Differential Forms on Riemannian

Annales de la Fondation Louis de Broglie, Volume 32 no 4, 2007 425

Diﬀerential Forms on Riemannian (Lorentzian) and Riemann-Cartan Structures and Some Applications to Physics

Waldyr Alves Rodrigues Jr.

Institute of Mathematics Statistics and Scientiﬁc Computation IMECC-UNICAMP CP 6065, 13083760 Campinas SP, Brazil email: [email protected], [email protected]

ABSTRACT. In this paper after recalling some essential tools concerning the theory of differential forms in the Cartan, Hodge and Clifford bundles over a Riemannian or Riemann-Cartan space or a Lorentzian or Riemann-Cartan spacetime we solve with details several exercises involving different grades of difficult. One of the problems is to show that a recent formula given in [10] for the exterior covariant derivative of the Hodge dual of the torsion 2-forms is simply wrong. We believe that the paper will be useful for students (and eventually for some experts) on applications of differential geometry to some physical problems. A detailed account of the issues discussed in the paper appears in the table of contents.

Contents

1 Introduction 428

2 Classiﬁcation of Metric Compatible Structures (M, g,D)429 2.1 Levi-Civita and Riemann-Cartan Connections ...... 430 2.2 Spacetime Structures ...... 430

3 Absolute Diﬀerential and Covariant Derivatives 431 ^ 4 Calculus on the Hodge Bundle ( T ∗M, ·, τg) 434 4.1 Exterior Product ...... 434 4.2 Scalar Product and Contractions ...... 435 4.3 Hodge Star Operator ? ...... 436 4.4 Exterior derivative d and Hodge coderivative δ ...... 437 426 W. A. Rodrigues Jr.

5 Clifford Bundles 437 5.1 Clifford Product ...... 438 5.2 Dirac Operators Acting on Sections of a Clifford Bundle C`(M, g)...... 440 5.2.1 The Dirac Operator ∂ Associated to D ...... 440

5.2.2 Cliﬀord Bundle Calculation of Dea A ...... 440 5.2.3 The Dirac Operator ∂| Associated to D˚ ...... 441

6 Torsion, Curvature and Cartan Structure Equations 441 6.1 Torsion and Curvature Operators ...... 441 6.2 Torsion and Curvature Tensors ...... 442 6.2.1 Properties of the Riemann Tensor for a Metric Compatible Connection ...... 442 6.3 Cartan Structure Equations ...... 444

7 Exterior Covariant Derivative D 445 7.1 Properties of D ...... 446 a 7.2 Formula for Computation of the Connection 1- Forms ωb 446

8 Relation Between the Connections D˚ and D 447

9 Expressions for d and δ in Terms of Covariant Derivative Operators D˚ and D 449

10 Square of Dirac Operators and D’ Alembertian, Ricci and Einstein Operators 449 10.1 The Square of the Dirac Operator ∂| Associated to D˚ . . . 449 10.2 The Square of the Dirac Operator ∂ Associated to D . . . 453

11 Coordinate Expressions for Maxwell Equations on Lorentzian and Riemann-Cartan Spacetimes 455 11.1 Maxwell Equations on a Lorentzian Spacetime ...... 455 11.2 Maxwell Equations on Riemann-Cartan Spacetime . . . . 458

12 Bianchi Identities 461 12.1 Coordinate Expressions of the First Bianchi Identity . . . 462 Diﬀerential Forms on Riemannian (Lorentzian). . . 427

13 A Remark on Evans 101th Paper on “ECE Theory” 467

14 Direct Calculation of D ? T a 468 14.1 Einstein Equations ...... 470

15 Two Counterexamples to Evans (Wrong) Equation a a b “D ? T = ?Rb ∧ θ ” 471 15.1 The Riemannian Geometry of S2 ...... 471 15.2 The Teleparallel Geometry of (S˚2, g,D) ...... 473

16 Conclusions 475 428 W. A. Rodrigues Jr.

1 Introduction

In this paper we first recall some essential tools concerning the theory of differential forms in the Cartan, Hodge and Clifford bundles over a n- 0 dimensional manifold M equipped with a metric tensor g ∈ sec T2 M of arbitrary signature (p, q), p+q = n and also equipped with metric compatible connections, the Levi-Civita (D˚) and a general Riemann-Cartan (D) one1. After that we solved with details some exercises involving different grades of difficult, ranging depending on the readers knowledge from kindergarten, intermediate to advanced levels. In particular we show how to express the derivative ( d) and coderivative (δ) operators as functions of operators related to the Levi-Civita or a Riemann-Cartan connection defined on a manifold, namely the standard Dirac operator (∂|) and general Dirac operator (∂). Those operators are then used to express Maxwell equations in both a Lorentzian and a Riemann-Cartan spacetime. We recall also important formulas (not well known as they deserve to be) for the square of the general Dirac and standard Dirac operators showing their relation with the Hodge D’Alembertian (♦), the ˚ ˚a a covariant D’ Alembertian () and the Ricci operators (R , R ) and ˚ Einstein operator () and the use of these operators in the Einstein- Hilbert gravitational theory. Finally, we study the Bianchi identities. a a b a Recalling that the first Bianchi identity is DT = Rb ∧ θ , where T a b and Rb are respectively the torsion and the curvature 2-forms and {θ } is a cotetrad we ask the question: Who is D? T a? We find the correct answer (Eq.(218)) using the tools introduced in previous sections of the a a b paper. Our result shows explicitly that the formula “D?T = ?Rb ∧θ ” recently found in [10] and claimed to imply a contradiction in Einstein- Hilbert gravitational theory is wrong. Two very simple counterexamples contradicting the wrong formula for D? T a are presented. A detailed account of the issues discussed in the paper appears in the table of con- tents2. We call also the reader attention that in the physical applications we use natural units for which the numerical values of c,h and the gravitational constant k (appearing in Einstein equations) are equal to 1.

1A spacetime is a special structure where the manifold is 4-dimensional, the metric has signature (1, 3) and which is equipped with a Levi-Civita or a Riemann-Cartan connection, orientability and time orientation. See below and, e.g., [22, 26] for more details, if needed. 2More on the subject may be found in, e.g., [22] and recent advanced material may be found in several papers of the author posted on the arXiv. Diﬀerential Forms on Riemannian (Lorentzian). . . 429

2 Classification of Metric Compatible Structures (M, g,D) 3 Let M denotes a n-dimensional manifold . We denote as usual by TxM ∗ and Tx M respectively the tangent and the cotangent spaces at x ∈ [ ∗ [ x M. By TM = TxM and T M = Tx M respectively x∈M x∈M r the tangent and cotangent bundles. By Ts M we denote the bundle L∞ r of r-contravariant and s-covariant tensors and by T M = r,s=0 Ts M ^r ^r the tensor bundle. By TM and T ∗M denote respectively the ^ bundles of r-multivector fields and of r-form fields. We call TM = Mr=n^r TM the bundle of (non homogeneous) multivector fields and r=0 ^ Mr=n^r call T ∗M = T ∗M the exterior algebra (Cartan) bundle. Of r=0 course, it is the bundle of (non homogeneous) form fields. Recall that Vr Vr ∗ n the real vector spaces are such that dim TxM = dim Tx M = r ^ and dim T ∗M = 2n. Some additional structures will be introduced or 4 0 mentioned below when needed. Let g ∈ sec T2 M a metric of signature (p, q) and D an arbitrary metric compatible connection on M, i.e., Dg = 0. We denote by R and T respectively the (Riemann) curvature and torsion tensors5 of the connection D, and recall that in general a given manifold given some additional conditions may admit many different metrics and many different connections. Given a triple (M, g,D): (a) it is called a Riemann-Cartan space if and only if

Dg = 0 and T 6= 0. (1)

(b) it is called Weyl space if and only if

Dg 6= 0 and T = 0. (2)

Dg = 0 and T = 0, (3)

3We left the toplogy of M unspecified for a while. 4We denote by sec(X(M)) the space of the sections of a bundle X(M). Note that all functions and differential forms are supposed smooth, unless we explicitly say the contrary. 5The precise definitions of those objects will be recalled below. 430 W. A. Rodrigues Jr. and in that case the pair (D, g) is called Riemannian structure. ( d) it is called Riemann-Cartan-Weyl space if and only if

Dg 6= 0 and T 6= 0. (4)

(e) it is called ( Riemann) ﬂat if and only if

Dg = 0 and R = 0,

(f) it is called teleparallel if and only if

Dg = 0, T 6= 0 and R =0. (5)

2.1 Levi-Civita and Riemann-Cartan Connections For each metric tensor defined on the manifold M there exists one and only one connection in the conditions of Eq.(3). It is is called Levi-Civita connection of the metric considered, and is denoted in what follows by D˚. A connection satisfying the properties in (a) above is called a Riemann- Cartan connection. In general both connections may be defined in a given manifold and they are related by well established formulas recalled below. A connection defines a rule for the parallel transport of vectors (more generally tensor fields) in a manifold, something which is conventional [20], and so the question concerning which one is more important is according to our view meaningless6. The author knows that this assertion may surprise some readers, but he is sure that they will be convinced of its correctness after studying Section 15. More on the subject in [22]. For implementations of these ideas for the theory of gravitation see [18] 2.2 Spacetime Structures Remark 1 When dim M = 4 and the metric g has signature (1, 3) we sometimes substitute Riemann by Lorentz in the previous definitions (c),(e) and (f).

Remark 2 In order to represent a spacetime structure a Lorentzian or a Riemann-Cartan structure (M, g,D) need be such that M is con- nected and paracompact [11] and equipped with an orientation deﬁned by

6Even if it is the case, that a particular one may be more convenient than others for some purposes. See the example of the Nunes connections in Section 15. Diﬀerential Forms on Riemannian (Lorentzian). . . 431

^4 the volume element τg ∈ sec T ∗M and a time orientation denoted by ↑. We omit here the details and ask to the interested reader to con- sult, e.g., [22]. A general spacetime will be represented by a pentuple (M, g, D, τg, ↑).

3 Absolute Differential and Covariant Derivatives Given a differentiable manifold M, let X,Y ∈ sec TM be any vector ∗ L∞ r fields and α ∈ sec T M any covector field . Let T M = r,s=0 Ts M be the tensor bundle of M and P ∈ sec T M any general tensor field. We now describe the main properties of a general connection D (also called absolute differential operator). We have

D : sec TM × sec T M → sec T M,

(X, P) 7→ DX P, (6)

where DX the covariant derivative in the direction of the vector field X satisfy the following properties: Given, differentiable functions f, g : M → R, vector fields X,Y ∈ sec TM and P, Q ∈ sec T M we have

DfX+gY P = fDX P+gDY P,

DX (P + Q) = DX P + DX Q,

DX (fP) = fDX (P)+X(f)P,

DX (P ⊗ Q) = DX P ⊗ Q + P⊗DX Q. (7)

r Given Q ∈ sec Ts M the relation between DQ, the absolute diﬀeren- tial of Q and DX Q the covariant derivative of Q in the direction of the vector ﬁled X is given by

r r D: sec Ts M → sec Ts+1M,

DQ(X,X1, ..., Xs, α1, ..., αr)

= DX Q(X1, ..., Xs, α1, ..., αr), ∗ X1, ..., Xs ∈ sec T M, α1, ...αr ∈ sec T M. (8)

Let U ⊂ M and consider a chart of the maximal atlas of M covering 7 µ 0 U coordinate functions {x }. Let g ∈ sec T2 M be a metric ﬁeld for 7If e ∈ M, then xµ(e) = xµ is the µ coordinate of e in the given chart. 432 W. A. Rodrigues Jr.

µ µ M. Let {∂µ} be a basis for TU, U ⊂ M and let {θ = dx } be the µ dual basis of {∂µ}. The reciprocal basis of {θ } is denoted {θµ}, and µ µ µ g(θ , θν ) := θ · θν = δν . Introduce next a set of diﬀerentiable functions a ν qµ, qb : U → R such that :

µ b b µ a µ qa qµ = δa , qa qν = δν . (9) It is trivial to verify the formulas

a b µν µ ν ab gµν = qµqν ηab , g = qa qbη , µ ν ab a b µν ηab = qa qbgµν , η = qµqν g , (10) with ηab = diag(1, ..., 1 −1, ... − 1) | {z } | {z } . (11) p times q times Moreover, deﬁning ν eb = qb∂ν

the set {ea} with ea ∈ sec TM is an orthonormal basis for TU. The a a a µ dual basis of TU is {θ }, with θ = qµdx . Also, {θb} is the reciprocal a a a basis of {θ }, i.e. θ · θb = δb.

Remark 3 When dim M = 4 the basis {ea} of TU is called a tetrad and the (dual) basis {θa} of T ∗U is called a cotetrad. The names are appropriate ones if we recall the Greek origin of the word.

