Appendix A May a Torus with Null Riemann Curvature Exist on E3?

1 1 1 We can also give a flat Riemann curvature for T3 S × S T1 T2. First parametrize 1 2 1 2 3 1 2 1 2 T3 by (x ,x ), x ,x ∈ R, such that its coordinates in R are x(x ,x )=x(x +2π, x +2π)and

x(x1,x2)=(h(x1)cosx2,h(x1)sinx2,l(x1), (A.1)

with h(x1)=R + r cos x1, l(x1)=r sin x1, (A.2) where here R and r are real positive constants and R>r(See Figure A.1).

3 Figure A.1: Some Possible GSS for the Torus T3 living in E where the grid defines the parallelism

1 Recall that T1 and T2 have been defined in Section 1.1. 110 Appendix A. May a Torus with Null Riemann Curvature Exist on E3?

1 2 Now, T T3 has a global coordinate {∂/∂x ,∂/∂x }, and the induced metric on T3 is easily found as g = r2dx1 ⊗ dx1 +(R + r cos x1)2dx2 ⊗ dx2 (A.3)

Now, let us introduce a ∇ on T3 such that

j ∇∂/∂xi ∂/∂x =0. (A.4)

With respect to this connection, it is immediately verified that its torsion and curvature are null, but the nonmetricity of the connection is non null. Indeed,

1 1 ∇∂/∂xi g = −2(R + r cos x )sinx . (A.5)

So, in this case, we have a particular Riemann-Cartan-Weyl GSS (T3, g, ∇)withA =0 , Θ=0, R =0. We can alternatively define the parallelism rule on T3 by introducing a metric compatible con-  nection ∇ as follows. First, we introduce the global orthonormal basis {ei} for T T3 with 1 ∂ 1 ∂ e1 = , e2 = , (A.6) r ∂x1 (R + r cos x1) ∂x2 and, then, postulate that  ∇ ej ei =0. (A.7) Since r sin x1 [e1, e2]= e2 (A.8) r(R + r cos x1) we immediately get that the torsion operator of ∇ is

1  R sin x τ (e1, e2)=[e1, e2]= e2, (A.9) r(R + r cos x1)

2 2 2 and the non null components of the are T12 and T21 = −T12,

 2 2 2 2 T12 = Θ(θ , e1, e2)=−θ (τ(e1, e2)) = −θ (−[e1, e2]) R sin x1 = − . (A.10) r(R + r cos x1)

Also, it is trivial to see that the curvature operator is null and that ∇g = 0, and, in this case,  we have a particular Riemann-Cartan MCGSS (T3, g, ∇ ), with Θ =0andwithanullcurvature tensor. Appendix B Levi-Civita and Nunes Connections on S˚2

First, consider S2, a sphere of radius R = 1 embedded in R3.Let(x1,x2)=(ϑ, ϕ)0<ϑ<π, 0 <ϕ<2π, be the standard spherical coordinates of S2, which cover all the open set U which is S2 with the exclusion of a semi-circle uniting the north and south poles. Introduce the coordinate bases μ μ {∂μ}, {θ = dx } (B.1) ∗ ∗ a ∗ ∗ for T U and T U. Next, introduce the orthonormal bases {ea}, {θ } for T U and T U with 1 e1 = ∂1, e2 = ∂2, (B.2a) sin x1 θ1 = dx1, θ2 =sinx1dx2. (B.2b)

Then,

k [ei, ej]=cijek, (B.3) 2 2 1 c12 = −c21 = − cot x .

0 2 Moreover, the metric g ∈ sec T2 S inherited from the ambient Euclidean metric is:

g = dx1 ⊗ dx1 +sin2 x1dx2 ⊗ dx2 = θ1 ⊗ θ1 + θ2 ⊗ θ2. (B.4)

ρ The Levi-Civita connection D of g has the following non null connections coefficients Γμν in the coordinate basis (just introduced):

∂ ρ ∂ D∂μ ν =Γμν ρ, 2 ϕ 2 ϕ 1 ϑ − Γ21 =Γθϕ =Γ12 =Γϕθ =cotϑ,Γ22 =Γϕϕ = cos ϑ sin ϑ. (B.5) 112 Appendix B. Levi-Civita and Nunes Connections on S˚2

{ } k Also, in the basis ea , Dei ej = ωijek and the non null coefficients are:

2 1 ω21 =cotϑ, ω22 = − cot ϑ. (B.6)

The torsion and the (Riemann) curvature tensors of D are k k k − − Θ(θ , ei, ej)=θ (τ(ei, ej)) = θ Dej ei Dei ej [ei, ej] , (B.7) a a − − R(ek,θ , ei, ej)=θ Dei Dej Dej Dei D[ei, ej] ek , (B.8) 1 1 2 2 which results in T = 0 and the non null components of R are R121= −R112 = R112= −R112 = −1. Since the is non null, the parallel transport of a given vector depends on the path to be followed. We say that a vector (say v0) is parallel transported along a generic 3 path R ⊃I → γ(s) ∈ R (say, from A = γ(0) to B = γ(1)) with tangent vector γ∗(s)(atγ(s)) if it determines a vector field V along γ satisfying

Dγ∗ V =0, (B.9)

1 and such that V(γ(0)) = v0. When the path is a of the connection D, i.e., a curve 3 R ⊃I → c(s) ∈ R with tangent vector c∗(s)(atc(s)) satisfying

Dc∗ c∗ =0, (B.10) the parallel transported vector along c forms a constant angle with c∗.Indeed,fromEq.(B.9)itis · γ∗ Dγ∗ V = 0. Then, taking into account Eq.(B.10) it follows that g

· Dγ∗ (γ∗ V)=0. g i.e., γ∗ · V = constant. This is clearly illustrated in Figure B.1 (adapted from [1]). g Consider next the S˚2 = {S2\north pole + south pole}⊂R3, which is a sphere of 0 2 unitary radius R = 1, but this time excluding the north and south poles. Let again g ∈ sec T2 S˚ be the metric field on S˚2 inherited from the ambient space R3 and introduce on S˚2 the Nunes (or navigator) connection ∇ defined by the following parallel transport rule: a vector at an arbitrary point of S˚2 is parallel transported along a curve γ, if it determines a vector field on γ, such that, at any point of γ the angle between the transported vector and the vector tangent to the latitude line passing through that point is constant during the transport. This is clearly illustrated in Figure B.2. And to distinguish the Nunes transport from the Levi-Civita transport, we also ask the reader to study the caption of Figure B.1.

We recall that, from the calculation of the Riemann tensor R, it follows that the structures ˚2 2 (S , g,D,τg)andalso(S , g,D,τg) are Riemann spaces of constant curvature. We now show that

1We recall that a geodesic of D also determines the minimal distance (as given by the metric g) between any two points on S2. Appendix B. Levi-Civita and Nunes Connections on S˚2 113

V a 2

r V a 1 s

a a V0 p q

Figure B.1: Levi-Civita and Nunes transport of a vector v0 starting at p through the paths psr and pqr. Levi-Civita transport through psr leads to v1 whereas Nunes transport leads to v2.Alongpqr both Levi-Civita and Nunes transport agree and leads to v2.

˚2 2 the structure (S , g, ∇,τg)isateleparallel space , with zero Riemann curvature tensor, but non zero torsion tensor. Indeed, from Figure B.2, it is clear that (a) if a vector is transported along the infinitesimal quadrilateral pqrs composed of latitudes and longitudes, first starting from p along pqr,andthen starting from p along psr, the parallel transported vectors that result in both cases will coincide (study also the caption of Figure B.2). 2 Now, the vector fields e1 and e2 in Eq.(B.2a) define a basis for each point p of TpS˚ and ∇ is clearly characterized by: ∇ ej ei =0. (B.11) The components of the curvature operator are:

 a a ∇ ∇ −∇ ∇ −∇ R(ek,θ , ei, ej)=θ ei ej ej ei [ei,ej] ek =0, (B.12) andthetorsionoperatoris

 ∇ −∇ − τ(ei, ej)= ej ei ei ej [ei, ej] =[ei, ej], (B.13)

2 2 2 which gives for the components of the torsion tensor, T12 = −T12 =cotϑ. It follows that S˚ , ˚2 considered as part of the structure (S , g, ∇,τg), is flat (but has torsion)! If you need more details to grasp this last result, consider Figure B.2 (b), which shows the standard parametrization of the points p, q, r, s in terms of the spherical coordinates introduced above. According to the geometrical meaning of torsion, its value at a given point is determined

2As recalled in Section 1, a teleparallel manifold M is characterized by the existence of global vector fields, which 2 is a basis for TxM for any x ∈ M. The reason for considering S˚ for introducing the Nunes connection is that, as is well known (see, e.g., [2]), S2 does not admit a continuous vector field that is nonnull at all its points. 114 Appendix B. Levi-Civita and Nunes Connections on S˚2

a b (J – dJ, j) (J – dJ, j + dj) s r r = 2 r 1

a

s r

p q

p q (J,j) (J,j + dj)

Figure B.2: Characterization of the Nunes connection.

