Tetrad in $SL(2,C) \times SU(2) \times U(1)$ Yang–Mills–Weyl Spacetimes

Abstract

A new set of tetrads is introduced within the framework of $SL(2,C) \times SU(2) \times U(1)$ Yang–Mills–Weyl field theories in curved spacetimes. A set of field differential equations is analyzed concerning the transformation properties of the tetrad vectors that can be constructed out of the fields satisfying these equations. In previous work it has been found the technique to build these tetrads. Here we are able to prove additional construction properties regarding the new “internal” groups of transformations involved in the formulation. In particular we show how to switch the bosonic “nesting” of tetrads associated to both groups, $SL(2,C)$ and $SU(2)$. We also show that the usual two vector fields ${{X}^{\mu }}$, ${{Y}^{\mu }}$, necessary to gauge the tetrads, can be constructed using currents, that is, Weyl spinors in curved spacetime. Employing our new tetrads we prove that the local group $SL(2,C)$ is isomorphic to local groups of tetrad transformations, equivalent to say that the gravitational field is a gauge field. A conjecture is raised in relation with the asymptotic properties of these tetrads. We conjecture that within the set of solutions to the classical field equations we are introducing, there could be one that we might be able to associate to or represent the geometry of a microparticle like the Neutrino or its antiparticle, for instance. We conjecture that we can associate spacetimes to microparticles since all the local symmetries of the standard model can be realized in four-dimensional curved Lorentzian spacetimes. The group isomorphisms between $U(1)$, $SU(2) \times U(1)$ or $SU(3) \times SU(2) \times U(1)$ on one hand, and local groups of tetrad transformations on the other hand have already been presented in previous manuscripts. In this regard, the asymptotic limit for this set of equations, in particular the Weyl equation on a Minkowskian background in the “far” region, would be the starting point for the standard Quantum Field Theory associated to this particular equation. Standard Quantum Field Theories are then interpreted as devices that deal with perturbative quantum “interactions” between geometries that radiate (create) and absorbe (annihilate) wave modes, but are otherwise never related to the spacetime background geometries that undergo the radiation or absorption processes. Quantum Field Theories just deal with perturbative interacting phenomena in the asymptotic limit to these hypothesized background spacetimes. Isomorphism theorems involving the group structure $SL(2,C) \times SU(2) \times U(1)$ are proved. A gauge invariant method to diagonalize the stress-energy tensor is discussed. Beyond the possible association of spacetimes to microparticles, the results found in this work relating gravitation to local internal transformations within the framework of $SL(2,C) \times SU(2) \times U(1)$ Yang–Mills–Weyl field theories, are worth being discussed by themselves as proper mathematical and geometrical results. This is a paper about grand Standard Model gauge theories – General Relativity gravity unification and grand group unification in four-dimensional curved Lorentzian spacetimes.

Appendices

APPENDIX A

We are introducing in this first appendix the necessary variables that we will need in order to construct the $SL(2,C)$ tetrads [20, 22]. Either the “skeleton” of the tetrad, or the gauge vector fields ${{X}^{\mu }}$ and ${{Y}^{\mu }}$. These are not the variables in terms of which the differential Eqs. (1)–(6) are written, but they do exist and we are using them. We follow the notation in [20]. We start with the Hermitian matrices $\sigma _{{AB'}}^{\mu }$ introduced by Infeld and van der Waerden,

$$\sigma _{{AB{\kern 1pt} '}}^{\mu }{{\sigma }^{{\nu AB{\kern 1pt} '}}} = {{g}^{{\mu \nu }}}.$$

(77)

Roman capital indices are the spinor indices taking values $0$ and $1$, while the primed Roman capital indices refer to the complex conjugate taking values $0{\kern 1pt} '$ and $1{\kern 1pt} '$. A detailed discussion is included in [20] chapter eight. Therefore we summarize the main relations and expressions involved in our work.

$$\sigma _{{AB{\kern 1pt} '}}^{\mu }\sigma _{{CD{\kern 1pt} '}}^{\nu }{{g}_{{\mu \nu }}} = {{g}_{{AB{\kern 1pt} 'CD{\kern 1pt} '}}},$$

(78)

$${{g}_{{AB{\kern 1pt} 'CD{\kern 1pt} '}}} = {{\epsilon }_{{AC}}}{{\epsilon }_{{B{\kern 1pt} 'D{\kern 1pt} '}}}.$$

(79)

${{\epsilon }_{{AC}}}$ and ${{\epsilon }_{{B{\kern 1pt} 'D{\kern 1pt} '}}}$ are the skew symmetric Levi–Civita metric spinors, both with components ${{\epsilon }_{{00}}} = {{\epsilon }_{{0{\kern 1pt} '0{\kern 1pt} '}}} = 0$, ${{\epsilon }_{{01}}} = {{\epsilon }_{{0{\kern 1pt} '1{\kern 1pt} '}}} = 1$, ${{\epsilon }_{{11}}} = {{\epsilon }_{{1{\kern 1pt} '1{\kern 1pt} '}}} = 0$ and ${{\epsilon }_{{10}}} = {{\epsilon }_{{1{\kern 1pt} '0{\kern 1pt} '}}} = - 1$.

$${{\zeta }_{{aA}}} = \zeta _{a}^{B}{{\epsilon }_{{BA}}},$$

(80)

$$\zeta _{a}^{B} = {{\epsilon }^{{BA}}}{{\zeta }_{{aA}}}.$$

(81)

$\zeta _{a}^{B}$ are local basis spinors, the index $a$ spanning a complex two dimensional vector space $\mathcal{C}$ at each spacetime point, see [20], chapter ten. In the complex conjugate to this vector space the indices are raised and lowered using the ${{\epsilon }_{{B{\kern 1pt} 'D{\kern 1pt} '}}}$ metric spinor,

$${{\eta }_{{aA{\kern 1pt} '}}} = \eta _{a}^{{B{\kern 1pt} '}}{{\epsilon }_{{B{\kern 1pt} 'A{\kern 1pt} '}}},$$

(82)

$$\eta _{a}^{{B{\kern 1pt} '}} = {{\epsilon }^{{B{\kern 1pt} 'A{\kern 1pt} '}}}{{\eta }_{{aA{\kern 1pt} '}}}.$$

(83)

Next, we write the normalization conditions,

$$\zeta _{a}^{A}{{\epsilon }_{{AB}}}\zeta _{b}^{B} = {{\epsilon }_{{ab}}},$$

(84)

and the completeness relations,

$$\zeta _{a}^{A}\,{{\epsilon }^{{ab}}}\zeta _{b}^{B} = {{\epsilon }^{{AB}}}.$$

(85)

Similar results follow for the basis spinors. The local basis spinors have the following transformation law,

$$\tilde {\zeta }_{a}^{A} = (S_{{sl}}^{{ - 1}})_{a}^{b}\zeta _{b}^{A},$$

(86)

where ${{S}_{{sl}}}$ is an $SL(2,C)$ gauge transformation. Since ${{\epsilon }^{{AB}}}$ belongs to the tensor product $\mathcal{C} \otimes \mathcal{C}$, it transforms under $SL(2,C)$ as,

$${{\epsilon }^{{'AB}}} = {{\epsilon }^{{CD}}}S_{C}^{A}S_{D}^{B} = (\det \,S){{\epsilon }^{{AB}}},$$

