---
abstract: 'A new tetrad introduced within the framework of geometrodynamics for non-null electromagnetic fields allows for the geometrical analysis of the Lorentz equation and its solutions. This tetrad, through the use of the Frenet-Serret formulae and Fermi-Walker transport, exhibits explicitly the set of solutions to the Lorentz equation in a curved spacetime.'
address: '1. Instituto de Física, Facultad de Ciencias, Iguá 4225, esq. Mataojo, Montevideo, Uruguay.'
author:
- 'Alcides Garat$^{1}$'
date: 'August 15th, 2006'
title: The Lorentz equation in geometrodynamics
---

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Introduction
============

The new tetrads introduced in [@A], yield maximum simplification in the expression of a non-null electromagnetic field in a curved spacetime, for instance. It is through this simplified expression of the electromagnetic field that we will get to the geometry associated with the Lorentz equation and its solutions. It is the purpose of this manuscript two-fold. On one hand we will prove that the Lorentz equation has its origin in the Fermi-Walker transport of the electromagnetic tetrads already mentioned. On the other hand we will study different kinds of solutions to the Lorentz equation from a geometric point of view through the use of the electromagnetic tetrads previously introduced. If $F_{\mu\nu}$ is the electromagnetic field and $f_{\mu\nu}= (G^{1/2} / c^2) \: F_{\mu\nu}$ is the geometrized electromagnetic field, then the Einstein-Maxwell equations can be written,

$$\begin{aligned}
f^{\mu\nu}_{\:\:\:\:\:;\nu} &=& 0 \label{EM1}\\
\ast f^{\mu\nu}_{\:\:\:\:\:;\nu} &=& 0 \label{EM2}\\
R_{\mu\nu} &=& f_{\mu\lambda}\:\:f_{\nu}^{\:\:\:\lambda}
+ \ast f_{\mu\lambda}\:\ast f_{\nu}^{\:\:\:\lambda}\ , \label{EM3}\end{aligned}$$

where $\ast f_{\mu\nu}={1 \over 2}\:\epsilon_{\mu\nu\sigma\tau}\:f^{\sigma\tau}$ is the dual tensor of $f_{\mu\nu}$. The symbol $``;''$ stands for covariant derivative with respect to the metric tensor $g_{\mu\nu}$. At every point in spacetime there is a duality rotation by an angle $-\alpha$ that transforms a non-null electromagnetic field into an extremal field,

$$\xi_{\mu\nu} = e^{-\ast \alpha} f_{\mu\nu}\ = \cos(\alpha)\:f_{\mu\nu} - \sin(\alpha)\:\ast f_{\mu\nu}.\label{dref}$$

The local scalar $\alpha$ is known as the complexion of the electromagnetic field. Extremal fields are essentially electric fields and they satisfy,

$$\xi_{\mu\nu} \ast \xi^{\mu\nu}= 0\ . \label{i0}$$

They also satisfy the identity given by equation (64) in [@MW],

$$\begin{aligned}
\xi_{\alpha\mu}\:\ast \xi^{\mu\nu} &=& 0\ .\label{i1}\end{aligned}$$

As antisymmetric fields the extremal fields also verify the identity,

$$\begin{aligned}
\xi_{\mu\alpha}\:\xi^{\nu\alpha} - 
\ast \xi_{\mu\alpha}\: \ast \xi^{\nu\alpha} &=& \frac{1}{2} 
\: \delta_{\mu}^{\:\:\:\nu}\ Q \ ,\label{i2}\end{aligned}$$

where $Q=\xi_{\mu\nu}\:\xi^{\mu\nu}=-\sqrt{T_{\mu\nu}T^{\mu\nu}}$ according to equations (39) in [@MW]. $Q$ is assumed not to be zero, because we are dealing with non-null electromagnetic fields. $T_{\mu\nu} = \xi_{\mu\lambda}\:\:\xi_{\nu}^{\:\:\:\lambda}
+ \ast \xi_{\mu\lambda}\:\ast \xi_{\nu}^{\:\:\:\lambda}$ is the electromagnetic stress-energy tensor [@MW]. In geometrodynamics, the Maxwell equations,

$$\begin{aligned}
f^{\mu\nu}_{\:\:\:\:\:;\nu} &=& 0 \label{L1}\nonumber\\
\ast f^{\mu\nu}_{\:\:\:\:\:;\nu} &=& 0 \ , \label{L2}\end{aligned}$$

are telling us that two potential vector fields exist,

$$\begin{aligned}
f_{\mu\nu} &=& A_{\nu ;\mu} - A_{\mu ;\nu}\label{ER}\nonumber\\
\ast f_{\mu\nu} &=& \ast A_{\nu ;\mu} - \ast A_{\mu ;\nu} \ .\label{DER}\end{aligned}$$

We can define then, a tetrad,

$$\begin{aligned}
U^{\alpha} &=& \xi^{\alpha\lambda}\:\xi_{\rho\lambda}\:A^{\rho} \:
/ \: (\: \sqrt{-Q/2} \: \sqrt{A_{\mu} \ \xi^{\mu\sigma} \ 
\xi_{\nu\sigma} \ A^{\nu}}\:) \label{UO}\\
V^{\alpha} &=& \xi^{\alpha\lambda}\:A_{\lambda} \:
/ \: (\:\sqrt{A_{\mu} \ \xi^{\mu\sigma} \ 
\xi_{\nu\sigma} \ A^{\nu}}\:) \label{VO}\\
Z^{\alpha} &=& \ast \xi^{\alpha\lambda} \: \ast A_{\lambda} \:
/ \: (\:\sqrt{\ast A_{\mu}  \ast \xi^{\mu\sigma}  
\ast \xi_{\nu\sigma}  \ast A^{\nu}}\:)
\label{ZO}\\
W^{\alpha} &=& \ast \xi^{\alpha\lambda}\: \ast \xi_{\rho\lambda}
\:\ast A^{\rho} \: / \: (\:\sqrt{-Q/2} \: \sqrt{\ast A_{\mu} 
\ast \xi^{\mu\sigma} \ast \xi_{\nu\sigma} \ast A^{\nu}}\:) \ .
\label{WO}\end{aligned}$$