3 by calculating the difference between the (infinitesimal) vectors pr1and pr2. If the vector pq is 1 transported along ps, one gets (recalling that R = 1) the vector v = sr1 such that |g(v, v)| 2 = sin ϑϕ. On the other hand, if the vector ps is transported along pq, one gets the vector qr2 = qr. Let w = sr. Then,

|g(w, w)| =sin(ϑ −ϑ)ϕ sin ϑϕ − cos ϑϑϕ, (B.14)

Also,

1 ∂ u = r1r2 = −u( ), u = |g(u, u)| =cosϑϑϕ. (B.15) sin ϑ ∂ϕ

˚2 ¯ Then, the connection ∇ of the structure (S , g, ∇,τg) has a non null torsion tensor T . Indeed, the ¯ϕ   component of u = r1r2 in the direction ∂/∂ϕ is precisely Tϑϕ ϑ ϕ. So, one gets (recalling that ∇ k ∂j∂i =Γji∂k) ¯ϕ ϕ − ϕ − Tϑϕ = Γϑϕ Γϕϑ = cot θ. (B.16)

To end this Appendix, it is worth showing that ∇ is metrically compatible, i.e., that ∇g =0. Indeed, we have:

∇ ∇ ∇ ∇ 0= ec g(ei, ej)=( ec g)(ei, ej)+ g( ec ei, ej)+ g(ei, ec ej) ∇ =( ec g)(ei, ej). (B.17)

3This wording, of course, means that those vectors are identified as elements of the appropriate tangent spaces. Appendix B. Levi-Civita and Nunes Connections on S˚2 115 References

[1] Bergmann, P. G., TheRiddleofGravitation, Dover, New York, 1992. [2] do Carmo, M. P., , Birkhh¨auser, Boston 1992. Appendix C Gravitational Theory for Independent h and Ω Fields

In this Appendix, we will present, for completeness1, a theory where the dynamics of the grav- itational field is such that it must be described by two independent fields [1], namely the plas- κ ♣ η tic distortion field h (more precisely h )andaκη-gauge rotation field Ω ≡ Ω defined on the canonical space U. These two independent fields distort the Lorentz vacuum, creating an effective † Minkowski-Cartan MCGSS (U, η, κ), or equivalently a Lorentz-Cartan MCGSS (U, g = h ηh,γ) which is viewed as gauge equivalent to (U, η, κ) as already discussed in the main text. From the mathematical point of view, our theory is then described by a Lagrangian, where the fields h and Ω are the dynamic variables to be varied in the action functional. The Lagrangian is postulated to be the scalar functional ♣ ♣ [h , Ω, ·∂Ω] → L[h , Ω, ·∂Ω] = R det[h], (C.1)

γh ♣ where R(= R)(givenbyEq.(4.114) ) is expressed as a functional of the fields h , Ωand·∂Ω, i.e., ♣ R = h (∂a ∧ ∂b) ·R2(a ∧ b) ♣ = h (∂a ∧ ∂b) · (a · ∂Ω(b) − b · ∂Ω(a) − Ω([a, b]) + Ω(a) × Ω(b)), (C.2) η ♣ ♣ and det[h], as in Section 5.1 is expressed as a functional of h , i.e., det[h]=(det[h ])−1. The Lagrangian formalism for extensor fields supposes the existence of an action functional ♣ A = L[h , Ω, ·∂Ω] τ = R det[h] τ (C.3) U U which according to the principle of stationary action, gives for any of the dynamic variables a contour problem.

1And also, as a justification for our construction in Section 6.4 of the spin of the gravitational field in our theory. 118 Appendix C. Gravitational Theory for Independent h and Ω Fields

♣ For the dynamic variable h ,itis w ♣ δ ♣ · h L[h , Ω, ∂Ω] τ =0, (C.4) U | for any smooth (1, 1)-extensor field w such that w ∂U =0. For the dynamic variable Ω, it is B ♣ δΩ L[h , Ω, ·∂Ω] τ =0, (C.5) U | for any smooth (1, 2)-extensor field B, such that B ∂U =0. ♣ We now show that Eqs.(C.4)and(C.5) lead to coupled differential equations for the fields h and Ω. Of course, the solution of the contour problems given by Eqs.(C.4)and(C.5) is obtained fol- lowing the traditional path, i.e., by using the proper variational formulas, the Gauss-Stokes and a fundamental lemma of integration theory (already mentioned in Section 5.1). First, we calculate the variation in Eq.(C.4). We have:

w ♣ w ♣ δ ♣ ∧ ·R ∧ δ ♣ ∧ ·R ∧ h (h (∂a ∂b) 2(a b)det[h]) = ( h h (∂a ∂b)) 2(a b)det[h] ♣ w + h (∂ ∧ ∂ ) ·R2(a ∧ b)(δ ♣ det[h]). (C.6) a b h For the scalar product in the first member of Eq.(C.6), an algebraic manipulation analogous to the one already used for deriving Eq.(5.15)gives

w ♣ δ ♣ ∧ ·R ∧ ·R ( h h (∂a ∂b)) 2(a b)=2w(∂a) 1(a), (C.7) where R(a) is the Ricci gauge field (in the Lorentz-Cartan MCGSS). w δ ♣ The calculation of h det[h] has already been obtained (see Eq.(5.22)). So, substituting Eqs.(C.7) γh and (5.22)inEq.(C.6) and recalling the definition of G≡G (Eq.(4.115)), we get that

w δ ♣ R ·R −R · h ( det[h]) = 2w(∂a) 1(a)det[h] w(∂a) h(a)det[h] 1 =2w(∂ ) · (R1(a) − h(a)R)det[h] a 2 w δ ♣ R ·G h ( det[h]) = 2w(∂a) (a)det[h]. (C.8)

♣ Substituting Eq.(C.8)inEq.(C.4), the contour problem for the dynamic variable h becomes

w(∂a) ·G(a)det[h] τ =0, (C.9) U for any w, and since w is arbitrary, recalling a fundamental lemma of integration theory, we get

G(a)=0. (C.10)

This field equation looks like the one already obtained for h in Section 6.1, but here, it is not a differential equation for the field h. To continue, recall that Eq.(C.10)isequivalenttoR(a)=0. Appendix C. Gravitational Theory for Independent h and Ω Fields 119

♣ Indeed, taking the scalar h -divergent of R1(a), we have

1 G(a)=0⇒R1(a) − h(a)R =0 2 ♣ 1 ♣ ⇒ h (∂ ) ·R1(a) − h (∂ ) · h(a)R =0 a 2 a 1 ⇒R− 4R =0 2 ⇒R=0, which implies

R1(a)=0. (C.11)

Also, it is quite clear that Eq.(C.11) implies Eq.(C.10). 0 3 Let us express now Eq.(C.11)intermsofthecanonical 1-forms of h (i.e., h(ϑ ),...,h(ϑ )) and the canonical biforms of Ω (i.e., Ω(ϑ0),...,Ω(ϑ3)),

R1(a)=0⇔R1(ϑν )=0 ♣ μ ⇔ h (ϑ )R2(ϑμ ∧ ϑν )=0, i.e.,

♣ μ h (ϑ )(ϑμ · ∂Ω(ϑν ) − ϑν · ∂Ω(ϑμ)+Ω(ϑμ) × Ω(ϑν )) = 0. (C.12) η

Eq.(C.12) is a set of four coupled multiform equations which are algebraic on the canonical 1-forms of h and differential on the canonical biforms of Ω. Let us, now, calculate the variation in Eq.(C.5). We have

B ♣ ♣ B δΩ (h (∂a ∧ ∂b) ·R2(a ∧ b)det[h]) = h (∂a ∧ ∂b) · (δΩ R2(a ∧ b)) det[h]. (C.13)

For the scalar product on the right side of Eq.(C.13), using the commutation property involving B B δΩ and a · ∂, the property δΩ Ω(a)=B(a) and the definition of the operator κη Da ≡ D a , i.e., DaX = a · ∂X +Ω(a) × X, we have η

♣ B ♣ μ ν B h (∂a ∧ ∂b) · (δΩ R2(a ∧ b)) = h (ϑ ∧ ϑ ) · (δΩ R2(ϑμ ∧ ϑν )) ♣ μ ν = h (ϑ ∧ ϑ ) · (ϑμ · ∂B(ϑν ) − ϑν · ∂B(ϑμ)

+ B(ϑμ) × Ω(ϑν )+Ω(ϑμ) × B(ϑν )) η−1 η−1 ♣ ϑμ ∧ ϑν ·D =2h ( ) ϑμ B(ϑν ). (C.14) 120 Appendix C. Gravitational Theory for Independent h and Ω Fields

After some algebraic manipulations, the right side of Eq.(C.14) may be written as ♣ ϑμ ∧ ϑν ·D ♣ ϑμ ∧ ϑν · · h ( ) ϑμ B(ϑν )=h ( ) ϑμ ∂B(ϑν ) ♣ μ ν + ϑμ · ∂h (ϑ ∧ ϑ ) · B(ϑν ) ♣ μ ν + h (ϑ ∧ ϑ ) · (Ω(ϑμ) × B(ϑν )) η−1 ♣ μ ν − ϑμ · ∂h (ϑ ∧ ϑ ) · B(ϑν ) ♣ μ ν ♣ μ ν = ϑμ · ∂(h (ϑ ∧ ϑ ) · B(ϑν )) − (ϑμ · ∂h (ϑ ∧ ϑ ) ♣ μ ν + η(Ω(ϑμ) × ηh (ϑ ∧ ϑ ))) · B(ϑν ) η ♣ μ ν ♣ μ ν = ϑμ · ∂(h (ϑ ∧ ϑ ) · B(ϑν )) − η(ϑμ · ∂ηh (ϑ ∧ ϑ ) ♣ μ ν +Ω(ϑμ) × ηh (ϑ ∧ ϑ )) · B(ϑν ), η i.e., ♣ ϑμ ∧ ϑν ·D · ♣ ϑμ ∧ ϑν · h ( ) ϑμ B(ϑν )=ϑμ ∂(h ( ) B(ϑν )) − D ♣ ϑμ ∧ ϑν · η ϑμ ηh ( ) B(ϑν ). (C.15) Also, the first member on the right side of Eq.(C.15) may be written as a scalar divergent of a 1-form field, i.e.,