(87)

where $\det S$ stands for the determinant of the $SL(2,C)$ transformation $S_{A}^{B}$, that is $\det S = 1$. ${{\epsilon }^{{AB}}}$ is a metric spinor, that is why it is used for raising and lowering spinor indices. The local coordinate representation of the covariant derivative is,

$${{\nabla }_{\mu }}{{\psi }^{A}} = {{\partial }_{\mu }}{{\psi }^{A}} + \Gamma _{{\mu \,B}}^{{\,A}}{{\psi }^{B}},$$

(88)

while the transformation properties of the $\Gamma _{{\mu D}}^{C}$ under $SL(2,C)$ are,

$$\tilde {\Gamma }_{{\mu A}}^{{\,B}} = ({{S}^{{ - 1}}})_{C}^{{\,B}}\Gamma _{{\mu \,D}}^{{\,C}}S_{A}^{{\,D}} + ({{S}^{{ - 1}}})_{C}^{{\,B}}{{\partial }_{\mu }}S_{A}^{{\,C}}.$$

(89)

Next, we formulate two conditions on the metric spinors,

$${{\nabla }_{\mu }}{{\epsilon }_{{AB}}} = 0,$$

(90)

$${{\nabla }_{\mu }}{{\epsilon }^{{AB}}} = 0.$$

(91)

The two conditions (90), (91) reduce the number of independent complex components of the spinor connection to 12 since they imply,

$$\Gamma _{{\mu A}}^{{\,B}} = \Gamma _{{\mu B}}^{{\,A}}.$$

(92)

Similar conditions are imposed on ${{\epsilon }_{{B{\kern 1pt} 'D{\kern 1pt} '}}}$ and ${{\epsilon }^{{B{\kern 1pt} 'D{\kern 1pt} '}}}$. We also demand that the operation of converting spinor to tensor indices and vice-versa commute with covariant differentiation,

$${{\nabla }_{\mu }}\sigma _{{AB{\kern 1pt} '}}^{\nu } = 0.$$

(93)

The above condition (93) implies that the relation between the components of the spinor connections and affine connections can be written as, see [20] chapter ten,

$$\Gamma _{{\mu \,A}}^{C} = \frac{1}{2}\sigma _{\nu }^{{CB{\kern 1pt} '}}(\sigma _{{AB{\kern 1pt} '}}^{\lambda }\Gamma _{{\lambda \mu }}^{\nu } + {{\partial }_{\mu }}\sigma _{{AB{\kern 1pt} '}}^{\nu }).$$

(94)

The gauge potentials in the $SL(2,C)$ gauge theory, are the dyad components of the spinor connection,

$${{\nabla }_{\mu }}\zeta _{a}^{A} = \Gamma _{{\mu \,B}}^{{\,A}}\zeta _{a}^{B},$$

(95)

$$({{B}_{\mu }})_{a}^{{\,b}} = \Gamma _{{\mu \,A}}^{{\,B}}\zeta _{a}^{A}\zeta _{B}^{b}.$$

(96)

Now, we consider the covariant derivatives of the basis vectors,

$${{\nabla }_{\mu }}\zeta _{a}^{A} = ({{B}_{\mu }})_{a}^{b}\zeta _{b}^{A}.$$

(97)

$$({{\tilde {B}}_{\mu }})_{a}^{b} = (S_{{sl}}^{{ - 1}})_{a}^{c}({{B}_{\mu }})_{c}^{d}S_{{sld}}^{{\,b}} - (S_{{sl}}^{{ - 1}})_{a}^{c}{{\partial }_{\mu }}S_{c}^{b}.$$

(98)

This gauge transformation (98) of the $SL(2,C)$ potential are the ones we will use when studying the gauge transformation of the tetrad vectors. The field strengths or curvature tensors are given by the expression,

$$F_{{\mu \nu A}}^{{\,B}} = {{\partial }_{\nu }}\Gamma _{{\mu A}}^{{\,B}} - {{\partial }_{\mu }}\Gamma _{{\nu A}}^{{\,B}} + \Gamma _{{\mu A}}^{{\,C}}\Gamma _{{\nu C}}^{{\,B}} - \Gamma _{{\nu A}}^{{\,C}}\Gamma _{{\mu C}}^{{\,B}},$$

(99)

Or in dyad components,

$$F_{{\mu \nu a}}^{{\,b}} = {{\partial }_{\nu }}B_{{\mu a}}^{{\,b}} - {{\partial }_{\mu }}B_{{\nu a}}^{{\,b}} + B_{{\mu a}}^{{\,c}}B_{{\nu c}}^{{\,b}} - B_{{\nu a}}^{{\,c}}B_{{\mu c}}^{{\,b}}.$$

(100)

Under a gauge $SL(2,C)$ transformation the field strength transforms homogeneously,

$$\tilde {F}_{{\mu \nu a}}^{{\,b}} = (S_{{sl}}^{{ - 1}})_{a}^{{\,c}}F_{{\mu \nu c}}^{{\,d}}S_{{sl\,d}}^{{\,b}}.$$

(101)

These fields (100) are the ones we will use in order to build the skeleton of the $SL(2,C)$ tetrads. Finally, we exhibit the relationship of the field strength and the Riemann tensor,

$$F_{{\mu \nu a}}^{{\,b}} = \frac{1}{2}R_{{\,\,\lambda \mu \nu }}^{\rho }{{\sigma }_{{\rho ac{\kern 1pt} '}}}{{\sigma }^{{\lambda bc{\kern 1pt} '}}}.$$

(102)

or

$$F_{{\mu \nu a{\kern 1pt} '}}^{{\,\,b{\kern 1pt} '}} = \frac{1}{2}R_{{\,\lambda \mu \nu }}^{\rho }{{\sigma }_{{\rho ca{\kern 1pt} '}}}{{\sigma }^{{\lambda cb{\kern 1pt} '}}}.$$

(103)

APPENDIX B

The second appendix is introducing the object ${{\Sigma }^{{\alpha \beta }}}$. The use of this object in the construction of our tetrads, allows for the local $SL(2,C)$ gauge transformations $S$, to get transformed into purely local geometrical transformations. That is, local proper Lorentz transformations of the $SU(2)$ tetrads $S_{\alpha }^{\rho }$ nested within the structure of the two vector fields ${{X}^{\mu }}$ and ${{Y}^{\mu }}$, for instance ${{X}^{\sigma }} = {{Y}^{\sigma }} = {\text{Tr}}[{{\Sigma }^{{\alpha \beta }}}S_{\alpha }^{{\,\,\rho }}S_{\beta }^{{\,\,\lambda }} * \epsilon _{\rho }^{{\,\,\,\sigma }} * {{\epsilon }_{{\lambda \tau }}}{{B}^{\tau }}]$ in Section 3. The object ${{\sigma }^{{\alpha \beta }}}$ is defined as ${{\sigma }^{{\alpha \beta }}} = \sigma _{ + }^{\alpha }\sigma _{ - }^{\beta } - \sigma _{ + }^{\beta }\sigma _{ - }^{\alpha }$, [20, 98–101]. The object $\sigma _{ \pm }^{\alpha }$ arises when building the Weyl representation for left handed and right handed spinors. According to [101], it is defined as $\sigma _{ \pm }^{\alpha } = (1, \pm {{\sigma }^{i}})$, where ${{\sigma }^{i}}$ are the Pauli matrices for $i = 1 \ldots 3$. Even though in [20] there is a different definition of these matrices, tensor expressions do not change. It must be stressed that we are suppressing either spinor or dyad indices for simplicity of notation. Under the $\left( {\frac{1}{2},0} \right)$ and $\left( {0,\frac{1}{2}} \right)$ spinor representations of the Lorentz group it transforms as,

$$S_{{(1/2)}}^{{ - 1}}\sigma _{ \pm }^{\alpha }{{S}_{{(1/2)}}} = \Lambda _{\gamma }^{\alpha }\sigma _{ \pm }^{\gamma }.$$

