# Unification of the Gauge Theories

Abolfazl<EMAIL_ADDRESS>Department of Physics, Faculty of
Science, Shahrekord University, Shahrekord, Iran

###### Abstract

Abstract

We take the Christoffel coefficients as an operator and introduce new mappings
for quaternionic products to reach the theory of electrodynamics in general
spacetime. With the help of the directional operator of the covariant
derivative, we generalize the quaternioic mechanism to the theory of gravity
and show that the Einstein equation has the freedom to choose the constant
term in agreement with the covariant derivative.

###### pacs:

03.67.Mn, 73.23.-b, 74.45.+c, 74.78.Na

Keywords: Quaternions, Theory of gravity, Christoffel connections, Riemann’s
curvature tensor, Ricci tensor

##  I Introduction

After general relativity succeeded in explaining the universe, numerous
attempts were made to develop geometric electrodynamics. Most of these
attempts were unsuccessful overduin ; ruiz ; wes ; cabral ; goenner ; lobo ;
wess ; gross ; ponce ; williams ; wesso . Our goal now is to attempt this
objective in a significantly different way. We want to incorporate the theory
of gravity into the mechanism of electrodynamics. This may help us with a
higher appreciation of quantum gravity. Since the realization of quantum
gravity is universally craved by physicists, it is possible to confirm the
results of this work by separating the components of Riemann’s curvature
tensor. This is because the realization of quantum gravity, as a worthwhile
goal, has forced us to propose a change in spacetime dimensions herbert ;
kiefer ; gianluca ; gianlu ; gianluc ; carlo ; ashteka ; ashtekar . In
tackling the enigma, we face local obstacles from quaternionic production,
such as the contraction of the indices of tangent and cotangent spaces, but
the result is crucially the same regardless of the space chosen. Such new
perspectives will help us change the theory of gravity. We approach two types
of structures, the spacetime metric and the quaternionic metric (quaternionic
production under new metric). With quaternions, the transition from
electrodynamics to gravity is possible, as will be shown. However, there is a
fundamental difference between the two theories. Systematically,
electrodynamics is a theory with two components which can be clarified as the
strength tensor in four-dimensional space. If one extends electrodynamics to
gravity theory using quaternions, there are four components in a four-
dimensional spacetime. Hence, we encounter the concept of component
separation. This change in the structure of the components leads to equations
of motion and finally to Bianchi’s identity weber ; misner ; wald ; weinberg ;
invarno ; malcolm .
The result of this article could be a modified gravitational theory (a
modification of the Einstein equation), which formally assumes additional
conditions gray ; parisi ; faisal ; sang . The additional conditions added to
the Einstein equation are seen strictly from a pure point of view, never
achieved by the concept of geometry. We will derive Einstein’s equation with
additional conditions and prove its necessity. The interpretation of
Einstein’s historical theorem is one of the consequences of the theory that
has the potential to be transformative misner ; wald ; invarno . It is an
achievement to induce the behavior of gauge fields in gravity. The last
theorem is complicated by Riemann’s four-component curvature tensor. Of
course, the theory of gravity has a gauge-theoretic property similar to that
of gauge fields theory. The definition of the directional covariant derivative
operator is an innovation in the present work. We will explain its action in a
subsection, but presently we will use its results only.

Clarification- It is necessary to emphasize some definitions and denotations:
(1) The vector and dual spaces (distinguished by $\varepsilon$ and
$\varepsilon^{\star}$) are ordered, and denoted by
$\varepsilon^{\star}\otimes\varepsilon$; other orders are not allowed.
$u^{\alpha}\ e_{\alpha}\in\varepsilon$ and $u_{\alpha}\
e^{\alpha}\in\varepsilon^{\star}$; $e_{\alpha}$ and $e^{\alpha}$ are also the
tangent and cotangent unit vectors. (2) The notation of
$\mathsf{u}^{\alpha\beta\cdots}_{\mu\nu\cdots}:=u^{\alpha\beta\cdots}_{\mu\nu\cdots}\
\hat{e}^{\alpha}\times\hat{e}^{\beta}\cdots\otimes\hat{e}_{\mu}\times\hat{e}_{\nu}\cdots$
shows the multicomponent member. The notation is changed as follows; the
$\mathsf{u}^{\alpha\beta}_{\mu\nu}$ parts are denoted by
$u^{\alpha\beta}_{\mu\nu}$. (3) The interaction of indices occurs only from a
vector to the dual space, left to right and vice versa, anyhow, from a vector
space into the dual one. (4) Metric similarity transformations have no
influence on the physical conditions.
Establishing- Introduction to ”Ast” Mapping: We define the new mapping
($\mathrm{ast}$ product) based on its result as the only direction-independent
action of the vector on its relevant dual space,