♣ μ ν ♣ μ ν ϑμ · ∂(h (ϑ ∧ ϑ ) · B(ϑν )) = ∂ · (ϑμh (ϑ ∧ ϑ ) · B(ϑν )), (C.16) whereweusedthefactthat∂ · (ϑμφ)=ϑμ · ∂φ, where φ is a scalar field. After putting Eq.(C.16)inEq.(C.15) and using the result obtained in Eq.(C.14), the Eq.(C.13) gives us finally the variational formula δB ♣ ∧ ·R ∧ − D ♣ ϑμ ∧ ϑν · Ω (h (∂a ∂b) 2(a b)det[h]) = 2(η ϑμ ηh ( ) B(ϑν ) ♣ μ ν − ∂ · (ϑμh (ϑ ∧ ϑ ) · B(ϑν ))) det[h]. (C.17) Putting Eq.(C.17)inEq.(C.5), the contour problem for the dynamic variable Ω becomes D ♣ ϑμ ∧ ϑν · η ϑμ ηh ( ) B(ϑν )det[h] τ U ♣ μ ν − ∂ · (ϑμh (ϑ ∧ ϑ ) · B(ϑν )) det[h] τ =0, (C.18) U | for all B such that B ∂U = 0. Now, the second member of Eq.(C.18) may be easily transformed in such a way that a scalar divergent of a 1-form field appears. Indeed, if we recall that and ∂ · (aφ)=a · ∂φ +(∂ · a)φ,wherea is a 1-form field and φ is a scalar field, we have ♣ μ ν ∂ · (ϑμh (ϑ ∧ ϑ ) · B(ϑν )) det[h] τ U ♣ μ ν = ∂ · (ϑμh (ϑ ∧ ϑ ) · B(ϑν )) det[h]) τ U ♣ μ ν − h (ϑ ∧ ϑ ) · B(ϑν )ϑμ · ∂ det[h] τ. (C.19) U Appendix C. Gravitational Theory for Independent h and Ω Fields 121

Putting Eq.(C.19)inEq.(C.18), we get D ♣ ϑμ ∧ ϑν ♣ ϑμ ∧ ϑν · · (η ϑμ ηh ( )det[h]+h ( )ϑμ ∂ det[h]) B(ϑν ) τ U ♣ μ ν − ∂ · (ϑμh (ϑ ∧ ϑ ) · B(ϑν )) det[h]) τ =0, (C.20) U | for all B such that B ∂U = 0. Next, using the Gauss-Stokes theorem with the boundary condition | B ∂U = 0, the second term of Eq.(C.20) may be integrated and gives ♣ μ ν ∂ · (ϑμh (ϑ ∧ ϑ ) · B(ϑν )) det[h]) τ U ρ ♣ μ ν = ϑ · ϑμh (ϑ ∧ ϑ ) · B(ϑν )det[h] τρ ∂U ♣ μ ν = h (ϑ ∧ ϑ ) · B(ϑν )det[h] τμ =0. (C.21) ∂U So, putting Eq.(C.21)inEq.(C.20)gives D ♣ ϑμ ∧ ϑν ♣ ϑμ ∧ ϑν · · (η ϑμ ηh ( )det[h]+h ( )ϑμ ∂ det[h]) B(ϑν ) τ =0, (C.22) U | for all B such that B ∂U = 0, and since B is arbitrary a fundamental lemma of integration theory finally yields D ♣ ϑμ ∧ ϑν ♣ ϑμ ∧ ϑν · η ϑμ ηh ( )det[h]+h ( )ϑμ ∂ det[h]=0, i.e., D ♣ ϑμ ∧ ϑν − · | | ∗ ϑμ ∧ ϑν ϑμ ηh ( )= (ϑμ ∂ ln det[h] )ηh ( ), (C.23) or yet,

♣ μ ν ♣ μ ν ♣ μ ν (ϑμ · ∂ ln |det[h]|)ηh (ϑ ∧ ϑ )+ϑμ · ∂ηh (ϑ ∧ ϑ )+Ω(ϑμ) × ηh (ϑ ∧ ϑ )=0. (C.24) η

This is the field equation for the κη-gauge rotation field Ω. It is a set of four coupled multiform ♣ ♣ 0 ♣ 3 equations, differential in the canonical 1-forms of h (i.e., h (ϑ ),...,h (ϑ )), and algebraic in the canonical biforms of Ω (i.e., Ω(ϑ0),...,Ω(ϑ3)). We end this Appendix, writing again the Euler-Lagrange equations for the fields h and Ω, in terms of the canonical 1-form and biform fields.

♣ μ h (ϑ )(ϑμ · ∂Ω(ϑν ) − ϑν · ∂Ω(ϑμ)+Ω(ϑμ) × Ω(ϑν )) = 0, (C.25) η

♣ μ ν ♣ μ ν ♣ μ ν (ϑμ · ∂ ln |det[h]|)ηh (ϑ ∧ ϑ )+ϑμ · ∂ηh (ϑ ∧ ϑ )+Ω(ϑμ) × ηh (ϑ ∧ ϑ )=0. (C.26) η References Fern´andez, V. V., Distortion and Rotation Extensors in Einstein’s Gravitational Theory, Ph.D. thesis (in Portuguese), IFGW-UNICAMP, 2000. Appendix D Proof of Eq.(6.13)

Proposition [1] The Einstein-Hilbert Lagrangian density Leh can be written as α Leh = −d g ∧ dgα + Lg, (D.1) g where 1 α 1 α 1 α α Lg = − dg ∧ dgα + δg ∧ δgα + dg ∧ gα ∧ (dg ∧ gα)(D.2) 2 g 2 g g g 4 g is the first order Lagrangian density (first introduced by Einstein).

Proof: We will establish that Lg may be written as 1 γ β α L = − τgg g ω ∧ ω , (D.3) g −1 −1 αβ γ 2 g g which is nothing more than the intrinsic form of the Einstein first order Lagrangian, in the gauge defined by the basis {gα}. To see this first recall Cartan’s structure equations for the Lorentzian structure (M R4, g,D), i.e., α − α ∧ β dg = ωβ g , (D.4a) Rα α α ∧ γ β = dωβ + ωγ ωβ, (D.4b) a a where ωb are the so called connection 1-form fields, which on the basis {g } are written as

α α γ ωβ = Lβγg . (D.5) Now, using one of the identities in Eq.(2.84), we easily get gγ gβ ω ∧ ωα = ηβκ Lδ Lγ − Lδ Lγ . (D.6) −1 −1 αβ γ κγ δβ δγ κβ g g 124 Appendix D. Proof of Eq.(6.13)

a a b Next, we recall that Eq.(D.4a) can easily be solved for ωb in terms of the dg ’s and the g ’s. We get 1 ωγδ = gδ dgγ − gγ dgδ + gγ gδ dg gα . (D.7) −1 −1 −1 −1 α 2 g g g g

We now have all ingredients to prove that Leh canbewrittenasinEq.(D.1). Indeed, using Cartan’s second structure equation (Eq.(D.4b)), we can write Eq.(6.11)as:

L 1 ∧ α ∧ β 1 ∧ γ ∧ α ∧ β eh = dωαβ (g g )+ ωαγ ωβ (g g ) 2 g 2 g 1 ∧ α ∧ β 1 ∧ α ∧ β 1 ∧ γ ∧ α ∧ β = d[ωαβ (g g )] + ωαβ d(g g )+ ωαγ ωβ (g g ) 2 g 2 g 2 g 1 ∧ α ∧ β − 1 ∧ α ∧ γ ∧ β = d[ωαβ (g g )] ωαβ ωγ (g g ). (D.8) 2 g 2 g

Next, using again some of the identities in Eq.(2.102) we get (recall that ωγδ = −ωδγ)

γ δ γδ (g ∧ g ) ∧ ωγδ = −  [ω (gγ ∧ gδ)] g g g γδ γδ γδ =  [(ω · gδ)θγ − (ω · gγ )gδ]=2 [(ω · gδ)gγ ], (D.9) g g g g g and from Cartan’s first structure equation (Eq.(D.4a)), we have

α α β βα g ∧ dgα = g ∧ (ωβα ∧ g )=−  [gα(ω ∧ gβ)] g g g g βα α β = −  [gα · ω )gβ]=−  [(g · ωβα)g ], (D.10) g g g g from where it follows that

1 γ δ α d[(g ∧ g ) ∧ ωγδ]=−d(g ∧ dgα). (D.11) 2 g g On the other hand, the second term in the last line of Eq.(D.8) can be written as