(104)

Equation (104) means that under the spinor representation of the Lorentz group, $\sigma _{ \pm }^{\alpha }$ transform the same way as vectors. In (104), the matrices ${{S}_{{(1/2)}}}$ are local, as well as [101] $\Lambda _{\gamma }^{\alpha }$. The $SL(2,C)$ spatial rotation elements can be considered to belong to the Weyl spinor representation of the Lorentz group. When we consider $SL(2,C)$ elements associated to proper transformations we have to be careful and therefore we analyze this case in Section 10. Since the group $SL(2,C)$ is homomorphic to the proper homogeneous Lorentz group, they all just represent local proper Lorentz transformations, again see Section 10 for the proper transformations case. It is also possible to define the object ${{\sigma }^{{\dag \alpha \beta }}} = \sigma _{ - }^{\alpha }\sigma _{ + }^{\beta } - \sigma _{ - }^{\beta }\sigma _{ + }^{\alpha }$, analogously.

We will show explicitly an according to matrix definitions in [98–101] the components of the following objects,

$$\begin{gathered} \imath \,\left( {{{\sigma }^{{\alpha \beta }}} + {{\sigma }^{{\dag \alpha \beta }}}} \right) \\ = \left\{ {\begin{array}{*{20}{l}} {0\,\,{\kern 1pt} {\text{if}}\,\,\,\,\alpha = 0\,\,\,{\text{and}}\,\,\,\,\beta = i{\kern 1pt} } \\ {4{{\epsilon }^{{ijk}}}{{\sigma }^{k}}\,\,\,{\kern 1pt} {\text{if}}\,\,\,\,\alpha = i\,\,\,{\text{and}}\,\,\,\,\beta = j,{\kern 1pt} } \end{array}} \right. \\ \end{gathered} $$

$${{\sigma }^{{\alpha \beta }}} - {{\sigma }^{{\dag \alpha \beta }}} = \left\{ {\begin{array}{*{20}{l}} { - 4{{\sigma }^{i}}\,\,\,\,{\text{if}}\,\,\,\,\alpha = 0\,\,\,{\text{and}}\,\,\,\,\beta = i{\kern 1pt} } \\ {0\,\,\,{\kern 1pt} {\text{if}}\,\,\,\alpha = i\,\,\,\,{\text{and}}\,\,\,\beta = j.{\kern 1pt} } \end{array}} \right.$$

Once again we remind ourselves that we are not writing either spinor or dyad indices like $\sigma _{ + }^{{\mu \,AA{\kern 1pt} '}}$, or $\sigma _{{ - \,A{\kern 1pt} 'A}}^{\mu }$, for the sake of simplicity. The reader can refer in this regard to [20]. We might then call $\Sigma _{{{\text{ROT}}}}^{{\alpha \beta }} = \imath \,\left( {{{\sigma }^{{\alpha \beta }}} + {{\sigma }^{{\dag \alpha \beta }}}} \right)$, and $\Sigma _{{{\text{BOOST}}}}^{{\alpha \beta }} = ({{\sigma }^{{\alpha \beta }}} - {{\sigma }^{{\dag \alpha \beta }}})$. Therefore, a possible choice for the object ${{\Sigma }^{{\alpha \beta }}}$ could be for instance, ${{\Sigma }^{{\alpha \beta }}} = \Sigma _{{{\text{ROT}}}}^{{\alpha \beta }} + \Sigma _{{{\text{BOOST}}}}^{{\alpha \beta }}$. In this case this object according to the matrix definitions introduced in the references is Hermitian. This a particularly suitable choice when we consider proper Lorentz transformations of the $SU(2)$ tetrad vectors nested within the structure of the gauge vectors ${{X}^{\mu }}$ and ${{Y}^{\mu }}$. However in our particular work we will consider for the analysis in Section 3 ${{\Sigma }^{{\alpha \beta }}} = \Sigma _{{SL2C}}^{{\alpha \beta }} = {{\sigma }^{{\alpha \beta }}}$. Then, we will have gauge vectors which are the addition of a structure plus the Hermitian conjugate like,

$$\begin{gathered} {{X}^{\sigma }} = {{Y}^{\sigma }} = {\text{Tr}}{\kern 1pt} \text{[}\Sigma _{{SL2C}}^{{\alpha \beta }}S_{\alpha }^{{\,\,\rho }}S_{\beta }^{{\,\,\lambda }} * \epsilon _{\rho }^{{\,\,\sigma }} * {{\epsilon }_{{\lambda \tau }}}{{B}^{\tau }}] \\ + \,\,{\text{Tr}}{\kern 1pt} [\Sigma _{{SL2C}}^{{\dag \alpha \beta }}S_{\alpha }^{{\,\,\rho }}S_{\beta }^{{\,\,\lambda }} * \epsilon _{\rho }^{{\,\,\sigma }} * {{\epsilon }_{{\lambda \tau }}}{{B}^{{\dag \tau }}}]. \\ \end{gathered} $$

(105)