$\ast:\ \varepsilon\vec{\times}\varepsilon^{\star}\sim\mathrm{is\ deffined}$

Also, the definition of the mapping as an action is:

$\ast:\
\varepsilon^{\star}\reflectbox{$\vec{\reflectbox{$\times$}}$}\varepsilon\sim\mathrm{is\
deffined}$

In this way, $\varepsilon^{\star}\vec{\times}\varepsilon$ and
$\varepsilon\reflectbox{$\vec{\reflectbox{$\times$}}$}\varepsilon^{\star}$
lead to invalid results. We can also represent this with $e_{\alpha}\times
e^{\beta}$ and so on. Now we follow the classical relation of the Christoffel
coefficients; we define $B^{\mu}_{\ \nu;\alpha}=B^{\mu}_{\
\nu,\alpha}+\Gamma^{\mu}_{\alpha\beta}B^{\beta}_{\nu}-\Gamma_{\alpha\nu}^{\beta}B^{\mu}_{\beta}$
and the $\mathsf{ast}$ mapping, and we define the directed gamma operator as
follows;

$\hat{\Gamma}\vec{\times}\varepsilon^{\star}\rightarrow\varepsilon^{\star},\ \
\ \
\varepsilon\reflectbox{$\vec{\reflectbox{$\times$}}$}\hat{\Gamma}\rightarrow\varepsilon,$

Here we will not mention the direction of the action. Therefore, operator
$\hat{\Gamma}$ obtains a structure of $\varepsilon^{\star}\times\varepsilon$
and consequently $\hat{\Gamma}\propto\varepsilon^{\star}\times\varepsilon$. We
point out that the operation of $\hat{\Gamma}$ is the contraction. In summary,
$\hat{\Gamma}$ is an operator in the space of indices.
Due to assumption 1, the contraction of the indices occurs up to the allowed
cases; $\Sigma_{\alpha}\
\hat{e}_{\alpha}\times\hat{e}^{\mu}\hat{e}^{\nu}=\Sigma_{\alpha}\
(\delta^{\mu}_{\alpha}\hat{e}^{\nu}+\hat{e}^{\mu}\delta^{\nu}_{\alpha})$. We
assign a Lorentz index of $\hat{\Gamma}_{\mu}$ to the operator (see appendix).
Metric- According to the definition of $\mathsf{ast}$ mapping, the metric is a
mapping of the form;

$\displaystyle\mathsf{g}^{\mathrm{up}}:\varepsilon^{2}\times\varepsilon^{\star}\rightarrow\varepsilon$
(1)

and

$\displaystyle\mathsf{g}_{{}_{\mathrm{down}}}:\varepsilon^{\star}\leftarrow\varepsilon\times\varepsilon^{\star
2}$ (2)

The symbol $"\mathbf{g}"$ is the general metric element and comprises
$\varepsilon\times\varepsilon$ or
$\varepsilon^{\star}\times\varepsilon^{\star}$. Equations 1 and 2 show that
the metric (with a repeated index) does not obey the rules of the operator,
and it is clear that any other interpretation of equations 1 and 2 would be
wrong for the action of the metric. It can be shown that the metric effect
does not play a role in the calculation of the components. Following the
definition of $\mathsf{ast}$ mapping, the operator $\hat{\Gamma}_{\kappa}$ is
represented as follows;

$\displaystyle\hat{\Gamma}_{\kappa}\equiv\Sigma_{\alpha\beta}(\Gamma^{\alpha}_{\beta\kappa}\
e^{\beta}e_{\alpha})^{\dagger}=\Gamma^{\alpha}_{\beta\kappa}\
e^{\alpha}e_{\beta}$ (3)

