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: walrod@ime.unicamp.br, walrod@mpc.com.br

ABSTRACT. In this paper after recalling some essential tools concern-

ing 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 deriva-
tive 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 physi-
cal 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 D̊ . . . . . . . 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
7.2 Formula for Computation of the Connection 1- Forms ωba 446

8 Relation Between the Connections D̊ and D 447

9 Expressions for d and δ in Terms of Covariant Derivative

Operators D̊ 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 D̊ . . . 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

“D ? T a = ?Rab ∧ θb ” 471
2
15.1 The Riemannian Geometry of S . . . . . . . . . . . . . . 471
15.2 The Teleparallel Geometry of (S̊ 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-
dimensional manifold M equipped with a metric tensor g ∈ sec T20 M of
arbitrary signature (p, q), p + q = n and also equipped with metric com-
patible connections, the Levi-Civita (D̊) 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
covariant D’ Alembertian () ˚ and the Ricci operators (R̊a , Ra ) and
˚
Einstein operator () and the use of these operators in the Einstein-
Hilbert gravitational theory. Finally, we study the Bianchi identities.
Recalling that the first Bianchi identity is DT a = Rab ∧ θb , where T a
and Rab are respectively the torsion and the curvature 2-forms and {θb }
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
paper. Our result shows explicitly that the formula “D ?T a = ?Rab ∧θb ”
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 grav-
itational constant k (appearing in Einstein equations) are equal to 1.

1 A 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.
2 More 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)

Let M denotes a n-dimensional manifold3 . We denote as usual by Tx M
and Tx∗ M respectively
[ the tangent and the [ cotangent spaces at x ∈
∗
M . By T M = Tx M and T M = Txx M respectively
x∈M x∈M
the tangent and cotangent bundles. By Tsr M we denoteLthe bundle
∞
of r-contravariant and s-covariant tensors and by T M = r,s=0 Tsr M
^r ^r
the tensor bundle. By T M and T ∗ M denote respectively the
^
bundles of r-multivector fields and of r-form fields. We call TM =
Mr=n ^r
T M 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)
Vr form fields.
Vr Recall that

the real vector spaces are such that dim Tx M = dim Tx∗ M = nr
^
and dim T ∗ M = 2n . Some additional structures will be introduced or
mentioned below when needed. Let4 g ∈ sec T20 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)

(c) it is called a Riemann space if and only if

Dg = 0 and T = 0, (3)
3 We left the toplogy of M unspecified for a while.
4 We 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.
5 The 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 D̊.
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 re-
called 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 im-
portant 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
6 Even 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 T MLbe any vector
∞
fields and α ∈ sec T ∗ M any covector field . Let T M = r,s=0 Tsr 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 T M × 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 T M and P, Q ∈ sec T M we have

Df X+gY P = f DX P+gDY P,
DX (P + Q) = DX P + DX Q,
DX (f P) = f DX (P)+X(f )P,
DX (P ⊗ Q) = DX P ⊗ Q + P⊗DX Q. (7)

Given Q ∈ sec Tsr 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

D: sec Tsr M → sec Ts+1r

M,
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

U coordinate functions7 {xµ }. Let g ∈ sec T20 M be a metric field for
7 If e ∈ M , then xµ (e) = xµ is the µ coordinate of e in the given chart.
432 W. A. Rodrigues Jr.

M . Let {∂ µ } be a basis for T U , U ⊂ M and let {θµ = dxµ } be the

dual basis of {∂ µ }. The reciprocal basis of {θµ } is denoted {θµ }, and
g(θµ , θν ) := θµ · θν = δνµ . Introduce next a set of differentiable functions
qµa , qbν : U → R such that :

qaµ qµb = δab , qaµ qνa = δνµ . (9)

It is trivial to verify the formulas

gµν = qµa qνb ηab , g µν = qaµ qbν η ab ,

ηab = qaµ qbν gµν , η ab = qµa qνb g µν , (10)

with
ηab = diag(1, ..., 1 −1, ... − 1)
| {z } | {z } . (11)
p times q times
Moreover, defining
eb = qbν ∂ ν
the set {ea } with ea ∈ sec T M is an orthonormal basis for T U . The
dual basis of T U is {θa }, with θa = qµa dxµ . Also, {θb } is the reciprocal
basis of {θa }, i.e. θa · θb = δba .

Remark 3 When dim M = 4 the basis {ea } of T U 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:

D∂ µ ∂ ν = Γρµν ∂ ρ , D∂ σ ∂ µ = −Γµσα ∂ α , (12)

c b b c c
D ea eb = ωab ec , D ea e = −ωac e , D∂ µ eb = ωµb ec ,
D∂ µ dxν = −Γνµα dxα , D∂ µ θν = Γρµν θρ , (13)
b b c
D ea θ = −ωac θ , D∂ µ θb = −ωµa
b a
θ (14)
b
D ea θ = −ωcab θc ,
d
ωabc = ηad ωbc = −ωcba , ωabc = η bk ωkal η cl , ωabc = −ωacb
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
D̊∂ µ ∂ ν = Γ̊ρµν ∂ ρ , D̊∂ µ dxν = −Γ̊νµρ dxρ . (16)

To understood how D works, consider its action, e.g., on the sections

of T11 M = T M ⊗ T ∗ M .

D(X ⊗ α) = (DX) ⊗ α + X ⊗ Dα. (17)

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

we have

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

D∂ µ V = D∂ µ (V α ∂ α ) = (D∂ µ V )α ∂ α
= (∂ µ V α )∂ α + V α D∂ µ ∂ α
∂V α
 
ρ α + α
= + V Γµρ ∂ α := (Dµ V )∂ α , (19)
∂xµ

where it is to be kept in mind that the symbol Dµ+ V α is a short notation

for
Dµ+ V α := (D∂ µ V )α (20)

Also, we have

D∂ µ C = D∂ µ (Cα θα ) = (D∂ µ C)α θα

 
∂Cα β α
= − C Γ
β µα θ ,
∂xµ
:= (Dµ− Cα )θα (21)

where it is to be kept in mind that 8 that the symbol Dµ− Cα is a short

notation for
Dµ− Cα := (D∂ µ C)α . (22)
8 Recallthat other authors prefer the notations (D∂ µ V )α := V:µ
α and (D
∂ µ C)α :=
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
Dµ− qνa := (D∂ µ θa )ν = (D∂ µ qνa dxν )ν = ∂µ qνa − Γρµν qρa = ωµb
a b
qν ,
Dµ+ qνa := (D∂ µ qνa ea )a = ∂µ qνa + ωµν
ρ a
qρ = Γρµν qρa ,

thus verifying that Dµ− qνa 6= Dµ+ qνa 6= 0 and that

∂µ qνa + ωµb
a b
qν − Γaµb qνb = 0. (23)
Moreover, if we define the object
q = ea ⊗ θa = qµa ea ⊗ dxµ ∈ sec T11 U ⊂ sec T11 M, (24)
which is clearly the identity endormorphism acting on sections of T U ,
we find
a
Dµ qνa := (D∂ µ q)ν = ∂µ qνa + ωµb
a b
qν − Γaµb qνb = 0. (25)

Remark 6 Some authors call q ∈ sec T11 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
meaning Dµ− qνa or Dµ+ qνa , 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
9
meaning of the exterior product of formVrfields and its main
Vs properties .
∗ ∗
We simply recall here that if Ar ∈ sec T M , Bs ∈ sec T M then
Ar ∧ Bs = (−1)rs Bs ∧ Ar . (26)
9 We 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
y : (Ar , Bs ) 7→ Ar yBs = i1 ....is (a1 ∧...∧ar )·(bi1 ∧...∧bir )∼ bir +1 ∧...∧bis
i1 <... <ir
(28)
where ∼ is the reverse mapping (reversion) defined by
^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
V0 ∗
α, β ∈ sec T M the contraction is the ordinary (pointwise) product
V0 ∗ Vr ∗
in the real
Vs ∗ field and that if α ∈ sec T M , Ar ∈ sec T M , Bs ∈
sec T M then (αAr )yBs = Ar y(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 = hAir yhBis , 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