We already know that $\Sigma _{{SL2C}}^{{\alpha \beta }} = {{\sigma }^{{\alpha \beta }}}$ and $\Sigma _{{SL2C}}^{{\dag \alpha \beta }} = {{\sigma }^{{\dag \alpha \beta }}}$. On another point to be commented, the gauge covariant derivatives are telling us the following [98–100]. The gauge covariant derivative ${{\mathcal{D}}_{\mu }}\psi = {{\partial }_{\mu }}{{\psi }_{{AM}}} - \Gamma _{{\mu M}}^{{\,Q}}{{\psi }_{{AQ}}}$ $- \imath gA_{{\mu A}}^{{\,Q}}{{\psi }_{{QM}}} + \imath \frac{{g{\kern 1pt} '}}{2}B_{\mu }^{Y}{{\psi }_{{AM}}} =$ ${{\partial }_{\mu }}{{\psi }_{{AM}}} + \frac{1}{2}({{\sigma }^{{\beta \gamma }}})_{M}^{{\,Q}}V_{\beta }^{\nu }{{V}_{{\gamma \nu ;\mu }}}{{\psi }_{{AQ}}} - \imath gA_{{\mu A}}^{Q}{{\psi }_{{QM}}} + \imath \frac{{g{\kern 1pt} '}}{2}B_{\mu }^{Y}{{\psi }_{{AM}}}$, commutes with gauge tranformations, in this case local $U(1)$, $SU(2)$ and $SL(2,C)$ gauge transformations [20, 102–105]. That means, it commutes with LB1 and LB2 transformations, or in other words, the generating vectors of blades one and two at each spacetime point are not distinguished by the derivative when “rotated” inside the blades they generate, because the derivative commutes with the local “rotation” itself which is in turn generated by local gauge transformations. That is the geometrical meaning of the gauge covariant derivative when commuting with either $U(1)$, $SU(2)$ or $SL(2,C)$ local gauge transformations. We must notice that $\Gamma _{{\mu M}}^{{\,Q}}$ are the spinor components of the $SL(2,C)$ potential. It can be written as in expression (94) or in the alternative form $\frac{1}{2}({{\sigma }^{{\beta \gamma }}})_{M}^{{\,Q}}V_{\beta }^{\nu }{{V}_{{\gamma \nu ;\mu }}}$ see [102, 103] (the signature in [102] is – + + +). The form we choose depends on the variables we are using. We can also think the two component $SU(2)$ basis spinor fields $\eta _{p}^{P}$, as right handed. We might analogously implement a formalism for left handed spinor fields [20, 21]. We must also stress that the spinors ${{\psi }_{{AM}}}$ are not necessarily the tensor product of a $SU(2)$ spinor ${{\eta }_{A}}$ and a $SL(2,C)$ spinor ${{\zeta }_{M}}$, that is ${{\psi }_{{AM}}} \ne {{\eta }_{A}}\,{{\zeta }_{M}}$ in general. $SL(2,C) \times SU(2)$ spinors are written in terms of the tensor product basis ${{\eta }_{A}}\,{{\zeta }_{M}}$ see chapter ten in [20].

APPENDIX C

The choice of local gauge vectors ${{X}^{\sigma }} = {{Y}^{\sigma }} = {\text{Tr}}{\kern 1pt} [{{\Sigma }^{{\alpha \beta }}}S_{\alpha }^{{\,\,\rho }}S_{\beta }^{{\,\,\lambda }} * \epsilon _{\rho }^{{\,\,\sigma }} * {{\epsilon }_{{\lambda \tau }}}{{B}^{\tau }}]$ has been made in this work. Nonetheless it is relevant in this appendix to see what happens with the gauge vectors when we proceed to carry out a local $SL(2,C)$ transformation associated either to boosts or spatial rotations like the connection transformation in Eq. (21). Let us remember that the object ${{\Sigma }^{{\alpha \beta }}}$ is built with the objects ${{\sigma }^{\alpha }} = \sigma _{ + }^{\alpha }$ and ${{\bar {\sigma }}^{\alpha }} = \sigma _{ - }^{\alpha }$ defined above in Section 9. The minus and the overbar are different but equivalent notations found in the literature. Local $SU(2)$ gauge transformations ${{S}_{{(1/2)}}}$ associated to local spatial rotations have already been analyzed in [8] and Section 9. Under the $\left( {\frac{1}{2},0} \right)$ and $\left( {0,\frac{1}{2}} \right)$ spinor representations of the Lorentz group they transform as,

$$S_{{(1/2)}}^{{ - 1}}{{\sigma }^{\alpha }}{{S}_{{(1/2)}}} = \Lambda _{{\,\gamma }}^{\alpha }{{\sigma }^{\gamma }},$$

(106)

$$S_{{(1/2)}}^{{ - 1}}{{\bar {\sigma }}^{\alpha }}{{S}_{{(1/2)}}} = \Lambda _{{\,\gamma }}^{\alpha }{{\bar {\sigma }}^{\alpha }}.$$

(107)

Equations (105), (107) mean that under the spinor representation of the Lorentz group, ${{\sigma }^{\alpha }} = \sigma _{ + }^{\alpha }$ and ${{\bar {\sigma }}^{\alpha }} = \sigma _{ - }^{\alpha }$ transform the same way as vectors. In (106), (107), the matrices ${{S}_{{(1/2)}}}$ are local, as well as the Lorentz transformations $\Lambda _{\gamma }^{\alpha }$ (see [106]). The $SU(2)$ elements ${{S}_{{(1/2)}}}$ can be considered to belong to the Weyl spinor representation of the Lorentz group. Since the group $SU(2)$ is homomorphic to $SO(3)$, the Lorentz $\Lambda _{\gamma }^{\alpha }$ just represent local space rotations. For $SL(2,C)$ group elements ${{S}_{{sl}}}$, the analogous Eqs. to (106), (107) are,

$$S_{{sl}}^{{ - 1}}{{\sigma }^{\alpha }}{{(S_{{sl}}^{{ - 1}})}^{\dag }} = \Lambda _{\gamma }^{\alpha }{{\sigma }^{\gamma }},$$

(108)

$$S_{{sl}}^{\dag }{{\bar {\sigma }}^{\alpha }}{{S}_{{sl}}} = \Lambda _{\gamma }^{\alpha }{{\bar {\sigma }}^{\alpha }}.$$

(109)

We follow in general the notation in [107] and since we will not write all these objects properties, we cite fundamentally [108–112]. The proof to Eqs. (108), (109) is a lengthy but straightforward proof. The results we need in Section 3 are the following,

$${{S}_{{sl}}}{{\sigma }^{\alpha }}{{({{S}_{{sl}}})}^{\dag }} = \tilde {\Lambda }_{{\,\gamma }}^{\alpha }{{\sigma }^{\gamma }},$$

(110)

$${{(S_{{sl}}^{{ - 1}})}^{\dag }}{{\bar {\sigma }}^{\alpha }}S_{{sl}}^{{ - 1}} = \tilde {\Lambda }_{{\,\gamma }}^{\alpha }{{\bar {\sigma }}^{\alpha }},$$

(111)

$${{(S_{{sl}}^{{ - 1}})}^{\dag }}{{\sigma }^{\alpha }}S_{{sl}}^{{ - 1}} = \Lambda _{{\,\,\gamma }}^{{(\dag )\alpha }}{{\sigma }^{\gamma }},$$

(112)

$${{S}_{{sl}}}{{\bar {\sigma }}^{\alpha }}S_{{sl}}^{\dag } = \Lambda _{{\,\,\gamma }}^{{(\dag )\alpha }}{{\bar {\sigma }}^{\alpha }}.$$

(113)

For the sake of simplicity we are using the notation, $\Lambda _{{\,\delta }}^{{( - 1){\kern 1pt} \alpha }} = \tilde {\Lambda }_{{\,\delta }}^{\alpha }$ where the local Lorentz transformation $\Lambda _{\gamma }^{{( - 1)\alpha }}$ is the inverse to $\Lambda _{\gamma }^{\alpha }$. The local Lorentz transformation $\Lambda _{{\,\gamma }}^{{(\dag )\alpha }}$ is a different one from $\Lambda _{\gamma }^{\alpha }$.