The four vectors (\[UO\]-\[WO\]) have the following algebraic properties,

$$-U^{\alpha}\:U_{\alpha}=V^{\alpha}\:V_{\alpha}
=Z^{\alpha}\:Z_{\alpha}=W^{\alpha}\:W_{\alpha}=1 \ .\label{TSPAUX}$$

Using the equations (\[i1\]-\[i2\]) it is simple to prove that (\[UO\]-\[WO\]) are orthogonal. When we make the transformation,

$$\begin{aligned}
A_{\alpha} \rightarrow A_{\alpha} + \Lambda_{,\alpha}\ , \label{G1}\end{aligned}$$

$f_{\mu\nu}$ remains invariant, and the transformation,

$$\begin{aligned}
\ast A_{\alpha} \rightarrow \ast A_{\alpha} + 
\ast \Lambda_{,\alpha}\ , \label{G2}\end{aligned}$$

leaves $\ast f_{\mu\nu}$ invariant, as long as the functions $\Lambda$ and $\ast \Lambda$ are scalars. Schouten defined what he called, a two-bladed structure in a spacetime [@SCH]. These blades are the planes determined by the pairs ($U^{\alpha}, V^{\alpha}$) and ($Z^{\alpha}, W^{\alpha}$). It was proved in [@A] that the transformation (\[G1\]) generates a “rotation” of the tetrad vectors ($U^{\alpha}, V^{\alpha}$) into ($\tilde{U}^{\alpha}, \tilde{V}^{\alpha}$) such that these “rotated” vectors ($\tilde{U}^{\alpha}, \tilde{V}^{\alpha}$) remain in the plane or blade one generated by ($U^{\alpha}, V^{\alpha}$). It was also proved in [@A] that the transformation (\[G2\]) generates a “rotation” of the tetrad vectors ($Z^{\alpha}, W^{\alpha}$) into ($\tilde{Z}^{\alpha}, \tilde{W}^{\alpha}$) such that these “rotated” vectors ($\tilde{Z}^{\alpha}, \tilde{W}^{\alpha}$) remain in the plane or blade two generated by ($Z^{\alpha}, W^{\alpha}$). Once we introduced all these elements found in [@A], we state the purpose of this manuscript. A particle of charge $q$ and mass $m$ in the presence of the electromagnetic field $f_{\mu\nu}$ is going to follow a timelike curve determined by the Lorentz equation [@ROH][@JAW][@DeWB][@DeWCM][@MC1]. It will be proved that the set of all curves that satisfy the Lorentz equation can be generated through Fermi-Walker transport.

Fermi-Walker transport in geometrodynamics
==========================================

The Frenet-Serret formulae for a tetrad associated for instance with a timelike curve $x^{\beta}(s)$ in spacetime [@SY][@HS] are,

$$\begin{aligned}
{\delta A^{\alpha} \over \delta s} &=& b \: B^{\alpha} \label{A}\\
{\delta B^{\alpha} \over \delta s} &=& c \: C^{\alpha} + b \: A^{\alpha} \label{B}\\
{\delta C^{\alpha} \over \delta s} &=& d \: D^{\alpha} - c \: B^{\alpha} \label{V}\label{C}\\
{\delta D^{\alpha} \over \delta s} &=& -d \: C^{\alpha} \label{D} \ ,\end{aligned}$$

where $ A^{\alpha}( x^{\beta}(s)) = {dx^{\alpha} \over ds}$ is the unit timelike tangent vector to the curve. The triad $( B^{\alpha}, C^{\alpha}, D^{\alpha})$ is formed by unit spacelike vector fields. We know that when the local scalar functions $b$, $c$ and $d$ are equal to zero, the curve is a geodesic [@MM]. If $b \neq 0$, then we can define the Fermi-Walker transport of a vector $F^{\alpha}$ as,

$${\delta F^{\alpha} \over \delta s} =  b \: F_{\rho} \: (A^{\alpha} \: B^{\rho} - A^{\rho} \: B^{\alpha}) \ , \label{FWT}\\$$

where $B^{\alpha}$ and $b$ are the first normal and first curvature to the timelike curve $x^{\beta}_{t}(s)$. The “absolute derivative” ${\delta \over \delta s}$ is defined in [@SY]. Analogously, Fermi-Walker transport of a vector $F^{\alpha}$ can be defined along a spacelike curve as [@SY],

$${\delta F^{\alpha} \over \delta s} =  - d \: F_{\rho} \: (C^{\alpha} \: D^{\rho} - C^{\rho} \: D^{\alpha}) \ . \label{FWS}\\$$