The connection coeﬃcients associated to the respective covariant derivatives in the respective basis will be denoted as:

ρ µ µ α D ∂ν = Γ ∂ρ,D ∂ = −Γ ∂ , (12) ∂µ µν ∂σ σα c b b c c D e eb = ω ec,D e e = −ω e ,D eb = ω ec, a ab a ac ∂µ µb ν ν α ρ D dx = −Γ dx ,D θν = Γ θρ, (13) ∂µ µα ∂µ µν b b c b b a D e θ = −ω θ ,D θ = −ω θ (14) a ac ∂µ µa b c D ea θ = −ωcabθ , d bc bk cl bc cb ωabc = ηadωbc = −ωcba, ωa = η ωkalη , ωa = −ωa etc... (15) Diﬀerential Forms on Riemannian (Lorentzian). . . 433

Remark 4 The connection coeﬃcients of the Levi-Civita Connection in a coordinate basis are called Christoﬀel symbols. We write in what follows ρ ν ν ρ D˚ ∂ν = ˚Γ ∂ρ, D˚ dx = −˚Γ dx . (16) ∂µ µν ∂µ µρ

To understood how D works, consider its action, e.g., on the sections 1 ∗ of T1 M = TM ⊗ T M. D(X ⊗ α) = (DX) ⊗ α + X ⊗ Dα. (17)

For every vector ﬁeld V ∈ sec TU and a covector ﬁeld C ∈ sec T ∗U we have

α α D V = D (V ∂α),D C = D (Cαθ ) (18) ∂µ ∂µ ∂µ ∂µ and using the properties of a covariant derivative operator introduced above, D V can be written as: ∂µ

α α D V = D (V ∂α) = (D V ) ∂α ∂µ ∂µ ∂µ α α = (∂µV )∂α + V D ∂α ∂µ ∂V α = + V ρΓα ∂ := (D+V α)∂ , (19) ∂xµ µρ α µ α

+ α where it is to be kept in mind that the symbol Dµ V is a short notation for D+V α := (D V )α (20) µ ∂µ Also, we have

α α D C = D (Cαθ ) = (D C)αθ ∂µ ∂µ ∂µ ∂C = α − C Γβ θα, ∂xµ β µα − α := (Dµ Cα)θ (21)

8 − where it is to be kept in mind that that the symbol Dµ Cα is a short notation for − D Cα := (D C)α. (22) µ ∂µ

8 α α Recall that other authors prefer the notations (D V ) := V and (D C)α := ∂µ :µ ∂µ Cα:µ. What is important is always to have in mind the meaning of the symbols. 434 W. A. Rodrigues Jr.

Remark 5 The necessity of precise notation becomes obvious when we calculate

− a a a ν a ρ a a b D q := (D θ )ν = (D q dx )ν = ∂µq − Γ q = ω q , µ ν ∂µ ∂µ ν ν µν ρ µb ν + a a a a ρ a ρ a D q := (D q ea) = ∂µq + ω q = Γ q , µ ν ∂µ ν ν µν ρ µν ρ − a + a thus verifying that Dµ qν 6= Dµ qν 6= 0 and that

a a b a b ∂µqν + ωµbqν − Γµbqν = 0. (23) Moreover, if we deﬁne the object

a a µ 1 1 q = ea ⊗ θ = qµ ea ⊗ dx ∈ sec T1 U ⊂ sec T1 M, (24) which is clearly the identity endormorphism acting on sections of TU, we ﬁnd

a a a a b a b Dµq := (D q) = ∂µq + ω q − Γ q = 0. (25) ν ∂µ ν ν µb ν µb ν

1 Remark 6 Some authors call q ∈ sec T1 U (a single object) a tetrad, thus forgetting the Greek meaning of that word. We shall avoid this nomenclature. Moreover, Eq.(25) is presented in many textbooks (see, e.g., [4, 13, 24]) and articles under the name ‘tetrad postulate’ and it is said that the covariant derivative of the “tetrad” vanish. It is obvious that Eq.(25) it is not a postulate, it is a trivial (freshman) identity. In those books, since authors do not distinguish clearly the derivative operators D+,D− and D, Eq.(25) becomes sometimes misunderstood as − a + a meaning Dµ qν or Dµ qν , thus generating a big confusion and producing errors (see below). ^ 4 Calculus on the Hodge Bundle ( T ∗M, ·, τg) ^ We call in what follows Hodge bundle the quadruple ( T ∗M, ∧, ·, τg). We now recall the meaning of the above symbols. 4.1 Exterior Product We suppose in what follows that any reader of this paper knows the meaning of the exterior product of form ﬁelds and its main properties9. Vr ∗ Vs ∗ We simply recall here that if Ar ∈ sec T M, Bs ∈ sec T M then rs Ar ∧ Bs = (−1) Bs ∧ Ar. (26) 9We use the conventions of [22]. Diﬀerential Forms on Riemannian (Lorentzian). . . 435

4.2 Scalar Product and Contractions Vr ∗ Vr ∗ Let be Ar = a1 ∧ ... ∧ ar ∈ sec T M, Br = b1 ∧ ... ∧ br ∈ sec T M V1 ∗ where ai, bj ∈ sec T M (i, j = 1, 2, ..., r).

(i) The scalar product Ar ·Br is deﬁned by

Ar ·Br = (a1 ∧ ... ∧ ar) · (b1 ∧ ... ∧ br)

a1 · b1 ....a1 · br

= ...... (27)

ar · b1 ....ar · br where ai · bj := g(ai, bj). We agree that if r = s = 0, the scalar product is simple the ordinary product in the real ﬁeld.

Also, if r 6= s, then Ar ·Bs = 0. Finally, the scalar product is extended ^ by linearity for all sections of T ∗M.

For r ≤ s, Ar = a1 ∧ ... ∧ ar, Bs = b1 ∧ ... ∧ bs we deﬁne the left contraction by

X i1....is ∼ :(Ar, Bs) 7→ Ar Bs = (a1∧...∧ar)·(b ∧...∧bi ) bi +1∧...∧bi y y i1 r r s i1 <...

p ^ ∗ ∼: sec T M 3 a1 ∧ ... ∧ ap 7→ ap ∧ ... ∧ a1 (29) ^ and extended by linearity to all sections of T ∗M. We agree that for α, β ∈ sec V0 T ∗M the contraction is the ordinary (pointwise) product V0 ∗ Vr ∗ in the real ﬁeld and that if α ∈ sec T M, Ar ∈ sec T M, Bs ∈ Vs ∗ sec T M then (αAr)yBs = Ary(αBs). Left contraction is extended ^ by linearity to all pairs of elements of sections of T ∗M, i.e., for A, B ∈ ^ sec T ∗M

X AyB = hAiryhBis, r ≤ s, (30) r,s r ^ ∗ where hAir means the projection of A in T M. It is also necessary to introduce the operator of right contraction denoted by x. The deﬁnition is obtained from the one presenting the 436 W. A. Rodrigues Jr. left contraction with the imposition that r ≥ s and taking into ac- Vr ∗ Vs ∗ count that now if Ar ∈ sec T M, Bs ∈ sec T M then BsyAr = s(r−s) (−1) ArxBs. 4.3 Hodge Star Operator ? The Hodge star operator is the mapping k n−k ^ ∗ ^ ∗ ? : sec T M → sec T M, Ak 7→ ?Ak

Vk ∗ where for Ak ∈ sec T M k ^ ∗ [Bk ·Ak]τg = Bk ∧ ?Ak, ∀Bk ∈ sec T M (31) n τg ∈ V T ∗M is the metric volume element. Of course, the Hodge star operator is naturally extended to an isomorphism ? : sec V T ∗M → sec V T ∗M by linearity. The inverse ?−1 : sec Vn−r T ∗M → sec Vr T ∗M of the Hodge star operator is given by: ?−1 = (−1)r(n−r)sgng?, (32) where sgn g = det g/| det g| denotes the sign of the determinant of the matrix (gαβ = g(eα, eβ)), where {eα} is an arbitrary basis of TU. We can show that (see, e.g., [22]) that

?Ak = Aekyτg, (33)

where as noted before, in this paper Aek denotes the reverse of Ak. α ∗ Let {ϑ } be the dual basis of {eα} (i.e., it is a basis for T U ≡ V1 ∗ α β αβ αβ β µ1...µp T U) then g(ϑ , ϑ ) = g , with g gαρ = δρ . Writing ϑ = ϑµ1 ∧ ... ∧ ϑµp , ϑνp+1...νn = ϑνp+1 ∧ ... ∧ ϑνn we have from Eq.(33) 1 p ? θµ1...µp = |det g|gµ1ν1 ...gµpνp ϑνp+1...νn . (34) (n − p)! ν1...νn Some identities (used below) involving the Hodge star operator, the exterior product and contractions are10:

Ar ∧ ?Bs = Bs ∧ ?Ar; r = s Ar · ?Bs = Bs · ?Ar; r + s = n r(s−1) ˜ Ar ∧ ?Bs = (−1) ? (AryBs); r ≤ s (35) rs ˜ Ary ?Bs = (−1) ? (Ar ∧ Bs); r + s ≤ n ?τg = sign g; ?1 = τg. 10See also the last formula in Eq.(45) which uses the Cliﬀord product. Diﬀerential Forms on Riemannian (Lorentzian). . . 437

4.4 Exterior derivative d and Hodge coderivative δ The exterior derivative is a mapping ^ ^ d : sec T ∗M → sec T ∗M,

satisfying: (i) d(A + B) = dA + dB; (ii) d(A ∧ B) = dA ∧ B + A¯ ∧ dB; (36) (iii) df(v) = v(f); (iv) d2 = 0, for every A, B ∈ sec V T ∗M, f ∈ sec V0 T ∗M and v ∈ sec TM. The Hodge codiﬀerential operator in the Hodge bundle is the mapping δ : sec Vr T ∗M → sec Vr−1 T ∗M, given for homogeneous multiforms, by:

δ = (−1)r ?−1 d?, (37) where ? is the Hodge star operator. The operator δ extends by linearity to all V T ∗M The Hodge Laplacian (or Hodge D’Alembertian) operator is the mapping ^ ∗ ^ ∗ ♦ : sec T M → sec T M given by: ♦ = −(dδ + δd). (38) The exterior derivative, the Hodge codiﬀerential and the Hodge D’ Alembertian satisfy the relations:

2 dd = δδ = 0; ♦ = (d − δ) d = d; δ = δ ♦ ♦ ♦ ♦ (39) δ? = (−1)r+1 ? d; ?δ = (−1)rd? dδ? = ?δd; ?dδ = δd?; ?♦ = ♦ ?.

5 Cliﬀord Bundles Let (M, g, ∇) be a Riemannian, Lorentzian or Riemann-Cartan struc- 11 2 ture . As before let g ∈ sec T0 M be the metric on the cotangent 0 ∗ p,q p,q bundle associated with g ∈ sec T2 M. Then Tx M ' R , where R

11∇ may be the Levi-Civita connection D˚ of g or an arbitrary Riemann-Cartan connection D. 438 W. A. Rodrigues Jr.

is a vector space equipped with a scalar product • ≡ g|x of signature (p, q). The Clifford bundle of differential forms C`(M, g) is the ∗ bundle of algebras, i.e., C`(M, g) = ∪x∈M C`(Tx M, •), where ∀x ∈ M, ∗ C`(Tx M, •) = Rp,q, a real Clifford algebra. When the structure (M, g, ∇) ∗ is part of a Lorentzian or Riemann-Cartan spacetime C`(Tx M, •) = R1,3 the so called spacetime algebra. Recall also that C`(M, g) is a vector bundle associated with the g-orthonormal coframe bundle P e (M, g), SO(p,q) i.e., C`(M, g) = P e (M, g)× (see more details in, e.g., [16, 22]). SO(p,q) adR1,3 ∗ For any x ∈ M, C`(Tx M, •) is a linear space over the real field R. More- ∗ over, C`(Tx M) is isomorphic as a real vector space to the Cartan alge- V ∗ bra Tx M of the cotangent space. Then, sections of C`(M, g) can be represented as a sum of non homogeneous differential forms. Let now a {ea} be an orthonormal basis for TU and {θ } its dual basis. Then, g(θa, θb) = ηab. 5.1 Clifford Product The fundamental Clifford product (in what follows to be denoted by juxtaposition of symbols) is generated by θaθb + θbθa = 2ηab (40) and if C ∈ C`(M, g) we have

1 1 C = s + v θa + b θaθb + a θaθbθc + pθn+1 , (41) a 2! ab 3! abc n+1 0 1 2 3 n where τg := θ = θ θ θ θ ...θ is the volume element and s, va, bab, V0 ∗ aabc, p ∈ sec T M,→ sec C`(M, g). Vr ∗ Vs ∗ Let Ar, ∈ sec T M,→ sec C`(M, g), Bs ∈ sec T M,→ sec C`(M, g). For r = s = 1, we deﬁne the scalar product as follows: For a, b ∈ sec V1 T ∗M,→ sec C`(M, g), 1 a · b = (ab + ba) = g(a, b). (42) 2 We identify the exterior product ((∀r, s = 0, 1, 2, 3, ..., n) of homogeneous forms (already introduced above) by

Ar ∧ Bs = hArBsir+s, (43)

Vk ∗ where hik is the component in T M (projection) of the Clifford field. The exterior product is extended by linearity to all sections of C`(M, g). Differential Forms on Riemannian (Lorentzian). . . 439

The scalar product, the left and the right are defined for homogeneous form fields that are sections of the Clifford bundle in exactly the same way as in the Hodge bundle and they are extended by linearity for all sections of C`(M, g). In particular, for A, B ∈ sec C`(M, g) we have

X AyB = hAiryhBis, r ≤ s. (44) r,s

The main formulas used in the present paper can be obtained (details may be found in [22]) from the following ones (where a ∈ sec V1 T ∗M,→ sec C`(M, g)):

aBs = ayBs + a ∧ Bs, Bsa = Bsxa + Bs ∧ a, 1 a B = (aB − (−1)sB a), y s 2 s s r(s−r) AryBs = (−1) BsxAr, 1 a ∧ B = (aB + (−1)sB a), s 2 s s ArBs = hArBsi|r−s| + hArBsi|r−s|+2 + ... + hArBsi|r+s| m X = hArBsi|r−s|+2k k=0

Ar ·Br = Br ·Ar = Aer yBr = ArxBer = hAerBri0 = hArBeri0, ?Ak = Aekyτg = Aekτg. (45)

Two other important identities to be used below are:

ˆ ay(X ∧ Y) = (ayX ) ∧ Y + X ∧ (ayY), (46) ^1 ^ for any a ∈ sec T ∗M and X , Y ∈ sec T ∗M, and

Ay(ByC) = (A ∧ B)yC, (47) for any A, B, C ∈ sec V T ∗M,→ C`(M, g) 440 W. A. Rodrigues Jr.