APPENDIX D

Let us review the geodesics through the origin of the $SU(2)$ $2\pi$ parameter sphere. We will see that they generate a set of tetrad transformations that does not belong to a subgroup of LB1, see Section 6 of [8]. Following the notation in [41] we write the elements in $SU(2)$ as,

$$\begin{gathered} S = {{\sigma }_{o}}\,\cos ({\theta \mathord{\left/ {\vphantom {\theta 2}} \right. \kern-0em} 2}) + \imath {{\sigma }_{j}}\,{{{\hat {\theta }}}^{j}}\sin ({\theta \mathord{\left/ {\vphantom {\theta 2}} \right. \kern-0em} 2}) \\ = \sum \frac{1}{{n!}}{{\left( {\sum\limits_{i = 1}^3 \frac{\imath }{2}{{\sigma }_{i}}{{\theta }^{i}}} \right)}^{n}}, \\ \end{gathered} $$

(114)

where ${{\sigma }_{o}}$ is the identity, ${{\sigma }_{j}}$ for $j = 1 \ldots 3$ are the usual Pauli matrices, and the summation convention is applied for $j = 1 \ldots 3$. We can then proceed to define the function $\theta$ as,

$${{\left( {\sum\limits_{i = 1}^3 \frac{\imath }{2}{{\sigma }_{i}}{{\theta }^{i}}} \right)}^{2}} = - {{\sigma }_{o}}{{\left( {\theta 2} \right)}^{2}},$$

(115)

$$\begin{gathered} {{\theta }^{2}} = \sum\limits_{i = 1}^3 {{({{\theta }^{i}})}^{2}},\,\,\,\,{\text{where}}\,\,\,\,{{{\hat {\theta }}}^{i}}\,\,\,{\text{is}}\,\,{\text{given}}\,\,{\text{by}},{\kern 1pt} \\ {{{\hat {\theta }}}^{i}} = {{{{\theta }^{i}}} \mathord{\left/ {\vphantom {{{{\theta }^{i}}} {\left| \theta \right|}}} \right. \kern-0em} {\left| \theta \right|}}. \\ \end{gathered} $$

(116)

Let us consider the $SU(2)$ $2\pi$ parameter sphere [41] in $\theta$, and let us evaluate $S$, and ${{\partial }_{\lambda }}S$ at $\theta = 0$. $S$ is just ${{\sigma }_{o}}$ at $\theta = 0$. The derivative ${{\partial }_{\lambda }}S$ can be written as,

$$\begin{gathered} {{\left. {{{\partial }_{\lambda }}S} \right|}_{{\theta = 0}}} = \left[ {({{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-0em} 2})\,{{\sigma }_{o}}\sin ({\theta \mathord{\left/ {\vphantom {\theta 2}} \right. \kern-0em} 2})\,{{\partial }_{\lambda }}\theta } \right. \\ {{\left. {\left. { + \,\,({\imath \mathord{\left/ {\vphantom {\imath 2}} \right. \kern-0em} 2}){{\sigma }_{j}}{{{\hat {\theta }}}^{j}}\cos ({\theta \mathord{\left/ {\vphantom {\theta 2}} \right. \kern-0em} 2}){{\partial }_{\lambda }}\theta + \imath {{\sigma }_{j}}{{\partial }_{\lambda }}{{{\hat {\theta }}}^{j}}\sin ({\theta \mathord{\left/ {\vphantom {\theta 2}} \right. \kern-0em} 2})} \right]} \right|}_{{\theta = 0}}}. \\ \end{gathered} $$

(117)

If we consider all possible geodesics through the $2\pi$ parameter sphere origin, then we can readily conclude that the four components of ${{\left. {{{\partial }_{\lambda }}\theta } \right|}_{{\theta = 0}}}$ can take on any value, ranging from $- \infty$ to $+ \infty$. Then, accordingly, the vector components of ${\text{Tr}}[\tilde {\Lambda }_{{\,\delta }}^{\alpha }\tilde {\Lambda }_{{\,\gamma }}^{\beta }{{\Sigma }^{{\delta \gamma }}}E_{\alpha }^{{\,\rho }}E_{\beta }^{{\,\lambda }} * {{\xi }_{{\rho \sigma }}} * {{\xi }_{{\lambda \tau }}}{{\partial }^{\tau }}(S){{S}^{{ - 1}}}]$, can take on any values ranging again, from $- \infty$ to $+ \infty$. Borrowing once more the notation and line of thinking from [1], and establishing a parallel with Section 3 of the present manuscript specially the section Gauge geometry, we can see that in correspondence to the scalars we named $C$ and $D$ in [1] we get the scalars $C{\kern 1pt} '$ and $D{\kern 1pt} '$. Once more this is the notation of [8] and similar notation in Section 3 of this manuscript.

Since the vector components of ${\text{Tr}}[\tilde {\Lambda }_{\delta }^{\alpha }\tilde {\Lambda }_{\gamma }^{\beta }{{\Sigma }^{{\delta \gamma }}}E_{\alpha }^{\rho }E_{\beta }^{\lambda } * {{\xi }_{{\rho \sigma }}} * {{\xi }_{{\lambda \tau }}}{{\partial }^{\tau }}(S){{S}^{{ - 1}}}]$, can take on any values, positive or negative, then we conclude that $1 + C{\kern 1pt} '$ and $D{\kern 1pt} '$ can take on any possible real values. Borrowing again the ideas from [1], we can study the case where $1 + C{\kern 1pt} ' > D{\kern 1pt} ' > 0$, and $0 > C{\kern 1pt} ' > - 1$. In addition let us suppose that ${{\partial }_{\rho }}\theta$, ${{\partial }_{\rho }}{{\hat {\theta }}^{i}}$ and ${{\hat {\theta }}^{i}}$ have finite components at the origin. We can always consider the geodesic through the origin of the $2\pi$ sphere, such that $\theta _{n}^{i} = n{{\theta }^{i}}$, where $n$ is a natural number. Now, ${{\theta }_{n}} = n\,\theta$ and $\hat {\theta }_{n}^{i} = {{\hat {\theta }}^{i}}$, but ${{\partial }_{\rho }}{{\theta }_{n}} = n\,{{\partial }_{\rho }}\theta$. Then, at the origin of the parameter sphere, ${{\theta }^{i}} = 0$, ${{\theta }_{n}} = 0$ and ${{\left. {{{\partial }_{\rho }}{{\theta }_{n}}} \right|}_{{\theta = 0}}} = {{\left. {n{{\partial }_{\rho }}\theta } \right|}_{{\theta = 0}}}$. Putting all this elements of analysis together we have that for the new geodesic, for $n$ sufficiently large, $D_{n}^{'} > 0 > 1 + C_{n}^{'}$ meaning that by changing the parameter $n$ we can “leak” or “transit” from one kind of transformation into another kind of transformation. Similar line of thinking for the other cases. This implies one important conclusion. The $SU(2)$ group of local gauge transformations, generates proper and improper LB1 transformations. Therefore the image of $SU(2)$ is not associated to a subgroup of LB1 (or tensor products of LB1).

We establish an absolute parallel of thinking for $SL(2,C)$ when we consider for instance the particular ${{S}_{{sl}}}$ gauge transformation given by the components $S_{{sl(1)}}^{{\,(1)}} = 1 + \imath \,\delta$, $S_{{sl(2)}}^{{\,(2)}} = 1 - \imath \,\delta$, $S_{{sl(1)}}^{{\,(2)}} = S_{{sl(2)}}^{{\,\,(1)}} = \delta$, where $\delta$ is a local scalar. For this particular gauge transformation, the derivatives are given by ${{\partial }_{\rho }}{{S}_{{sl}}} = {{\partial }_{\rho }}\left( {\delta \,(\imath \,{{\sigma }^{3}} + {{\sigma }^{1}})} \right) = {{\partial }_{\rho }}\delta \,(\imath \,{{\sigma }^{3}} + {{\sigma }^{1}})$, where ${{\sigma }^{1}}$, ${{\sigma }^{2}},{{\sigma }^{3}}$ are the standard Pauli matrices. With all these elements, we can pursue an exact replica of the arguments offered in Section 5 of [8] and reproduced above to prove the surjectivity, in essence, using the element ${{S}_{{sl}}}$ and geodesics through the origin.