Fermi-Walker transport given by (\[FWS\]) is along the spacelike curve $x^{\beta}_{s}(s)$, whose unit spacelike tangent vector is $D^{\alpha}$. In [@A] we defined a new orthonormal tetrad, expressions (\[UO\]-\[WO\]) for spacetimes where non-null electromagnetic fields are present $ (U^{\alpha}, V^{\alpha}, Z^{\alpha}, W^{\alpha})$. Written in terms of these tetrad vectors, the electromagnetic vector field is,

$$f_{\alpha\beta} = -2\:\sqrt{-Q/2}\:\:\cos\alpha\:\:U_{[\alpha}\:V_{\beta]} +
2\:\sqrt{-Q/2}\:\:\sin\alpha\:\:Z_{[\alpha}\:W_{\beta]}\ .\label{EMF}$$

It is precisely the tetrad structure of (\[EMF\]) that allows for a “connection” to the Fermi-Walker transport. It is also the possibility of “rotating” through a scalar angle $\phi$ on blade one, the tetrad vectors $U^{\alpha}$ and $V^{\alpha}$, such that $U_{[\alpha}\:V_{\beta]}$ remains invariant [@A], that we have the freedom to build timelike curve congruences as a function of the “angle” $\phi$ on blade one. Similar on blade two. A “rotation” of the tetrad vectors $Z^{\alpha}$ and $W^{\alpha}$ through an “angle” $\varphi$, such that $Z_{[\alpha}\:W_{\beta]}$ remains invariant [@A], entails the freedom to build spacelike curve congruences as a function of the “angle” $\varphi$ on blade two. We must remember that to each local function $\phi$ there corresponds, through an isomorphic relationship, an electromagnetic gauge transformation local scalar $\Lambda$ as a counterpart to $\phi$ on blade one [@A]. Similar for $\varphi$ on blade two. That is, an electromagnetic local gauge transformation of the vectors $(U^{\alpha}, V^{\alpha})$ generates a “rotation” of these vectors on blade one at every point in spacetime, such that these vectors remain on blade one after the “rotation”. We called the Abelian group of “rotations” on blade one, $LB1$. Similar for $(Z^{\alpha}, W^{\alpha})$ on blade two, under the group $LB2$.

Summarizing the above ideas, we are going to find for each tetrad, solution to the Einstein-Maxwell equations, congruences of timelike curves, solutions to the Lorentz equation.

The Lorentz equation {#Lorentzequation}
====================

We are going to consider a timelike curve in spacetime, such that at each point, the tangent to this curve is given by the unit timelike vector $ u^{\alpha} = u\:U^{\alpha}_{(\phi)} + v\:V^{\alpha}_{(\phi)} + z\:Z^{\alpha}_{(\varphi)} + w\:W^{\alpha}_{(\varphi)}$; $\:\:\:u^{\alpha}\: u_{\alpha}  = -1$. The functions $u$, $v$, $z$ and $w$ are local scalars. The “angles” $(\phi,\varphi)$ are also local scalars and we are introducing them in order to express $u^{\alpha}$ in the most general way. The tetrad vectors $ (U^{\alpha}, V^{\alpha}, Z^{\alpha}, W^{\alpha})$ are provided through (\[UO\]-\[WO\]). Simultaneously, we are going to consider the “rotated” tetrad $(\tilde{U}^{\alpha}, \tilde{V}^{\alpha}, \tilde{Z}^{\alpha}, \tilde{W}^{\alpha})$ defined below through expressions (\[uUO\]-\[uWO\]). For the sake of simplicity we are going to assume that the transformation of the two vectors $(U^{\alpha},\:V^{\alpha})$ on blade one, given in (\[UO\]-\[VO\]) by the “angle” $\phi$ is a proper transformation. For improper transformations the result follows the same lines [@A]. Therefore we can write,

$$\begin{aligned}
U^{\alpha}_{(\phi)}  &=& \cosh(\phi)\: U^{\alpha} +  \sinh(\phi)\: V^{\alpha} \label{UT} \\
V^{\alpha}_{(\phi)} &=& \sinh(\phi)\: U^{\alpha} +  \cosh(\phi)\: V^{\alpha} \label{VT} \ .\end{aligned}$$

The transformation of the two tetrad vectors $(Z^{\alpha},\:W^{\alpha})$ on blade two, given in (\[ZO\]-\[WO\]) by the “angle” $\varphi$, can be expressed as,

$$\begin{aligned}
Z^{\alpha}_{(\varphi)}  &=& \cos(\varphi)\: Z^{\alpha} -  \sin(\varphi)\: W^{\alpha} \label{ZT} \\
W^{\alpha}_{(\varphi)}  &=& \sin(\varphi)\: Z^{\alpha} +  \cos(\varphi)\: W^{\alpha} \label{WT} \ .\end{aligned}$$

We can define then, a “rotated” orthonormal tetrad with respect to the tetrad (\[UO\]-\[WO\]), along the integral timelike curve as,

$$\begin{aligned}
\tilde{U}^{\alpha} &=& \xi^{\alpha\lambda}\:\xi_{\rho\lambda}\:u^{\rho} \:
/ \: (\: \sqrt{-Q/2} \: \sqrt{u_{\mu} \ \xi^{\mu\sigma} \ 
\xi_{\nu\sigma} \ u^{\nu}}\:) \label{uUO}\\
\tilde{V}^{\alpha} &=& \xi^{\alpha\lambda}\:u_{\lambda} \:
/ \: (\:\sqrt{u_{\mu} \ \xi^{\mu\sigma} \ 
\xi_{\nu\sigma} \ u^{\nu}}\:) \label{uVO}\\
\tilde{Z}^{\alpha} &=& \ast \xi^{\alpha\lambda} \:  u_{\lambda} \:
/ \: (\:\sqrt{ u_{\mu}  \ast \xi^{\mu\sigma}  
\ast \xi_{\nu\sigma}  u^{\nu}}\:)
\label{uZO}\\
\tilde{W}^{\alpha} &=& \ast \xi^{\alpha\lambda}\: \ast \xi_{\rho\lambda}
\: u^{\rho} \: / \: (\:\sqrt{-Q/2} \: \sqrt{ u_{\mu} 
\ast \xi^{\mu\sigma} \ast \xi_{\nu\sigma}  u^{\nu}}\:) \ ,
\label{uWO}\end{aligned}$$