5.2 Dirac Operators Acting on Sections of a Cliﬀord Bundle C`(M, g)

5.2.1 The Dirac Operator ∂ Associated to D

The Dirac operator associated to a general Riemann-Cartan structure (M, g,D) acting on sections of C`(M, g) is the invariant ﬁrst order differential operator a α ∂ = θ Dea = ϑ Deα . (48)

For any A ∈ sec V T ∗M,→ sec C`(M, g) we deﬁne

∂A = ∂ ∧ A + ∂yA a a ∂ ∧ A = θ ∧ (Dea A), ∂yA = θ y(Dea A). (49)

5.2.2 Cliﬀord Bundle Calculation of Dea A

b Recall that the reciprocal basis of {θ } is denoted {θa} with θa ·θb = ηab (ηab = diag(1, ..., 1, −1, ..., −1)) and that

b b c bc Dea θ = −ωacθ = −ωa θc, (50)

bc cb bc bk cl d with ωa = −ωa , and ωa = η ωkalη , ωabc = ηadωbc = −ωcba. Deﬁning

1 ^2 ω = ωbcθ ∧ θ ∈ sec T ∗M,→ sec C`(M, g), (51) a 2 a b c we have (by linearity) that [16] for any A ∈ sec V T ∗M,→ sec C`(M, g)

1 D A = ∂ A + [ω , A], (52) ea ea 2 a

1 i1 .ip where ∂ea is the Pfaﬀ derivative, i.e., for any A = p! Ai1...ip θ ...θ ∈ sec Vp T ∗M,→ sec C`(M, g) it is:

1 ∂ A = [e (A )]θi1 ...θ.ip . (53) ea p! a i1...ip Diﬀerential Forms on Riemannian (Lorentzian). . . 441

5.2.3 The Dirac Operator ∂| Associated to D˚

Using Eq.(52) we can show that for the case of a Riemannian or Lorentzian structure (M, g, D˚) the standard Dirac operator deﬁned by:

a ˚ α ˚ ∂| = θ Dea = ϑ Deα , ∂| A = ∂| ∧A + ∂|yA (54)

for any A ∈ sec V T ∗M,→ sec C`(M, g) is such that

∂| ∧A = dA , ∂|yA = −δA (55) i.e., ∂| = d − δ (56)

6 Torsion, Curvature and Cartan Structure Equations As we said in the beginning of Section 1 a given structure (M, g) may admit many different metric compatible connections. Let then D˚ be the Levi-Civita connection of g and D a Riemann-Cartan connection acting on the tensor fields defined on M. Let U ⊂ M and consider a chart of the maximal atlas of M covering µ U with arbitrary coordinates {x }. Let {∂µ} be a basis for TU, U ⊂ M µ µ and let {θ = dx } be the dual basis of {∂µ}. The reciprocal basis of µ µ µ µ µ {θ } is denoted {θ }, and g(θ , θν ) := θ · θν = δν .

Let also {ea} be an orthonormal basis for TU ⊂ TM with eb = ν a a a µ qb∂ν . The dual basis of TU is {θ }, with θ = qµdx . Also, {θb} is a a a the reciprocal basis of {θ }, i.e. θ · θb = δb. An arbitrary frame on TU ⊂ TM, coordinate or orthonormal will be denote by {eα}. Its dual ρ ρ ρ frame will be denoted by {ϑ } (i.e., ϑ (eα) = δα ). 6.1 Torsion and Curvature Operators Deﬁnition 7 The torsion and curvature operators τ and ρ of a connection D, are respectively the mappings:

τ(u, v) = Duv − Dvu − [u, v], (57)

ρ(u, v) = DuDv − DvDu − D[u,v], (58) for every u, v ∈ sec TM. 442 W. A. Rodrigues Jr.

6.2 Torsion and Curvature Tensors

Deﬁnition 8 The torsion andcurvature tensors of a connection D, are respectively the mappings:

T(α, u, v) = α (τ(u, v)) , (59) R(w, α, u, v) = α(ρ(u, v)w), (60) for every u, v, w ∈ sec TM and α ∈ sec V1 T ∗M.

We recall that for any diﬀerentiable functions f, g and h we have

τ(gu,hv) = ghτ(u, v), ρ(gu,hv)fw=ghfρ(u, v)w (61)

6.2.1 Properties of the Riemann Tensor for a Metric Com- patible Connection

Note that it is quite obvious that

R(w, α, u, v) = R(w, α, v, u). (62)

Deﬁne the tensor ﬁeld R0 as the mapping such that for every a, u, v, w ∈ sec TM and α ∈ sec V1 T ∗M.

R0(w, a, u, v) = R(w, α, v, u). (63)

It is quite ovious that

R0(w, a, u, v) = a·(ρ(u, v)w), (64)

where α = g(a, ), a = g(α, ) (65) We now show that for any structure (M, g,D) such that Dg = 0 we have for c, u, v ∈ sec TM,

R0(c, c, u, v) = c·(ρ(u, v)c) = 0. (66) Diﬀerential Forms on Riemannian (Lorentzian). . . 443

We start recalling that for every metric compatible connection it holds:

u(v(c · c)= u(Dvc · c + c·Dvc) =2u(Dvc · c)

= 2(DuDvc) · c + 2(Duc) · Dvc, (67)

Exachanging u ↔ v in the last equation we get

v(u(c · c) =2(DvDuc) · c + 2(Dvc) · Duc. (68)

Subtracting Eq.(67) from Eq.(68) we have

[u, v](c · c) =2([Du,Dv]c) · c (69)

But since [u, v](c · c) =D[u,v](c · c) = 2(D[u,v]c) · c, (70) we have from Eq.(69) that

([Du,Dv]c−D[u,v]c) · c = 0 , (71) and it follows that R0(c, c, u, v) = 0 as we wanted to show.

Exercise 9 Prove that for any metric compatible connection,

R0(c, d, u, v) = R0(d, c, u, v). (72)

ρ Given an arbitrary frame {eα} on TU ⊂ TM, let {ϑ } be the dual frame. We write: ρ [eα,eβ]=cαβeρ ρ (73) Deα eβ =Lαβeρ, ρ ρ where cαβ are the structure coeﬃcientsof the frame {eα} and Lαβ are the connection coeﬃcientsin this frame. Then, the components of the torsion and curvature tensors are given, respectively, by:

ρ ρ ρ ρ ρ T(ϑ , eα,eβ) = Tαβ = Lαβ − Lβα − cαβ ρ ρ R(eµ, ϑ , eα,eβ) = Rµ αβ ρ ρ ρ σ ρ σ σ ρ = eα(Lβµ) − eβ(Lαµ) + LασLβµ − LβσLαµ − cαβLσµ. (74) 444 W. A. Rodrigues Jr.

It is important for what follows to keep in mind the deﬁnition of the 0 (symmetric) Ricci tensor, here denoted Ric ∈ sec T2 M and which in an arbitrary basis is written as

µ ν ρ µ ν Ric =Rµν ϑ ⊗ ϑ :=Rµ ρν ϑ ⊗ ϑ (75)

It is crucial here to take into account the place where the contraction in the Riemann tensor takes place according to our conventions. We also have: ρ 1 ρ α β dϑ = − 2 cαβϑ ∧ ϑ ρ ρ β (76) Deα ϑ = −Lαβϑ

ρ V1 ∗ ρ where ωβ ∈ sec T M are the connection 1-forms, Lαβ are said to be the connection coeﬃcients in the given basis, and the T ρ ∈ sec V2 T ∗M ρ V2 ∗ are the torsion 2-forms and the Rβ ∈ sec T M are the curvature 2-forms, given by:

ρ ρ α ωβ = Lαβϑ , 1 T ρ = T ρ ϑα ∧ θβ (77) 2 αβ 1 Rρ = R ρ ϑα ∧ ϑβ. µ 2 µ αβ

1 α β Multiplying Eqs.(74) by 2 ϑ ∧ ϑ and using Eqs.(76) and (77), we get:

6.3 Cartan Structure Equations

ρ ρ β ρ dϑ + ωβ ∧ ϑ = T , ρ ρ β ρ (78) dωµ + ωβ ∧ ωµ = Rµ.

We can show that the torsion and (Riemann) curvature tensors can be written as

α T = eα ⊗ T , (79) µ ρ R = eρ ⊗ e ⊗ Rµ. (80) Diﬀerential Forms on Riemannian (Lorentzian). . . 445

7 Exterior Covariant Derivative D Sometimes, Eqs.(78) are written by some authors [27] as:

Dϑρ = T ρ, (81) ρ ρ “ Dωµ = Rµ. ” (82) and D : sec V T ∗M → sec V T ∗M is said to be the exterior covariant derivativerelated to the connection D. Now, Eq.(82) has been printed with quotation marks due to the fact that it is an incorrect equation. Indeed, a legitimate exterior covariant derivative operator12 is a concept that can be deﬁned for (p+q)-indexed r-form ﬁelds13 as follows. Suppose r+q that X ∈ sec Tp M and let ^r Xµ1....µp ∈ sec T ∗M, (83) ν1....νq such that for vi ∈ sec T M, i = 0, 1, 2, .., r,

Xµ1....µp (v , ..., v ) = X(v , ..., v , e , ..., e , ϑµ1 , ..., ϑµp ). (84) ν1....νq 1 r 1 r ν1 νq

µ1....µp The exterior covariant diﬀerential D of Xν1....νq on a manifold with a general connection D is the mapping:

^r ^r+1 D : sec T ∗M → sec T ∗M , 0 ≤ r ≤ 4, (85)

such that14

µ1....µp (r + 1)DXν1....νq (v0, v1, ..., vr) r X ν µ1 µp = (−1) Deν X(v0, v1, ..., vˇν , ...vr, eν1 , ..., eνq , ϑ , ..., ϑ ) ν=0

X ν+ς µ1 µp − (−1) X(T(vλ, vς ), v0, v1, ..., vˇλ, ..., vˇς , ..., vr, eν1 , ..., eνq , ϑ , ..., ϑ ). 0≤λ,ς ≤r (86)

12Sometimes also called exterior covariant diﬀerential. 13 α Which is not the case of the connection 1-forms ωβ , despite the name. More α precisely, the ωβ are not true indexed forms, i.e., there does not exist a tensor ﬁeld α α ω such that ω(ei, eβ , ϑ ) = ωβ (ei). 14As usual the inverted hat over a symbol (in Eq.(86)) means that the corresponding symbol is missing in the expression. 446 W. A. Rodrigues Jr.

Then, we may verify that

DXµ1....µp = dXµ1....µp + ωµ1 ∧ Xµs....µp + ... + ωµ1 ∧ Xµ1....µp (87) ν1....νq ν1....νq µs ν1....νq µs ν1....νq − ωνs ∧ Xµ1....µp − ... − ωµ1 ∧ Xµ1....µp . ν1 νs....νq µs ν1....νs

Remark 10 Note that if Eq.(87) is applied on any one of the connection µ µ µ µ α α µ 1-forms ων we would get Dων = dων +ωα ∧ων −ων ∧ωα. So, we see that µ the symbol Dων in Eq.(82), supposedly deﬁning the curvature 2-forms is simply wrong despite this being an equation printed in many Physics textbooks and many professional articles15!.