1 ∧ α ∧ γ ∧ β ωαβ ωγ (g g ) 2 g −1 β · γ · α − β · α γ · =  [(g ωαβ)(g ωγ ) (g ωγ )(g ωαβ)]. 2 g g g g g Now,

β · γ · α (g ωαβ)(g ωγ ) g g · γ · α β = ωαβ [(g ωγ )g ] g g · γ α ∧ β αβ = ωαβ [g (ωγ g )+ω ] g g ∧ γ α ∧ β · γδ =(ωαβ g ) (ωγ g )+ωγδ ω ] g g Appendix D. Proof of Eq.(6.13) 125

α − α ∧ β −1 α − α · β − −1 and taking into account that dg = ωβ g ,  dg = ωβ g ,andthatδgα =  dgα, g g g g g g we have

1 ∧ α ∧ γ ∧ β 1 − α ∧ − α ∧ ∧ γδ ωαβ ωγ (g g )= [ dg dgα δg δgα + ωγδ ω ]. (D.12) 2 g 2 g g g g g Next, using Eq.(D.7) the last term in the last equation can be written, after some algebra, as

1 γδ α 1 α α ωγδ ∧ ω = dg ∧ dgα − (dg ∧ gα) ∧  (dg ∧ gα) 2 g g 4 g and we finally get 1 α 1 α 1 α β Lg = − dg ∧ dgα + δg ∧ δgα + (dg ∧ gα) ∧  dg ∧ gβ , (D.13) 2 g 2 g g g 4 g and the proposition is proved.

References

[1] Rodrigues, W. A. Jr., and Capelas de Oliveira, E., The Many Faces of Maxwell, Dirac and Ein- stein Equations. A Clifford Bundle Approach. Lecture Notes in 722, Springer, Heidelberg, 2007. Appendix E Derivation of the Field Equations from Leh

1 Let X ∈ sec p T ∗U. A multiform functional F of X (not depending explicitly on x ∈ U)isa mapping p r F :sec T ∗U → sec T ∗U

As, in Section 3, when no confusion arises, we use a sloppy notation and denote the image F (X) ∈ sec rT ∗U simply by F . Eventually, we also denote a functional F by F (X). Which object we are talking about is always obvious from the context of the equations where they appear. p ∗ Let w := δX ∈ sec TU. As in Exercise 3.6. we write the variation of F in the direction of δX as the functional δF : p U → r U given by F (X + λδX) − F (X) δF = lim . (E.1) λ→0 λ ∂F Recall from Remark 3.1 that the algebraic derivative of F relative to X, ∂X is such that: ∂F δF = δX ∧ . (E.2) ∂X Moreover, given the functionals F : p U → r U → and G : pU → sU the variation δ satisfies

δ(F ∧ G)=δF ∧ G + F ∧ δG, (E.3) and the algebraic derivative (as is trivial to verify) satisfies ∂ ∂F ∂G (F ∧ G)= ∧ G +(−1)rpF ∧ . (E.4) ∂X ∂X ∂X An important property of δ is that it commutes with the operator d, i.e., for any given functional F dδF = δdF. (E.5) 128 Appendix E. Derivation of the Field Equations from Leh

In general, we may have functionals depending on several different multiform fields, say, F : sec( pT ∗U × qT ∗U) → sec rT ∗U,with(X, Y ) → F (X, Y ) ∈ sec rT ∗U. In this case, we have:

∂F ∂F δF = δX ∧ + δY ∧ . (E.6) ∂X ∂Y p ∗ × An important case is the one where the functional F is such that F (X, dX):sec( T U p+1 ∗ 4 ∗ T U) → sec T U. We then can write, supposing that the variation δX is chosen to be   ⊂ ∂F null on the boundary ∂U , U U (or that ∂dX ∂U = 0) and taking into account Stokes theorem, ∂F ∂F δ F = δF = δX ∧ + δdX ∧    ∂X ∂dX U U U ∂F ∂F ∂F = δX ∧ − (−1)pd + d δX ∧  ∂X ∂dX ∂dX U ∂F ∂F ∂F = δX ∧ − (−1)pd + δX ∧ U  ∂X ∂dX ∂U ∂dX δ δ ∧ F = X δ , (E.7) U  X

δ +1 4− where F (X, dX) : sec( pT ∗U × p T ∗U) → pT ∗U is called the functional derivative of δX F , and we have: δF ∂F ∂F = − (−1)pd . (E.8) δX ∂X ∂dX

δL When F = L is a Lagrangian density in field theory is the Euler-Lagrange functional. δX We now obtain the variation of the Einstein-Hilbert Lagrangian density Leh, given by Eq.(6.11), i.e.,

1 κ ι 1 κ ι Leh = (g ∧ g ) ∧ Rκι = Rκι ∧ (g ∧ g ), 2 g 2 g

κ by varying the ωγκ and the g independently. We have1

1 γ δ δLeh = δ[Rγδ ∧ (g ∧ g )] 2 g 1 γ δ 1 γ δ = δRγδ ∧ (g ∧ g )+ Rγδ ∧ δ  (g ∧ g ). (E.9) 2 g 2 g

1We use only constrained variations of the gα that do no change the metric field g. Appendix E. Derivation of the Field Equations from Leh 129

From Cartan’s second structure equation, we can write

γ δ δRγδ ∧ (g ∧ g ) g δ ∧ γ ∧ δ δ ∧ κ ∧ γ ∧ δ ∧ δ κ ∧ γ ∧ δ = dωγδ (g g )+ ωγκ ωδ (g g )+ωγκ ωδ (g g ) g g γ δ = δdωγδ ∧ (g ∧ g ) (E.10) g γ δ γ δ = d[δωγδ ∧ (g ∧ g )] − δωγδ ∧ d[(g ∧ g )]. g g δ ∧ γ ∧ δ − δ ∧ − α ∧ κ ∧ κ − δ ∧ γ ∧ κ = d[ ωγδ (g g )] ωγδ [ ωκ (g g ) ωκ (g g )] g g g γ δ = d[δωγδ ∧ (g ∧ g )]. g

Moreover, using the definition of algebraic derivative (Eq.(E.1)), we have immediately

∂[(gγ ∧ gδ)] δ γ ∧ δ δ κ ∧ g  (g g )= g κ (E.11) g ∂g

Now, recalling Eq.(2.103) of Chapter 4, we can write

γ δ 1 γκ δι μ ν δ  (g ∧ g )=δ( η η κιμνg ∧ g ) g 2 μ γκ δι ν = δg ∧ (η η κιμνg ), (E.12) from where we get ∂(gγ ∧ gδ) g = ηγκηδι gν . (E.13) ∂gm κιμν On the other hand we have

γ δ 1 γκ δι ρ σ gμ  (g ∧ g )=gμ( η η κιρσg ∧ g ) g g 2 γκ δι ν = η η κιμνg . (E.14)

Moreover, using the fourth formula in Eq.(2.102), we can write

∂[(gγ ∧ gδ)] g γ ∧ δ κ = gκ  (g g ) ∂g g g γ δ γ δ = [gκ ∧ (g ∧ g )] = (g ∧ g ∧ gκ). (E.15) g g

Finally, γ δ κ γ δ δ  (g ∧ g )=δg ∧ (g ∧ g ∧ gκ). (E.16) g g Then, using Eq.(E.10)andEq.(E.16)inEq.(E.9)weget

1 γ δ κ 1 α β δLeh = d[δωγδ ∧ (g ∧ g )] + δg ∧ [ Rαβ ∧ (g ∧ g ∧ gκ)]. (E.17) 2 g 2 g 130 Appendix E. Derivation of the Field Equations from Leh

1 γ κ Now, the curvature 2-form fields are given by Rαβ = 2 Rαβγκg ∧ g where Rαβγκ are the components of the Riemann tensor, and then

1 α β 1 α β Rαβ ∧ (g ∧ g ∧ gμ)=−  [Rαβ (g ∧ g ∧ gμ)] 2 g 2 g g 1 γ κ α β = − Rαβγκ  [(g ∧ g )(g ∧ g ∧ gμ)] 4 g g 1 = −  (R − Rg )=−  G , (E.18) μ 2 μ μ where the Rμ are the Ricci 1-form fields and Gμ are the Einstein 1-form fields. So, finally, we can write ∂L δ(L + L )= δgα ∧ (−  G + m )=0. (E.19) eh m α ∂gα To have the equations of motion in the form of Eq.(6.25)(herewithm = 0), i.e.,

dSγ + tγ = −  T γ (E.20) g g g it remains to prove that dSγ + tγ = −  Gγ . (E.21) g g

In order to do that we recall that Leh may be written (recall Eq.(6.13)andEq.(6.17)) as

α Leh = Lg − d(g ∧ dgβ) g

1 α 1 α 1 α β α = − dg ∧ dgα + δg ∧ δgα + dg ∧ gα ∧ (dg ∧ gβ) − d(g ∧ dgβ) 2 g 2 g g g 4 g g 1 α β 1 α β α = − dg ∧ gβ ∧ (dg ∧ gα)+ dg ∧ gα ∧ (dg ∧ gβ) − d(g ∧ dgβ) 2 g 4 g g 1 α α = dgα ∧ S − d(g ∧ dgβ) (E.22) 2 g g Then L L δL δ α ∧ ∂ g δ α ∧ ∂ g g = g α + dg α ∂g ∂dg L L L δ α ∧ ∂ g ∂ g δ α ∧ ∂ g g α + d α + d g α ∂g ∂dg ∂dg α α = δg ∧ tα + dSα + d(δg ∧ Sα), (E.23) g g g with L L ∂ g S ∂ g tα := α ,α := α (E.24) g ∂g g ∂dg