Vr ∗ that r ≥Vs and taking into ac-
s ∗
count that now if Ar ∈ sec T M , Bs ∈ sec T M then Bs yAr =
s(r−s)
(−1) Ar xBs .
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)
Vn
τg ∈ T ∗ M is the metric volume element. Of course, the V ∗Hodge
starVoperator is naturally extended to an isomorphism
Vn−r ∗ ? : sec T M→
r ∗
sec T ∗ M by linearity. The inverse ?−1 : sec
V
T M → sec 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 T U .
We can show that (see, e.g., [22]) that
?Ak = Aek yτg , (33)
where as noted before, in this paper Aek denotes the reverse of Ak .
Let {ϑα } be the dual basis of {eα } (i.e., it is a basis for T ∗ U ≡
V1 ∗
T U ) then g(ϑα , ϑβ ) = g αβ , with g αβ gαρ = δρβ . Writing ϑµ1 ...µp =
ϑ ∧ ... ∧ ϑµp , ϑνp+1 ...νn = ϑνp+1 ∧ ... ∧ ϑνn we have from Eq.(33)
µ1

1 p
? θµ1 ...µp = |det g|g µ1 ν1 ...g µp νp ν1 ...νn ϑνp+1 ...νn . (34)
(n − p)!
Some identities (used below) involving the Hodge star operator, the ex-
terior product and contractions are10 :

Ar ∧ ?Bs = Bs ∧ ?Ar ; r = s
Ar · ?Bs = Bs · ?Ar ; r + s = n
Ar ∧ ?Bs = (−1)r(s−1) ? (Ãr yBs ); r ≤ s (35)
Ar y ? Bs = (−1)rs ? (Ãr ∧ Bs ); r + s ≤ n
?τg = sign g; ?1 = τg .
10 See 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 + Ā ∧ dB;
(36)
(iii) df (v) = v(f );
(iv) d2 = 0,
V0 ∗
for every A, B ∈ sec T ∗ M , f ∈ sec
V
T M and v ∈ sec T M .
TheVHodge codifferential operator in the Hodge bundle is the mapping
r ∗ Vr−1 ∗
δ : sec T M → sec T M , given for homogeneous multiforms, by:

δ = (−1)r ?−1 d?, (37)

where V
? is the Hodge star operator. The operator δ extends by linearity
to all T ∗ M
The Hodge Laplacian (or Hodge D’Alembertian) operator is the map-
ping ^ ^
♦ : 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:

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

5 Clifford Bundles
Let (M, g, ∇) be a Riemannian, Lorentzian or Riemann-Cartan struc-
ture11 . As before let g ∈ sec T02 M be the metric on the cotangent
bundle associated with g ∈ sec T20 M . Then Tx∗ M ' Rp,q , where Rp,q
11 ∇ may be the Levi-Civita connection D̊ 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 signa-

ture (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 bun-
dle associated with the g-orthonormal coframe bundle PSOe(p,q) (M, g),
i.e., C`(M, g) = PSOe(p,q) (M, g)×ad R1,3 (see more details in, e.g., [16, 22]).
For any x ∈ M , C`(Tx∗ M, •) is a linear space over the real field R. More-
over,VC`(Tx∗ M ) is isomorphic as a real vector space to the Cartan alge-
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
{ea } be an orthonormal basis for T U and {θa } its dual basis. Then,
g(θa , θb ) = η ab .
5.1 Clifford Product
The fundamental Clifford product (in what follows to be denoted by
juxtaposition of symbols) is generated by
θa θb + θb θa = 2η ab (40)
and if C ∈ C`(M, g) we have

1 1
C = s + va θa + bab θa θb + aabc θa θb θc + pθn+1 , (41)
2! 3!
where τg := θn+1 = θ0 θ1 θ2 θ3 ...θn 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:
V1 ∗
For a, b ∈ sec 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 = hAr Bs ir+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 = hAir yhBis , r ≤ s. (44)
r,s

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

aBs = ayBs + a ∧ Bs , Bs a = Bs xa + Bs ∧ a,
1
ayBs = (aBs − (−1)s Bs a),
2
Ar yBs = (−1)r(s−r) Bs xAr ,
1
a ∧ Bs = (aBs + (−1)s Bs a),
2
Ar Bs = hAr Bs i|r−s| + hAr Bs i|r−s|+2 + ... + hAr Bs i|r+s|
Xm
= hAr Bs i|r−s|+2k
k=0

Ar · Br = Br · Ar = Aer yBr = Ar xBer = hAer Br i0 = hAr Ber i0 ,

?Ak = Aek yτg = Aek τg . (45)

Two other important identities to be used below are:

ay(X ∧ Y) = (ayX ) ∧ Y + X̂ ∧ (ayY), (46)

^1 ^
for any a ∈ sec T ∗ M and X , Y ∈ sec T ∗ M , and

Ay(ByC) = (A ∧ B)yC, (47)

T ∗ M ,→ C`(M, g)
V
for any A, B, C ∈ sec
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 dif-
ferential operator
∂ = θa Dea = ϑα Deα . (48)

T ∗ M ,→ sec C`(M, g) we define

V
For any A ∈ sec

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

5.2.2 Clifford Bundle Calculation of Dea A

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

Dea θb = −ωac
b c
θ = −ωabc θc , (50)

with ωabc = −ωacb , and ωabc = η bk ωkal η cl , ωabc = ηad ωbc

d
= −ωcba .
Defining

1 bc ^2
ωa = ωa θb ∧ θc ∈ sec T ∗ M ,→ sec C`(M, g), (51)
2

we have (by linearity) that [16] for any A ∈ sec T ∗ M ,→ sec C`(M, g)
V

1
Dea A = ∂ea A + [ωa , A], (52)
2
1
where ∂ea is the Pfaff derivative, i.e., for any A = p! Ai1 ...ip θ ...θ
i1 .ip
∈
Vp ∗
sec T M ,→ sec C`(M, g) it is:

1
∂ea A = [ea (Ai1 ...ip )]θi1 ...θ.ip . (53)
p!
Differential Forms on Riemannian (Lorentzian). . . 441

5.2.3 The Dirac Operator ∂| Associated to D̊

Using Eq.(52) we can show that for the case of a Riemannian or

Lorentzian structure (M, g, D̊) the standard Dirac operator defined by:

∂| = θa D̊ea = ϑα D̊eα ,
∂| A = ∂| ∧A + ∂|yA (54)

T ∗ M ,→ sec C`(M, g) is such that

V
for any A ∈ sec

∂| ∧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 D̊ 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
U with arbitrary coordinates {xµ }. Let {∂ µ } be a basis for T U , U ⊂ M
and let {θµ = dxµ } be the dual basis of {∂ µ }. The reciprocal basis of
{θµ } is denoted {θµ }, and g(θµ , θν ) := θµ · θν = δνµ .
Let also {ea } be an orthonormal basis for T U ⊂ T M with eb =
qbν ∂ ν . The dual basis of T U is {θa }, with θa = qµa dxµ . Also, {θb } is
the reciprocal basis of {θa }, i.e. θa · θb = δba . An arbitrary frame on
T U ⊂ T M , coordinate or orthonormal will be denote by {eα }. Its dual
frame will be denoted by {ϑρ } (i.e., ϑρ (eα ) = δαρ ).

6.1 Torsion and Curvature Operators

Definition 7 The torsion and curvature operators τ and ρ of a connec-
tion D, are respectively the mappings:

τ (u, v) = Du v − Dv u − [u, v], (57)

ρ(u, v) = Du Dv − Dv Du − D[u,v] , (58)

for every u, v ∈ sec T M .

442 W. A. Rodrigues Jr.

6.2 Torsion and Curvature Tensors

Definition 8 The torsion and curvature 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)
V1
for every u, v, w ∈ sec T M and α ∈ sec T ∗M .