APPENDIX E

Let us introduce for the purpose of illustration the electromagnetic Abelian tetrad,

$${{U}^{\alpha }} = {{\xi }^{{\alpha \lambda }}}{{\xi }_{{\rho \lambda }}}{{{{A}^{\rho }}} \mathord{\left/ {\vphantom {{{{A}^{\rho }}} {(\sqrt {{{ - Q} \mathord{\left/ {\vphantom {{ - Q} 2}} \right. \kern-0em} 2}} \sqrt {{{A}_{\mu }}{{\xi }^{{\mu \sigma }}}{{\xi }_{{\nu \sigma }}}{{A}^{\nu }}} )}}} \right. \kern-0em} {(\sqrt {{{ - Q} \mathord{\left/ {\vphantom {{ - Q} 2}} \right. \kern-0em} 2}} \sqrt {{{A}_{\mu }}{{\xi }^{{\mu \sigma }}}{{\xi }_{{\nu \sigma }}}{{A}^{\nu }}} )}},$$

(118)

$${{V}^{\alpha }} = {{{{\xi }^{{\alpha \lambda }}}{{A}_{\lambda }}} \mathord{\left/ {\vphantom {{{{\xi }^{{\alpha \lambda }}}{{A}_{\lambda }}} {(\sqrt {{{A}_{\mu }}{{\xi }^{{\mu \sigma }}}{{\xi }_{{\nu \sigma }}}{{A}^{\nu }}} )}}} \right. \kern-0em} {(\sqrt {{{A}_{\mu }}{{\xi }^{{\mu \sigma }}}{{\xi }_{{\nu \sigma }}}{{A}^{\nu }}} )}},$$

(119)

$${{Z}^{\alpha }} = * {{\xi }^{{\alpha \lambda }}} * {{{{A}_{\lambda }}} \mathord{\left/ {\vphantom {{{{A}_{\lambda }}} {(\sqrt { * {{A}_{\mu }} * {{\xi }^{{\mu \sigma }}} * {{\xi }_{{\nu \sigma }}} * {{A}^{\nu }}} )}}} \right. \kern-0em} {(\sqrt { * {{A}_{\mu }} * {{\xi }^{{\mu \sigma }}} * {{\xi }_{{\nu \sigma }}} * {{A}^{\nu }}} )}},$$

(120)

$$\begin{gathered} {{W}^{\alpha }} = * {{\xi }^{{\alpha \lambda }}} * {{\xi }_{{\rho \lambda }}} * {{{{A}^{\rho }}} \mathord{\left/ {\vphantom {{{{A}^{\rho }}} {(\sqrt {{{ - Q} \mathord{\left/ {\vphantom {{ - Q} 2}} \right. \kern-0em} 2}} }}} \right. \kern-0em} {(\sqrt {{{ - Q} \mathord{\left/ {\vphantom {{ - Q} 2}} \right. \kern-0em} 2}} }} \\ \times \,\,\sqrt { * {{A}_{\mu }} * {{\xi }^{{\mu \sigma }}} * {{\xi }_{{\nu \sigma }}} * {{A}^{\nu }}} ). \\ \end{gathered} $$

(121)

The tensor ${{\xi }_{{\mu \nu }}}$ is the extremal electromagnetic field and is obtained through a local duality transformation of the geometrized electromagnetic field ξ_μν = $\cos (\alpha ){{f}_{{\mu \nu }}} - \sin (\alpha ) * {{f}_{{\mu \nu }}}$, where ${{f}_{{\mu \nu }}} = ({{{{G}^{{1/2}}}} \mathord{\left/ {\vphantom {{{{G}^{{1/2}}}} {{{c}^{2}}}}} \right. \kern-0em} {{{c}^{2}}}}){{F}_{{\mu \nu }}}$ is the geometrized electromagnetic field, see [1–7] for the details. The transformation generated in the electromagnetic potential ${{A}_{\alpha }}$ where ${{f}_{{\mu \nu }}} = {{A}_{{\nu ;\mu }}} - {{A}_{{\mu ;\nu }}}$ by ${{X}_{\alpha }} = {{A}_{\alpha }} \to {{X}_{\alpha }} = {{A}_{\alpha }} + {{\Lambda }_{{,\alpha }}}$ of the two tetrad vectors $({{U}^{\alpha }},\,{{V}^{\alpha }})$ on blade one, given in (118), (119), by the “hyperbolic angle” $\phi$, can be expressed as,

$$U_{{(\phi )}}^{\alpha } = \cosh (\phi )\,{{U}^{\alpha }} + \sinh (\phi )\,{{V}^{\alpha }},$$

(122)

$$V_{{(\phi )}}^{\alpha } = \sinh (\phi )\,{{U}^{\alpha }} + \cosh (\phi )\,{{V}^{\alpha }}\;.$$

(123)

The transformation generated in the second electromagnetic potential $* {{A}_{\alpha }}$ where $* {{f}_{{\mu \nu }}} = * {{A}_{{\nu ;\mu }}} - * {{A}_{{\mu ;\nu }}}$ by ${{Y}_{\alpha }} = * {{A}_{\alpha }} \to {{Y}_{\alpha }} = * {{A}_{\alpha }} + * {{\Lambda }_{{,\alpha }}}$ of the two tetrad vectors $({{Z}^{\alpha }},\,{{W}^{\alpha }})$ on blade two, given in (120), (121), by the “spatial angle” $\varphi$, can be expressed as,

$$Z_{{(\varphi )}}^{\alpha } = \cos (\varphi )\,{{Z}^{\alpha }} - \sin (\varphi )\,{{W}^{\alpha }},$$

(124)

$$W_{{(\varphi )}}^{\alpha } = \sin (\varphi )\,{{Z}^{\alpha }} + \cos (\varphi )\,{{W}^{\alpha }}\;.$$

(125)