7.1 Properties of D The exterior covariant derivative D satisfy the following properties: (a) For any XJ ∈ sec Vr T ∗M and Y K ∈ sec Vs T ∗M are sets of indexed forms16, then

D(XJ ∧ Y K ) = DXJ ∧ Y K + (−1)rsXJ ∧ DY K . (88)

r (b) For any Xµ1....µp ∈ sec V T ∗M then

DDXµ1....µp = dXµ1....µp + Rµ1 ∧ Xµs....µp + ...Rµp ∧ Xµ1....µs . (89) µs µs

µ ν (c) For any metric-compatible connection D if g = gµν ϑ ⊗ ϑ then,

Dgµν = 0. (90)

a 7.2 Formula for Computation of the Connection 1- Forms ωb In an orthonormal cobasis {θa} we have (see, e.g., [22]) for the connection 1-forms 1 ωcd = θd dθc − θc dθd + θc (θd dθ )θa , (91) 2 y y y y a a 1 a j k or taking into account that dθ = − 2 cjkθ ∧ θ , 1 ωcd = (−cc ηdj + cd ηcj − ηcaη ηdjcb )θk. (92) 2 jk jk bk ja 15 ρ The authors of reference [27] knows exactly what they are doing and use “Dωµ = ρ Rµ” only as a short notation. Unfortunately this is not the case for some other authors. 16Multi indices are here represented by J and K. Diﬀerential Forms on Riemannian (Lorentzian). . . 447

8 Relation Between the Connections D˚ and D As we said above a given structure (M, g) in general admits many diﬀer- ent connections. Let then D˚ and D be the Levi-Civita connection of g on M and D and arbitrary Riemann-Cartan connection. Given an arbi- ρ trary basis {eα} on TU ⊂ TM, let {ϑ } be the dual frame. We write for the connection coeﬃcients of the Riemann-Cartan and the Levi-Civita ρ connections in the arbitrary bases {eα},{ϑ }:

ρ ρ ρ β Deα eβ = Lαβeρ,Deα ϑ = −Lαβϑ , ˚ ˚ρ ˚ ρ ˚ρ β Deα eβ = Lαβeρ, Deα ϑ = −Lαβϑ . (93)

Moreover, the structure coeﬃcients of the arbitrary basis {eα} are:

ρ [eα,eβ] = cαβeρ. (94) Let moreover, ρ bαβ = −(£eρ g)αβ, (95) ρ where £eρ is the Lie derivative in the direction of the vector ﬁeld e . Then, we have the noticeable formula (for a proof, see, e.g., [22]): 1 1 Lρ = ˚Lρ + T ρ + Sρ , (96) αβ αβ 2 αβ 2 αβ ρ where the tensor Sαβ is called the strain tensor of the connection and can be decomposed as:

2 Sρ = S˘ρ + sρg (97) αβ αβ n αβ ˘ρ where Sαβ is its traceless part, is called the shear of the connection, and 1 sρ = gµν Sρ (98) 2 µν is its trace part, is called the dilation of the connection. We also have that connection coeﬃcients of the Levi-Civita connection can be written as: 1 ˚Lρ = (bρ + cρ ). (99) αβ 2 αβ αβ Moreover, we introduce the contorsion tensor whose components in an arbitrary basis are deﬁned by 448 W. A. Rodrigues Jr.

1 Kρ = Lρ − ˚Lρ = (T ρ + Sρ ), (100) αβ αβ αβ 2 αβ αβ and which can be written as 1 Kρ = − gρσ(g T µ + g T µ − g T µ ). (101) αβ 2 µα σβ µβ σα µσ αβ We now present the relation between the Riemann curvature ten- ρ sor Rµ αβ associated with the Riemann-Cartan connection D and the ρ Riemann curvature tensor R˚µ αβ of the Levi-Civita connection D˚. ρ ˚ ρ ρ Rµ αβ = Rµ αβ + Jµ [αβ], (102) where: ρ ˚ ρ ρ σ ρ ρ σ σ ρ Jµ αβ = DαKβµ − KβσKαµ = DαKβµ − KασKβµ + KαβKσµ. (103) 1 α β Multiplying both sides of Eq.(102) by 2 θ ∧ θ we get: ρ ˚ρ ρ Rµ = Rµ + Jµ, (104) where 1 Jρ = J ρ θα ∧ θβ. (105) µ 2 µ [αβ] From Eq.(102) we also get the relation between the Ricci tensors of the connections D and D˚. We write for the Ricci tensor of D µ ν Ric = Rµαdx ⊗ dx ρ Rµα := Rµ αρ (106) Then, we have Rµα = R˚µα + Jµα, (107) with ˚ ρ ˚ ρ ρ σ ρ σ Jµα = DαKρµ − DρKαµ + KασKρµ − KρσKαµ ρ ρ ρ σ ρ σ = DαKρµ − DρKαµ − KσαKρµ + KρσKαµ. (108) Observe that since the connection D is arbitrary, its Ricci tensor will be not be symmetric in general. Then, since the Ricci tensor R˚µα of D˚ is necessarily symmetric, we can split Eq.(107) into:

R[µα] = J[µα], (109) ˚ R(µα) = R(µα) + J(µα). Diﬀerential Forms on Riemannian (Lorentzian). . . 449

9 Expressions for d and δ in Terms of Covariant Derivative Operators D˚ and D We have the following noticeable formulas whose proof can be found in, ^ e.g., [22]. Let Q ∈ sec T ∗M. Then as we already know

α ˚ dQ = ϑ ∧ (Deα Q) = ∂| ∧Q, α ˚ | δQ = −ϑ y(Deα Q) = ∂yQ. (110) We have also the important formulas

α α α dQ = ϑ ∧ (Deα Q) − T ∧ (ϑαyQ) =∂ ∧ Q − T ∧ (ϑαyQ), α α α δQ = −ϑ y(Deα Q) − T y(ϑα ∧ Q) = −∂yQ − T y(ϑα ∧ Q). (111)

10 Square of Dirac Operators and D’ Alembertian, Ricci and Einstein Operators We now investigate the square of a Dirac operator. We start recalling that the square of the standard Dirac operator can be identiﬁed with the Hodge D’ Alembertian and that it can be separated in some inter- esting parts that we called in [22] the D’Alembertian, Ricci and Einstein operators of (M, g, D˚).

10.1 The Square of the Dirac Operator ∂| Associated to D˚

The square of standard Dirac operator ∂| is the operator, ∂|2 = ∂| ∂| : sec Vp T ∗M,→ sec C`(M, g) → sec Vp T ∗M,→ sec C`(M, g) given by:

2 ∂| = (∂| ∧ + ∂|y)(∂| ∧ + ∂|y) = (d − δ)(d − δ) (112)

It is quite obvious that

∂| 2 = −(dδ + δd), (113)

2 and thus we recognize that ∂| ≡ ♦ is the Hodge D’Alembertian of the manifold introduced by Eq.(38) On the other hand, remembering the standard Dirac operator is ∂| = α ˚ α ϑ Deα , where {ϑ } is the dual basis of an arbitrary basis {eα} on TU ⊂ 450 W. A. Rodrigues Jr.

TM and D˚ is the Levi-Civita connection of the metric g, we have:

2 α ˚ β ˚ α β ˚ ˚ ˚ β ˚ ∂| = (ϑ Deα )(ϑ Deβ ) = ϑ (ϑ Deα Deβ + (Deα ϑ )Deβ ) αβ ˚ ˚ ˚ρ ˚ α β ˚ ˚ ˚ρ ˚ = g (Deα Deβ − LαβDeρ ) + ϑ ∧ ϑ (Deα Deβ − LαβDeρ ).

Then deﬁning the operators:

(a) ∂| · ∂| =gαβ(D˚ D˚ − ˚Lρ D˚ ) eα eβ αβ eρ (114) (b) | | α β ˚ ˚ ˚ρ ˚ ∂∧ ∂=ϑ ∧ ϑ (Deα Deβ − LαβDeρ ), we can write: 2 ♦ = ∂| = ∂| · ∂| + ∂| ∧ ∂| (115) or,

2 ∂| = (∂|y + ∂| ∧)(∂|y + ∂| ∧) = ∂|y ∂| ∧ + ∂| ∧ ∂|y (116)

It is important to observe that the operators ∂| · ∂| and ∂| ∧ ∂| do not have anything analogous in the formulation of the diﬀerential geometry in the Cartan and Hodge bundles. The operator ∂| · ∂| can also be written as:

1 h i ∂| · ∂| = gαβ D˚ D˚ + D˚ D˚ − bρ D˚ . (117) 2 eα eβ eβ eα αβ eρ Applying this operator to the 1-forms of the frame {θα}, we get:

1 (∂| · ∂|)ϑµ = − gαβM˚ µ θρ, (118) 2 ρ αβ where:

˚ µ ˚µ ˚µ ˚µ ˚σ ˚µ ˚σ σ ˚µ Mρ αβ = eα(Lβρ) + eβ(Lαρ) − LασLβρ − LβσLαρ − bαβLσρ. (119)

The proof that an object with these components is a tensor may be found in [22]. In particular, for every r-form ﬁeld ω ∈ sec Vr T ∗M, 1 α1 αr ω = r! ωα1...αr θ ∧ ... ∧ θ , we have: 1 (∂| · ∂|)ω = gαβD˚ D˚ ω θα1 ∧ ... ∧ θαr , (120) r! α β α1...αr Diﬀerential Forms on Riemannian (Lorentzian). . . 451

˚ ˚ where DαDβωα1...αr are the components of the covariant derivative of ˚ 1 ˚ α1 αr ω, i.e., writing Deβ ω = r! Dβωα1...αr θ ∧ ... ∧ θ , it is: D˚ ω = e (ω )−˚Lσ ω −· · ·−˚Lσ ω . (121) β α1...αr β α1...αr βα1 σα2...αr βαr α1...αr−1σ

˚ In view of Eq.(120), we call the operator = ∂| · ∂| the covariant D’Alembertian. Note that the covariant D’Alembertian of the 1-forms ϑµ can also be written as: 1 (∂| · ∂|)ϑµ = ˚gαβD˚ D˚ δµϑρ = ˚gαβ(D˚ D˚ δµ + D˚ D˚ δµ)ϑρ α β ρ 2 α β ρ β α ρ and therefore, taking into account the Eq.(118), we conclude that: ˚ µ ˚ ˚ µ ˚ ˚ µ Mρ αβ = −(DαDβδρ + DβDαδρ ). (122) By its turn, the operator ∂| ∧ ∂| can also be written as: 1 h i ∂| ∧ ∂| = ϑα ∧ ϑβ D˚ D˚ − D˚ D˚ − cρ D˚ . (123) 2 α β β α αβ ρ Applying this operator to the 1-forms of the frame {ϑµ}, we get: 1 (∂| ∧ ∂|)ϑµ = − R˚ µ (ϑα ∧ ϑβ)ϑρ = −R˚µϑρ, (124) 2 ρ αβ ρ µ where R˚ρ αβ are the components of the curvature tensor of the connection D˚. Then using the second formula in the first line of Eq.(45) we have ˚µ ρ ˚µ ρ ˚µ ρ Rρ θ = Rρ xθ + Rρ ∧ θ . (125) The second term in the r.h.s. of this equation is identically null because due to the first Bianchi identity which for the particular case of the µ ˚µ ρ Levi-Civita connection (T = 0) is Rρ ∧ θ = 0 . The first term in Eq.(125) can be written 1 R˚µ θρ = R˚ µ (θα ∧ θβ) θρ ρ x 2 ρ αβ x 1 = R˚ µ θρ (θα ∧ θβ) 2 ρ αβ y 1 = − R˚ µ (˚gραθβ − ˚gρβθα) 2 ρ αβ ρα ˚ µ β ˚µ β = −˚g Rρ αβθ = −Rβθ , (126) 452 W. A. Rodrigues Jr.