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

τ (gu,hv) = ghτ (u, v),

ρ(gu,hv)f w=ghf ρ(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)

0
V1R ∗as the mapping such that for every a, u, v, w ∈
Define the tensor field
sec T M and α ∈ sec 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 T M ,

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(Dv c · c + c·Dv c) =2u(Dv c · c)

= 2(Du Dv c) · c + 2(Du c) · Dv c, (67)

Exachanging u ↔ v in the last equation we get

v(u(c · c) =2(Dv Du c) · c + 2(Dv c) · Du c. (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)

Given an arbitrary frame {eα } on T U ⊂ T M , let {ϑρ } be the dual

frame. We write:
[eα ,eβ ]=cραβ eρ
(73)
Deα eβ =Lραβ eρ ,
where cραβ are the structure coefficientsof the frame {eα } and Lραβ are
the connection coefficientsin this frame. Then, the components of the
torsion and curvature tensors are given, respectively, by:
ρ
T(ϑρ , eα ,eβ ) = Tαβ = Lραβ − Lρβα − cραβ
R(eµ , ϑρ , eα ,eβ ) = Rµ ραβ
= eα (Lρβµ ) − eβ (Lραµ ) + Lρασ Lσβµ − Lρβσ Lσαµ − cσαβ Lρσµ .
(74)
444 W. A. Rodrigues Jr.

It is important for what follows to keep in mind the definition of the

(symmetric) Ricci tensor, here denoted Ric ∈ sec T20 M and which in an
arbitrary basis is written as

Ric =Rµν ϑµ ⊗ ϑν :=Rµ ρρν ϑµ ⊗ ϑν (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:
dϑρ = − 21 cραβ ϑα ∧ ϑβ
(76)
Deα ϑρ = −Lραβ ϑβ
V1 ∗
where ωβρ ∈ sec T M are the connection 1-forms, Lραβ are said to be
V2 ∗
the connection coefficients in the given basis, and the T ρ ∈ sec T M
ρ V 2 ∗
are the torsion 2-forms and the Rβ ∈ sec T M are the curvature
2-forms, given by:

ωβρ = Lραβ ϑα ,
1 ρ α
T ρ = Tαβ ϑ ∧ θβ (77)
2
1
Rρµ = Rµ ραβ ϑα ∧ ϑβ .
2

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

6.3 Cartan Structure Equations

dϑρ + ωβρ ∧ ϑβ = T ρ ,
(78)
dωµρ + ωβρ ∧ ωµβ = Rρµ .

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

T = eα ⊗ T α , (79)
R = eρ ⊗ eµ ⊗ Rρµ . (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)
“ Dωµρ = Rρµ . ” (82)

and D : sec T ∗ M → sec T ∗ M is said to be the exterior covariant

V V
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
that X ∈ sec Tpr+q M and let
^r
Xνµ11....ν
....µp
q
∈ sec T ∗ M, (83)

such that for vi ∈ sec T M, i = 0, 1, 2, .., r,

Xνµ11....ν
....µp
q
(v1 , ..., vr ) = X(v1 , ..., vr , eν1 , ..., eνq , ϑµ1 , ..., ϑµp ). (84)

µ ....µ
The exterior covariant differential D of Xν11....νqp 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
µ ....µ
(r + 1)DXν11....νqp (v0 , v1 , ..., vr )
r
X
= (−1)ν Deν X(v0 , v1 , ..., v̌ν , ...vr , eν1 , ..., eνq , ϑµ1 , ..., ϑµp )
ν=0
X
− (−1)ν+ς X(T(vλ , vς ), v0 , v1 , ..., v̌λ , ..., v̌ς , ..., vr , eν1 , ..., eνq , ϑµ1 , ..., ϑµp ).
0≤λ,ς ≤r
(86)

12 Sometimes also called exterior covariant differential.

13 Which is not the case of the connection 1-forms ωβα , despite the name. More
precisely, the ωβα are not true indexed forms, i.e., there does not exist a tensor field
ω such that ω(ei , eβ , ϑα ) = ωβα (ei ).
14 As 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

DXνµ11....ν
....µp
q
= dXνµ11....ν
....µp
q
+ ωµµs1 ∧ Xνµ1s....ν
....µp
q
+ ... + ωµµs1 ∧ Xνµ11....ν
....µp
q
(87)
− ωνν1s ∧ Xνµs1....ν
....µp
q
− ... − ωµµs1 ∧ Xνµ11....ν
....µp
s
.

Remark 10 Note that if Eq.(87) is applied on any one of the connection

1-forms ωνµ we would get Dωνµ = dωνµ +ωαµ ∧ωνα −ωνα ∧ωαµ . So, we see that
the symbol Dωνµ 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 articles 15 ! .

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

D(X J ∧ Y K ) = DX J ∧ Y K + (−1)rs X J ∧ DY K . (88)

Vr
(b) For any X µ1 ....µp ∈ sec T ∗ M then

DDX µ1 ....µp = dX µ1 ....µp + Rµµ1s ∧ X µs ....µp + ...Rµµps ∧ X µ1 ....µs . (89)

(c) For any metric-compatible connection D if g = gµν ϑµ ⊗ ϑν then,

Dgµν = 0. (90)

7.2 Formula for Computation of the Connection 1- Forms ωba

In an orthonormal cobasis {θa } we have (see, e.g., [22]) for the connection
1-forms
1 d
ω cd = θ ydθc − θc ydθd + θc y(θd ydθa )θa ,

(91)
2
or taking into account that dθa = − 12 cajk θj ∧ θk ,

1
ω cd = (−ccjk η dj + cd cj ca dj b k
jk η − η ηbk η cja )θ . (92)
2
15 The authors of reference [27] ρ
knows exactly what they are doing and use “Dωµ =
Rρµ ” only as a short notation. Unfortunately this is not the case for some other
authors.
16 Multi indices are here represented by J and K.
Differential Forms on Riemannian (Lorentzian). . . 447

8 Relation Between the Connections D̊ and D

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

Deα eβ = Lραβ eρ , Deα ϑρ = −Lραβ ϑβ ,

D̊eα eβ = L̊ραβ eρ , D̊eα ϑρ = −L̊ραβ ϑβ . (93)

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

[eα ,eβ ] = cραβ eρ . (94)

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

ρ ρ 2 ρ
Sαβ = S̆αβ + s gαβ (97)
n
ρ
where S̆αβ is its traceless part, is called the shear of the connection, and

1 µν ρ
sρ = g Sµν (98)
2
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
L̊ραβ = (bραβ + cραβ ). (99)
2
Moreover, we introduce the contorsion tensor whose components in
an arbitrary basis are defined by
448 W. A. Rodrigues Jr.

1 ρ
Kραβ = Lραβ − L̊ραβ = ρ
(T + Sαβ ), (100)
2 αβ
and which can be written as
1
Kραβ = − g ρσ (gµα Tσβ
µ µ
+ gµβ Tσα µ
− gµσ Tαβ ). (101)
2
We now present the relation between the Riemann curvature ten-
sor Rµ ραβ associated with the Riemann-Cartan connection D and the
Riemann curvature tensor R̊µ ραβ of the Levi-Civita connection D̊.
Rµ ραβ = R̊µ ραβ + Jµ ρ[αβ] , (102)
where:
Jµ ραβ = D̊α Kρβµ − Kρβσ Kσαµ = Dα Kρβµ − Kρασ Kσβµ + Kσαβ Kρσµ . (103)

Multiplying both sides of Eq.(102) by 21 θα ∧ θβ we get:

Rρµ = R̊ρµ + Jρµ , (104)

where
1 ρ
Jρµ = Jµ [αβ] θα ∧ θβ . (105)
2
From Eq.(102) we also get the relation between the Ricci tensors of
the connections D and D̊. We write for the Ricci tensor of D
Ric = Rµα dxµ ⊗ dxν
Rµα := Rµ ραρ (106)
Then, we have
Rµα = R̊µα + Jµα , (107)
with
Jµα = D̊α Kρρµ − D̊ρ Kραµ + Kρασ Kσρµ − Kρρσ Kσαµ
= Dα Kρρµ − Dρ Kραµ − Kρσα Kσρµ + Kρρσ Kσαµ . (108)
Observe that since the connection D is arbitrary, its Ricci tensor will be
not be symmetric in general. Then, since the Ricci tensor R̊µα of D̊ is
necessarily symmetric, we can split Eq.(107) into:
R[µα] = J[µα] ,
(109)
R(µα) = R̊(µα) + J(µα) .
Differential Forms on Riemannian (Lorentzian). . . 449

9 Expressions for d and δ in Terms of Covariant Derivative

Operators D̊ 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

dQ = ϑα ∧ (D̊eα Q) = ∂| ∧Q,
δQ = −ϑα y(D̊eα Q) = ∂|yQ. (110)

We have also the important formulas

dQ = ϑα ∧ (Deα Q) − T α ∧ (ϑα yQ) =∂ ∧ Q − T α ∧ (ϑα yQ),

δQ = −ϑα y(Deα Q) − T α y(ϑα ∧ Q) = −∂yQ − T α 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, D̊).