where $u^{\alpha}$ is the tangent along the timelike curve. We must be careful about the definition of tetrad vectors (\[uZO\]-\[uWO\]). There is the possibility that the timelike tangent vector $u^{\alpha}$ does not have non-zero components on blade two. We solve this particular problem as an example in the next section. As a comment we can say that if we replace the two tetrad vectors (\[uZO\]-\[uWO\]) for the vectors (\[ZO\]-\[WO\]), we would have an orthonormal tetrad too. We proceed then, to write the Frenet-Serret expressions associated to the timelike curve whose tangent is $u^{\alpha}$.

$$\begin{aligned}
{\delta u^{\alpha} \over \delta s}  &=& A\:\left(u_{\beta}\: \xi^{\alpha\beta}\right) + B\:\left(u_{\beta}\: \ast \xi^{\alpha\beta}\right) + C\: \left( (u_{\beta}\:\tilde{W}^{\beta})\:\tilde{U}^{\alpha} - (u_{\beta}\:\tilde{U}^{\beta})\:\tilde{W}^{\alpha}  \right)
\label{FSE} \ ,\end{aligned}$$

where $ \tilde{U}^{\alpha}$ and $ \tilde{W}^{\beta}$ are provided by expressions (\[uUO\]) and (\[uWO\]). The functions $A$, $B$ and $C$ are local scalars. It is very simple to prove that $u_{\alpha}$ is orthogonal to $u_{\beta}\: \xi^{\alpha\beta}$, $u_{\beta}\: \ast \xi^{\alpha\beta}$, and $(u_{\beta}\:\tilde{W}^{\beta})\:\tilde{U}^{\alpha} - (u_{\beta}\:\tilde{U}^{\beta})\:\tilde{W}^{\alpha}$. It is also very simple to prove using the tetrad vectors (\[uUO\]-\[uWO\]) and the identities (\[i1\]-\[i2\]) that the three vectors $u_{\beta}\: \xi^{\alpha\beta}$, $u_{\beta}\: \ast \xi^{\alpha\beta}$, and $(u_{\beta}\:\tilde{W}^{\beta})\:\tilde{U}^{\alpha} - (u_{\beta}\:\tilde{U}^{\beta})\:\tilde{W}^{\alpha}$ are orthogonal among them. The extremal field and its dual can be written as [@A],

$$\begin{aligned}
\xi_{\alpha\beta} &=& -2\:\sqrt{-Q/2}\:U_{[\alpha}\:V_{\beta]}\label{ET}\\
\ast \xi_{\alpha\beta} &=& 2\:\sqrt{-Q/2}\:Z_{[\alpha}\:W_{\beta]}\ .\label{DET}\end{aligned}$$

Because of gauge invariance we also know that [@A],

$$\begin{aligned}
U^{[\alpha}_{(\phi)}\:V^{\beta]}_{(\phi)} = U^{[\alpha}\:V^{\beta]} = \tilde{U}^{[\alpha}\:\tilde{V}^{\beta]} \label{GEQ1}\\
Z^{[\alpha}_{(\varphi)}\:W^{\beta]}_{(\varphi)} = Z^{[\alpha}\:W^{\beta]} = \tilde{Z}^{[\alpha}\:\tilde{W}^{\beta]} \label{GEQ2} \ .\end{aligned}$$

We can consider the curvature associated scalars to be, $A = (q/m)\:\cos(\alpha)$, $B = (q/m)\:\sin(\alpha)$ and $C = 0$ where $q$ is the charge and $m$ is the mass of the particle following the integral timelike curve to which $u^{\alpha}$ is tangent. We would like to emphasize that expression $\left( \tilde{W}^{\beta}\:\tilde{U}^{\alpha} - \tilde{W}^{\alpha}\:\tilde{U}^{\beta} \right)$ is not invariant under gauge transformations of the tetrad vectors. Therefore, we have one further justification for considering the case where the curvature scalar $C = 0$. Then, we can rewrite (\[FSE\]) as,

$$\begin{aligned}
{\delta u^{\alpha} \over \delta s}  &=& -(q/m)\:\sqrt{-Q/2}\:\cos(\alpha) \:u_{\beta}\: \left(U^{\alpha}\:V^{\beta}- U^{\beta}\:V^{\alpha}\right) + (q/m)\:\sqrt{-Q/2}\:\sin(\alpha)\:u_{\beta}\: \left(Z^{\alpha}\:W^{\beta}- Z^{\beta}\:W^{\alpha}\right) \label{FSESIMPALPHA} \ .\end{aligned}$$

Making use of the expression for the electromagnetic field (\[EMF\]) we can write (\[FSESIMPALPHA\]) as,

$$\begin{aligned}
{\delta u^{\alpha} \over \delta s}  = (q/m)\:u_{\beta}\:f^{\alpha\beta}  \label{FSEMF} \ .\end{aligned}$$