˚µ where Rβ are the components of the Ricci tensor of the Levi-Civita connection D˚ of g. Thus we have a really beautiful result:

(∂| ∧ ∂|)θµ = R˚µ, (127)

˚µ ˚µ β where R = Rβθ are the Ricci 1-forms of the manifold. Because of this relation, we call the operator ∂| ∧ ∂| the Ricci operatorof the manifold associated to the Levi-Civita connection D˚ of g. We can show [22] that the Ricci operator ∂| ∧ ∂| satisﬁes the relation:

σ ρσ ∂| ∧ ∂| = R˚ ∧ iσ + R˚ ∧ iρiσ, (128)

˚ρσ 1 ˚ρσ α β where the curvature 2-forms are R = 2 R αβϑ ∧ ϑ and

iσω := ϑσyω. (129)

Observe that applying the operator given by the second term in the r.h.s. of Eq.(128) to the dual of the 1-forms ϑµ, we get:

˚ρσ µ ˚ ρ σ µ R ∧ iρiσ ? ϑ = Rρσ ? ϑ y(ϑ y ? ϑ )) ρ σ µ = −R˚ρσ ∧ ?(ϑ ∧ ϑ ? ϑ ) (130) ˚ ρ σ µ = ?(Rρσy(ϑ ∧ ϑ ∧ ϑ )), where we have used the Eqs.(35). Then, recalling the deﬁnition of the curvature forms and using the Eq.(28), we conclude that:

1 R˚ρσ ∧ (ϑ ϑ ? ϑµ) = 2 ? (R˚µ − Rϑ˚ µ) = 2 ? G˚µ, (131) ρy σy 2 where R˚ is the scalar curvature of the manifold and the G˚µ may be called the Einstein 1-form ﬁelds. That observation motivate us to introduce in [22] the Einstein operator of the Levi-Civita connection D˚ of g on the manifold M as the ˚ mapping : sec C`(M, g) → sec C`(M, g) given by: 1 ˚ = ?−1 (R˚ρσ ∧ i i ) ?. (132) 2 ρ σ Diﬀerential Forms on Riemannian (Lorentzian). . . 453

Obviously, we have: 1 ˚θµ = G˚µ = R˚µ − Rϑ˚ µ. (133) 2 In addition, it is easy to verify that ?−1(∂| ∧ ∂|)? = − ∂| ∧ ∂| and ?−1(R˚σ ∧ ˚σ iσ)? = R yjσ. Thus we can also write the Einstein operator as: 1 ˚ = − (∂| ∧ ∂| +R˚σ j ), (134) 2 y σ where jσA = ϑσ ∧ A, (135) ^ for any A ∈ sec T ∗M,→ sec C`(M, g). µ We recall [22] that if ˚ωρ are the Levi-Civita connection 1-forms ﬁelds in an arbitrary moving frame {ϑµ} on (M, g, D˚) then:

µ µ σ µ ρ (a) (∂| · ∂|)ϑ =−(∂| ·˚ωρ − ˚ωρ · ˚ωσ )ϑ µ µ σ µ ρ (136) (b) (∂| ∧ ∂|)ϑ =−(∂| ∧˚ωρ − ˚ωρ ∧ ˚ωσ )ϑ , and 2 µ µ σ µ ρ ∂| ϑ = −(∂|˚ωρ − ˚ωρ ˚ωσ )ϑ . (137)

ρσ Exercise 11 Show that ϑρ ∧ ϑσR˚ = −R,˚ where R˚ is the curvature scalar.

10.2 The Square of the Dirac Operator ∂ Associated to D Consider the structure (M, g,D), where D is an arbitrary Riemann- Cartan-Weyl connection and the Cliﬀord algebra C`(M, g). Let us now α compute the square of the (general) Dirac operator ∂ = ϑ Deα . As in the earlier section, we have, by one side,

2 ∂ = (∂y + ∂∧)(∂y + ∂∧) = ∂y∂y + ∂y∂∧ + ∂ ∧ ∂y + ∂ ∧ ∂∧

2 2 and we write ∂y∂y ≡ ∂ y, ∂∧ ∂∧ ≡ ∂ ∧ and

L+ = ∂y∂ ∧ + ∂ ∧ ∂y, (138) so that: 2 2 2 ∂ = ∂ y + L+ + ∂ ∧ . (139) 454 W. A. Rodrigues Jr.

The operator L+ when applied to scalar functions corresponds, for the case of a Riemann-Cartan space, to the wave operator introduced by Rapoport [23] in his theory of Stochastic Mechanics. Obviously, for the case of the standard Dirac operator, L+ reduces to the usual Hodge D’ Alembertian of the manifold, which preserve graduation of forms. For more details see [18]. On the other hand, we have also: 2 α β α β β ∂ = (ϑ Deα )(ϑ Deβ ) = ϑ (ϑ Deα Deβ + (Deα ϑ )Deβ ) αβ ρ α β ρ = g (Deα Deβ − LαβDeρ ) + ϑ ∧ ϑ (Deα Deβ − LαβDeρ ) and we can then define: αβ ρ ∂ · ∂ =g (Deα Deβ − LαβDeρ ) α β ρ (140) ∂ ∧ ∂=θ ∧ θ (Deα Deβ − LαβDeρ ) in order to have: ∂2= ∂∂ = ∂ · ∂ + ∂ ∧ ∂ . (141) The operator ∂ · ∂ can also be written as: 1 1 ∂ · ∂ = θα · θβ(D D − Lρ D ) + θβ · θα(D D − Lρ D ) 2 eα eβ αβ eρ 2 eβ eα βα eρ 1 = gαβ[D D + D D − (Lρ + Lρ )D ] (142) 2 eα eβ eβ eα αβ βα eρ or, 1 ∂ · ∂ = gαβ(D D + D D − bρ D ) − sρD , (143) 2 eα eβ eβ eα αβ eρ eρ where sρ has been defined in Eq.(98). By its turn, the operator ∂ ∧ ∂ can also be written as: 1 1 ∂ ∧ ∂ = ϑα ∧ ϑβ(D D − Lρ D ) + ϑβ ∧ ϑα(D D − Lρ D ) 2 eα eβ αβ eρ 2 eβ eα βα eρ 1 = ϑα ∧ ϑβ[D D − D D − (Lρ − Lρ )D ] 2 eα eβ eβ eα αβ βα eρ or, 1 ∂ ∧ ∂ = ϑα ∧ ϑβ(D D − D D − cρ D ) − T ρD . (144) 2 eα eβ eβ eα αβ eρ eρ Remark 12 For the case of a Levi-Civita connection we have similar formulas for ∂| · ∂| (Eq.(142)) and ∂| ∧ ∂| (Eq.(144)) with D 7→ D˚, and of course, T ρ = 0, as follows directly from Eq.(114). Differential Forms on Riemannian (Lorentzian). . . 455

11 Coordinate Expressions for Maxwell Equations on Lorentzian and Riemann-Cartan Spacetimes 11.1 Maxwell Equations on a Lorentzian Spacetime

We now take (M, g) as a Lorentzian manifold, i.e., dim M = 4 and the signature of g is (1, 3). We consider moreover a Lorentzian spacetime structure on (M, g), i.e., the pentuple (M, g, D,˚ τg, ↑) and a Riemann- Cartan spacetime structure (M, g, D, τg, ↑). Now, in both spacetime structures, Maxwell equations in vacuum read: dF = 0, δF = −J, (145) ^2 where F ∈ sec T ∗M is the Faraday tensor (electromagnetic ﬁeld) and ^1 J ∈ sec T ∗M is the current. We observe that writing

1 1 1 F = F dxµ ∧ dxν = F θµ ∧ θν = F θµν , (146) 2 µν 2 µν 2 µν

we have using Eq.(34) that

1 1 1 1 ?F = F (?θµν ) = ?F ϑρσ = (F p|det g|gµαgνβ )ϑρσ 2 µν 2 ρσ 2 µν 2 αβρσ (147) Thus 1 ?F = (?F) = F p|det g|gµαgνβ . (148) ρσ ρσ 2 µν αβρσ

The homogeneous Maxwell equation dF = 0 can be writing as δ?F = 0. The proof follows at once from the deﬁnition of δ (Eq.(37)). Indeed, we can write

0 = dF = ??−1 d ? ?−1F = ?δ ?−1 F = − ? δ ? F = 0.

Then ?−1 ? δ ? F = 0 and we end with

δ ? F = 0.

(a) We now express the equivalent equations dF = 0 and δ ? F = 0 in arbitrary coordinates {xµ} covering U ⊂ M using ﬁrst the Levi-Civita 456 W. A. Rodrigues Jr. connection and noticeable formula in Eq.(110). We have

dF = θα ∧ (D˚ F ) ∂α 1 α h µ ν i = θ ∧ D˚ (Fµν θ ∧ θ ) 2 ∂α 1 h i = θα ∧ (∂ F )θµ ∧ θν − F ˚Γµ θρ ∧ θν − F ˚Γν θµ ∧ θρ 2 α µν µν αρ µν αρ 1 h i = θα ∧ (D˚ F )θµ ∧ θν 2 α µν 1 = D F θα ∧ θµ ∧ θν 2 α µν 1 » 1 1 1 – = D˚ F θα ∧ θµ ∧ θν + D˚ F θµ ∧ θν ∧ θα + D˚ F θν ∧ θα ∧ θµ 2 3 α µν 3 µ να 3 ν αµ 1 » 1 1 1 – = D˚ F θα ∧ θµ ∧ θν + D˚ F θα ∧ θµ ∧ θν + D˚ F θα ∧ θµ ∧ θν 2 3 α µν 3 µ να 3 ν αµ 1 “ ” = D˚ F + D˚ F + D˚ F θα ∧ θµ ∧ θν . 6 α µν µ να ν αµ

So,

dF = 0 ⇔ D˚αFµν + D˚µFνα + D˚ν Fαµ = 0. (149)

If we calculate dF = 0 using the deﬁnition of d we get:

1 dF = (∂ F )θα ∧ θµ ∧ θν (150) 2 α µν 1 = (∂ F + ∂ F + ∂ F ) θα ∧ θµ ∧ θν , 6 α µν µ να ν αµ from where we get that

dF = 0 ⇐⇒ ∂αFµν +∂µFνα+∂ν Fαµ = 0 ⇐⇒ D˚αFµν +D˚µFνα+D˚ν Fαµ = 0. (151) Diﬀerential Forms on Riemannian (Lorentzian). . . 457

Next we calculate δ ? F = 0. We have

δ ? F = −θα (D˚ ? F) y ∂α 1 α n˚ ? µ ν o = − θ y D [ Fµν θ ∧ θ ] 2 ∂α 1 n o = − θα (∂ ?F )θµ ∧ θν − ?F ˚Γµ θρ ∧ θν − ?F ˚Γν θµ ∧ θρ 2 y α µν µν αρ µν αρ 1 n o = − θα (∂ ?F )θµ ∧ θν − ?F ˚Γρ θµ ∧ θν − ?F ˚Γρ θµ ∧ θν 2 y α µν ρν αµ µρ αν 1 n o = − θα (D˚ ?F )θµ ∧ θν 2 y α µν 1 n o = − (D˚ ?F )gαµθν − (D˚ ?F )gαν θµ 2 α µν α µν ? αµ ν = −(D˚α Fµν )g θ ? αµ ν = −[D˚α( Fµν g )]θ ˚ ? α ν = −(Dα F ν )]θ . (152)

Then we get that

˚ ˚ ˚ ˚ ? α DαFµν + DµFνα + Dν Fαµ = 0 ⇔ dF = 0 ⇔ δ ? F = 0 ⇐⇒ Dα F ν = 0. (153) (b) Also, the non homogenoeous Maxwell equation δF = −J can be written using the deﬁnition of δ (Eq.(37)) as d ? F = − ? J:

δF = −J, (−1)2 ?−1 d ? F = −J, ??−1 d ? F = − ? J, d ? F = − ? J. (154)

We now express δF = − J in arbitrary coordinates17 using ﬁrst the Levi-Civita connection. We have following the same steps as in Eq.(152)

1 α n˚ µ ν o ν δF + J = − θ y D [Fµν θ ∧ θ ] + Jν θ (155) 2 ∂α ˚ α ν = (−DαF ν + Jν )θ .

17We observe that in terms of the “classical” charge and “vector” current densities 0 i we have J =ρθ − jiθ . 458 W. A. Rodrigues Jr.

