wiley, New York (1972) 25. Weinberg, S.: Cosmology. Oxford University Press, Oxford (2008)

26. West, P.C.: Introduction to Supersymmetry and Supergravity. World Scientiﬁc, Singapore (1990) 27. Zel’dovich, Y.B., Novikov, I.D.: Relativistic Astrophysics, vol. II. Chicago Press, Chicago (1983) 28. Zwiebach, B.: A First Course in String Theory. Cambridge University Press, Cambridge (2009) Index

A trace, 61 Adiabatic evolution, 22 Clifford algebra, 262 Affine connection, 54 Closed differential form, 305 Anholonomic indices, 244 Closed string, 356 Anti-de Sitter manifold, 343, 351, 353 CMB radiation, 188 Antisymmetric tensor fields, 341 B-mode polarization, 193 Atlas, 42 E-mode polarization, 193 Autoparallel curve, 57 polarization, 191, 193 Commutator of covariant derivatives, 100, 254, 311 B Compactification scale, 339 Barotropic fluid, 233 Conformal invariance, 77, 354 Basis one-form, 309 Conformal time, 183, 200 Belinfante–Rosenfeld procedure, 14 Conformal transformation, 333 Bianchi Congruence transformations, 42 contracted identity, 101, 321 Connection one-form, 309 identity, 99, 316 Constant-curvature manifold, 102 models, 231 Contortion tensor, 59, 253 Birkhoff theorem, 206 Contravariant tensors, 43, 44 Black hole, 213, 218 Cosmic gravitational waves, 182, 189 Boundary condition amplification, 184 of Dirichlet type, 356 mode equation, 183 of Neumann type, 356 spectral amplitude, 185 Brane-world scenario, 345 spectral energy density, 187 Bulk manifold, 345 spectral index, 186 Cosmological constant, 129, 130 Covariant C d’Alembert operator, 64 Canonical differential, 53 angular momentum tensor, 11, 13 divergence, 63 energy-momentum tensor, 6, 8 tensors, 43, 44 Cartan algebra, 304 Covariant conservation Chart, 42 of the electric charge, 72 Chiral fermions, 343 of the energy-momentum tensor, 124, Christoffel 132, 321 connection, 60 Covariant derivative symbol, 59 along a curve, 57 © Springer International Publishing AG 2017 369 M. Gasperini, Theory of Gravitational Interactions, UNITEXT for Physics, DOI 10.1007/978-3-319-49682-5 370 Index

of a contravariant vector, 55 for perfect fluids, 21 of a covariant vector, 56 for point-like particles, 19, 27 of a mixed tensor, 56 for scalar fields, 16 of a tensor density, 62 symmetrization, 14 of the totally antisymmetric tensor, 68 Equations of motion Critical density, 187 for a free string, 355 Curvature two-form, 311, 315 for a p-brane, 349 for point-like test bodies, 133 for spinning test bodies, 134, 136, 137 D Eternal black hole, 218 De Donder gauge, 143 Euler–Lagrange equations, 3 De Sitter manifold, 40, 104, 106, 113, 114, Exact differential form, 305 131, 352 Experiment Diffeomorphism, 42 of Pound and Rebka, 89 Dilaton field, 344 of Reasenberg and Shapiro, 151 Dimensional reduction, 236, 331, 334, 362 Exterior Dirac equation co-derivative operator, 308 from a symmetrised action, 268 derivative, 305 in curved space-time, 266, 268, 271 product, 305 in flat space-time, 262 Exterior covariant derivative, 309, 313 in the Einstein–Cartan theory, 326 of contravariant vectors, 310 Dirac matrices, 262 of mixed tensors, 310 Dirichlet brane, 345, 354 of spinor fields, 325 Dixon–Mathisson–Papapetrou equation, of the Minkowski metric, 310 137 Extrinsic Dynamical energy-momentum tensor, 123 curvature, 121 conservation, 123, 132 geometry, 34 for electromagnetic fields, 126 for gravitational waves, 168, 195 for p-branes, 349 F for perfect fluids, 129 Finsler geometry, 35 for point-like particles, 128 Five-dimensional gravity, 330 for scalar fields, 126 Free particle action for spinor fields, 273 in curved space-time, 81 in Minkowski space-time, 25, 26 Freund–Rubin mechanism, 343, 344 E Einstein angle, 148 Einstein–Cartan equations, 318, 320 G Einstein equations, 122 Gauge for a spherically symmetric field, 205 covariant derivative, 247 in linearized form, 142, 143 invariance of the electromagnetic action, in the language of exterior forms, 319, 72 323 potential, 247 in the vierbein formalism, 257 Gauge theory, 246 with cosmological constant, 129 for the Lorentz group, 255 Einstein–Hilbert action, 116, 118 for the Poincarè group, 313 in the language of exterior forms, 317 Gauss in the vierbein formalism, 256 curvature, 105 Einstein tensor, 101 theorem, 63 Energy-momentum conservation, 7, 9 Geodesic Energy-momentum tensor in Minkowski completeness, 215 space-time deviation, 96, 98 for electromagnetic fields, 16 equation of motion, 83, 133 Index 371