A particular case {#anotherview}
=================

We are going to consider a curve in spacetime, such that at each point, the projection of its tangent vector on blade one is given by ${dx_{1}^{\alpha}(s) \over ds}$, and on blade two by ${dx_{2}^{\alpha}(s) \over ds} = 0$. It is our objective to find special congruences of solutions to the Lorentz equation through Fermi-Walker transport, if they exist at all. Then, we are going to proceed as follows. For a “generic” scalar “angle” $\phi_{1}$ on blade one, and a “generic” scalar “angle” $\varphi_{1}$ on blade two, we are going to study the Frenet-Serret formulae associated to the timelike curve on blade one, whose tangent at every point is given by the unit timelike vector $U^{\alpha}_{(\phi_{1})} = {dx_{1}^{\alpha}(s) \over ds}= (u\:U^{\alpha}_{(\phi)} + v\:V^{\alpha}_{(\phi)})/\sqrt{u^2-v^2} $. Therefore, we are going to analyze the Frenet-Serret formulae for the tetrad $(U^{\alpha} _{(\phi_{1})},\:V^{\alpha} _{(\phi_{1})},\:Z^{\alpha} _{(\varphi_{1})},\:W^{\alpha} _{(\varphi_{1})})$. For the sake of simplicity we are going to assume that the transformation of the two vectors $(U^{\alpha},\:V^{\alpha})$ on blade one, given in (\[UTAV\]-\[VTAV\]) by the “angle” $\phi_{1}$ is a proper transformation. For improper transformations the result follows the same lines [@A]. Therefore we can write,

$$\begin{aligned}
U^{\alpha}_{(\phi_{1})}  &=& \cosh(\phi_{1})\: U^{\alpha} +  \sinh(\phi_{1})\: V^{\alpha} \label{UTAV} \\
V^{\alpha}_{(\phi_{1})} &=& \sinh(\phi_{1})\: U^{\alpha} +  \cosh(\phi_{1})\: V^{\alpha} \label{VTAV} \ .\end{aligned}$$

The transformation of the two tetrad vectors $(Z^{\alpha},\:W^{\alpha})$ on blade two, given in (\[ZTAV\]-\[WTAV\]) by the “angle” $\varphi_{1}$, can be expressed as,

$$\begin{aligned}
Z^{\alpha}_{(\varphi_{1})}  &=& \cos(\varphi_{1})\: Z^{\alpha} -\sin(\varphi_{1})\: W^{\alpha} \label{ZTAV} \\
W^{\alpha}_{(\varphi_{1})}  &=& \sin(\varphi_{1})\: Z^{\alpha} +  \cos(\varphi_{1})\: W^{\alpha} \label{WTAV} \ .\end{aligned}$$

We proceed then, to write the Frenet-Serret expressions associated to the timelike curve on blade one $x_{1}^{\alpha}(s)$ whose tangent is $U^{\alpha}_{(\phi_{1})}$,

$$\begin{aligned}
{\delta U^{\alpha}_{(\phi_{1})} \over \delta s}  &=& a_{1} \: V^{\alpha}_{(\phi_{1})} + b_{1}\: Z^{\alpha}_{(\varphi_{1})} + q_{1}\:W^{\alpha}_{(\varphi_{1})}   \label{FSU1}\\
{\delta V^{\alpha}_{(\phi_{1})}  \over \delta s} &=& a_{1} \: U^{\alpha}_{(\phi_{1})}  + p_{1}\: Z^{\alpha}_{(\varphi_{1})} + d_{1}\: W^{\alpha}_{(\varphi_{1})} \label{FSV1}\\
{\delta Z^{\alpha}_{(\varphi_{1})}  \over \delta s} &=& c_{1} \: W^{\alpha}_{(\varphi_{1})}  - p_{1} \: V^{\alpha}_{(\phi_{1})} +  b_{1} \: U^{\alpha}_{(\phi_{1})}   \label{FSZ1}\\
{\delta W^{\alpha}_{(\varphi_{1})}  \over \delta s} &=& -c_{1} \: Z^{\alpha}_{(\varphi_{1})} - d_{1} \: V^{\alpha}_{(\phi_{1})} + q_{1}\:U^{\alpha}_{(\phi_{1})}  \label{FSW1} \ .\end{aligned}$$

The functions $a_{1}$, $b_{1}$, $c_{1}$, $d_{1}$, $p_{1}$ and $q_{1}$ are local scalars. In addition, we are also going to introduce the auxiliary scalar variables, $$\begin{aligned}
A_{z} &=& Z_{\alpha}\:U^{\alpha}_{\:\:\:;\beta}\:U^{\beta} \label{ASz} \\
B_{z} &=& Z_{\alpha}\:U^{\alpha}_{\:\:\:;\beta}\:V^{\beta} \label{BSz} \\
C_{z} &=& Z_{\alpha}\:V^{\alpha}_{\:\:\:;\beta}\:U^{\beta} \label{CSz} \\
D_{z} &=& Z_{\alpha}\:V^{\alpha}_{\:\:\:;\beta}\:V^{\beta} \label{DSz} \\
A_{w} &=& W_{\alpha}\:U^{\alpha}_{\:\:\:;\beta}\:U^{\beta} \label{ASw} \\
B_{w} &=& W_{\alpha}\:U^{\alpha}_{\:\:\:;\beta}\:V^{\beta} \label{BSw} \\
C_{w} &=& W_{\alpha}\:V^{\alpha}_{\:\:\:;\beta}\:U^{\beta} \label{CSw} \\
D_{w} &=& W_{\alpha}\:V^{\alpha}_{\:\:\:;\beta}\:V^{\beta} \label{DSw} \ .\end{aligned}$$