10.1 The Square of the Dirac Operator ∂| Associated to D̊

2
TheVsquare of standard Dirac operator | | ||
p ∗ Vp ∗ ∂ is the operator, ∂ = ∂ ∂ :
sec T M ,→ sec C`(M, g) → sec 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)

and thus we recognize that ∂| 2 ≡ ♦ is the Hodge D’Alembertian of the

manifold introduced by Eq.(38)
On the other hand, remembering the standard Dirac operator is ∂| =
ϑα D̊eα , where {ϑα } is the dual basis of an arbitrary basis {eα } on T U ⊂
450 W. A. Rodrigues Jr.

T M and D̊ is the Levi-Civita connection of the metric g, we have:

∂| 2 = (ϑα D̊eα )(ϑβ D̊eβ ) = ϑα (ϑβ D̊eα D̊eβ + (D̊eα ϑβ )D̊eβ )

= g αβ (D̊eα D̊eβ − L̊ραβ D̊eρ ) + ϑα ∧ ϑβ (D̊eα D̊eβ − L̊ραβ D̊eρ ).

Then defining the operators:

ρ
(a) ∂| · ∂| =g αβ (D̊eα D̊eβ − L̊αβ D̊eρ )
(114)
(b) ∂| ∧ ∂|=ϑα ∧ ϑβ (D̊eα D̊eβ − L̊ραβ D̊eρ ),

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 αβ h i
∂| · ∂| = g D̊eα D̊eβ + D̊eβ D̊eα − bραβ D̊eρ . (117)
2
Applying this operator to the 1-forms of the frame {θα }, we get:

1
(∂| · ∂|)ϑµ = − g αβ M̊ρ µ αβ θρ , (118)
2
where:

M̊ρ µ αβ = eα (L̊µβρ ) + eβ (L̊µαρ ) − L̊µασ L̊σβρ − L̊µβσ L̊σαρ − bσαβ L̊µσρ . (119)

The proof that an object with these components is a tensorVmay be

r ∗
found in [22]. In particular, for every r-form field ω ∈ sec T M,
1 α1 αr
ω = r! ωα1 ...αr θ ∧ . . . ∧ θ , we have:

1 αβ
(∂| · ∂|)ω = g D̊α D̊β ωα1 ...αr θα1 ∧ . . . ∧ θαr , (120)
r!
Differential Forms on Riemannian (Lorentzian). . . 451

where D̊α D̊β ωα1 ...αr are the components of the covariant derivative of
1
ω, i.e., writing D̊eβ ω = r! D̊β ωα1 ...αr θα1 ∧ . . . ∧ θαr , it is:

D̊β ωα1 ...αr = eβ (ωα1 ...αr ) − L̊σβα1 ωσα2 ...αr − · · · − L̊σβαr ωα1 ...αr−1 σ . (121)

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 = g̊ αβ D̊α D̊β δρµ ϑρ = g̊ αβ (D̊α D̊β δρµ + D̊β D̊α δρµ )ϑρ
2
and therefore, taking into account the Eq.(118), we conclude that:
M̊ρ µ αβ = −(D̊α D̊β δρµ + D̊β D̊α δρµ ). (122)

By its turn, the operator ∂| ∧ ∂| can also be written as:

1 h i
∂| ∧ ∂| = ϑα ∧ ϑβ D̊α D̊β − D̊β D̊α − cραβ D̊ρ . (123)
2
Applying this operator to the 1-forms of the frame {ϑµ }, we get:
1
(∂| ∧ ∂|)ϑµ = − R̊ρ µ αβ (ϑα ∧ ϑβ )ϑρ = −R̊µρ ϑρ , (124)
2
where R̊ρ µ αβ are the components of the curvature tensor of the connec-
tion D̊. Then using the second formula in the first line of Eq.(45) we
have
R̊µρ θρ = R̊µρ xθρ + R̊µρ ∧ θρ . (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
Levi-Civita connection (T µ = 0) is R̊µρ ∧ θρ = 0 . The first term in
Eq.(125) can be written
1 µ
R̊µρ xθρ = R̊ρ αβ (θα ∧ θβ )xθρ
2
1
= R̊ρ µ αβ θρ y(θα ∧ θβ )
2
1
= − R̊ρ µ αβ (g̊ ρα θβ − g̊ ρβ θα )
2
= −g̊ ρα R̊ρ µ αβ θβ = −R̊βµ θβ , (126)
452 W. A. Rodrigues Jr.

where R̊βµ are the components of the Ricci tensor of the Levi-Civita
connection D̊ of g. Thus we have a really beautiful result:

(∂| ∧ ∂|)θµ = R̊µ , (127)

where R̊µ = R̊βµ θβ are the Ricci 1-forms of the manifold. Because of
this relation, we call the operator ∂| ∧ ∂| the Ricci operator of the manifold
associated to the Levi-Civita connection D̊ of g.
We can show [22] that the Ricci operator ∂| ∧ ∂| satisfies the relation:

∂| ∧ ∂| = R̊σ ∧ iσ + R̊ρσ ∧ iρ iσ , (128)

where the curvature 2-forms are R̊ρσ = 21 R̊ρσ αβ ϑα ∧ ϑβ and

iσ ω := ϑσ yω. (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:

R̊ρσ ∧ iρ iσ ? ϑµ = R̊ρσ ? ϑρ y(ϑσ y ? ϑµ ))

= −R̊ρσ ∧ ?(ϑρ ∧ ϑσ ? ϑµ ) (130)
= ?(R̊ρσ y(ϑρ ∧ ϑσ ∧ ϑµ )),

where we have used the Eqs.(35). Then, recalling the definition of the
curvature forms and using the Eq.(28), we conclude that:

1
R̊ρσ ∧ (ϑρ yϑσ y ? ϑµ ) = 2 ? (R̊µ − R̊ϑµ ) = 2 ? G̊ µ , (131)
2

where R̊ is the scalar curvature of the manifold and the G̊ µ may be

called the Einstein 1-form fields.
That observation motivate us to introduce in [22] the Einstein op-
erator of the Levi-Civita connection D̊ of g on the manifold M as the
mapping  ˚ : sec C`(M, g) → sec C`(M, g) given by:

˚ = 1 ?−1 (R̊ρσ ∧ iρ iσ ) ? .
 (132)
2
Differential Forms on Riemannian (Lorentzian). . . 453

Obviously, we have:

˚ µ = G̊ µ = R̊µ − 1 R̊ϑµ .
θ (133)
2
In addition, it is easy to verify that ?−1 (∂| ∧ ∂|)? = − ∂| ∧ ∂| and ?−1 (R̊σ ∧
iσ )? = R̊σ yjσ . Thus we can also write the Einstein operator as:

˚ = − 1 (∂| ∧ ∂| +R̊σ yjσ ),

 (134)
2
where
jσ A = ϑσ ∧ A, (135)
^
for any A ∈ sec T ∗ M ,→ sec C`(M, g).
We recall [22] that if ω̊ρµ are the Levi-Civita connection 1-forms fields
in an arbitrary moving frame {ϑµ } on (M, g, D̊) then:

(a) (∂| · ∂|)ϑµ =−(∂| ·ω̊ρµ − ω̊ρσ · ω̊σµ )ϑρ

(136)
(b) (∂| ∧ ∂|)ϑµ =−(∂| ∧ω̊ρµ − ω̊ρσ ∧ ω̊σµ )ϑρ ,

and
2 µ
∂| ϑ = −(∂| ω̊ρµ − ω̊ρσ ω̊σµ )ϑρ . (137)

Exercise 11 Show that ϑρ ∧ ϑσ R̊ρσ = −R̊, where R̊ 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
compute the square of the (general) Dirac operator ∂ = ϑα Deα . As in
the earlier section, we have, by one side,

∂ 2 = (∂y + ∂∧)(∂y + ∂∧)

= ∂y∂y + ∂y∂∧ + ∂ ∧ ∂y + ∂ ∧ ∂∧

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

L+ = ∂y∂ ∧ + ∂ ∧ ∂y, (138)

so that:
∂ 2 = ∂ 2y + L+ + ∂ 2 ∧ . (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 = (ϑα Deα )(ϑβ Deβ ) = ϑα (ϑβ Deα Deβ + (Deα ϑβ )Deβ )
= g αβ (Deα Deβ − Lραβ Deρ ) + ϑα ∧ ϑβ (Deα Deβ − Lραβ Deρ )
and we can then define:
∂ · ∂ =g αβ (Deα Deβ − Lραβ Deρ )
(140)
∂ ∧ ∂=θα ∧ θβ (Deα Deβ − Lραβ Deρ )
in order to have:
∂ 2 = ∂∂ = ∂ · ∂ + ∂ ∧ ∂ . (141)
The operator ∂ · ∂ can also be written as:
1 α β 1
∂·∂ = θ · θ (Deα Deβ − Lραβ Deρ ) + θβ · θα (Deβ Deα − Lρβα Deρ )
2 2
1 αβ
= g [Deα Deβ + Deβ Deα − (Lαβ + Lρβα )Deρ ]
ρ
(142)
2
or,
1 αβ
∂·∂ = g (Deα Deβ + Deβ Deα − bραβ Deρ ) − sρ Deρ , (143)
2
where sρ has been defined in Eq.(98).
By its turn, the operator ∂ ∧ ∂ can also be written as:
1 α 1
∂∧∂ = ϑ ∧ ϑβ (Deα Deβ − Lραβ Deρ ) + ϑβ ∧ ϑα (Deβ Deα − Lρβα Deρ )
2 2
1 α
= ϑ ∧ ϑ [Deα Deβ − Deβ Deα − (Lραβ − Lρβα )Deρ ]
β
2
or,
1 α
∂∧∂ = ϑ ∧ ϑβ (Deα Deβ − Deβ Deα − cραβ Deρ ) − T ρ Deρ . (144)
2
Remark 12 For the case of a Levi-Civita connection we have similar
formulas for ∂| · ∂| (Eq.(142)) and ∂| ∧ ∂| (Eq.(144)) with D →
7 D̊, 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, D̊, 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= Fµν dxµ ∧ dxν = Fµν θµ ∧ θν = Fµν θµν , (146)
2 2 2

we have using Eq.(34) that

1 1 1 1p
?F = Fµν (?θµν ) = ?
Fρσ ϑρσ = (Fµν |det g|g µα g νβ αβρσ )ϑρσ
2 2 2 2
(147)
Thus
? 1 p
Fρσ = (?F)ρσ = Fµν |det g|g µα g νβ αβρσ . (148)
2

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 ? ?−1 F = ?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 ∧ (D̊∂ α F )
1 h i
= θα ∧ D̊∂ α (Fµν θµ ∧ θν )
2
1 α h i
= θ ∧ (∂α Fµν )θµ ∧ θν − Fµν Γ̊µ ρ ν ν µ
αρ θ ∧ θ − Fµν Γ̊αρ θ ∧ θ
ρ
2
1 h i
= θα ∧ (D̊α Fµν )θµ ∧ θν
2
1
= Dα Fµν θα ∧ θµ ∧ θν
2» –
1 1 1 1
= D̊α Fµν θα ∧ θµ ∧ θν + D̊µ Fνα θµ ∧ θν ∧ θα + D̊ν Fαµ θν ∧ θα ∧ θµ
2 3 3 3
» –
1 1 α µ ν 1 α µ ν 1 α µ ν
= D̊α Fµν θ ∧ θ ∧ θ + D̊µ Fνα θ ∧ θ ∧ θ + D̊ν Fαµ θ ∧ θ ∧ θ
2 3 3 3
1 “ ”
= D̊α Fµν + D̊µ Fνα + D̊ν Fαµ θα ∧ θµ ∧ θν .
6

So,

dF = 0 ⇔ D̊α Fµν + D̊µ Fνα + D̊ν Fαµ = 0. (149)

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

1
dF = (∂α Fµν )θα ∧ θµ ∧ θν (150)
2
1
= (∂α Fµν + ∂µ Fνα + ∂ν Fαµ ) θα ∧ θµ ∧ θν ,
6

from where we get that

dF = 0 ⇐⇒ ∂α Fµν +∂µ Fνα +∂ν Fαµ = 0 ⇐⇒ D̊α Fµν +D̊µ Fνα +D̊ν Fαµ = 0.
(151)
Differential Forms on Riemannian (Lorentzian). . . 457

Next we calculate δ ? F = 0. We have

δ ? F = −θα y(D̊∂ α ? F)
1 n o
= − θα y D̊∂ α [? Fµν θµ ∧ θν ]
2
1 n o
= − θα y (∂α ? Fµν )θµ ∧ θν − ? Fµν Γ̊µαρ θρ ∧ θν − ? Fµν Γ̊ναρ θµ ∧ θρ
2
1 α n ? o
= − θ y (∂α Fµν )θµ ∧ θν − ? Fρν Γ̊ραµ θµ ∧ θν − ? Fµρ Γ̊ραν θµ ∧ θν
2
1 α n o
= − θ y (D̊α ? Fµν )θµ ∧ θν
2
1n o
=− (D̊α ? Fµν )g αµ θν − (D̊α ? Fµν )g αν θµ
2
= −(D̊α ? Fµν )g αµ θν
= −[D̊α (? Fµν g αµ )]θν
= −(D̊α ? F να )]θν . (152)

Then we get that

D̊α Fµν + D̊µ Fνα + D̊ν Fαµ = 0 ⇔ dF = 0 ⇔ δ ? F = 0 ⇐⇒ D̊α ? F να = 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,
2 −1
(−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 n o
δF + J = − θα y D̊∂ α [Fµν θµ ∧ θν ] + Jν θν (155)
2
= (−D̊α F να + Jν )θν .
17 We observe that in terms of the “classical” charge and “vector” current densities

we have J =ρθ0 − ji θi .
458 W. A. Rodrigues Jr.