We have considered the inverted version of $\hat{\Gamma}_{\kappa}$ as the
operator due to the compatibility of the Christoffel connections with the role
of the operator. Furthermore, except for the $"\kappa"$ index in
$\hat{\Gamma}_{\kappa}$, its other two indices are repeated and dummy. Anyway,
according to assumption 2, $\hat{\Gamma}_{\kappa}\equiv\hat{\Gamma}_{\kappa}\
e^{\kappa}$ (without summation), which is the missing part of relation
equation 3. From this follows (a connection coefficient of the first kind):

$\displaystyle\Gamma^{\alpha}_{\beta\kappa}\mathsf{g}_{{}_{\mathrm{down}}}\rightarrow\Gamma^{\mu}_{\beta\kappa}\
g_{\mu\nu}\equiv\Gamma_{\beta\kappa\nu},$

the mapping is, of course, not an operator. Due to the specificity of
contracting indices, we will remove the indicator from the top of the letters.
With the notation of

$\displaystyle e_{\alpha}\rightarrow\ <\alpha|$ $\displaystyle
e^{\alpha}\rightarrow\ |\alpha>$

The above statements and results become easier to understand with the new
representation, because the $\hat{\Gamma}_{\zeta}$ operator yields
$"\hat{\Gamma}_{\kappa}=\Sigma_{\alpha,\beta}\Gamma^{\alpha}_{\beta\kappa}|\alpha><\beta|\equiv\Gamma^{\alpha}_{\beta\kappa}|\alpha><\beta|"$.
The $\mathsf{ast}$ mapping becomes more formal when referring to the ”bra-
ket” representations. Moreover, the representation corresponding to the ”bra-
ket” extends our possibilities for future calculations. The $\alpha$ and
$\beta$ are dummy indices in the
$\Gamma^{\alpha}_{\beta\kappa}|\alpha><\beta|$-operator. They are used, among
other things, for the gamma operator effect;

$\displaystyle\hat{\Gamma}_{\kappa}\ast\mathsf{u}_{\lambda}$
$\displaystyle=\Gamma^{\alpha}_{\beta\kappa}\
|\alpha><\beta|\lambda>u_{\lambda}=\Gamma^{\alpha}_{\beta\kappa}\
\delta_{\beta\lambda}\ u_{{}_{\Box}}|\alpha>$
$\displaystyle=\Gamma^{\alpha}_{\lambda\kappa}u_{\alpha},$ (4)

In this process, the box is mated with the alpha vector. That is, the box
takes the alpha index. Hence,

$\displaystyle\mathsf{u}^{\lambda}\ast\hat{\Gamma}_{\kappa}=u^{\lambda}\Gamma^{\alpha}_{\beta\kappa}\
<\lambda|\alpha><\beta|=\Gamma^{\lambda}_{\beta\kappa}u^{\beta}.$ (5)

We emphasize that in changing the $\mathsf{u}_{\lambda}$ notation, the
position of the index $\lambda$ in $"u_{\lambda}"$ is the main, but the name
of the index changes in ”$\mathsf{ast}$” multiplication. However, we should
use the $\mathsf{ast}$ mapping relations as simply as possible.
Quaternions- The relations that exist between the quaternions’ generators
$\hat{q}_{1}=I$, $\hat{q}_{2}=J$ and $\hat{q}_{3}=K$ hold under a Euclidean
metric and satisfy the following relations:
$\hat{q}_{i}\cdot\hat{q}_{j}=\delta_{ij}$ and
$\hat{q}_{i}\times\hat{q}_{j}=\Sigma_{k}\epsilon_{ijk}\hat{q}_{k}$. If we add
$\hat{q}_{0}=\Im$ (as identity), we obtain the quaternion expressed by the
Lorentz quantities (parameters of the Lorentz index). They are coupled and are
called “$q$ vector”. The Lorentz quantities (quantities carrying the Lorentz
index) are represented by the quaternions. Thus, we have two representations
of the Lorentz index: the four-vector and the q-vector. We will use the
notation $\mathtt{A},\ \mathtt{B},\cdots$ to represent the quaternions and
normally, we construct quaternions from four vectors:
$\mathtt{A}=A_{0}\Im+A_{k}\hat{q}^{k}\equiv(\mathtt{A}_{0},\mathtt{A}_{k}\hat{q}^{k})_{q}$.
Bold letters are used for the space part of the four vectors. The four-vector
representation has the basis of the spacetime metric, and the Euclidean metric
is for the quaternion representation. This configuration has consequences for
quaternionic products. We also consider the Lorentz index in coupling the
physical parameters to the quaternion generator. In this way, we can
generalize the range of the coupled quantities from four vectors to tensors.
For the $q$-vectors $\hat{\Gamma}$ and $\mathtt{C}$, the extended Grassman
quaternion multiplication $"\triangleright"$ (with $\ast\ $-product), is
defined as follows andre ; walker ; kansu ; stefano