∂Lg To calculate ∂gα , we first recall from Eq.(2.23)thatwecanwrite

1 α 1 α 1 α gκ( dgα ∧ S )= (gκdgα) ∧ S + dgα ∧ (gκ  S ) (E.25) g 2 g 2 g g 2 g g Appendix E. Derivation of the Field Equations from Leh 131

1 α and thus since from Eq.(E.22)itisLg = 2 dgα ∧ S ,wehave g α 1 α α gκLg − (gκdgα) ∧ S = dgα ∧ (gκ  S ) − (gκdgα) ∧ S (E.26) g g g 2 g g g g Next, we recall that

δSα 1Sα δ κ ∧ ι δ κ ∧ Sα = κι (g g )= g (gκ ), 2 g α κ α δ  S = δg ∧ (gκ  S ) (E.27) g g g and, so we have δ, Sα = δ  Sα − δSα g g g κ α κ α = δg ∧ (gκS ) − δg ∧ (gκ  S ). (E.28) g g g

κ α Multiplying Eq.(E.28)ontheleftbydgα∧ and, moreover adding δdg ∧ S to both sides, we get g 1 1 1 δ( gκ ∧ Sα)=δdgκ ∧  Sα + dgκ ∧ δSα 2 g 2g 2 g κ 1 α α + δg ∧ dgα ∧ (gκ  S ) − (gκdgα) ∧ S 2 g g g g or

κ 1 α 1 κ α δLg = δdg ∧  S + dg ∧ δS 2 g 2 g κ 1 α α + δg ∧ dgα ∧ (gκ  S ) − (gκdgα) ∧ S . (E.29) 2 g g g g

Thus, comparing the coefficient of δgκ in Eq.(E.29) with the one appearing on the first line in Eq.(E.23), and taking into account the identity given by Eq.(E.26), we get

L ∂ g − κ ∧ S tα := α = gα Lg (gα dg )  κ. (E.30) g ∂g g g g

Also, take into account that

L S ∂ g − ∧ κ ∧ 1 ∧ κ ∧  α := α = gκ (dg gα)+ 2 gα (dg gκ), (E.31) g ∂dg g g or κ κ κ α 1 κ α S = −dg − (g  g ) ∧ dgα + g ∧ (dg ∧ gα). (E.32) g g g g g g 2 g Finally, comparing the coefficient of δgκ in Eq.(E.29) with the one appearing on Eq.(E.17), we get the proof of Eq.(E.21). Remark E1 Before proceeding, recall that in our theory the tα are 3-form fields defined directly as functions of the gravitational potentials gα and the fields dgα (Eqs.(6.26)and(6.27)), where the 132 Appendix E. Derivation of the Field Equations from Leh set {gα} defines by chance a basis for the g-orthonormal bundle. However, from Remark 6.4, we know that the cotetrad fields {gα} defining the gravitational potentials are associated with the extensor field h modulus, an arbitrary local Lorentz rotation Λ which is hidden in the definition of † † g = h ηh = h Λ†ηΛh. Thus the energy-momentum 3-forms tγ are gauge dependent. However, once g a gauge is chosen the tγ are, of course, independent on the basis (coordinate or otherwise) where they are expressed, i.e., they are legitimate tensor fields. However, in GRT, where the Sγ : U → g 2 U are called superpotentials2 and the tγ are called the gravitational pseudo energy-momentum 3-forms this is not the case. The reason is implicit in the name pseudo 3-formsfor tγ. Indeed, in GRT, those objects are directly associated with the connection 1-forms ωαβ associated with a particular section of the g-orthonormal bundle and thus, besides being gauge dependent, their components in any basis do not define a true tensor field. Taking into account Remark E1, we provide alternative expressions for tγ and Sγ given by g g Eqs.(E.30)and(E.31)as

γ −1 ∧ γ ∧ α ∧ β ∧ δ β ∧ α ∧ β ∧ γ t = ωαβ [ωδ (g g g )+ωδ (g g g )], g 2 g g γ 1 α β γ S = ωαβ ∧ (g ∧ g ∧ g ), (E.33) g 2 g whereweuseEq.(D.7) for packing part of the complicated formulas involving gα and dgα,inthe form of ωαβ. With Eqs.(E.33), we can provide a simple proof of Eq.(E.21). Indeed, let us compute −2  Gγ using Eq.(E.18) and Cartan’s second structure equation. g

− Gδ ω ∧ α ∧ β ∧ δ ω ∧ ωγ ∧ α ∧ β ∧ δ 2  = d αβ (g g g )+ αγ β (g g g ) g g g α β δ α β δ = d[ωαβ ∧ (g ∧ g ∧ g )] + ωαβ ∧ d(g ∧ g ∧ g ) g g ω ∧ ωγ ∧ α ∧ β ∧ δ + αγ β (g g g ) g

ω ∧ α ∧ β ∧ δ − ω ∧ ωα ∧ ρ ∧ β ∧ δ = d[ αβ (g g g )] αβ ρ (g g g ) g g − ω ∧ ωβ ∧ α ∧ ρ ∧ δ − ω ∧ ωδ ∧ α ∧ β ∧ ρ αβ ρ (g g g ) αβ ρ (g g g ) g g ω ∧ ωγ ∧ α ∧ β ∧ δ + αγ β (g g g ) g ω ∧ α ∧ β ∧ δ − ω ∧ ωδ ∧ α ∧ β ∧ ρ = d[ αβ (g g g )] αβ [ ρ (g g g ) g g ωβ ∧ α ∧ ρ ∧ δ + ρ (g g g )] g =2(dSδ + tδ). (E.34) g g

2These objects are called in [1], the Sparling forms [2]. However, they were already used much earlier by Thirring and Wallner in [3]. Appendix E. Derivation of the Field Equations from Leh 133 References

[1] Szabados, L. B., Quasi-Local Energy-Momentum and Angular Momentum in GR: A Review Article, Living Reviews in Relativity. [http://www.livingreviews.org/lrr-2004-4] [2] Sparling, G. A. J., Twistors, and the Einstein Vacuum Equations (unknown status), University of Pittsburg preprint (1982). [3] Thirring, W., and Wallner, R., The Use of Exterior Forms in Einstein’s Gravitational Theory, Brazilian J. Phys. 8, 686-723 (1978). Appendix F Comment on the LDG Gauge Theory of Gravitation

In the footnote 1 of Chapter 1, we recalled that in GRT, a gravitational field is defined by an equiv- ↑     ↑ alence class of pentuples, where (M,g,D,τg, )and(M , g ,D ,τg , ) are said to be equivalent if →   ∗  ∗  ∗ ↑ ∗ ↑ there is a diffeomorphism f : M M , such that g = f g, D = f D, τg = f τg, = f , (where f ∗ here denotes the pullback mapping)1.Moreover,inGRT, when one is studying the coupling of the 2 matter fields, represented, say by some sections of the bundle ψ1,...,ψn,whichsat- 3 isfy certain coupled differential equations (the equations of motion ), two models {(M,g,D,τg, ↑), } {     ↑   } ψ1,...,ψn and (M , g ,D ,τg , ),ψ1,...,ψn are said to be dynamically equivalent iff there is a →  ↑     ↑ diffeomorphism f : M M , such that the structures (M,g,D,τg, )and(M , g ,D ,τg , )are  ∗ diffeormorphic and moreover ψi = f ψi, i =1, ..., n. This kind of equivalence is not a peculiarity of GRT. It is an obvious mathematical requirement that any theory formulated with tensor fields existing in an arbitrary manifold must satisfy4.This fact is sometimes confused with the fact that, for a particular theory, say T, it may happen that → { ↑ } there are diffeomorphisms f : M M such that for a given model (M,g,D,τg, ),ψ1,...,ψn it  ∗  ∗  ∗ ↑ ∗ ↑  ∗ is g = f g, D = f D, τg = f τg, = f , but ψi = f ψi. When this happens, the set of