Then αν ν δF + J = 0 ⇔ D˚αF = J . (156) We also observe that using the symmetry of the connection coeﬃ- αν ˚ν αρ ˚ν αρ cients and the antisymmetry of the F that ΓαρF = −ΓαρF = 0. Also, α p 1 ˚Γ = ∂ρ ln |det g| = ∂ρ |det g| , αρ p|det g| and ˚ αν αν ˚α ρν ˚ν αρ DαF = ∂αF + ΓαρF + ΓαρF αν ˚α ρν = ∂αF + ΓαρF

ρν 1 p ρν = ∂ρF + ∂ρ( |det g|)F . p|det g| Then

1 p ρν ν ∂ρ( |det g|F ) = J , (157) p|det g| and

αν ν 1 p ρν ν δF = 0 ⇔ D˚αF = J ⇔ ∂ρ( |det g|F ) = J . (158) p|det g|

Exercise 13 Show that in a Lorentzian spacetime Maxwell equations become Maxwell equation, i.e., ∂| F = J. (159)

11.2 Maxwell Equations on Riemann-Cartan Spacetime From time to time we see papers (e.g., [19, 25]) writing Maxwell equations in a Riemann-Cartan spacetime using arbitrary coordinates and (of course) the Riemann-Cartan connection. As we shall see such enter- prises are simple exercises, if we make use of the noticeable formulas of Eq.(111). Indeed, the homogeneous Maxwell equation dF = 0 reads

α α dF = θ ∧ (D F) − T ∧ (θα F) = 0 (160) ∂α y Diﬀerential Forms on Riemannian (Lorentzian). . . 459

1 (D F + D F + D F )θα ∧ θµ ∧ θν 6 α µν µ να ν αµ 1 1 − T α θρ ∧ θσ ∧ [θ F (θµ ∧ θν )] 2 2 ρσ αy µν 1 = (D F + D F + D F )θα ∧ θµ ∧ θν 6 α µν µ να ν αµ 1 − T α F θρ ∧ θσ ∧ δµθν 2 ρσ µν α 1 = (D F + D F + D F )θα ∧ θµ ∧ θν 6 α µν µ να ν αµ 1 − T σ F θα ∧ θµ ∧ θν 2 αµ σν 1 = (D F + D F + D F )θα ∧ θµ ∧ θν 6 α µν µ να ν αµ 1 − (T σ F + T σ F + T σ F )θα ∧ θµ ∧ θν 6 αµ σν µν σα να σµ 1 = (D F + D F + D F )θα ∧ θµ ∧ θν 6 α µν µ να ν αµ 1 + (F T σ + F T σ + F T σ )θα ∧ θµ ∧ θν . 6 ασ µν µσ να νσ αµ

i.e.,

σ σ σ dF = 0 ⇐⇒ DαFµν +DµFνα +Dν Fαµ +FσαTµν +FµσTνα +FνσTαµ = 0. (161) Also, taking into account that dF = 0 ⇐⇒ δ ? F = 0 we have using the second noticeable formula in Eq.(111) that

α α δ ? F =−θ y(Deα ? F) − T y(θα ∧ ?F) = 0. (162)

Now,

α ? α ν ? αν θ y(Deα ? F) = (Dα F ν )θ = (Dα F )θν (163) 460 W. A. Rodrigues Jr. and

α T y(θα ∧ ?F) 1 = T α (θβ ∧ θρ) (θ ∧ (?F θµ ∧ θν ) 4 βρ y α µν 1 = T α ?F (θβ ∧ θρ) (θ ∧ θµ ∧ θν ) 4 βρ µν y α 1 = T α ?F µν θβ [θρ (θ ∧ θ ∧ θ )] 4 βρ y y α µ ν 1 = T α ?F µν θβ (δρ θ ∧ θ − δρ θ ∧ θ + δρθ ∧ θ ) 4 βρ y α µ ν µ α ν ν α µ 1 1 1 = T α ?F µν θβ (θ ∧ θ ) − T α ?F µν θβ (θ ∧ θ ) + T α ?F µν θβ (θ ∧ θ ) 4 βα y µ ν 4 βµ y α ν 4 βν y α µ

1 1 1 = T α ?F µν θβ (θ ∧ θ ) − T µ ?F ρν θβ (θ ∧ θ ) + T µ ?F νρθβ (θ ∧ θ ) 4 βα y µ ν 4 βρ y µ ν 4 βρ y µ ν 1 = (T α ?F µν − T µ ?F ρν + T µ ?F νρ)θβ (θ ∧ θ ) 4 βα βρ βρ y µ ν 1 = (T α ?F µν − T µ ?F ρν + T µ ?F νρ)(δβ θ − δβ θ ) 4 βα βρ βρ µ ν ν µ 1 1 = (T α ?F µν − T µ ?F ρν + T µ ?F νρ)θ − (T α ?F νµ − T ν ?F ρµ + T ν ?F µρ)θ 4 µα µρ µρ ν 4 µα µρ µρ ν 1 = (T α ?F µν − T µ ?F ρν + T ν ?F µρ)θ (164) 2 µα µρ µρ ν

Using Eqs.(163) and (164) in Eq.(162) we get

1 D ?F αν + (T α ?F µν − T µ ?F ρν + T ν ?F µρ) = 0 (165) α 2 µα µρ µρ and we have 1 dF = 0 ⇔ δ ?F =0 ⇔ D ?F αν + (T α ?F µν −T µ ?F ρν +T ν ?F µρ) = 0. α 2 µα µρ µρ (166) Finally we express the non homogenous Maxwell equation δF = −J in arbitrary coordinates using the Riemann-Cartan connection. We have

α α δF = −θ y(Deα F) − T y(θα ∧ F) 1 = −[D F αν + (T α ?F µν − T µ ?F ρν + T ν ?F µρ)]θ = −J ν θ , α 2 µα µρ µρ ν ν (167) Diﬀerential Forms on Riemannian (Lorentzian). . . 461

i.e.,

1 D F αν + (T α ?F µν − T µ ?F ρν + T ν ?F µρ) = J ν . (168) α 2 µα µρ µρ

Exercise 14 Show (use Eq.(111)) that in a Riemann-Cartan spacetime Maxwell equations become Maxwell equation, i.e.,

a a ∂F = J + T y(θa ∧ F) − T ∧ (θayF). (169)

12 Bianchi Identities

We rewrite Cartan’s structure equations for an arbitrary Riemann- Cartan structure (M, g, D, τg) where dim M = n and g is a metric of signature (p, q), with p + q = n using an arbitrary cotetrad {θa} as

a a a b a T = dθ + ωb ∧ θ = Dθ , a a a c (170) Rb = dωb + ωc ∧ ωb where

a a c ωb = ωcbθ , 1 T a = T a θb ∧ θc (171) 2 bc 1 Ra = R a θc ∧ θd. (172) b 2 b cd

a a Since the T and the Rb are index form ﬁelds we can apply to those objects the exterior covariant diﬀerential (Eq.(87)). We get

a a a b 2 a a b a b DT = dT + ωb ∧ T = d θ + d(ωb ∧ θ ) + ωb ∧ T a b a b a b = dωb ∧ θ − ωb ∧ dθ + ωb ∧ T a b a b b c a b = dωb ∧ θ − ωb ∧ (T − ωc ∧ θ ) + ωb ∧ T a a c b = (dωb + ωc ∧ ωb) ∧ θ a b = Rb ∧ θ (173) 462 W. A. Rodrigues Jr.

Also,

a a a c c a DRb = dRb + ωc ∧ Rb − ωb ∧ Rc 2 a a c c a a c c a = d ωb + dωc ∧ ωb − dωb ∧ ωc − Rc ∧ ωb + Rb ∧ ωc a c a a d c a c c c d a = dωc ∧ ωb − (dωc + ωd ∧ ωc ) ∧ ωb − dωc ∧ ωb + (dωb + ωd ∧ ωb ) ∧ ωc a d c c d a = −ωd ∧ ωc ∧ ωb + ωd ∧ ωb ∧ ωc a d c d c a = −ωd ∧ ωc ∧ ωb + ωc ∧ ωb ∧ ωd a d c a d c = −ωd ∧ ωc ∧ ωb + ωd ∧ ωc ∧ ωb = 0. (174) So, we have the general Bianchi identities which are valid for any one of the metrical compatible structures18 classiﬁed in Section 2,

a a b DT = Rb ∧ θ , a DRb = 0. (175)

12.1 Coordinate Expressions of the First Bianchi Identity Taking advantage of the calculations we done for the coordinate expressions of Maxwell equations we can write in a while:

a a a b DT = dT + ωb ∧ T 1 “ ” = ∂ T a + ωa T b + ∂ T a + ωa T b + ∂ T a + ωa T b θµ ∧ θα ∧ θβ . 3! µ αβ µb αβ α βµ αb βµ β µα βb µα (176) Now, a a ρ a ρ ∂µTαβ = (∂µqρ )Tαβ + qρ ∂µTαβ, (177) and using the freshman identity (Eq.(23)) we can write

a b a b ρ a b ρ a ρ ωµbTαβ = ωµbqρ Tαβ = Lµbqρ Tαβ − (∂µqρ )Tαβ. (178) So,

a a b ∂µTαβ + ωµbTαβ a ρ a b ρ = qρ ∂µTαβ + Γµbqρ Tαβ a ρ κ ρ κ ρ = qρ (DµTαβ + ΓµαTκβ + ΓµβTακ). (179)

18For non metrical compatible structures we have more general equations than the Cartan structure equations and thus more general identities, see [22]. Diﬀerential Forms on Riemannian (Lorentzian). . . 463

κ κ κ Now, recalling that Tµα = Γµα − Γαµ we can write

a κ ρ κ ρ µ α β qρ (ΓµαTκβ + ΓµβTακ)θ ∧ θ ∧ θ (180) a κ ρ µ α β = qρ TµαTκβθ ∧ θ ∧ θ . Using these formulas we can write DT a = 1 qa ˘D T ρ + D T ρ + D T ρ + T κ T ρ + T κ T ρ + T κ T ρ ¯ θµ ∧ θα ∧ θβ . 3! ρ µ αβ α βµ β µα µα κβ αβ κµ βµ κα (181)

a b Now, the coordinate representation of Rb ∧ θ is: 1 Ra ∧ θb = qa(R ρ + R ρ + R ρ )θµ ∧ θα ∧ θβ, (182) b 3! ρ µ αβ α βµ β µα and thus the coordinate expression of the ﬁrst Bianchi identity is:

ρ ρ ρ DµTαβ + DαTβµ + DβTµα ρ ρ ρ κ ρ κ ρ κ ρ = (Rµ αβ + Rα βµ + Rβ µα) − (TµαTκβ + TαβTκµ + TβµTκα), (183) which we can write as

X ρ X ρ κ ρ Rµ αβ = DµTαβ − TµβTκα , (184) (µαβ) (µαβ) X with denoting as usual the sum over cyclic permutation of the (µαβ) indices (µαβ). For the particular case of a Levi-Civita connection D˚ ρ since the Tαβ = 0 we have the standard form of the ﬁrst Bianchi identity in classical Riemannian geometry, i.e.,

ρ ρ ρ Rµ αβ + Rα βµ + Rβ µα = 0. (185)

If we now recall the steps that lead us to Eq.(166) we can write for the torsion 2-form ﬁelds T a,

dT a = ??−1 d ? ?−1T a = (−1)n−2 ? δ ?−1 T a = (−1)n−2(−1)n−2sgng ? δ ? T a = (−1)n−2 ?−1 δ ? T a. (186) 464 W. A. Rodrigues Jr. with sgng = det g/ |det g|. Then we can write the ﬁrst Bianchi identity as

a n−2 a b a b δ ? T = (−1) ? [Rb ∧ θ − ωb ∧ T ], (187)

and taking into account that

a b b a b a ?(Rb ∧ θ ) = ?(θ ∧ Rb) = θ y ? Rb, a b a b ?(ωb ∧ T ) = ωby ? T , (188)

we end with

a n−2 b a a b δ ? T = (−1) (θ y ? Rb − ωby ? T ). (189)

This is the ﬁrst Bianchi identity written in terms of duals. To calculate its coordinate expression, we recall the steps that lead us to Eq.(166) and write directly for the torsion 2-form ﬁelds T a

δ ? T a 1 = −(D ?T aαν + (T α ?T aµν − T µ ?T aρν + T ν ?T aµρ))θ . (190) α 2 µα µρ µρ ν

Also, writing 1 ?Ra = ∗R a θc ∧ θd, (191) b 2 b cd

we have:

a b b a ?(Rb ∧ θ ) = θ y ? Rb 1 = θb (?R a θc ∧ θd) 2 y b cd ? a bc d = Rb cdη θ ? ca d ? ca d ? ca d ν = R cdθ = R c θd = R c qdθν . (192) Diﬀerential Forms on Riemannian (Lorentzian). . . 465

On the other hand we can also write:

a b b a ?(Rb ∧ θ ) = θ y ? Rb 1 1 = ϑb ( R akl θm ∧ θn) 2 y (n − 2)! b klmn 1 1 = (R akl ηbm ∧ θn − R akl ηbn ∧ θm) 2 (n − 2)! b klmn b klmn 1 1 = R akl ηbmθn = Rmakl θn (n − 2)! b klmn (n − 2)! klmn 1 = R akl mnθ (n − 2)! m kl n 1 = R akl mnqν θ . (n − 2)! m kl n ν from where we get in agreement with Eq.(34) the formula 1 ?Rca = Rmakl , (193) cd (n − 2)! mkld ? ca which shows explicitly that R cd are not the components of the Ricci tensor. Moreover,

a b ωby ? T (194) 1 = ωa θα (?T bµν θ ∧ θ ) 2 αb y µ ν ? bµν a = T ωαbθν . Collecting the above formulas we end with 1 D ?T aαν + (T α ?T aµν −T µ ?T aρν +T ν ?T aµρ) = (−1)n−1(?Rca dqν −ωa? T bαν ), α 2 µα µρ µρ c d αb (195) which is another expression for the ﬁrst Bianchi identity written in terms of duals.

? aαν Remark 15 Consider, e.g., the term Dα T in the above equation and write ? aαν a? ραν Dα T = Dα(qρ T ). (196) We now show that

a? ραν a ? ραν Dα(qρ T ) 6= qρ Dα T . (197) 466 W. A. Rodrigues Jr.