$\displaystyle\hat{\Gamma}\triangleright\mathtt{C}=(\hat{\Gamma}_{0}\ast\mathtt{C}_{0}-\hat{\Gamma}\ast\mathtt{C}\mathtt{C},\hat{\Gamma}\vee\mathtt{C}+\hat{\Gamma}\wedge\mathtt{C}\mathtt{C})_{q},$
(6)

where $\wedge$ and $\vee$ are the vector products under the $\mathsf{ast}$
mapping:

$\displaystyle\hat{\Gamma}\wedge\mathtt{C}\mathtt{C}=\epsilon^{ijk}(\hat{\Gamma}_{i}\ast\mathtt{C}_{j})\hat{q}_{k}.$

and

$\displaystyle(\hat{\Gamma}\vee\mathtt{C})_{k}=\hat{\Gamma}_{0}\ast\mathtt{C}_{k}-\hat{\Gamma}_{k}\ast\mathtt{C}_{0}.$

According to the above, $\vee$ and $\wedge$ add the spatial and temporal
components to the above two commutators;
i) The commutator associated with the product $"\ast"$ is denoted by
$[\hat{\Gamma},\mathtt{C}]_{\ast}=\hat{\Gamma}\ast\mathtt{C}-\mathtt{C}\ast\hat{\Gamma}$.
ii) Moreover, we can introduce a new commutator, i.e. the commutator
associated with the multiplication of the quaternion
$[\mathtt{B},\mathtt{C}]_{\triangleright}=\mathtt{B}\triangleright\mathtt{C}-\mathtt{C}\triangleright\mathtt{B}$
containing the product $"\ast"$, when at least one of them is operator.
Based on the operator rule for the connection coefficients $\hat{\Gamma}$, we
introduce a new covariant derivative,

$\displaystyle\Delta_{\mu}=\partial_{\mu}\Im-e\hat{\Gamma}_{\mu}$ (7)

where $"e"$ is the coupling coefficient constant with a value of one. As
mentioned earlier, $\hat{\Gamma}_{\mu}$ contains two coupling indices. Due to
the operators $\hat{\Gamma}$’s and the $"\ast"$-product $\Delta_{\mu}$ is a
semi-complete operator. Equation 7 with the $"\ast"$-product has the expected
effects

$\displaystyle\Delta_{\mu}\ast\mathtt{B}^{\lambda}_{\kappa}$
$\displaystyle=\partial_{\mu}\mathtt{B}^{\lambda}_{\kappa}-e\Gamma^{\alpha}_{\mu\kappa}\mathtt{B}^{\lambda}_{\alpha}$
$\displaystyle=:\mathtt{B}^{\lambda}_{\kappa,\mu}$

The multiplications used in this paper refer to the $"\ast"$-product.
Accordingly, the effect described above dictates the following action,

$\displaystyle[\Delta_{\mu},(\mathtt{B}\mathtt{C})]_{\ast}$
$\displaystyle=\mathtt{B}[\Delta_{\mu},\mathtt{C}]_{\ast}+[\Delta_{\mu},\mathtt{B}]_{\ast}\mathtt{C}$
(8)

It follows that we have

$\displaystyle[\Delta_{\mu},\mathtt{B}]_{\ast}$
$\displaystyle=\partial_{\mu}\mathtt{B}-e[\hat{\Gamma}_{\mu},\mathtt{B}]_{\ast}$
$\displaystyle\equiv\mathtt{B}_{;\mu}$ (9)