1 Details may be found in [1,2,3]. 2If the exterior algebra bundle T ∗M is viewed as embedded in the Clifford algebra bundle C(M,g)( T ∗M→ C(M,g)) then even fields can be represented in that formalism as some appropriate equivalence classes of sums of even non homogeneous differential forms. Details may be found in [1]. 3The equations of motion, for any particular problem, satisfy appropriate initial and boundary conditions. 4Some people now call this property background independence, although we do not think that to be a very appropriate name. Others call this property general covariance. Eventually it is most appropriate to simply say that the theory is invariant under diffeomorphisms. Moreover, we recall that there is an almost trivial mathematical theorem saying that the action functional for the matter fields represented by sections of the exterior algebra bundle plus the gravitational field is invariant under the action of a one-parameter group of diffeomorphisms for Lagrangian densities that vanish on the boundary of the region where the action functional is defined. Details, for example, in [1]. 136 Appendix F. Comment on the LDG Gauge Theory of Gravitation diffeomorphisms satisfying that property defines a group, called a symmetry group of T,andof course, knowing the possible symmetry groups of a given theory facilitates the finding of solutions for the equations of motion 5. Having said that, we now analyze first the motivations for the ‘gauge theory of the gravitational field’ in Minkowski spacetime of references [4] and [5] formulated on a vector manifold U 6.Those authors, taking advantage of the obvious identification of the tangent spaces of U, write that for a diffeomorphism f : U → U, x → f(x) we have some particularly useful representations for the ∗ ∗ ∗ ∗   → →  pullback mapping (f ,fx : Tx U Tx U and pushforward mapping (f∗,f∗x : TxU Tx U). Indeed, for representing the pushforward mapping [1] and [2], let’s introduce the operator f, such that 1 1 f :sec TU → sec TU, (F.1) a → f(a)=a · ∂xf η where ∂ ∂ = γμ , (F.2) x ∂xμ with {xμ} coordinates for U in the Einstein-Lorentz-Poincar´e gauge and {γμ} the dual basis of the 7 basis {ϑμ}, as introduced in Chapter 5. For representing the pullback mapping [1] and [2] introduce the operator f† such that8 † f :sec TxU → sec TxU, →  † ψi(x) ψi(x)=(f ψi)(x)=ψi(f(x)) (F.3)

 Next, those authors argue that the description of physics in terms of ψi or ψi is not covariant. To 1 1 cure this ‘deficiency’, they propose to introduce a gauge field h :sec TU → sec TU,such that, † ψi(x)=h [Ψi(x)],  † ψi(x)=h [Ψi(x)], (F.4) where Ψi(x) is said to be the covariant description of ψi(x). This implies the following transforma- tion law for h under a diffeomorphism f : U → U, † † h = f†h . (F.5) From this point, using heuristic arguments, those authors say that this permits the introduction of a new metric extensor field on U, that they write as † g = h h, (F.6) because, different from what we did in this book they choose as basic algebra for performing calculations C(U, η), instead of C(U, ·). Also, take notice that in [4,5] it is h = h−1.

5See some examples, e.g., in Chapter 5 of [1]. 6 The vector manifod U is as defined in Chapter 5, but with the elements of U being vectors instead of forms, and the elements of TU being fields instead of multiform fields. 7 μ ν ν Take notice that the reciprocal basis of {γ } is the basis {γμ} such that γμ · γ = δ . η μ 8Recall that f† is the extended of f. Appendix F. Comment on the LDG Gauge Theory of Gravitation 137 The reasoning behind Eqs.(F.4) is that the field Ψi ∈ sec TU has been selected as a representa- † tive of the class of the diffeomorphically equivalent fields {f ψi},forf ∈ DiffU, the diffeomorphism group of U. This is a very important point to keep in mind. Indeed, in [4] and [5] in a region V ⊂ U the gravitational potentials (called displacement gauge invariant frame {gμ}) are introduced as follows. First, take arbitrary coordinates {xμ},coveringV , and introduce the coordinate vector fields ∂x e := = ∂ x, (F.7) μ ∂xμ μ μ α μ where the position vector x := x (x )γα and the reciprocal basis of {eμ} is the basis {e } such that9 μ μ e := ∂xx . (F.8) Then, † α gμ := h (eμ)=hμeα, (F.9) μ −1 μ −1 μ g = h (e )=(h ∂x)x (F.10) and † † h (eμ) · h (eν )=g(eμ) · eν = gμν , (F.11) η η −1 −1 h (eμ) · h (eν )=g−1(eμ) · eν = gμν . (F.12) η η

From its definition, the commutator [eμ,eν ]:=eμ · ∂xeν − eν · ∂xeμ = 0, but, in general ,we have η η that

† † † † † † [gμ,gν ]=[h eμ, h eν ]:=(h eμ · ∂x)h eν − (h eν · ∂x)h eμ η η α − α α =(∂μhv ∂ν hμ)eα = cμν eα =0, (F.13) { } α i.e., gμ is not a coordinate basis, because the structure coefficients cμν of the basis gμ are non null. Remark F1 Before proceeding, we recall that on page 477 of [4] it is stated that the main † property that must be required from a nontrivial field strength h is that we should not have h (a)= † † f (a)wheref is obtained from a diffeomorphism f : U → U.Thisimpliesthat∂x ∧ h (eμ) =0, α − α i.e., (∂μhv ∂ν hμ)eα =0. The next step in [4] towards the formulation of their gravitational theory is the introduction of what those authors call a ‘gauge covariant derivative’ that ‘converts ∂μ into a covariant derivative’. If M ∈ sec TU they define

DμM = ∂μ +Ωμ × M, (F.14) 1 2 with Ω : sec TU → sec TU,Ωμ =Ω(eμ). Before going on we must observe that the correct 10 notation for Dμ,, in order to avoid confusion ,mustbeDeμ , and we shall use it in what follows,

9 μ ∂ μ ∂ μ α ∂xμ Recall that ∂x = γ ∂xμ = e ∂xμ ,withe = γ ∂xα . 10 † In [5] Eq.(F.15) is written DμM = ∂μ + ω(gμ) × M. We must immediately conclude that ω(gμ)=ω(h (eμ)) = ♣ Ω(eμ), i.e., ω = h Ω. Moreover from the notation used in [5] we should not infer that Dμ is representing Dgμ , α × because if that was the case then the defining equation for Dgμ should read Dgμ M = hμ ∂μ +Ω(gμ) M. 138 Appendix F. Comment on the LDG Gauge Theory of Gravitation thus to start, we write × Deμ M = ∂μ +Ωμ M. (F.15) μ ∗ μ We also introduce the basis {θ } of T U dual of the basis {eμ} of TU and also the basis {ζ } of ∗ T U dual of the basis {gμ} of TU and write as in [4,5]: α α − α ν α − α ν Deμ gν := Lμν gα, Deμ g = Lμνg , Deμ ζ = Lμν ζ (F.16) We also write α α − α ν α − α ν Deμ eν := Lμν gα, Deμ e = Lμν e , Deμ θ = Lμν θ . (F.17) α 11 { } Note now that, whereas the indices μ and ν in Lμν refer to the same basis ,namely eμ the α 12 { } indices μ and ν in Lμν are hybrid , i.e., the first (μ) corresponds to the basis eμ whereas the second (ν) corresponds to the basis {gν}. This observation is very important, for indeed, if we write explicitly η and g, the metric tensors which correspond to the metric extensors η−1 and g−1,we have immediately from gμ · gν = gμν and eμ · eν = gμν that η g μ ν μ ν η = gμν ζ ⊗ ζ , g = gμν θ ⊗ θ , (F.18)

μ μ with {ζ } the dual basis of {gμ} and {θ } the dual basis of {eμ}. We then get η α ⊗ β Deμ = Deμ (gαβζ ζ ) − ρ − ρ α ⊗ β =(∂μgαβ gρβ Lμα gαρLμβ )ζ ζ , (F.19) i.e., the connection D will be metric compatible with η iff − ρ − ρ ∂μgαβ gρβ Lμα gαρLμβ =0. (F.20) Now, in [4,5] Eq.(F.20) is trivially true13, but it is interpreted by those authors as meaning that D is compatible with g. The error in those papers arises due to the confusion with the hybrid indices ρ (holonomic and non holonomic) in Lμα, and we already briefly mentioned in [6]. And, indeed, a trivial calculation shows immediately that

Deμ g =0, (F.21) i.e., the connection D has a non null nonmetricity tensor relative to g. Indeed, we know from the results of Section 4.9.1 that the connection which is metrically compatible with g is D,the 1 h-deformation of D such that if a, b ∈ sec TM,  −1 Dab = h (Dah(b)). (F.22) Another serious equivocation in [4,5] is the following. Those authors introduce a Dirac like operator, here written as D = gαDeα . Then, it is stated (see, e.g., Eq.(131) in [5]) that 1 D∧gα = (Lα − Lα )gν ∧ gμ (F.23) 2 μν νμ

11 Those indices are holonomic, since {eμ} is a coordinate basis. 12The index μ is holonomic and the index ν is non holonomic. This has already been observed in [6]. 13 Indeed,wehaveforonesidethatDe (gα · gβ)=∂μgαβ and on the other side De (gα · gβ)=De gα · gβ μ η μ η μ η ρ ρ + gα · De gβ = gρβLμα + gαρL . η μ μβ Appendix F. Comment on the LDG Gauge Theory of Gravitation 139 is the torsion tensor (more precisely the torsion 2-forms Θα). However, since the structure coeffi- α cients of the basis {gμ} are non null, the correct expression for the torsion 2-form fields Θ are, as well known (see. e.g., [1] ) 1 Θα = (Lα − Lα − cα )gν ∧ gμ. (F.24) 2 μν νμ μν α α It follows that even assuming Lμν = Lνμ the connection D has torsion, contrary to what is claimed in [5]. Of course, a simple calculation shows that the Riemann curvature tensor of D is also non null. The above comments clearly show that the gravitational theory presented in [4,5] introduces a connection D that is never compatible with g and moreover does not reduce to GRT even when it happens that Θα = 0, and as such that theory seems to be unfortunately inconsistent concerning its claims14.