The $*$ in $* {{A}_{\nu }}$ is just a name, not the Hodge map meaning that $* {{A}_{{\mu ;\nu }}} = {{( * {{A}_{\mu }})}_{{;\nu }}}$. We must stress that the local transformations (122), (123) are not local boosts imposed on the vectors that define the local plane one. They are the result of genuine local gauge transformations of the vectors (${{U}^{\alpha }},{{V}^{\alpha }}$). Once we make the choice ${{X}^{\alpha }} = {{A}^{\alpha }}$ and ${{Y}^{\alpha }} = * {{A}^{\alpha }}$ the question about the geometrical implications of electromagnetic gauge transformations of the tetrad vectors (118)–(121) arises. We first notice that a local electromagnetic gauge transformation of the “gauge vectors” ${{X}^{\alpha }} = {{A}^{\alpha }}$ and ${{Y}^{\alpha }} = * {{A}^{\alpha }}$ can be just interpreted as a new choice for the gauge vectors ${{X}_{\alpha }} = {{A}_{\alpha }} + {{\Lambda }_{{,\alpha }}}$ and ${{Y}_{\alpha }} = * {{A}_{\alpha }} + * {{\Lambda }_{{,\alpha }}}$. It is valid to ask how the tetrad vectors (118), (119) will transform under ${{A}_{\alpha }} \to {{A}_{\alpha }} + {{\Lambda }_{{,\alpha }}}$ and (120), (121) under $* {{A}_{\alpha }} \to * {{A}_{\alpha }} + * {{\Lambda }_{{,\alpha }}}$. For example, from [1] one case on blade one could be a boost as the result of a gauge transformation,

$$\begin{gathered} \frac{{\tilde {V}_{{(1)}}^{\alpha }}}{{\sqrt {\tilde {V}_{{(1)}}^{\beta }{{{\tilde {V}}}_{{(1)\beta }}}} }} = \frac{{(1 + C)}}{{\sqrt {{{{(1 + C)}}^{2}} - {{D}^{2}}} }}\frac{{V_{{(1)}}^{\alpha }}}{{\sqrt { - V_{{(1)}}^{\beta }\,{{V}_{{(1)\beta }}}} }} \\ + \,\,\frac{D}{{\sqrt {{{{(1 + C)}}^{2}} - {{D}^{2}}} }}\frac{{V_{{(2)}}^{\alpha }}}{{\sqrt {V_{{(2)}}^{\beta }\,{{V}_{{(2)\beta }}}} }}, \\ \end{gathered} $$

(126)

$$\begin{gathered} \frac{{\tilde {V}_{{(2)}}^{\alpha }}}{{\sqrt {\tilde {V}_{{(2)}}^{\beta }{{{\tilde {V}}}_{{(2)\beta }}}} }} = \frac{D}{{\sqrt {{{{(1 + C)}}^{2}} - {{D}^{2}}} }}\frac{{V_{{(1)}}^{\alpha }}}{{\sqrt { - V_{{(1)}}^{\beta }\,{{V}_{{(1)\beta }}}} }} \\ + \,\,\frac{{(1 + C)}}{{\sqrt {{{{(1 + C)}}^{2}} - {{D}^{2}}} }}\frac{{V_{{(2)}}^{\alpha }}}{{\sqrt {V_{{(2)}}^{\beta }\,{{V}_{{(2)\beta }}}} }}. \\ \end{gathered} $$

(127)

In Eqs. (126), (127) the following notation has been used, $C = ({{ - Q} \mathord{\left/ {\vphantom {{ - Q} 2}} \right. \kern-0em} 2}){{V}_{{(1)\sigma }}}{{{{\Lambda }^{\sigma }}} \mathord{\left/ {\vphantom {{{{\Lambda }^{\sigma }}} {({{V}_{{(2)\beta }}}V_{{(2)}}^{\beta })}}} \right. \kern-0em} {({{V}_{{(2)\beta }}}V_{{(2)}}^{\beta })}}$, D = $({{ - Q} \mathord{\left/ {\vphantom {{ - Q} 2}} \right. \kern-0em} 2}){{V}_{{(2)\sigma }}}{{{{\Lambda }^{\sigma }}} \mathord{\left/ {\vphantom {{{{\Lambda }^{\sigma }}} {({{V}_{{(1)\beta }}}V_{{(1)}}^{\beta })}}} \right. \kern-0em} {({{V}_{{(1)\beta }}}V_{{(1)}}^{\beta })}}$ and $[{{(1 + C)}^{2}} - {{D}^{2}}] > 0$ must be satisfied. ${{U}^{\alpha }} = \frac{{V_{{(1)}}^{\alpha }}}{{\sqrt { - V_{{(1)}}^{\beta }\,{{V}_{{(1)\beta }}}} }}$ and ${{V}^{\alpha }} = \frac{{V_{{(2)}}^{\alpha }}}{{\sqrt {V_{{(2)}}^{\beta }\,{{V}_{{(2)\beta }}}} }}$. We would like to calculate the norm of the transformed vectors $\tilde {V}_{{(1)}}^{\alpha }$ and $\tilde {V}_{{(2)}}^{\alpha }$ as well,

$$\tilde {V}_{{(1)}}^{\alpha }\,{{\tilde {V}}_{{(1)\alpha }}} = [{{(1 + C)}^{2}} - {{D}^{2}}]\,V_{{(1)}}^{\alpha }\,{{V}_{{(1)\alpha }}},$$

(128)

$$\tilde {V}_{{(2)}}^{\alpha }\,{{\tilde {V}}_{{(2)\alpha }}} = [{{(1 + C)}^{2}} - {{D}^{2}}]\,V_{{(2)}}^{\alpha }\,{{V}_{{(2)\alpha }}},$$

(129)

where the relation $V_{{(1)}}^{\alpha }\,{{V}_{{(1)\alpha }}} = - V_{{(2)}}^{\alpha }\,{{V}_{{(2)\alpha }}}$ has been used. According to the notation used in [1],

$$V_{{(1)}}^{\alpha } = {{\xi }^{{\alpha \lambda }}}\,{{\xi }_{{\rho \lambda }}}\,{{A}^{\rho }},$$

(130)

$$V_{{(2)}}^{\alpha } = \sqrt {{{ - Q} \mathord{\left/ {\vphantom {{ - Q} 2}} \right. \kern-0em} 2}} \,{{\xi }^{{\alpha \lambda }}}\,{{A}_{\lambda }},$$

(131)

$$V_{{(3)}}^{\alpha } = \sqrt {{{ - Q} \mathord{\left/ {\vphantom {{ - Q} 2}} \right. \kern-0em} 2}} * {{\xi }^{{\alpha \lambda }}} * {{A}_{\lambda }},$$

(132)

$$V_{{(4)}}^{\alpha } = * {{\xi }^{{\alpha \lambda }}} * {{\xi }_{{\rho \lambda }}} * {{A}^{\rho }}.$$

(133)