Now comes the core on the idea to find this special solution to the Lorentz equation. Let us demand or impose in (\[FSU1\]) the following equations,

$$\begin{aligned}
Z_{\alpha(\varphi_{1})} \:{\delta U^{\alpha}_{(\phi_{1})} \over \delta s} &=& 0 \label{eq1b1} \\
W_{\alpha(\varphi_{1})} \:{\delta U^{\alpha}_{(\phi_{1})} \over \delta s} &=& 0 \ .\label{eq2b1} \end{aligned}$$

These equations (\[eq1b1\]-\[eq2b1\]) represent equations for the two local scalar functions $(\phi_{1},\varphi_{1})$. If we replace in (\[eq1b1\]-\[eq2b1\]) expressions (\[UTAV\]-\[VTAV\]) and (\[ZTAV\]-\[WTAV\]), work out the algebraic equations, and make use of (\[ASz\]-\[DSw\]) we find,

$$\begin{aligned}
\tan(\varphi_{1}) &=& {\left(\cosh^{2}(\phi_{1})\:A_{z} + \sinh^{2}(\phi_{1})\:D_{z} + \cosh(\phi_{1})\:\sinh(\phi_{1})\:(B_{z} + C_{z})\right) \over \left(\cosh^{2}(\phi_{1})\:A_{w} + \sinh^{2}(\phi_{1})\:D_{w} + \cosh(\phi_{1})\:\sinh(\phi_{1})\:(B_{w} + C_{w})\right)} \label{eqvarphi1b1} \end{aligned}$$

$$\begin{aligned}
\tan(\varphi_{1}) &=& - { \left(\cosh^{2}(\phi_{1})\:A_{w} + \sinh^{2}(\phi_{1})\:D_{w} + \cosh(\phi_{1})\:\sinh(\phi_{1})\:(B_{w} + C_{w})\right) \over \left(\cosh^{2}(\phi_{1})\:A_{z} + \sinh^{2}(\phi_{1})\:D_{z} + \cosh(\phi_{1})\:\sinh(\phi_{1})\:(B_{z} + C_{z})\right) } \ .\label{eqvarphi2b1}\end{aligned}$$

Now, in order for both equations (\[eqvarphi1b1\]-\[eqvarphi2b1\]) to be compatible, it can be readily noticed that the following equality would have to be true, $\tan(\varphi_{1}) = - 1 / \tan(\varphi_{1})$. Then, there are no solutions to the “pure” blade one Lorentz equation problem.

A more general example {#anothergenview}
======================

It is our objective to find special congruences of solutions to the Lorentz equation through Fermi-Walker transport, as a more general example. We are increasing the level of complexity following a pedagogical scheme. Thus, we are going to proceed as follows. For a “generic” scalar “angle” $\phi_{1}$ on blade one, and a “generic” scalar “angle” $\varphi_{1}$ on blade two, we are going to study the Frenet-Serret formulae associated to the timelike curve, whose projection on blade one at every point is given by the timelike vector $(M / \sqrt{M^2 - N^2}\:)\:U^{\alpha}_{(\phi_{1})} = (M / \sqrt{M^2 - N^2}\:)\: (u\:U^{\alpha}_{(\phi)} + v\:V^{\alpha}_{(\phi)})/\sqrt{u^2-v^2} $ and its projection on blade two is given by $(N / \sqrt{M^2 - N^2}\:)\:W^{\alpha}_{(\varphi_{1})} = (N / \sqrt{M^2 - N^2}\:)\: (z\:Z^{\alpha}_{(\varphi)} + w\:W^{\alpha}_{(\varphi)})/\sqrt{z^2+w^2} $. Both $M$ and $N$ are local scalars. We have already analyzed the Frenet-Serret formulae for the tetrad $(U^{\alpha} _{(\phi_{1})},\:V^{\alpha} _{(\phi_{1})},\:Z^{\alpha} _{(\varphi_{1})},\:W^{\alpha} _{(\varphi_{1})})$. We are simply going to take advantage of this previous analysis to find solutions to the Lorentz equation of a more general nature. For brevity we are going to call $A = M / \sqrt{M^2 - N^2}$ and $B = N / \sqrt{M^2 - N^2}$. We proceed then, to write the Frenet-Serret expressions associated to the projections of the curve both on blade one and two,