References

[1] Rodrigues, W. A. Jr., and Capelas de Oliveira, E., The Many Faces of Maxwell, Dirac and Ein- stein Equations. A Clifford Bundle Approach. Lecture Notes in Physics 722, Springer, Heidelberg, 2007. [2]Sachs,R.K.,andWu,H., for Mathematicians, Springer-Verlag, New York 1977. [3] Rodrigues, W. A., Jr., and Rosa, M. A. F., The Meaning of Time in Relativity and Einstein’s Later View of the Twin Paradox, Found. Phys. 19, 705–724 (1989). [4] Doran, C., and Lasenby, A., Geometric Algebra for Physicists, Cambridge Univ. Press, Cambridge 2003. [5] Hestenes, D., Gauge Theory Gravity with Geometrical Calculus, Found. Phys. 35, 903-969 (2005). [6] Fern´andez, V. V, Moya, A. M., and Rodrigues, W. A. Jr., Covariant Derivatives on Minkowski , in R. Ablamowicz and Fauser, B. (eds.), Clifford Algebras and their Applications in Mathematical Physics (Ixtapa-Zihuatanejo, Mexico 1999), vol.1, Algebra and Physics, Progress in Physics 18, pp 373-398, Birkh¨auser, Boston, Basel and Berlin, 2000.

14At http://www.mrao.cam.ac.uk/˜clifford/publications/index.html there is a list of articles (published in very good journals) written by the authors of [4] and collaborators and based on their gravitational theory may be found. Appendix G Gravitational Field as a Nonmetricity

In this Appendix to further illustrate the point of view used in this book, we give a model for gravitational field of a point mass, where that field is represented by the nonmetricity tensor of a special connection. To start with, suppose that the world manifold M is a 4-dimensional manifold diffeomorphic to R4. Let ,moreover, (t, x, y, z)=(x0,x1,x2,x3) be global Cartesian coordinates for M. Next, introduce on M two metric fields:

η = dt ⊗ dt − dx1 ⊗ dx1 − dx2 ⊗ dx2 − dx3 ⊗ dx3, (G.1) and 2m g = 1 − dt ⊗ dt r 3 1 2m − (xi)2dxi ⊗ dxi r2 r − 2m i=1 −1 2m − dx1 ⊗ dx1 + 1 − (dx2 ⊗ dx2 − dx3 ⊗ dx3). (G.2) r

In Eq. (G.2) r = (x1)2 +(x2)2 +(x3)2. (G.3)

Now, introduce (t, r, ϑ, ϕ)=(x0,x1,x2,x3) as the usual spherical coordinates for M. Recall that x1 = r sin ϑ cos ϕ, x2 = r sin ϑ sin ϕ, x3 = r cos ϑ (G.4) 142 Appendix G. Gravitational Field as a Nonmetricity Tensor Field and the range of these coordinates in η are r>0, 0 <ϑ<π,0<ϕ<2π.Forg, the range of the r variable must be (0, 2m) ∪ (2m, ∞). As it can be easily verified, the metric g in spherical coordinates is: −1 2m 2m g = 1 − dt ⊗ dt − 1 − dr ⊗ dr − r2(dϑ ⊗ dϑ +sin2 ϑdϕ ⊗ dϕ), (G.5) r r which we immediately recognize as the Schwarzschild metric of GR. Of course, η is a Minkowski metric on M. As next step, we introduce two distinct connections, D˚ and D on M. We assume that D˚ is the Levi-Civita connection of η in M and D is the Levi-Civita connection of g in M. Then, by definition (see, e.g., [1] for more details) the nonmetricities tensors of D˚ relative to η and of D relative to g are null, i.e.,

D˚η =0, Dg =0. (G.6)

η 0 However, the nonmetricity tensor A ∈ sec T3 M of D˚ relative to g is non null, i.e., D˚g = Aη=0 , (G.7)

g 0 and also the nonmetricity tensor A ∈ sec T3 M of D relative to η is non null, i.e., Dη = Ag=0 . (G.8)

η ν We, now, calculate the components of A in the coordinated bases {∂μ} for TM and {dx } for T ∗M associated with the coordinates (x0,x1,x2,x3)ofM.SinceD˚ is the Levi-Civita connection of the Minkowski metric η,wehavethat ˚ ∂ ρ ∂ D∂μ ν = Lμν ρ =0, ˚ α − α ν D∂μ dx = Lμν dx =0. (G.9)

ρ ˚ η α ⊗ β ⊗ μ i.e., the connection coefficients Lμν of D in this basis are null. Then, A = Qμαβ dx dx dx is given by η ˚ ˚ α ⊗ β ⊗ μ A = Dg = D∂μ gαβdx dx dx ∂g =( αβ )dxα ⊗ dxβ ⊗ dxμ. (G.10) ∂xμ

To fix ideas, recall that for Q100,itis, ∂ 2m Q100 = 1 − ∂x1 (x1)2 +(x2)2 +(x3)2 ∂ 1 = −2m ∂x1 (x1)2 +(x2)2 +(x3)2 2mx1 2mx1 = 1 2 2 2 3 2 3 = 3 , (G.11) [(x ) +(x ) +(x ) ] 2 r Appendix G. Gravitational Field as a Nonmetricity Tensor Field 143

1 which is non null for x =0.Notealsothat Q010 = Q001 =0. Remark G1 Note that using Riemann normal coordinates {ξμ} covering V ⊂ U ⊂ M, naturally 1 adapted to a reference frame Z ∈ sec TV in free fall (DZZ =0,dα∧ α =0,α = g(Z, )), it is possible, as it is well known, to put the connection coefficients of the Levi-Civita connection D of g equal to zero in all points of the world line of a free fall observer (an observer is here modelled as an integral line σ of a reference frame Z,whereZ is a time like vector field pointing to the future | such that Z σ = σ∗). In the Riemann normal coordinates system covering U ⊂ M, not all the connection coefficients of the non metric connection D˚ (relative to g ) are null. Also (of course) the nonmetricity tensor Aη of D˚ (relative to g) representing the true gravitational field is not null. An observer following σ does not feel any force along its world line because the gravitational force represented by the nonmetricity 2 ρ field is compensated by an inertial force represented by the non null connection coefficients Lμν of ˚ { ∂ } D in the basis ∂ξν . The situation is somewhat analogous to what happens in any non inertial reference frame which, of course, may be conveniently used in any problem (as e.g., in a rotating disc [2]), where the connection coefficients of the Levi-Civita connection of η are not all null. Standard clocks and standard rulers (standard according to GRT)arepredictedtohaveabe- havior that seems to be (at least with the precision of the GPS system) exactly the observed one. However, take into account that those objects are not the standard clocks and standard rulers of the model proposed here3. We would say that the gravitational field distorts the period of the standard clocks and the length of the standard rulers of GRT relative to the standard clocks and standard rulers of the proposed model, which has non null nonmetricity tensor4. But, who are the devices that materialize those concepts in the proposed model? Well, they are paper concepts, like the notion of time in some Newtonian theories. They are defined and calculated in order to make correct predictions. However, given the status of present technology, we can easily imagine how to build devices to directly realize the standard rulers and clocks of the proposed model.

References

[1] Rodrigues, W. A. Jr., and Capelas de Oliveira, E., The Many Faces of Maxwell, Dirac and Ein- stein Equations. A Clifford Bundle Approach, Lecture Notes in Physics 722, Springer, Heidelberg, 2007. [2] Rodrigues, W. A. Jr., and Sharif, M., Rotating Frames in RT: Sagnac’s Effect in SRT and other Related Issues, Found. Phys. 31, 1767-1784 (2001). [3] Coll, B., A Universal Law of Gravitational Deformation for General Relativity, Proc. of the Spanish Relativistic Meeting, EREs, Salamanca, Spain,1998.

1For the mathematical definitions of reference frames, naturally adapted coordinates to a reference frame and observers, see. e.g., Chapter 5 of [1]. 2 ∂ ρ ∂ ρ ∇ ∂ ν = Lμν ν . The explicit form of the coefficients Lμν may be found in Chapter 5 of [1]. ∂ξμ ∂ξ ∂ξ 3Of course, our intention was only to illustrate a statement we made in the Introduction, namely that some gravitational fields may have many (equivalent) geometrical interpretations. 4On his respect see also [3]. Acronyms and Abbreviations

ADM Arnowitt-Deser-Misner GRT General Relativity Theory GSS Geometrical Space Structure MCGSS Metrical Compatibale Geometrical Space Structure MCPS Metrical Compatible Parallelism Structure DHSF Dirac-Hestenes Spinor Field MECM Multiform and Extensor Calculus on Manifolds iff if and only if i.e. Id Est e.g. Exempli Gratia List of Symbols

M manifold h distortion field † h adjoint of h ♣ h inverse of the adjoint of h η Minkowski metric extensor η Minkowski g Lorentzian metric extensor g Lorentzian metric tensor T ∗M bundle of non homogenous differential forms k T ∗M bundle of homogeneous differential k-forms sec section D, ∇ connections

τg volume element {ϑμ} orthonormal cobasis of Minkowski spacetime {gμ} gravitational potentials ↑ time orientation Tp particles energy-momentum tensor Tf fields energy-momentum tensor A nonmetricity tensor R Riemann curvature tensor Θ torsion tensor An n-dimensional affine space {xμ} coordinates in Einstein-Lorentz-Poincar´e gauge £ξ in the direction of the ξ vector field V n-dimensional 148 List of Symbols