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

1 p
= ∂ρ F ρν + p ∂ρ ( |det g|)F ρν .
|det g|
Then
D̊α F αν = J ν ,
p p p
|det g|∂ρ F ρν + ∂ρ ( |det g|)F ρν = |det g|J ν ,
p p
∂ρ ( |det g|F ρν ) = |det g|J ν ,
1 p
p ∂ρ ( |det g|F ρν ) = J ν , (157)
|det g|
and
1 p
δF = 0 ⇔ D̊α F αν = J ν ⇔ p ∂ρ ( |det g|F ρν ) = J ν . (158)
|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 equa-
tions 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
dF = θα ∧ (D∂ α F) − T α ∧ (θα yF) = 0 (160)
Differential Forms on Riemannian (Lorentzian). . . 459

or

1
(Dα Fµν + Dµ Fνα + Dν Fαµ )θα ∧ θµ ∧ θν
6
11 α ρ
− T θ ∧ θσ ∧ [θα yFµν (θµ ∧ θν )]
2 2 ρσ
1
= (Dα Fµν + Dµ Fνα + Dν Fαµ )θα ∧ θµ ∧ θν
6
1 α
− Tρσ Fµν θρ ∧ θσ ∧ δαµ θν
2
1
= (Dα Fµν + Dµ Fνα + Dν Fαµ )θα ∧ θµ ∧ θν
6
1 σ
− Tαµ Fσν θα ∧ θµ ∧ θν
2
1
= (Dα Fµν + Dµ Fνα + Dν Fαµ )θα ∧ θµ ∧ θν
6
1 σ σ σ
− (Tαµ Fσν + Tµν Fσα + Tνα Fσµ )θα ∧ θµ ∧ θν
6
1
= (Dα Fµν + Dµ Fνα + Dν Fαµ )θα ∧ θµ ∧ θν
6
1 σ σ σ
+ (Fασ Tµν + Fµσ Tνα + Fνσ Tαµ )θα ∧ θµ ∧ θν .
6

i.e.,

σ σ σ
dF = 0 ⇐⇒ Dα Fµν +Dµ Fνα +Dν Fαµ +Fσα Tµν +Fµσ Tνα +Fνσ Tαµ = 0.
(161)
Also, taking into account that dF = 0 ⇐⇒ δ ? F = 0 we have using the
second noticeable formula in Eq.(111) that

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

Now,

θα y(Deα ? F) = (Dα ? F αν )θν = (Dα ? F αν )θν (163)

460 W. A. Rodrigues Jr.

and

T α y(θα ∧ ?F)
1 α β
= Tβρ (θ ∧ θρ )y(θα ∧ (? Fµν θµ ∧ θν )
4
1 α?
= Tβρ Fµν (θβ ∧ θρ )y(θα ∧ θµ ∧ θν )
4
1 α ? µν β ρ
= Tβρ F θ y[θ y(θα ∧ θµ ∧ θν )]
4
1 α ? µν β ρ
= Tβρ F θ y(δα θµ ∧ θν − δµρ θα ∧ θν + δνρ θα ∧ θµ )
4
1 α ? µν β 1 α ? µν β 1 α ? µν β
= Tβα F θ y(θµ ∧ θν ) − Tβµ F θ y(θα ∧ θν ) + Tβν F θ y(θα ∧ θµ )
4 4 4

1 α ? µν β 1 µ ? ρν β 1 µ ? νρ β
= Tβα F θ y(θµ ∧ θν ) − Tβρ F θ y(θµ ∧ θν ) + Tβρ F θ y(θµ ∧ θν )
4 4 4
1 α ? µν µ ? ρν µ ? νρ β
= (Tβα F − Tβρ F + Tβρ F )θ y(θµ ∧ θν )
4
1 α ? µν µ ? ρν µ ? νρ
= (Tβα F − Tβρ F + Tβρ F )(δµβ θν − δνβ θµ )
4
1 α ? µν µ ? ρν µ ? νρ 1 α ? νµ ν ? ρµ ν ? µρ
= (Tµα F − Tµρ F + Tµρ F )θν − (Tµα F − Tµρ F + Tµρ F )θν
4 4
1 α ? µν µ ? ρν ν ? µρ
= (Tµα F − Tµρ F + Tµρ F )θν (164)
2

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

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

δF = −θα y(Deα F) − T α y(θα ∧ F)

1 α ? µν
= −[Dα F αν + (Tµα F − Tµρ µ ? ρν ν ? µρ
F + Tµρ F )]θν = −J ν θν ,
2
(167)
Differential Forms on Riemannian (Lorentzian). . . 461

i.e.,

1 α ? µν
Dα F αν + (Tµα µ ? ρν
F − Tµρ ν ? µρ
F + Tµρ F ) = Jν. (168)
2

Exercise 14 Show (use Eq.(111)) that in a Riemann-Cartan spacetime

Maxwell equations become Maxwell equation, i.e.,

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

T a = dθa + ωba ∧ θb = Dθa ,

(170)
Rab = dωba + ωca ∧ ωbc

where

ωba = ωcb
a c
θ ,
1 a b
T a = Tbc θ ∧ θc (171)
2
1
Rab = Rb acd θc ∧ θd . (172)
2

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

DT a = dT a + ωba ∧ T b = d2 θa + d(ωba ∧ θb ) + ωba ∧ T b

= dωba ∧ θb − ωba ∧ dθb + ωba ∧ T b
= dωba ∧ θb − ωba ∧ (T b − ωcb ∧ θc ) + ωba ∧ T b
= (dωba + ωca ∧ ωbc ) ∧ θb
= Rab ∧ θb (173)
462 W. A. Rodrigues Jr.

Also,
DRab = dRab + ωca ∧ Rcb − ωbc ∧ Rac
= d2 ωba + dωca ∧ ωbc − dωbc ∧ ωca − Rac ∧ ωbc + Rcb ∧ ωca
= dωca ∧ ωbc − (dωca + ωda ∧ ωcd ) ∧ ωbc − dωca ∧ ωbc + (dωbc + ωdc ∧ ωbd ) ∧ ωca
= −ωda ∧ ωcd ∧ ωbc + ωdc ∧ ωbd ∧ ωca
= −ωda ∧ ωcd ∧ ωbc + ωcd ∧ ωbc ∧ ωda
= −ωda ∧ ωcd ∧ ωbc + ωda ∧ ωcd ∧ ωbc = 0. (174)

So, we have the general Bianchi identities which are valid for any one
of the metrical compatible structures18 classified in Section 2,

DT a = Rab ∧ θb ,
DRab = 0. (175)

12.1 Coordinate Expressions of the First Bianchi Identity

Taking advantage of the calculations we done for the coordinate expres-
sions of Maxwell equations we can write in a while:
DT a = dT a + ωba ∧ T b
1 “ a a b a a b a a b
”
= ∂µ Tαβ + ωµb Tαβ + ∂α Tβµ + ωαb Tβµ + ∂β Tµα + ωβb Tµα θµ ∧ θα ∧ θβ .
3!
(176)

Now,
a ρ ρ
∂µ Tαβ = (∂µ qρa )Tαβ + qρa ∂µ Tαβ , (177)
and using the freshman identity (Eq.(23)) we can write
a b a b ρ ρ ρ
ωµb Tαβ = ωµb qρ Tαβ = Laµb qρb Tαβ − (∂µ qρa )Tαβ . (178)

So,

a a b
∂µ Tαβ + ωµb Tαβ
ρ ρ
= qρa ∂µ Tαβ + Γaµb qρb Tαβ
ρ ρ
= qρa (Dµ Tαβ + Γκµα Tκβ + Γκµβ Tακ
ρ
). (179)
18 For 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
Now, recalling that Tµα = Γκµα − Γκαµ we can write
ρ
qρa (Γκµα Tκβ + Γκµβ Tακ
ρ
)θµ ∧ θα ∧ θβ (180)
ρ µ
= qρa Tµα
κ
Tκβ θ α
∧θ ∧θ .β

Using these formulas we can write

DT a =
1 a˘ ρ ρ ρ κ ρ κ ρ κ ρ ¯ µ
qρ Dµ Tαβ + Dα Tβµ + Dβ Tµα + Tµα Tκβ + Tαβ Tκµ + Tβµ Tκα θ ∧ θα ∧ θβ .
3!
(181)

Now, the coordinate representation of Rab ∧ θb is:

1 a
Rab ∧ θb = q (Rµ ρ αβ + Rα ρ βµ + Rβ ρ µα )θµ ∧ θα ∧ θβ , (182)
3! ρ
and thus the coordinate expression of the first Bianchi identity is:
ρ ρ ρ
Dµ Tαβ + Dα Tβµ + Dβ Tµα
ρ
= (Rµ ρ αβ + Rα ρβµ + Rβ ρ µα ) − (Tµα
κ
Tκβ κ
+ Tαβ ρ
Tκµ κ
+ Tβµ ρ
Tκα ),
(183)
which we can write as
X X  ρ