If we define $g^{\alpha\beta}\hat{\Gamma}_{\beta}$ directly, we can write

$\displaystyle
g^{\alpha\beta}\Delta_{\beta}\ast\mathtt{B}_{\lambda}=g^{\alpha\beta}\mathtt{B}_{\lambda,\beta}=g^{\alpha\beta}\mathtt{B}_{\lambda;\beta}\equiv\mathtt{B}_{\lambda}^{\
;\alpha}$

The covariant derivative is not complete. The incompleteness of the derivative
operator is evident from equations I and 5.
$\mu$ and $\zeta$ are free indices in
$\hat{\Gamma}_{\mu}\ast\mathtt{B}_{\zeta}$. This means that
$\hat{\Gamma}_{\mu}\ast\mathtt{B}_{\zeta}$ ends up being a second rank tensor;
In other words,
$\hat{\Gamma}_{\mu}\ast\mathtt{B}_{\zeta}\propto\Gamma^{\alpha}_{\mu\zeta}B_{\alpha}$
actually has the two free indices $\mu$ and $\zeta$ that can be considered as
free indices;
$\Gamma^{\alpha}_{\mu\zeta}B_{\alpha}\equiv\Gamma^{\alpha}_{\mu\zeta}B_{\alpha}\
|\mu,\zeta>$. Equation 3, and our calculations in the ”bra-ket” notation show
that,

$\displaystyle\hat{\Gamma}_{\nu}\ast\hat{\Gamma}_{\mu}$
$\displaystyle=\Gamma^{\alpha}_{\nu\beta}|\beta><\alpha|\Gamma^{\kappa}_{\mu\lambda}|\lambda><\kappa|+\Gamma^{\alpha}_{\nu\beta}|\alpha><\beta|\mu>$
$\displaystyle\ \ \ \ \times\Gamma^{\kappa}_{\mu\lambda}|\lambda><\kappa|$
$\displaystyle=\Gamma^{\alpha}_{\nu\beta}\Gamma^{\kappa}_{\mu\lambda}|\beta>(\delta_{\alpha}^{\lambda})<\kappa|+\Gamma^{\alpha}_{\nu\beta}|\alpha>(\delta_{\beta\mu})$
$\displaystyle\ \ \ \ \times\Gamma^{\kappa}_{\mu\lambda}|\lambda><\kappa|$
$\displaystyle=\Gamma^{\alpha}_{\nu\beta}\Gamma^{\kappa}_{\mu\alpha}|\beta><\kappa|+\Gamma^{\alpha}_{\nu\mu}\Gamma^{\kappa}_{\alpha\lambda}|\lambda><\kappa|,$

furthermore

$\displaystyle(\hat{\Gamma}_{\nu}\ast\hat{\Gamma}_{\mu})^{\dagger}$
$\displaystyle=(\Gamma^{\alpha}_{\nu\beta}\Gamma^{\kappa}_{\mu\alpha}+\Gamma^{\alpha}_{\nu\mu}\Gamma^{\kappa}_{\alpha\beta})|\beta><\kappa|)^{\dagger}$
$\displaystyle=(\Gamma^{\alpha}_{\nu\beta}\Gamma^{\kappa}_{\mu\alpha}+\Gamma^{\alpha}_{\nu\mu}\Gamma^{\kappa}_{\alpha\beta})|\kappa><\beta|,$

so inverted version becomes

$\displaystyle(\hat{\Gamma}_{\nu}\ast\hat{\Gamma}_{\mu})\ast\mathtt{B}_{\zeta}=\Gamma^{\alpha}_{\nu\zeta}\Gamma^{\kappa}_{\mu\alpha}B_{\kappa}+\Gamma^{\alpha}_{\nu\mu}\Gamma^{\kappa}_{\alpha\zeta}B_{\kappa}.$

It also has the free indices, $\mu$, $\nu$ and $\zeta$. Also,

$\displaystyle\hat{\Gamma}_{\nu}\ast(\hat{\Gamma}_{\mu}\ast\mathtt{B}_{\zeta})$
$\displaystyle=\Gamma^{\alpha}_{\nu\beta}|\alpha><\beta|\ast(|\mu,\zeta>\Gamma^{\kappa}_{(\mu\zeta)}B_{\kappa})$
$\displaystyle=\Gamma^{\alpha}_{\nu\zeta}\Gamma^{\kappa}_{\mu\alpha}B_{\kappa}+\Gamma^{\alpha}_{\nu\mu}\Gamma^{\kappa}_{\alpha\zeta}B_{\kappa}.$
(10)