Indeed, recall that we already found that

1 (D ?T aαν )θ = −δ?T a + (T α ?T aµν −T µ ?T aρν +T ν ?T aµρ)θ , (198) α ν 2 µα µρ µρ ν

and taking into account the second formula in Eq.(111) we can write

1 θα (D ?T a) = −δ?T a+ (T α ?T aµν −T µ ?T aρν +T ν ?T aµρ)θ . (199) y ∂α 2 µα µρ µρ ν

a 1 a ? ρµν Now, writing ?T = 2 qρ T θµ ∧ θν and get

α a θ y(D∂α ? T ) 1 = θα [D (qa?T ρµν θ ∧ θ )] 2 y ∂α ρ µ ν 1 = ϑα [∂ (qa?T ρµν )θ ∧ θ + qa?T ρµν D (θ ∧ θ )] 2 y α ρ µ ν ρ ∂α µ ν 1 = ϑα [(∂ qa)?T ρµν θ ∧ θ + qa∂ (?T ρµν )θ ∧ θ + qa?T ρµν D (θ ∧ θ )] 2 y α ρ µ ν ρ α µ ν ρ ∂α µ ν 1 = ϑα [(∂ qa)?T ρµν θ ∧ θ + qaD (?T ρµν )θ ∧ θ ] 2 y α ρ µ ν ρ α µ ν a ? ρµν α a ? ρµν α = (∂αqρ ) T δµ θν + qρ Dα( T )δµ θν . (200)

Comparing the Eq.(198) with Eq.(199) using Eq.(200) we get

? aαν a? aαν a ? ρµν a ? ρµν Dα T θν = Dα(qρ T θν ) = (∂αqρ ) T + qρ Dα( T ), (201)

thus proving our statement and showing the danger of applying a so called “tetrad postulate” asserting without due care on the meaning of the symbols that “ the covariant derivative of the tetrad is zero, and thus a using “Dαqρ = 0”.”

Exercise 16 Show that the coordinate expression of the second Bianchi a identity DRb = 0 is

X α X α α DµRβ νρ = TνµRβ αρ. (202) (µνρ) (µνρ)

a b Exercise 17 Calculate ?Rb ∧ θ in an orthonormal basis. Diﬀerential Forms on Riemannian (Lorentzian). . . 467

a b b a Solution: First we recall the ?Rb ∧ θ = θ ∧ ?Rb and then use the formula in the third line of Eq.(35) to write:

b a b a θ ∧ ?Rb = − ? (θ yRb) 1 = − ? θb (R a θc ∧ θd) 2 y b cd a bc d = − ? [Rb cdη θ ] ca d ac d = − ? [R cdθ ] = − ? [R dcθ ] a d a = − ? [Rdθ ] = − ? R (203) Of course, if the connection is the Levi-Civita one we get

b ˚a b ˚a ˚a b ˚a θ ∧ ?Rb = − ? (θ yRb) = −Rbθ = − ? R . (204)

13 A Remark on Evans 101th Paper on “ECE Theory” Eq. (195) or its equivalent Eq.(201) is to be compared with a wrong one derived by Evans from where he now claims that the Einstein-Hilbert (gravitational) theory which uses in its formulation the Levi-Civita connection D˚ is incompatible with the first Bianchi identity. Evans con- clusion follows because he thinks to have derived “from first principles” that a a b D?T = ?Rb ∧ θ , (205) an equation that if true implies as we just see from Eq.(203) that for the Levi-Civita connection for which T a = 0 the Ricci tensor of the connection D˚ is null. We show below that Eq.(205) is a false one in two different ways, firstly by deriving the correct equation for D?T a and secondly by showing explicit counterexamples for some trivial structures. Before doing that let us show that we can derive from the first Bianchi identity that ˚ a Ra cd = 0, (206) an equation that eventually may lead Evans in believing that for a Levi- Civita connection the first Bianchi identity implies that the Ricci tensor is null. As we know, for a Levi-Civita connection the first Bianchi iden- a ˚a tity gives (with Rb 7−→ Rb): ˚a b Rb ∧ θ = 0. (207) 468 W. A. Rodrigues Jr.

Contracting this equation with θa we get ˚a b b ˚a θay(Rb ∧ θ ) = θay(θ ∧ Rb) b ˚a b ˚a = δa Rb − θ ∧ (θayRb) 1 = R˚a − θb ∧ [θ (R˚ a θc ∧ θd)] a 2 ay b cd ˚a ˚ a b d = Ra − Rb adθ ∧ θ Now, the second term in this last equation is null because according to a a the Eq.(106), −Rb ad = Rb da = Rbd are the components of the Ricci tensor, which is a symmetric tensor for the Levi-Civita connection. For the first term we get ˚ a c d Ra cdθ ∧ θ = 0, (208) which implies that as we stated above that ˚ a Ra cd = 0. (209) ˚ a But according to Eq.(106) the Ra cd are not the components of the Ricci tensor, and so there is not any contradiction. As an additional ver- ification recall that the standard form of the first Bianchi identity in Riemannian geometry is

a a a R˚b cd + R˚c db + R˚d bc. = 0 (210) Making b = a we get

a a a R˚a cd + R˚c da + R˚d ac a a a = R˚a cd − R˚c ad + R˚d ac a = R˚a cd + R˚cd − R˚dc a = R˚a cd = 0. (211)

14 Direct Calculation of D ? T a We now present using results of Cliﬀord bundle formalism, recalled above (for details, see, e.g., [22]) a calculation of D ? T a. We start from Cartan ﬁrst structure equation

a a a b T = dθ + ωb ∧ θ . (212) Diﬀerential Forms on Riemannian (Lorentzian). . . 469

By deﬁnition a a a b D ? T = d ? T + ωb ∧ ?T . (213) ^2 Now, if we recall Eq.(39), since the T a ∈ sec T ∗M,→ sec C`(M, g) we can write d ? T a = ?δT a. (214)

We next calculate δT a. We have:

a a a b δT = δ dθ + ωb ∧ θ a a a a b = δdθ + dδθ − dδθ + δ(ωb ∧ θ ) . (215)

Next we recall the deﬁnition of the Hodge D’Alembertian which, recalling Eq.(112) permit us to write the ﬁrst two terms in Eq.(215) as the negative of the square of the standard Dirac operator (associated with the Levi-Civita connection)19. We then get:

a 2 a a a b δT = − ∂| θ − dδθ + δ(ωb ∧ θ ) Eq.(115) ˚ a a a a b = −θ − (∂| ∧ ∂|)θ − dδθ + δ(ωb ∧ θ ) Eq.(127) ˚ a ˚a a a b = −θ − R − dδθ + δ(ωb ∧ θ ) ˚ a a a a a b = −θ − R + J − dδθ + δ(ωb ∧ θ ) (216) where we have used Eq(107) to write

a a b ˚a a b R = Rbθ = (Rb + Jb)θ . (217)

So, we have

a ˚ a a a a a b d ? T = − ? θ − ?R + ?J − ?dδθ + ?δ(ωb ∧ θ ) and ﬁnally

a ˚ a a a a a b a b D? T = − ? θ − ?R + ?J − ?dδθ + ?δ(ωb ∧ θ ) + ωb ∧ ?T (218) or equivalently recalling Eq.(35)

a ˚ a a a a a b a b D?T = −?θ −?R +?J −?dδθ +?δ(ωb ∧θ )−?(ωbyT ) (219) 19Be patient, the Riemann-Cartan connection will appear in due time. 470 W. A. Rodrigues Jr.

a a b Remark 18 Eq.(219) does not implies that D? T = ?Rb ∧ θ because taking into account Eq.(203) a b a a ˚ a a a a a b a b ?Rb∧θ = −?R 6= D?T = −?θ −?R +?J −?dδθ +?δ(ωb∧θ )−?(ωbyT ) (221) in general.

So, for a Levi-Civita connection we have that D? T a = 0 and then Eq.(218) implies a ˚ a ˚a a a b D? T = 0 ⇔ −θ − R − dδθ + δ(˚ωb ∧ θ ) = 0 (220) a b b or since ˚ωb ∧ θ = −dθ for a Levi-Civita connection, a ˚ a ˚a a a D? T = 0 ⇔ −θ − R − dδθ − δdθ = 0 (221) or yet ˚ a ˚a 2 a a a −θ − R = − ∂| θ = dδθ + δdθ , (222) an identity that we already mentioned above (Eq.(113)). 14.1 Einstein Equations The reader can easily verify that Einstein equations in the Clifford bundle formalism is written as: 1 R˚a − Rθ˚ a = Ta, (223) 2 ˚ a a b where R is the scalar curvature and T = −Tb θ are the energy- momentum 1-form fields. Comparing Eq.(221) with Eq.(223). We immediately get the “wave equation” for the cotetrad fields: 1 Ta = − Rθ˚ a − ˚θa − dδθa − δdθa, (224) 2 which does not implies that the Ricci tensor is null.

Remark 19 We see from Eq.(224) that a Ricci ﬂat spacetime is characterized by the equality of the Hodge and covariant D’ Alembertians acting on the coterad ﬁelds, i.e., ˚ a a θ = ♦θ , (225) a non trivial result.

˚ a Exercise 20 Using Eq.(120) and Eq.(121 ) write θ in terms of the connection coeﬃcients of the Riemann-Cartan connection. Diﬀerential Forms on Riemannian (Lorentzian). . . 471

15 Two Counterexamples to Evans (Wrong) Equation a a b “D ? T = ?Rb ∧ θ ” 15.1 The Riemannian Geometry of S2 Consider the well known Riemannian structure on the unit radius sphere [12] {S2, g, D˚}. Let {xi}, x1 = ϑ , x2 = ϕ, 0 < ϑ < π, 0 < ϕ < 2π, be spherical coordinates covering U = {S2 − l}, where l is the curve joining the north and south poles. µ A coordinate basis for TU is then {∂µ} and its dual basis is {θ = µ 2 dx }. The Riemannian metric g ∈ sec T0 M is given by

g = dϑ ⊗ dϑ + sin2 ϑdϕ ⊗ dϕ (226)

0 and the metric g∈ sec T2 M of the cotangent space is 1 g = ∂1 ⊗ ∂1 + ∂2 ⊗ ∂2. (227) sin2 ϑ

An orthonormal basis for TU is then {ea} with 1 e = ∂ , e = ∂ , (228) 1 1 2 sin ϑ 2 with dual basis {θa} given by

θ1 = dϑ, θ2 = sin ϑdϕ. (229)

The structure coeﬃcients of the orthonormal basis are

k [ei, ej] = cijek (230)

i 1 i j k and can be evaluated, e.g., by calculating dθ = − 2 cjkθ ∧ θ . We get immediately that the only non null coeﬃcients are

2 2 c12 = −c21 = − cot θ. (231)

c To calculate the connection 1-form ωd we use Eq.(92), i.e., 1 ωcd = (−cc ηdj + cd ηcj − ηcaη ηdjcb )θk. 2 jk jk bk ja Then, 472 W. A. Rodrigues Jr.

1 ω21 = (−c2 η11 − η22η η11c2 )θ2 = cot ϑθ2. (232) 2 12 22 12 Then

ω21 = −ω12 = cot ϑθ2, 2 1 2 ω1 = −ω2 = cot ϑθ , (233)

2 2 ˚ω21 = cot ϑ , ˚ω11 = 0. (234) Now, from Cartan’ s second structure equation we have

˚1 1 1 1 1 2 1 R2 = d˚ω2 + ˚ω1 ∧ ˚ω1 + ˚ω2 ∧ ˚ω2 = d˚ω2 (235) = θ1 ∧ θ2 and20 1 R˚ 1 = −R˚ 1 = −R˚ 2 = R˚ 2 = . (236) 2 12 2 21 1 12 1 21 2 0 1 ^ ∗ Now, let us calculate ?R2 ∈ sec T M. We have

1 1 1 2 1 2 1 2 1 2 ?R2 = Rf2yτg = −(θ ∧ θ )y(θ ∧ θ ) = −θ θ θ θ 2 2 = θ1 θ2 = 1 (237)

and 1 a 1 2 2 ?Ra ∧ θ = R2 ∧ θ = θ 6= 0. (238) 1 1 Now, Evans equation implies that ?Ra ∧ θ = 0 for a Levi-Civita connection and thus as promised we exhibit a counterexample to his wrong equation.

Remark 21 We recall that the ﬁrst Bianchi identity for (S2, g, D˚), i.e., a a b DT = Rb ∧ θ = 0 which translate in the orthonormal basis used above a a a in R˚b cd + R˚c db + R˚d bc. = 0 is rigorously valid. Indeed, we have

1 1 1 1 1 R˚2 12 + R˚1 21 + R˚2 21 = R˚2 12 − R˚2 12 = 0, 2 2 2 2 2 R˚1 12 + R˚1 21 + R˚2 21 = R˚1 12 − R˚1 12 = 0. (239)

20 ˚ ˚1 ˚2 Observe that with our deﬁnition of the Ricci tensor it results that R = R1 +R2 = −1. Diﬀerential Forms on Riemannian (Lorentzian). . . 473

15.2 The Teleparallel Geometry of (S˚2, g,D)

Consider the manifold S˚2 = {S2\north pole} ⊂ R3, it is an sphere of 0 ˚2 unitary radius excluding the north pole. Let g ∈ sec T2 S be the standard Riemann metric ﬁeld for S˚2 (Eq.(226)). Now, consider besides the Levi-Civita connection another one, D, here called the Nunes (or navi- gator [17]) connection21. It is deﬁned by the following parallel transport rule: a vector is parallel transported along a curve, if at any x ∈ S˚2 the angle between the vector and the vector tangent to the latitude line passing through that point is constant during the transport (see Figure 1)

!"#$%& '( )&*+&,%"-./ 01.%.-,&%"2.,"*3 *4 ,1& 5$3&6 0*33&-,"*37 Figure 1: Geometrical Characterization of the Nunes Connection.