motion in the Schwarzschild geometry, Higher-curvature corrections, 117 209 Hodge duality, 306 Geometric object, 42 Holonomic indices, 244 Global infinitesimal Homogeneous anisotropic metric, 232 Lorentz transformations, 11 Hubble horizon, 184 translations, 6, 26 Hypersphere Global supersymmetry with four space-time dimensions, 38, 113 and space-time translations, 279, 295 with n space-like dimensions, 105, 111 in the graviton-gravitino system, 284, 295 in the spin 0-spin 1/2system,277 I in the Wess–Zumino model, 280, 295 Induced metric, 34 Grassmann algebra, 276 Inflation, 195 Gravitational antennas Intrinsic geometry, 34 interferometric detectors, 181 Isometries, 50, 64, 65 resonant detectors, 178, 180 Gravitational coupling in five dimensions, 334 J in four dimensions, 116, 144 Jacobian matrix, 41 in higher dimensions, 339 of massive modes, 362, 364 Gravitational deflection K of a massive particle, 154, 156 Kaluza–Klein of light, 146, 147 gravity, 330, 331 Gravitational effects zero modes, 332 Kasner solution, 235 frequency shift, 88–91 Killing vectors, 51, 64–66, 134, 137, 140, lensing, 148 335 time delay, 149 Kruskal time dilatation, 88, 223 coordinates, 216 velocity shift, 153 plane, 218 Gravitational radiation from a binary system, 172 from a harmonic oscillator, 195 L radiated power, 170, 195 Lagrangian density, 1 radiation zone, 166 Levi-Civita Gravitational waves antisymetric symbol, 47 helicity, 163, 195 connection, 253 interaction with test masses, 175 Lie algebra polarization states, 162, 176, 195 of Killing vectors, 335 quadrupole approximation, 169 of SUSY generators, 279 retarded solutions, 164 of the Lorentz group, 12 wave equation, 160, 182 of the Poincaré group, 314 Gravitino, 282 Lie algebra of the Lorentz group, 248 consistency condition, 292 Lie derivative, 51 motion in curved space-time, 292 Local motion in flat space-time, 283 supersymmetry and supergravity, 285 Graviton, 164 symmetries, 246 Gravity confinement, 357, 360 Weyl transformations, 354 Local infinitesimal coordinate transformations, 49 H second-order transformations, 52 Hamiltonian density, 9 translations, 123 Harmonic gauge, 64, 143, 153 Local Lorentz 372 Index

invariance, 246 P transformations, 246, 248 Palatini Locally inertial system, 37, 54 formalism, 255, 290, 317 Lorentz identity, 119 connection, 248, 252 Pauli–Lyubarskii spin vector, 24 curvature, 254 Pauli matrices, 263 group, 248 P-brane, 346 Lorentz covariant derivative, 248 action, 348 commutator, 254 equation of motion, 349 of controvariant vectors, 249 tension, 347 of covariant vectors, 250 Perihelion precession of mixed tensors, 250 in a Newtonian field, 30 of the gravitino, 286 in the Schwarzschild field, 212 of the totally antisymmetric tensor, 258 Planck experiment, 194 of the vierbein, 251 Poincaré transformations, 38, 51, 279 Lorentz generators Polyakov action, 82, 348 for spinor representations, 263 Principle for vector representations, 12, 249 of equivalence, 36, 37, 84, 96 of general covariance, 32 Pulsars, 188 M Majorana spinors, 276, 295 Q Massive modes Quadrupole moment, 167 in brane-world gravity, 360, 363 in Kaluza–Klein gravity, 332 Maximal analytical extension, 215 R Maximally symmetric manifolds, 104, 341 Radar-echo delay, 149 Maxwell equations in curved space–times, Randall–Sundrum model, 351, 357 74 Rarita–Schwinger action, 282 Metric-compatible connection, 60 Ricci Metric determinant, 61, 67 rotation coefficients, 252 Metricity condition, 251 tensor, 100 Milne space-time, 237 Riemann Minimal-action principle, 3 geometry, 33 Minimal coupling principle, 69, 288 manifold, 41 Mixed tensors, 43 metric, 33, 45 tensor, 98, 99 Riemann–Cartan geometry, 60, 286, 315 N Rindler space-time, 105, 107, 216, 226, 238 Nambu–Goto action, 347 Newtonian approximation, 84–86 S Non-Abelian Scalar curvature, 101 gauge theory, 247 Scale factor, 182, 234 gauge transformation, 247, 337 Schwarzschild isometries, 231, 335, 336 horizon, 214 Non-metricity tensor, 59 radius, 208 Nöther theorem, 5 singularity, 215 Nucleosynthesis, 188 solution, 207 solution in isotropic form, 208 Shapiro effect, 151 O Short-range gravitational corrections, 361, Open string, 356 363, 364 Index 373

Similarity transformations, 41 Tensor densities, 46 Simple supergravity model, 289 Torsion, 54 Spherically symmetric geometry, 204 for the Dirac field, 326 Spinning test body, 134 for the gravitino field, 291 Spinor currents, 268 Torsion two-form, 312, 314 Spontaneous compactification, 340, 342 Totally antisymmetric tensor, 47, 48 Static geometry, 206, 226 TT gauge, 162 Stationary geometry, 206 Twin paradox, 221 Stereographic coordinates, 102, 105, 109 inthepresenceofgravity,222 Stokes parameters, 192 in the Schwarzschild geometry, 224 Structure equations, 312 Supergravity, 275 Supergravity equations V for the gravitino, 292 Vacuum energy density, 130 for the metric, 291 Vielbeins, 244 for the torsion, 290 Vierbeins, 244 in the language of exterior form, 327 Volcano-like potential, 359 local SUSY properties, 289, 295 Supersymmetry, 275 Symmetry and conserved currents, 6 W transformation, 4 Warp factor, 335 Weak-field approximation, 141, 145 White hole, 220 T Tangent manifold, 37, 244 Tangent space projection, 245 Y Tedrads, 244 York-Gibbons-Hawking action, 118, 120