In addition,

$\displaystyle\hat{\Gamma}_{\kappa}\ast(\hat{\Gamma}_{\lambda}\ast\mathtt{B}_{\mu})=(\hat{\Gamma}_{\kappa}\ast\hat{\Gamma}_{\lambda})\ast\mathtt{B}_{\mu}.$

confirms that $\Delta_{\mu}$ is an incomplete operator, satisfying the
condition:

$\displaystyle\Delta_{\mu}\ast(\mathtt{B}_{\lambda}\mathtt{C}_{\kappa})=(\Delta_{\mu}\ast\mathtt{B}_{\lambda})\mathtt{C}_{\kappa}+\mathtt{B}_{\lambda}(\Delta_{\mu}\ast\mathtt{C}_{k}).$

In particular,

$[\hat{\Gamma}_{\mu},\hat{\Gamma}_{\nu}]_{\ast}=(\Gamma^{\zeta}_{\mu\lambda}\Gamma^{\kappa}_{\nu\zeta}-\Gamma^{\zeta}_{\nu\lambda}\Gamma^{\kappa}_{\mu\zeta})(|\kappa><\lambda|).$

It is clear that the free indices after transposition can be seen. In this
way,
$(\Gamma^{\zeta}_{\mu\lambda}\Gamma^{\kappa}_{\nu\zeta}-\Gamma^{\zeta}_{\nu\lambda}\Gamma^{\kappa}_{\mu\zeta})(|\kappa><\lambda|)$
as an operator contains the two free indices $\kappa$ and $\lambda$. It
becomes a simplicial $[\hat{\Gamma}_{\mu},\hat{\Gamma}_{\nu}]^{\kappa}_{\ast\
\lambda}$ which has free $\lambda$ and $\kappa$ indices. Next, we follow the
computation of $[\Delta_{\alpha},\Delta_{\beta}]_{\ast}$. We compute the case
without torsion,

$\displaystyle\hat{\Gamma}_{\kappa;\alpha;\beta}$
$\displaystyle=\partial_{\beta}\partial_{\alpha}\hat{\Gamma}_{\kappa}-e\partial_{\beta}(\hat{\Gamma}_{\alpha}\ast\hat{\Gamma}_{\kappa})-e\Gamma^{\mu}_{\beta\alpha}\partial_{\mu}\hat{\Gamma}_{\kappa}$
$\displaystyle-e\hat{\Gamma}_{\beta}\partial_{\alpha}\ast\hat{\Gamma}_{\kappa}+e^{2}\hat{\Gamma}_{\beta}\ast(\hat{\Gamma}_{\alpha}\ast\hat{\Gamma}_{\kappa}),$

also

$\displaystyle\hat{\Gamma}_{\kappa;\beta;\alpha}$
$\displaystyle=\partial_{\alpha}\partial_{\beta}\hat{\Gamma}_{\kappa}-e\partial_{\alpha}(\hat{\Gamma}_{\beta}\ast\hat{\Gamma}_{\kappa})-e\Gamma^{\mu}_{\alpha\beta}\partial_{\mu}\hat{\Gamma}_{\kappa}$
$\displaystyle-e\hat{\Gamma}_{\alpha}\partial_{\beta}\ast\hat{\Gamma}_{\kappa}+e^{2}\hat{\Gamma}_{\alpha}\ast(\hat{\Gamma}_{\beta}\ast\hat{\Gamma}_{\kappa}),$

consequently

$\displaystyle\hat{\Gamma}_{\kappa;\alpha;\beta}-\hat{\Gamma}_{\kappa;\beta;\alpha}=e\hat{R}_{\alpha\beta}\ast\hat{\Gamma}_{\kappa}.$
(11)

Eq.11 means

$\displaystyle[\Delta_{\alpha},\Delta_{\beta}]_{\ast}=e\hat{R}_{\alpha\beta}.$
(12)