$$\begin{aligned}
{\delta \left( A\:U^{\alpha}_{(\phi_{1})}\right) \over \delta s}  &=& {\delta \left( A \right) \over \delta s}\: U^{\alpha}_{(\phi_{1})} + A\:{\delta U^{\alpha}_{(\phi_{1})} \over \delta s}\label{GEN1}\\
{\delta U^{\alpha}_{(\phi_{1})} \over \delta s}  &=& a_{1} \: V^{\alpha}_{(\phi_{1})} + b_{1}\: Z^{\alpha}_{(\varphi_{1})} + q_{1}\:W^{\alpha}_{(\varphi_{1})}   \label{GEN11}\\
{\delta V^{\alpha}_{(\phi_{1})}  \over \delta s} &=& a_{1} \: U^{\alpha}_{(\phi_{1})}  + p_{1}\: Z^{\alpha}_{(\varphi_{1})} + d_{1}\: W^{\alpha}_{(\varphi_{1})} \label{GEN2}\\
{\delta Z^{\alpha}_{(\varphi_{1})}  \over \delta s} &=& c_{1} \: W^{\alpha}_{(\varphi_{1})}  - p_{1} \: V^{\alpha}_{(\phi_{1})} +  b_{1} \: U^{\alpha}_{(\phi_{1})}   \label{GEN3}\\
{\delta W^{\alpha}_{(\varphi_{1})}  \over \delta s} &=& -c_{1} \: Z^{\alpha}_{(\varphi_{1})} - d_{1} \: V^{\alpha}_{(\phi_{1})} + q_{1}\:U^{\alpha}_{(\phi_{1})}  \label{GEN44}\\
{\delta \left( B\:W^{\alpha}_{(\varphi_{1})}\right)  \over \delta s} &=& {\delta \left( B \right)  \over \delta s}\: W^{\alpha}_{(\varphi_{1})} + B\:{\delta W^{\alpha}_{(\varphi_{1})}  \over \delta s} \label{GEN4} \ .\end{aligned}$$

The functions $a_{1}$, $b_{1}$, $c_{1}$, $d_{1}$, $p_{1}$ and $q_{1}$ are again local scalars.

Once more comes the core on the idea to find these more general solutions to the Lorentz equation. Let us demand or impose in (\[GEN1\]-\[GEN4\]) the following equations,

$$\begin{aligned}
{\delta \left( A \right) \over \delta s} &=& 0 \label{SCALAR1} \\
{\delta \left( B \right)  \over \delta s} &=& 0 \label{SCALAR2} \\
W_{\alpha(\varphi_{1})} \:{\delta U^{\alpha}_{(\phi_{1})} \over \delta s} &=& 0 \ .\label{ORTHO2} \end{aligned}$$

These equations (\[SCALAR1\]-\[SCALAR2\]) represent equations for the two local scalar functions $(M, N)$. Expression (\[ORTHO2\]) is just one equation for $(\phi_{1},\varphi_{1})$. In order to find one solution for the pair $(\phi_{1},\varphi_{1})$ we need another equation. It can be supplemented by different additional equations, for instance, $Z_{\alpha(\varphi_{1})} \:{\delta V^{\alpha}_{(\phi_{1})} \over \delta s} = 0$. Each possible choice for an additional equation yields different solutions $(\phi_{1},\varphi_{1})$ to the Lorentz equation.

The problem is posed as follows. First, we are supposed to know the tetrad vectors and their covariant derivatives from solutions to the Einstein-Maxwell equations (\[EM1\]-\[EM3\]). Second, we solve equations (\[SCALAR1\],\[SCALAR2\]) for the scalars $(M, N)$. Then we have available equation (\[ORTHO2\]) for the pair $(\phi_{1},\varphi_{1})$. As mentioned above we can supplement this last equation (\[ORTHO2\]) with another one in order to find $(\phi_{1},\varphi_{1})$. We finally see that Frenet-Serret equations (\[GEN11\]) and (\[GEN44\]) turn out to be,

$$\begin{aligned}
{\delta U^{\alpha}_{(\phi_{1})} \over \delta s}  &=& a_{1} \: V^{\alpha}_{(\phi_{1})} + b_{1}\: Z^{\alpha}_{(\varphi_{1})} \label{RED11} \\
{\delta W^{\alpha}_{(\varphi_{1})}  \over \delta s} &=& -c_{1} \: Z^{\alpha}_{(\varphi_{1})} - d_{1} \: V^{\alpha}_{(\phi_{1})}   \label{RED44}
 \ . \end{aligned}$$

Our intention is to Fermi-Walker transport the timelike vector $A\:U^{\alpha}_{(\phi_{1})} + B \: W^{\alpha}_{(\varphi_{1})} $ along the integral timelike curve whose tangent is this same vector that in turn satisfies equations (\[SCALAR1\]-\[SCALAR2\]),

$$\begin{aligned}
{\delta \left( A\: U^{\alpha}_{(\phi_{1})} + B\: W^{\alpha}_{(\varphi_{1})} \right) \over \delta s} &=& A \: {\delta U^{\alpha}_{(\phi_{1})} \over \delta s} + B \: {\delta W^{\alpha}_{(\varphi_{1})}  \over \delta s}   \ . \label{GENEQ1}\end{aligned}$$

Now, using equations (\[RED11\]-\[RED44\]) we can write,

$$\begin{aligned}
{\delta \left( A\: U^{\alpha}_{(\phi_{1})} + B\: W^{\alpha}_{(\varphi_{1})} \right) \over \delta s} &=& A \: \left( a_{1} \: V^{\alpha}_{(\phi_{1})} + b_{1}\: Z^{\alpha}_{(\varphi_{1})} \right) + B \: \left( -c_{1} \: Z^{\alpha}_{(\varphi_{1})} - d_{1} \: V^{\alpha}_{(\phi_{1})} \right)  \ . \label{GENEQ2}\end{aligned}$$