Rµ ρ αβ = Dµ Tαβ κ
− Tµβ ρ
Tκα , (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 D̊
ρ
since the Tαβ = 0 we have the standard form of the first Bianchi identity
in classical Riemannian geometry, i.e.,
Rµ ρ αβ + Rα ρ βµ + Rβ ρ 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 ? ?−1 T a
= (−1)n−2 ? δ ?−1 T a = (−1)n−2 (−1)n−2 sgng ? 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
δ ? T a = (−1)n−2 ? [Rab ∧ θb − ωba ∧ T b ], (187)

and taking into account that

?(Rab ∧ θb ) = ?(θb ∧ Rab ) = θb y ? Rab ,

?(ωba ∧ T b ) = ωba y ? T b , (188)

we end with

δ ? T a = (−1)n−2 (θb y ? Rab − ωba y ? T b ). (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?Ta
1 α ? aµν
= −(Dα ? T aαν + (Tµα T µ ? aρν
− Tµρ T ν ? aµρ
+ Tµρ T ))θν . (190)
2

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

we have:

?(Rab ∧ θb ) = θb y ? Rab
1
= θb y(? Rba cd θc ∧ θd )
2
= ? Rba cd η bc θd
= ? Rcacd θd = ? Rcac d θd = ? Rcac d qdν θν . (192)
Differential Forms on Riemannian (Lorentzian). . . 465

On the other hand we can also write:

?(Rab ∧ θb ) = θb y ? Rab
1 1
= ϑb y( R akl klmn θm ∧ θn )
2 (n − 2)! b
1 1
= (R akl klmn η bm ∧ θn − Rbakl klmn η bn ∧ θm )
2 (n − 2)! b
1 1
= Rbakl klmn η bm θn = Rmakl klmn θn
(n − 2)! (n − 2)!
1
= R akl  mn θn
(n − 2)! m kl
1
= R akl  mn qnν θν .
(n − 2)! m kl
from where we get in agreement with Eq.(34) the formula
? 1
Rcacd = Rmakl mkld , (193)
(n − 2)!
?
which shows explicitly that Rcacd are not the components of the Ricci
tensor.
Moreover,
ωba y ? T b (194)
1 a α ? bµν
= ωαb θ y( T θµ ∧ θν )
2
=? T bµν ωαba
θν .
Collecting the above formulas we end with
1 α ? aµν
Dα ? T aαν + (Tµα T µ ? aρν
−Tµρ T ν ? aµρ
+Tµρ T ) = (−1)n−1 (? Rcac d qdν −ωαb
a? bαν
T ),
2
(195)
which is another expression for the first Bianchi identity written in terms
of duals.

Remark 15 Consider, e.g., the term Dα ? T aαν in the above equation

and write
Dα ? T aαν = Dα (qρa ? T ραν ). (196)
We now show that
Dα (qρa ? T ραν ) 6= qρa Dα ? T ραν . (197)
466 W. A. Rodrigues Jr.

Indeed, recall that we already found that

1 α ? aµν
(Dα ? T aαν )θν = −δ?T a + (Tµα T µ ? aρν
−Tµρ T ν ? aµρ
+Tµρ T )θν , (198)
2

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

1 α ? aµν
θα y(D∂α ?T a ) = −δ?T a + (Tµα T µ ? aρν
−Tµρ T ν ? aµρ
+Tµρ T )θν . (199)
2

Now, writing ?T a = 21 qρa ? T ρµν θµ ∧ θν and get

θα y(D∂α ? T a )
1
= θα y[D∂α (qρa ? T ρµν θµ ∧ θν )]
2
1
= ϑα y[∂α (qρa ? T ρµν )θµ ∧ θν + qρa ? T ρµν D∂α (θµ ∧ θν )]
2
1
= ϑα y[(∂α qρa )? T ρµν θµ ∧ θν + qρa ∂α (? T ρµν )θµ ∧ θν + qρa ? T ρµν D∂α (θµ ∧ θν )]
2
1
= ϑα y[(∂α qρa )? T ρµν θµ ∧ θν + qρa Dα (? T ρµν )θµ ∧ θν ]
2
= (∂α qρa )? T ρµν δµα θν + qρa Dα (? T ρµν )δµα θν . (200)

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

Dα ? T aαν θν = Dα (qρa ? T aαν θν ) = (∂α qρa )? T ρµν + qρa Dα (? T ρµν ), (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
using “Dα qρa = 0”.”

Exercise 16 Show that the coordinate expression of the second Bianchi

identity DRab = 0 is
X X
Dµ Rβανρ = α
Tνµ Rβααρ . (202)
(µνρ) (µνρ)

Exercise 17 Calculate ?Rab ∧ θb in an orthonormal basis.

Differential Forms on Riemannian (Lorentzian). . . 467

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

θb ∧ ?Rab = − ? (θb yRab )

 
1 b a c d
= − ? θ y(Rb cd θ ∧ θ )
2
= − ? [Rba cd η bc θd ]
= − ? [Rcacd θd ] = − ? [Racdc θd ]
= − ? [Rda θd ] = − ? Ra (203)

Of course, if the connection is the Levi-Civita one we get

θb ∧ ?R̊ab = − ? (θb yR̊ab ) = −R̊ba θb = − ? R̊a . (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 con-
nection D̊ is incompatible with the first Bianchi identity. Evans con-
clusion follows because he thinks to have derived “from first principles”
that
D?T a = ?Rab ∧ θb , (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 D̊ 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 show-
ing explicit counterexamples for some trivial structures.
Before doing that let us show that we can derive from the first Bianchi
identity that
R̊aacd = 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-
tity gives (with Rab 7−→ R̊ab ):

R̊ab ∧ θb = 0. (207)
468 W. A. Rodrigues Jr.

Contracting this equation with θa we get

θa y(R̊ab ∧ θb ) = θa y(θb ∧ R̊ab )

= δab R̊ab − θb ∧ (θa yR̊ab )
1
= R̊aa − θb ∧ [θa y(R̊ba cd θc ∧ θd )]
2
= R̊aa − R̊ba ad θb ∧ θd

Now, the second term in this last equation is null because according to
the Eq.(106), −Rba ad = Rba 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
R̊aacd θc ∧ θd = 0, (208)
which implies that as we stated above that

R̊aacd = 0. (209)

But according to Eq.(106) the R̊aacd 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

R̊b acd + R̊c adb + R̊d abc . = 0 (210)

Making b = a we get

R̊a acd + R̊c ada + R̊d aac

= R̊a acd − R̊c aad + R̊d aac
= R̊a acd + R̊cd − R̊dc
= R̊a acd = 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

T a = dθa + ωba ∧ θb . (212)

Differential Forms on Riemannian (Lorentzian). . . 469

By definition
D ? T a = d ? T a + ωba ∧ ?T b . (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:

δT a = δ dθa + ωba ∧ θb

= δdθa + dδθa − dδθa + δ(ωba ∧ θb ) . (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:

δT a = − ∂| 2 a
θ − dδθa + δ(ωba ∧ θb )
Eq.(115) ˚ a
= −θ − (∂| ∧ ∂|)θa − dδθa + δ(ωba ∧ θb )
Eq.(127) ˚ a
= −θ − R̊a − dδθa + δ(ωba ∧ θb )
˚ a − Ra + J a − dδθa + δ(ωba ∧ θb )
= −θ (216)

where we have used Eq(107) to write

Ra = Rba θb = (R̊ba + Jba )θb . (217)

So, we have
˚ a − ?Ra + ?J a − ?dδθa + ?δ(ωba ∧ θb )
d ? T a = − ? θ

and finally
˚ a − ?Ra + ?J a − ?dδθa + ?δ(ωba ∧ θb ) + ωba ∧ ?T b (218)
D ? T a = − ? θ

or equivalently recalling Eq.(35)

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

Remark 18 Eq.(219) does not implies that D ? T a = ?Rab ∧ θb because

taking into account Eq.(203)
˚ a −?Ra +?J a −?dδθa +?δ(ωba ∧θb )−?(ωba yT b )
?Rab ∧θb = −?Ra 6= D?T a = −?θ
(221)
in general.