For the particular case when $1 + C > 0$, the transformations (126), (127) report that an electromagnetic gauge transformation on the vector field ${{A}^{\alpha }} \to {{A}^{\alpha }} + {{\Lambda }^{\alpha }}$, that leaves invariant the electromagnetic field ${{f}_{{\mu \nu }}}$, generates a boost transformation on the normalized tetrad vector fields $\left( {\frac{{V_{{(1)}}^{\alpha }}}{{\sqrt { - V_{{(1)}}^{\beta }\,{{V}_{{(1)\beta }}}} }},\frac{{V_{{(2)}}^{\alpha }}}{{\sqrt {V_{{(2)}}^{\beta }\,{{V}_{{(2)\beta }}}} }}} \right)$. The notation ${{\Lambda }^{\alpha }}$ has been used for ${{\Lambda }^{{,\alpha }}}$ where $\Lambda$ is a local scalar. In this case $\cosh (\phi ) = (1 + C)\sqrt {{{{(1 + C)}}^{2}} - {{D}^{2}}}$. This was just one of the possible cases in LB1. Similar analysis for the vector transformations (124), (125) in the local plane two generated by (${{Z}^{\alpha }},{{W}^{\alpha }}$). See [1] for the detailed analysis of all possible cases. It is simple to see that the equalities $U_{{(\phi )}}^{{[\alpha }}\,V_{{(\phi )}}^{{\beta ]}} = {{U}^{{[\alpha }}}\,{{V}^{{\beta ]}}}$ and $Z_{{(\varphi )}}^{{[\alpha }}\,W_{{(\varphi )}}^{{\beta ]}} = {{Z}^{{[\alpha }}}\,{{W}^{{\beta ]}}}$ hold. These equalities report to us that these antisymmetric tetrad objects are gauge invariant. We remind ourselves that it was proved in [1–6] that the group of local electromagnetic gauge transformations is isomorphic to the group LB1 of boosts plus discrete transformations on blade one, and independently to LB2, the group of spatial rotations on blade two. Equations (122), (123) represent a local electromagnetic gauge transformation of the vectors $({{U}^{\alpha }},{{V}^{\alpha }})$. Equations (124), (125) represent a local electromagnetic gauge transformation of the vectors $({{Z}^{\alpha }},{{W}^{\alpha }})$. If the case is that $[{{(1 + C)}^{2}} - {{D}^{2}}] < 0$, the vectors $V_{{(1)}}^{\alpha }$ and $V_{{(2)}}^{\alpha }$ will change their timelike or spacelike character,

$$\tilde {V}_{{(1)}}^{\alpha }\,{{\tilde {V}}_{{(1)\alpha }}} = [ - {{(1 + C)}^{2}} + {{D}^{2}}]\,( - V_{{(1)}}^{\alpha }\,{{V}_{{(1)\alpha }}}),$$

(134)

$$( - \tilde {V}_{{(2)}}^{\alpha }\,{{\tilde {V}}_{{(2)\alpha }}}) = [ - {{(1 + C)}^{2}} + {{D}^{2}}]\,V_{{(2)}}^{\alpha }\,{{V}_{{(2)\alpha }}}.$$

(135)

These are improper transformations on blade one. They have the property of being a composition of boosts and a discrete transformation given by $\Lambda _{0}^{0} = 0$, $\Lambda _{1}^{0} = 1$, $\Lambda _{0}^{1} = 1$, $\Lambda _{1}^{1} = 0$. We notice that this discrete transformation is not a Lorentz transformation because it is a reflection. They might also be composed with a full inversion, see reference [1] for the whole analysis. On blade or plane two, the choice ${{Y}_{\alpha }} = * {{A}_{\alpha }} + * {{\Lambda }_{{,\alpha }}}$ induces just local spatial rotation tetrad vector transformations,

$$\begin{gathered} \frac{{\tilde {V}_{{(3)}}^{\alpha }}}{{\sqrt {\tilde {V}_{{(3)}}^{\beta }\,{{{\tilde {V}}}_{{(3)\beta }}}} }} = \frac{{(1 + N)}}{{\sqrt {{{{(1 + N)}}^{2}} + {{M}^{2}}} }}\frac{{V_{{(3)}}^{\alpha }}}{{\sqrt {V_{{(3)}}^{\beta }\,{{V}_{{(3)\beta }}}} }} \\ - \,\,\frac{M}{{\sqrt {{{{(1 + N)}}^{2}} + {{M}^{2}}} }}\frac{{V_{{(4)}}^{\alpha }}}{{\sqrt {V_{{(4)}}^{\beta }\,{{V}_{{(4)\beta }}}} }}, \\ \end{gathered} $$

(136)

$$\begin{gathered} \frac{{\tilde {V}_{{(4)}}^{\alpha }}}{{\sqrt {\tilde {V}_{{(4)}}^{\beta }\,{{{\tilde {V}}}_{{(4)\beta }}}} }} = \frac{M}{{\sqrt {{{{(1 + N)}}^{2}} + {{M}^{2}}} }}\,\frac{{V_{{(3)}}^{\alpha }}}{{\sqrt {V_{{(3)}}^{\beta }\,{{V}_{{(3)\beta }}}} }} \\ + \,\,\frac{{(1 + N)}}{{\sqrt {{{{(1 + N)}}^{2}} + {{M}^{2}}} }}\,\frac{{V_{{(4)}}^{\alpha }}}{{\sqrt {V_{{(4)}}^{\beta }\,{{V}_{{(4)\beta }}}} }}, \\ \end{gathered} $$

(137)

where,

$$\tilde {V}_{{(3)}}^{\alpha }\,{{\tilde {V}}_{{(3)\alpha }}} = [{{(1 + N)}^{2}} + {{M}^{2}}]\,V_{{(3)}}^{\alpha }\,{{V}_{{(3)\alpha }}},$$

(138)

$$\tilde {V}_{{(4)}}^{\alpha }\,{{\tilde {V}}_{{(4)\alpha }}} = [{{(1 + N)}^{2}} + {{M}^{2}}]\,V_{{(4)}}^{\alpha }\,{{V}_{{(4)\alpha }}}\;,$$

(139)

and where the relation $V_{{(3)}}^{\alpha }\,{{V}_{{(3)\alpha }}}V_{{(4)}}^{\alpha }\,{{V}_{{(4)\alpha }}}$ has been used with the following notation,

$$M = ({{ - Q} \mathord{\left/ {\vphantom {{ - Q} 2}} \right. \kern-0em} 2}){{V}_{{(3)\sigma }}} * {{{{\Lambda }^{\sigma }}} \mathord{\left/ {\vphantom {{{{\Lambda }^{\sigma }}} {({{V}_{{(4)\beta }}}V_{{(4)}}^{\beta })}}} \right. \kern-0em} {({{V}_{{(4)\beta }}}V_{{(4)}}^{\beta })}},$$

(140)

$$N = ({{ - Q} \mathord{\left/ {\vphantom {{ - Q} 2}} \right. \kern-0em} 2}){{V}_{{(4)\sigma }}} * {{{{\Lambda }^{\sigma }}} \mathord{\left/ {\vphantom {{{{\Lambda }^{\sigma }}} {({{V}_{{(3)\beta }}}V_{{(3)}}^{\beta })}}} \right. \kern-0em} {({{V}_{{(3)\beta }}}V_{{(3)}}^{\beta })}}.$$

(141)

As long as $[{{(1 + N)}^{2}} + {{M}^{2}}] > 0$ the transformations (136), (137) are telling us that an electromagnetic gauge transformation on the vector field $* {{A}^{\alpha }}$ that leaves invariant the dual electromagnetic field $* {{f}_{{\mu \nu }}}$, generates a spatial rotation on the normalized tetrad vector fields $\left( {\frac{{V_{{(3)}}^{\alpha }}}{{\sqrt {V_{{(3)}}^{\beta }\,{{V}_{{(3)\beta }}}} }},\frac{{V_{{(4)}}^{\alpha }}}{{\sqrt {V_{{(4)}}^{\beta }\,{{V}_{{(4)\beta }}}} }}} \right)$. We reiterate that local tetrad electromagnetic gauge transformations can be interpreted as new or different gauge choices ${{X}_{\alpha }} = {{A}_{\alpha }} + {{\Lambda }_{{,\alpha }}}$ and ${{Y}_{\alpha }} = * {{A}_{\alpha }} + * {{\Lambda }_{{,\alpha }}}$.