V = V∗ dual space of V R real field C complex field k V space of homogenous k-forms V space of non homogenous forms ∧ exterior product k k-part operator principal involution operator reversion operator · canonical euclidean scalar product · Minkowski scalar product η · metric scalar product g , canonical contractions , metric contractions g g extV space of general extensor over V (p, q)-extV space of (p, q)-extensors over V t exterior power extension of a (1, 1)-extensor t+ symmetric part of a (1, 1)-extensor t− antisymmetric part of a (1, 1)-extensor tr[t]traceofa(1, 1)-extensor det[t] determinant of a (1, 1)-extensor T generalization of a (1, 1)-extensor t bif biform mapping × canonical commutator ×metric commutator g C(V,·) canonical Clifford algebra C(V,g) metric Clifford algebra  canonical  metric Hodge star operator g A · ∂X A-directional derivative of multiform functions ∂X derivative mapping of multiform functions t(∂X ) t-distortion of ∂X F (X1,...,Xk)[t] multiform functional A · ∂t A-directional derivative of a functional ∂t derivative of a functional ∂t ∗∗-product derivative of a functional w δt , δ variational operator γ(x),γ connection 2-extensor field γa connection (1, 1)-extensor field Γa generalization of γa ω(a) rotation gauge field ∇ ∇− a, a covariant derivatives of multivector and extensor fields γ τ torsion 2-extensor field associated to a given connection γ List of Symbols 149

γ T torsion (2, 1)-extensor field associated to a given connection γ γ ρ curvature operator associated to a given connection γ γ R1 Riemann (curvature) 4-extensor field associated to given a connection γ γ R2 Ricci 2-extensor field γ R1 Ricci (1, 1)-extensor field [a, b, c] first Christofell operator [a, b] Lie bracket of 1-form fields c second Christofell operator a, b λ Levi-Civita connection on Uo γg ω(a) γg-gauge rotation field γg R3 Riemann (curvature) 4-extensor field associated to structure (Uo, g,γ) γg R2 Ricci 2-extensor field associated to structure (Uo, g,γ) γg R Ricci scalar field γg R2 Riemann (2, 2)-extensor field associated to structure (Uo, g,γ) γg R1 Ricci (1, 1)-extensor field associated to structure (Uo, g,γ) γg G Einstein (1, 1)-extensor field associated to structure (Uo, g,γ) − Da, Da Levi-Civita covariant derivatives associated to λ R2 Riemann (2, 2)-extensor field associated to structure (Uo, g,λ) R1 Ricci (1, 1)-extensor field associated to structure (Uo, g,λ) − D g-gradient associated to Da g − D g-divergent associated to Da g ∧ − D rotational associated to Da χ connection compatible with Minkowski metric η χη χη− D a, D a η-metric compatible covariant derivatives χη ⊗(a) χη-gauge rotation field associated to structure (U0,η,χ) χη χη− χη χη− ∇a, ∇ a h-distortions of the D a, D acovariant derivatives χh R 2 Riemann ((2, 2)-extensor) gauge field for (U0, g,γ) χh R 1 Ricci ((1, 1)-extensor) gauge field for (U0, g,γ) χh R Ricci (scalar) gauge field for (U0, g,γ) χh G 1 Einstein ((1, 1)-extensor) gauge field for (U0, g,γ) Ω(a) η-rotation operator for the Minkowski-Cartan structure (U0,η,μ) Leh Einstein-Hilbert Lagrangian Leh Einstein-Hilbert Lagrangian density Lg Lagrangian density for the gravitational field in terms of potentials Lh h-deformation of Lg Sα superpotential 2-form fields T α energy-momentum 1-form fields 150 List of Symbols ta gravitational energy-momnetum 1-form fields d differential operator δ Hodge coderivative operator g Tα energy-momentum 3-forms fields g Lαβ orbital angular-momentum 3-forms fields g Sαβ spin angular-momentum 3-forms fields g Jαβ total angular-momentum 3-forms fields g ◦ ♦ Hodge Laplacian = ∂·∂ covariant D’Alembertian operator g ∂ ∧ ∂ Ricci operator Index

(1,1)-extensors metric Clifford algebra, 13 extension operation, 20 metric Clifford algebra, 24 properties Clifford product associated extensors, 20 canonical Clifford product, 18 (p, q)-extV metric Clifford product, 26 (p, q)-extensors, 19 connection 2-extensor field, 50 2-exform torsion field, 54 associated extensor fields, 50 2-extensor coupling, 68 conservation law angular momentum, 89 contraction energy-momentum, 88 metric contraction contraction left and rigth, 25 canonical, 17 left contraction, 17 adjoint operator, 19 right contraction, 17 ADM energy, 100 cosmological constant, 81 affine spaces, 3 covariant derivative angular momentum Levi-Civita, 61 orbital, 89 metric compatible spin, 89 Christoffel operators, 56 gravitational field, 90 covariant derivative metric compatible, 56 canonical space, 48 curvature and bending, 3 Cartan curvature 2-form fields, 85 curvature operator characteristic biform curvature extensor fields, 54 bif[t], 23 curvature tensor, 1 Clifford algebra canonical Clifford algebra Deformation of MCGSS Structures 152 Index

Deformation of MCGSS plastic distor- golden rule, 30 tion field h, 64 gravitational potentials gα, 83 deformation of MCGSS structures, 64 gravitational theory deformations independent h and Ω fields, 117 elastic, 65 graviton mass, 86 Cauchy-Green tensor, 65 GRT, 1 Minkowski-Cartan →Lorentz-Cartan, 66 plastic, 65 Hamilton’s equations, 98 determinant of of (1,1)-extensor Hamiltonian 3-form, 93 det[t], 21 Hamiltonian formalism, 93 distortions Hodge coderivative operator, 85 of covariant derivatives, 67 Hodge star operators, 30 duality identity, 19 involutions, 14 Einstein (1,1)-extensor field, 60 , 2 k-part operator, 14 Einstein-Lorentz-Poincar´e gauge, 136 Killing vector fields Einstein-Lorentz-Poincare gauge, 75 symmetries energy-momentum tensor, 2 conservation laws, 6 extensor calculus on manifolds, 47 extensor field, 49 Lagrangian covariant derivative, 52 gauge fixing term, 86 extensors, 13, 19 gravitational field plus matter field, 81 exterior product, 15 Lagrangian density Einstein-Hilbert, 123 flat spaces, 3 Leh, 127 free gravitational field LDG Gauge Theory, 135 equation of motion, 77 Legendre transformation, 94 Lagrangian, 76 Levi-Civita connection, 2 functional Levi-Civita differential operators, 63 algebraic derivatives, 127 divergent, 63 Euler-Lagrange, 128 gradient, 63 rotational, 63 gauge extensor fields, 70 Lorentz-Cartan MGSS, 57 gauge freedom, 29 Lorentzian metric, 2 gauge Ricci field, 69 gauge Riemann field, 69 MECM, 1 γg-gauge rotation extensor field, 58 metric extensor η, 27 κη-gauge rotation extensor field, 66 Metric Extensorg, 28 generalization operator, 23 multiform and extensor calculus geometrical space structure, 1 notable identities, 53 GSS, 3 multiform calculus on manifolds, 47 metric compatible multiform field, 49 MCGSS, 3 covariant derivative, 50 Riemann, 3 multiform functionals, 33, 41 Riemann-Cartan, 3 A-directional derivative A · ∂t, 41 Riemann-Cartan-Weyl, 3 derivative of induced functionals, 41 Index 153

δw variational derivative t , 44 quasi local energy, 97 multiform functions, 33 derivative, 34 Riemann (2,2)-extensor field, 60 limit and continuity, 34 Riemann 4-extensor field, 59 multiform variables, 35 Riemann curvature tensor, 3 chain rules, 36 Riemann-Cartan MGSS, 57 rotation gauge field, 50 derivative mapping ∂X , 37 differentiablility, 35 scalar product directional derivative A · ∂ , 35 X canonical scalar product, 15 limit and continuity, 35 metric scalar product, 24 real variable, 33 shape tensor, 4 multiforms, 13 space extV , 19 mutiform functionals spacetime operators ∂ ∗, 42 t Lorentz-Cartan, 6 Lorentz-Cartan-Weyl, 6 Noether current, 95 Lorentzian, 6 nonmetricity tensor, 1, 3 Minkowski, 6 nonmetricity tensor field, 141 superpotentials, 132 normal (1,1)-extensors, 24 Nunes connection teleparallel connection navigator connection teleparallel MCGSS, 6 Columbus connection, 5 time oriented, 2 torsion tensor, 1, 3 operator torus covariant D’Alembertian, 91 null Riemann curvature, 109 Dirac operator, 91 torus geometry, 4 Hodge D’Alembertian, 91 trace of (1,1)-extensor Ricci operator, 91 tr[t], 21 Ricci operator ∂| ∧ ∂|, 91 operators ∂X ∗ useful identities, 31 t-distortions, 40 volume element, 2 parallelism rule, 3 vorticities, 83 parallelism structure wave equations, 91 covariant derivative, 50 parallelizable, 6 Yang-Mills, 83 plastic distortion Lorentz vacuum gravitation, 75 position 1-form, 49 pseudo energy-momentum 3-forms, 132 pseudo-euclidean metric extensor, 27 pseudo-euclidean space, 2 punctured sphere Nunes connection, 111 Riemann connection, 111