It is simple to see that equation (\[GENEQ2\]) can be rewritten,

$$\begin{aligned}
{\delta \left( A\: U^{\alpha}_{(\phi_{1})} + B\: W^{\alpha}_{(\varphi_{1})} \right) \over \delta s} = (q/m) \: \left( A \: U_{\rho(\phi_{1})} + B\: W_{\rho(\varphi_{1})} \right)    \nonumber \\ \nonumber \\ \left( - \sqrt{-Q/2}\:\:\cos\alpha\:\: \left( U^{\alpha}_{(\phi_{1})}\:V^{\rho}_{(\phi_{1})} - U^{\rho}_{(\phi_{1})}\:V^{\alpha}_{(\phi_{1})} \right) +
\sqrt{-Q/2}\:\:\sin\alpha\:\:\left( Z^{\alpha}_{(\varphi_{1})}
\:W^{\rho}_{(\varphi_{1})} - Z^{\rho}_{(\varphi_{1})}
\:W^{\alpha}_{(\varphi_{1})} \right)\right)\ , \label{GENEQ3}\end{aligned}$$

as long as we ask for the scalar functions $a_{1}$, $b_{1}$, $c_{1}$ and $d_{1}$ to satisfy the following algebraic equations,

$$\begin{aligned}
-\:\sqrt{-Q/2}\:\:\cos\alpha\:A\:q/m &=& A\:a_{1} - B\:d_{1} \label{algeb1}\\
\:\sqrt{-Q/2}\:\:\sin\alpha\:B\:q/m &=& A\:b_{1} - B\:c_{1} \label{algeb2}\ .\end{aligned}$$

We can also remind ourselves of the gauge invariant equalities [@A],

$$\begin{aligned}
U^{[\alpha}_{(\phi_{1})}\:V^{\beta]}_{(\phi_{1})} = U^{[\alpha}\:V^{\beta]} = U^{[\alpha}_{(\phi)}\:V^{\beta]}_{(\phi)}  \label{GEQphi1}\\
Z^{[\alpha}_{(\varphi_{1})}\:W^{\beta]}_{(\varphi_{1})} = Z^{[\alpha}\:W^{\beta]} = Z^{[\alpha}_{(\varphi)}\:W^{\beta]}_{(\varphi)}  \label{GEQvarphi2} \ .\end{aligned}$$

Therefore, equation (\[GENEQ3\]) can be expressed as,

$$\begin{aligned}
{\delta \left( A\: U^{\alpha}_{(\phi_{1})} + B\: W^{\alpha}_{(\varphi_{1})} \right) \over \delta s} &=& \left( A \: U_{\rho(\phi_{1})} + B\: W_{\rho(\varphi_{1})} \right) \: \left( -2\:\sqrt{-Q/2}\:\:\cos\alpha\:\:U^{[\alpha}\:V^{\rho]} +
2\:\sqrt{-Q/2}\:\:\sin\alpha\:\:Z^{[\alpha}\:W^{\rho]}\right)\ . \label{GENEQ4}\end{aligned}$$

Calling $u^{\alpha} = A\: U^{\alpha}_{(\phi_{1})} + B\: W^{\alpha}_{(\varphi_{1})}$, reminding equation (\[EMF\]), and taking into account the orthogonality relations, $U_{\rho}\:Z^{\rho} =  U_{\rho}\:W^{\rho} = W_{\rho}\:V^{\rho} = 0$, we can rewrite (\[GENEQ4\]) as,

$${\delta u^{\alpha} \over \delta s} = (q/m)\: u_{\rho}\: f^{\alpha\rho}\ , \label{Lorentz}\\$$

that is, the Lorentz “force” equation for the particle of mass $m$ and charge $q$, through Fermi-Walker transport of the timelike vector $u^{\alpha}$ along the integral timelike curve whose tangent is $u^{\alpha} = A\: U^{\alpha}_{(\phi_{1})} + B\: W^{\alpha}_{(\varphi_{1})}$.

Conclusions
===========

The introduction of new tetrads for non-null electromagnetic fields in [@A] allows for the geometrical analysis of the Lorentz equation. Through Fermi-Walker transport and Frenet-Serret expressions it is possible to identify curvature scalars in terms of the local scalar functions associated to the non-null electromagnetic field present in the curved spacetime. A blade one “pure” solutions are studied through the use of Frenet-Serret expressions, and more general solutions are found for solutions with non-trivial projections on both blades. The new tetrads allow for a better insight regarding the relationship between traditional Abelian gauge theories on one hand, and traditional Riemannian geometry on the other hand. The key lies upon the two theorems proved in [@A] relating through isomorphisms the groups $U(1)$ and LB1, LB2. The local “internal” transformations that leave invariant the electromagnetic field are proved to be isomorphic to the local tetrad transformations either on blade one or two. In other words, the local “internal” group of transformations is isomorphic to a subgroup of the local group of Lorentz tetrad transformations. This was not known before. It is precisely this “gauge geometry” the tool that allows for a new understanding of a traditional equation like the Lorentz equation. Other future applications are possible. For instance, the study of the kinematics in these spacetimes [@RG][@RW][@MC2][@EJKS], that is, the search for a “connection” to the theory of embeddings, time slicings [@KK][@JDBKK][@FM][@FAEP], the initial value formulation [@JWY1][@JWY2][@NOMJWY][@HY][@JY][@LICH][@YCB][@CMDWYCB], the Cauchy evolution [@ADM][@LSJWY], etc. If there is a geometry such that the gradient of the complexion is a timelike vector field, then there would be the possibility of finding a slicing such that the leaves of constant complexion foliate spacetime, and potentially could be a simplification tool.

Simplicity is the most sought after quality in all these geometrical descriptions, and the previously introduced tetrads, blades, isomorphisms, prove to be practical and useful. We quote from [@MW] “The question poses itself insistently to find a point of view which will make Rainich-Riemannian geometry seem a particularly natural kind of geometry to consider”.

I am grateful to R. Gambini and J. Pullin for reading this manuscript and for many fruitful discussions.

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