5*89 ", "6 *:;"*$6 4%*+ 81., 1.6 :&&3 6."< .:*;& ,1., *$% -*33&-,"*3 "6 -1.%.-,&%"2&< := ! ! ! "" >?@?A 1 2 !! " AsB1&3 before,.C"3# "3,* (x.--*,$ x3, ),1& =<&D (3ϑ,","*3 ϕ*)4 , 01& -<$%;. ϑ,$% <& ,&3 π6*,% 8 0& 1<.;& ϕ < 2π, denote the standard spherical coordinates of a S˚2 of unitary radius, which covers

˚2 $ $ ! U = {"S#! −# $ l# }!",# ! where%$ ! $ l!!is!! the! curve! !! ! joining!# the! "" north>?@E andA south poles. # " ! ! ! " ! !!""!!" !" # $ Now,F/6*9 ,.C it"3# is"3, obvious* .--*$3, ,1 from& <&D3" what,"*3 *4 , has1& ,*% been6"*3 *G& said%.,"*3 above8& 1.;& that our connection

is characterized by # %#!"# !%$ ! & !# ! !! !" !! !% %!"# !%& "% ! ! " ! # De ei = 0. (240) ! %!"# !%& ! '"%!## j >?@@A

21 & & ! & & ! '() (9 & ' ! & ' ! "" >?@HA See some historical&' ! details'& in [22].&' ! '& I, 4*//*86 ,1., ,1& $3"J$& 3*3 3$// ,*%6"*3 *K4*%+ "6(

& ! '() ($' $&" " ! # I4 =*$ 6,"// 3&&< +*%& <&,."/69 -*3-&%3"3# ,1"6 /.6, %&6$/,9 -*36"<&% !"#$%& '>:A 81"-1 61*86 ,1& 6,.3<.%< G.%.+&,%"2.,"*3 *4 ,1& G*"3,6 )# *# +# , "3 ,&%+6

@? 474 W. A. Rodrigues Jr.

Then taking into account the deﬁnition of the curvature tensor we have

h i R(e , θa, e , e ) = θa D D − D D − Dc e = 0. (241) k i j ei ej ej ei [ei,ej] k

Also, taking into account the deﬁnition of the torsion operation we have

k τ(ei, ej) = Tij ek = Dej ei − Dei ej − [ei, ej] k = [ei, ej] = cijek, (242)

2 2 1 1 T21 = −T12 = cot ϑ , T21 = −T12 = 0. (243)

It follows that the unique non null torsion 2-form is:

T 2 = − cot ϑθ1 ∧ θ2.

If you still need more details, concerning this last result, consider Fig- ure 1(b) which shows the standard parametrization of the points p, q, r, s in terms of the spherical coordinates introduced above [17]. According to the geometrical meaning of torsion, we determine its value at a given point by calculating the diﬀerence between the (inﬁnitesimal)22 vectors pr1 and pr2 determined as follows. If we transport the vector pq along ps 1 2 we get the vector ~v = sr1 such that |g(~v,~v)| = sin ϑ4ϕ. On the other hand, if we transport the vector ps along pr we get the vector qr2 = qr. Let ~w = sr. Then,

1 |g(~w, ~w)| 2 = sin(ϑ − 4ϑ)4ϕ ' sin ϑ4ϕ − cos ϑ4ϑ4ϕ, (244)

Also,

1 ~u = r r = −u( ∂ ), u = |g(~u,~u)| = cos ϑ4ϑ4ϕ. (245) 1 2 sin ϑ 2

22This wording, of course, means that this vectors are identiﬁed as elements of the appropriate tangent spaces. Diﬀerential Forms on Riemannian (Lorentzian). . . 475

Then, the connection D of the structure (S˚2, g,D) has a non null torsion tensor Θ. Indeed, the component of ~u = r1r2 in the direction ∂2 is ϕ k precisely T 4ϑ4ϕ. So, we get (recalling that D ∂i = Γ ∂k) ϑϕ ∂j ji

ϕ ϕ ϕ Tϑϕ = Γϑϕ − Γϕϑ = − cot ϑ. (246)

Exercise 22 Show that D is metrical compatible, i.e., Dg = 0.

Solution:

0 = Dec g(ei, ej) = (Dec g)(ei, ej) + g(Dec ei, ej) + g(ei,Dec ej)

= (Dec g)(ei, ej) (247)

Remark 23 Our counterexamples that involve the parallel transport rules deﬁned by a Levi-Civita connection and a teleparallel connection in S˚2 show clearly that we cannot mislead the Riemann curvature tensor of a connection deﬁned in a given manifold with the fact that the manifold may be bend as a surface in an Euclidean manifold where it is embedded. Neglecting this fact may generate a lot of wishful thinking.

16 Conclusions In this paper after recalling the main definitions and a collection of tricks of the trade concerning the calculus of differential forms on the Cartan, Hodge and Clifford bundles over a Riemannian or Riemann- Cartan space or a Lorentzian or Riemann-Cartan spacetime we solved with details several exercises involving different grades of difficult and which we believe, may be of some utility for pedestrians and even for experts on the subject. In particular we found using technology of the Clifford bundle formalism the correct equation for D?T a. We show that a a b the result found by Dr. Evans [10], “D ? T = ?Rb ∧ T ” because it contradicts the right formula we found. Besides that, the wrong formula is also contradicted by two simple counterexamples that we exhibited in Section 15 . The last sentence before the conclusions is a crucial remark, which each one seeking truth must always keep in mind: do not confuse the Riemann curvature tensor23 of a connection defined in a given manifold with the fact that the manifold may be bend as a surface in an Euclidean manifold where it is embedded. 23The remark applies also to the torsion of a connection. 476 W. A. Rodrigues Jr.

We end the paper with a necessary explanation. An attentive reader may ask: Why write a bigger paper as the present one to show wrong a result not yet published in a scientific journal? The justification is that Dr. Evans maintain a site on his (so called) “ECE theory” which is read by thousand of people that thus are being continually mislead, thinking that its author is creating a new Mathematics and a new Physics. Besides that, due to the low Mathematical level of many referees, Dr. Evans from time to time succeed in publishing his papers in SCI journals, as the recent ones., [8, 9]. In the past we already showed that several published papers by Dr. Evans and colleagues contain serious flaws (see, e.g., [5, 21]) and recently some other authors spent time writing papers to correct Mr. Evans claims (see, e.g.,[1, 2, 3, 14, 15, 28]) It is our hope that our effort and of the ones by those authors just quoted serve to counterbalance Dr. Evans influence on a general public24 which being anxious for novelties may be eventually mislead by people that claim among other things to know [6, 7, 8, 9] how to project devices to withdraw energy from the vacuum.

Acknowledgement 24 The author is grateful to Prof. E. A. Notte- Cuello, for have checking all calculations and discovered several misprints, that are now corrected. Moreover, the author will be grateful to any one which point any misprints or eventual errors.

References [1] Bruhn,G.W., No Energy to be Extracted From the Vacuum, Phys.Scripta 74, 535–536( 2006). [2] Bruhn G.W, Hehl F.W., Jadczyk A. , Comments on “Spin Connection Resonance in Gravitational General Relativity”, Acta Physica Polonia B 39 1001-1008 (2008). [arXiv:0707.4433v2] [3] Bruhn, G. W., Evans’ Central Claim in his Paper #100, http://www.mathematik.tu-darmstadt.de/˜bruhn/onEvans100-101.html [4] Carroll, S. M., Lecture Notes in Relativity, http://arxiv.org/PS cache/gr- qc/pdf/9712/9712019.pdf [5] Carvalho A. L. T. , and Rodrigues, W. A. Jr, The Non Sequitur Mathematics and Physics of the ’New Electrodynamics’ of the AIAS Group, Random Operators and Stochastic Equations 9, 161-206 (2001) [arXiv:physics/0302016v5]

24And we hope also on many scientists, see a partial list in [6, 7]! Diﬀerential Forms on Riemannian (Lorentzian). . . 477

[6] Anastasovski, P K , Bearden, T E, Ciubotariu, C., Coffey W. T. , Crowell, B. , Evans,G. J. , Evans, M. W., Flower, R., Jeffers, S., Labounsky, A Lehnert, B , Mészáros, M., MolnárP. R., Vigier ,J. P. and Roy, S., Classical Electrodynamics Without the Lorentz Condition: Extracting Energy from the Vacuum, Phys. Scripta. 61, 513-517 (2000) [7] Anastasovski, P. K., Bearden, T. E. , Ciubotariu, C. , Coffey, W. T.,Crowell, L. B. , Evans, G. J., Evans, M. W., Flower, R., Laboun- sky, A.,(10), Lehnert, B., Mészáros,M., Molnar, P. R., . Moscicki,J. K , Roy, S., and Vigier, J. P., Explanation of the Montionless Electromag- netic Generator with O(3)Electrodynamics, Found. Phys. Lett. 14., 87-94 (2001) [8] Evans, M. W,, Spin Connection Resonance in Gravitational General Rel- ativity, Acta Physica Polonica B 38 2211-2220 (2007) [9] Evans, M. W, and Eckardt, H., Spin Connection Resonance in Magnetic Motors, Physica B 400 175-179, (2007). [10] Evans, M. W., The Incompatibility of the Christoffel Connection with the Bianchi Identity, http://www.aias.us/documents/uft/a101stpaper.pdf [11] Hawking, S. W. and Ellis, G. F. .R, The Large Scale Structure of Space- time, Cambridge University Press, Cambridge, 1973. [12] Göckeler, M. and Schückler, Differential Geometry, Gauge Theories, and Gravity, Cambridge University Press, Cambridge, 1987. [13] Green, M. B., Schwarz, J. H. and Witten, E., Superstring Theory, volume 2, Cambridge University Press, Cambridge, 1987. [14] Hehl, F. W., An Assessment of Evans’ Unified Field Theory, Found. Phys. 38, 7-37 (2008) [arXiv:physics/0703116v1] [15] Hehl, F. W. and Obukhov, An Assessment of Evans’ Unified Field Theory II,Found. Phys. 38, 38-46 (2008) [arXiv:physics/0703117v1 ] [16] Mosna, R. A. and Rodrigues, W. A., Jr., The Bundles of Algebraic and Dirac-Hestenes Spinor Fields, J. Math. Phys 45, 2945-2966 (2004). [arXiv:math-ph/0212033v5] [17] Nakahara, M., Geometry, Topology and Physics, Inst. Phys. Publ. Bristol and Philadelphia, 1990. [18] Notte-Cuello, E. A., Rodrigues, W. A. Jr., and de Souza, Q. A. G. The Squares of the Dirac and Spin-Dirac Operators on a Riemann- Cartan Spacetime, Rep. Math. Phys. 60, 135-157 (2007). [arXiv:math- ph/0703052v2]

[19] Prasana, A. R., Maxwell Equations in Riemann-Cartan Space U4, Phys. Lett. A 54, 17-18 (1975). [20] Poincar´e, H., Science and Hypothesis25, Dover Publ. Inc., New York,1952.

25First published in English in 1905 by Walter Scoot Publishing Co., Ltd. 478 W. A. Rodrigues Jr.

[21] Rodrigues, W. A Jr. and de Souza, Q. A. G, An ambiguous statement called ‘tetrad postulate’ and the correct field equations satis- fied by the tetrad fields, Int. J. Mod.Phys. D 14 2095-2150 (2005) [http://arXiv.org/math-ph/0411085] [22] Rodrigues, W. A. Jr. and Capelas de Oliveira, E., The Many Faces of Maxwell, Dirac and Einstein Equations. A Clifford Bundle Approach., Lecture Notes in Physics 722, Sringer, Heidelberg, 2007. [23] Rapoport, D. L., Cartan-Weyl Dirac and Laplacian Operators, Brownian Motions: The Quantum Potential and Scalar Curvature, Maxwell’s and Dirac-Hestenes Equations, and Supersymmetric Systems, Found. Phys. 35 , 1383-1431 (2005). [24] Rovelli, C., Loop Gravity, Cambrige University Press, Cambridge, 2004. [25] de Sabbata, V. and Gasperini, M., On the Maxwell Equations in a Riemann-Cartan Space, Phys. Lett. A 77, 300-302 (1980). [26] Sachs, R. K., and Wu, H., General Relativity for Mathematicians, Springer- Verlag, New York 1977. [27] Thirring, W. and Wallner, R., The Use of Exterior Forms in Einstein’s Gravitational Theory, Brazilian J. Phys. 8, 686-723 (1978). [28] ’t Hooft, G., Editorial Note, Found. Phys. 38, 1-2(2008). [http://www.springerlink.com/content/l1008h127565m362/]

(Manuscrit re¸cule 10 janvier 2008)