So, for a Levi-Civita connection we have that D ? T a = 0 and then

Eq.(218) implies
˚ a − R̊a − dδθa + δ(ω̊ba ∧ θb ) = 0
D ? T a = 0 ⇔ −θ (220)
or since ω̊ba b b
∧ θ = −dθ for a Levi-Civita connection,
˚ a − R̊a − dδθa − δdθa = 0
D ? T a = 0 ⇔ −θ (221)
or yet
˚ a − R̊a = − ∂| 2 θa = dδθa + δdθa ,
−θ (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
R̊a − R̊θa = Ta , (223)
2
where R̊ is the scalar curvature and Ta = −Tba θb are the energy-
momentum 1-form fields. Comparing Eq.(221) with Eq.(223). We im-
mediately get the “wave equation” for the cotetrad fields:
1 ˚ a − dδθa − δdθa ,
Ta = − R̊θa − θ (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 charac-

terized by the equality of the Hodge and covariant D’ Alembertians acting
on the coterad fields, i.e.,
˚ a = ♦θa ,
θ (225)
a non trivial result.

Exercise 20 Using Eq.(120) and Eq.(121 ) write θ ˚ a in terms of the

connection coefficients of the Riemann-Cartan connection.
Differential Forms on Riemannian (Lorentzian). . . 471

15 Two Counterexamples to Evans (Wrong) Equation

“D ? T a = ?Rab ∧ θb ”
15.1 The Riemannian Geometry of S 2
Consider the well known Riemannian structure on the unit radius sphere
[12] {S 2 , g, D̊}. Let {xi }, x1 = ϑ , x2 = ϕ, 0 < ϑ < π, 0 < ϕ < 2π, be
spherical coordinates covering U = {S 2 − l}, where l is the curve joining
the north and south poles.
A coordinate basis for T U is then {∂ µ } and its dual basis is {θµ =
dx }. The Riemannian metric g ∈ sec T02 M is given by
µ

g = dϑ ⊗ dϑ + sin2 ϑdϕ ⊗ dϕ (226)

and the metric g∈ sec T20 M of the cotangent space is

1
g = ∂1 ⊗ ∂1 + ∂2 ⊗ ∂2. (227)
sin2 ϑ
An orthonormal basis for T U is then {ea } with

1
e1 = ∂ 1 , e2 = ∂2, (228)
sin ϑ
with dual basis {θa } given by

θ1 = dϑ, θ2 = sin ϑdϕ. (229)

The structure coefficients of the orthonormal basis are

[ei , ej ] = ckij ek (230)

and can be evaluated, e.g., by calculating dθi = − 21 cijk θj ∧ θk . We get

immediately that the only non null coefficients are

c212 = −c221 = − cot θ. (231)

To calculate the connection 1-form ωdc we use Eq.(92), i.e.,

1
ω cd = (−ccjk η dj + cd cj ca dj b k
jk η − η ηbk η cja )θ .
2
Then,
472 W. A. Rodrigues Jr.

1
ω 21 = (−c212 η 11 − η 22 η22 η 11 c212 )θ2 = cot ϑθ2 . (232)
2
Then

ω 21 = −ω 12 = cot ϑθ2 ,
ω12 = −ω21 = cot ϑθ2 , (233)
2 2
ω̊21 = cot ϑ , ω̊11 = 0. (234)
Now, from Cartan’ s second structure equation we have

R̊12 = dω̊21 + ω̊11 ∧ ω̊11 + ω̊21 ∧ ω̊22 = dω̊21 (235)

1 2
=θ ∧θ

and20
1
R̊2112 = −R̊2121 = −R̊1212 = R̊1221 = . (236)
2
^0
Now, let us calculate ?R12 ∈ sec T ∗ M . We have

?R12 = R
f1 yτ = −(θ1 ∧ θ2 )y(θ1 ∧ θ2 ) = −θ1 θ2 θ1 θ2
2 g

2 2
2
= θ1 θ =1 (237)

and
?R1a ∧ θa = R12 ∧ θ2 = θ2 6= 0. (238)
Now, Evans equation implies that ?R1a ∧ θ1 = 0 for a Levi-Civita con-
nection and thus as promised we exhibit a counterexample to his wrong
equation.

Remark 21 We recall that the first Bianchi identity for (S 2 , g, D̊), i.e.,
DT a = Rab ∧ θb = 0 which translate in the orthonormal basis used above
in R̊b acd + R̊c adb + R̊d abc . = 0 is rigorously valid. Indeed, we have

R̊2 112 + R̊1 121 + R̊2 121 = R̊2 112 − R̊2 112 = 0,
R̊1 212 + R̊1 221 + R̊2 221 = R̊1 212 − R̊1 212 = 0. (239)
20 Observe 1 + R̊2 =
that with our definition of the Ricci tensor it results that R̊ = R̊1 2
−1.
Differential Forms on Riemannian (Lorentzian). . . 473

15.2 The Teleparallel Geometry of (S̊ 2 , g, D)

Consider the manifold S̊ 2 = {S 2 \north pole} ⊂ R3 , it is an sphere of
unitary radius excluding the north pole. Let g ∈ sec T20 S̊ 2 be the stan-
dard Riemann metric field for S̊ 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 ∈ S̊ 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
As before (x1 , x2 ) = (ϑ, ϕ) 0 < ϑ < π, 0 < ϕ < 2π, denote the
B1&3 ,.C"3# "3,* .--*$3, ,1& <&D3","*3 *4 ,1& -$%;.,$%& ,&36*% 8& 1.;&
standard spherical coordinates of a S̊ 2 of unitary radius, which covers
!" # $
U = {"#!S̊ 2#−
# $$l},
# !" # !where
%$ ! $
$ l!!is" !!the curve joining
!
! ! !!! !!" ! !!!" "!! " !#
the
! "" north
>?@EAand south poles.

Now, it is"3,*
F/6*9 ,.C"3# obvious
.--*$3, ,1& from what
<&D3","*3 has
*4 ,1& been
,*%6"*3 said above
*G&%.,"*3 8& 1.;& that our connection
is characterized by #
% #!" # !% $ ! &"% !# ! !! !" ! !! !% ! %!" # !% &
! "

! %!" # !% & ! '#"% !# # Dej ei = 0. >?@@A (240)

21 See & & ' '
&&' ! !&
some historical '& ! '()in
details (9 [22].
&&' ! !&'& ! "" >?@HA
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(ek , θa , ei , ej ) = θa c
Dei Dej − Dej Dei − D[e ,e
i j ] ek = 0. (241)

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

have

τ (ei , ej ) = Tijk ek = Dej ei − Dei ej − [ei , ej ]

= [ei , ej ] = ckij ek , (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
we get the vector ~v = sr1 such that |g(~v , ~v )| 2 = 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, ~ 2 = sin(ϑ − 4ϑ)4ϕ ' sin ϑ4ϕ − cos ϑ4ϑ4ϕ,
~ w)| (244)

Also,

1
~u = r1 r2 = −u( ∂2) , u = |g(~u, ~u)| = cos ϑ4ϑ4ϕ. (245)
sin ϑ
22 This 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 (S̊ 2 , g, D) has a non null torsion
tensor Θ. Indeed, the component of ~u = r1 r2 in the direction ∂ 2 is
ϕ
precisely Tϑϕ 4ϑ4ϕ. So, we get (recalling that D∂ j ∂ i = Γkji ∂ k )
 
ϕ
Tϑϕ = Γϕ
ϑϕ − Γ ϕ
ϕϑ = − 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 S̊ 2 show clearly that we cannot mislead the Riemann curvature ten-
sor 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
the result found by Dr. Evans [10], “D ? T a = ?Rab ∧ T b ” 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.
23 The 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 mis-
prints, that are now corrected. Moreover, the author will be grateful to
any one which point any misprints or eventual errors.

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[http://www.springerlink.com/content/l1008h127565m362/]

(Manuscrit reçu le 10 janvier 2008)