We are in a situation where the four components of the Riemann’s curvature
tensor have become a single operator with two distinct subspaces. Then, we
will look at the roles of the components. With the action role and equation I,
and for a space with torsion, the new covariant derivative satisfies the
following commutation relation,
$[\Delta_{\alpha},\Delta_{\beta}]_{\ast}=e\hat{R}_{\alpha\beta}+e^{2}(\Gamma^{\lambda}_{\alpha\beta}-\Gamma^{\lambda}_{\beta\alpha})\hat{\Gamma}_{\lambda}$.
But the effect of $\hat{R}_{\alpha\beta}$ on variable $\mathtt{B}_{\mu}$ is
now,

$\displaystyle\hat{R}_{\alpha\beta}\ast\mathtt{B}_{\mu}$
$\displaystyle=(\partial_{\alpha}\hat{\Gamma}_{\beta}-\partial_{\beta}\hat{\Gamma}_{\alpha}-e\hat{\Gamma}_{\alpha}\ast\hat{\Gamma}_{\beta}+e\hat{\Gamma}_{\beta}\ast\hat{\Gamma}_{\alpha})\ast\mathtt{B}_{\mu}$
$\displaystyle=R^{\lambda}_{\ \mu\alpha\beta}B_{\lambda}|\mu,\alpha,\beta>.$
(13)

Another result is accessible; $[\Delta_{\mu},\mathsf{A}]_{\ast}$ reach to

$\displaystyle[\Delta_{\mu},A^{\alpha}_{\beta}]_{\ast}=\partial_{\mu}A^{\alpha}_{\beta}-g\Gamma^{\lambda}_{\beta\mu}A^{\alpha}_{\lambda}+g\Gamma^{\alpha}_{\lambda\mu}A^{\lambda}_{\beta},$

and

$\displaystyle[\Delta_{\mu},\mathsf{A}]^{\alpha}_{\ast\ \beta}$
$\displaystyle=(\partial_{\mu}\mathsf{A}-g\hat{\Gamma}_{\mu}\ast\mathsf{A}+g\mathsf{A}\ast\hat{\Gamma}_{\mu})^{\alpha}_{\beta}$
$\displaystyle=\partial_{\mu}A^{\alpha}_{\beta}-g\Gamma^{\kappa}_{\mu\zeta_{i}}A^{\cdots\zeta_{j}\cdots}_{..\zeta_{i-1}\kappa\zeta_{i+1}..}<\beta|\vec{\zeta}_{i}><\vec{\zeta}_{j}|\alpha>$
$\displaystyle-
A_{\cdots\zeta_{i}\cdots}^{..\zeta_{j-1}\kappa\zeta_{j+1}..}\Gamma^{\zeta_{i}}_{\kappa\mu}<\beta|\vec{\zeta}_{i}><\vec{\zeta}_{j}|\alpha>.$

Now, considering relations I and I, we can write:
lemma\- The covariance derivative effect is independent of entering the
components.
Since
$[\hat{\Gamma}_{\mu},\mathtt{A}_{\nu}]_{\ast}+[\mathtt{A}_{\mu},\hat{\Gamma}_{\nu}]_{\ast}=0$
, we take $\mathtt{A}$ to be as the gauge fields, then the change of the
tensor of the field strength is

$\displaystyle\mathcal{E}_{\mu\nu}$
$\displaystyle=\partial_{\mu}\mathtt{A}_{\nu}-\partial_{\nu}\mathtt{A}_{\mu}$
$\displaystyle=\Delta_{\mu}\ast\mathtt{A}_{\nu}-\Delta_{\nu}\ast\mathtt{A}_{\mu}$
(14)

##  II Electrodynamics

In this section we present a mechanism according to which, first, the
components of gauge fields $\mathbf{A}$ are coupled with the generators of the
quaternions ($\mathbf{A}\rightarrow\mathtt{A}$). Then, a coupling between the
components of the covariant derivative and the quaternion generators is
established in the same way. The quaternion variables obtained in this way
help us to reconstruct the strength tensor of the fields. Due to the semi
completeness of the covariant derivative operator, we will introduce a new
quaternion commutator to obtain the equation of motion and conclude the
theory.
Based on the above, we will define new commonly used quatrain variables and
operators. Starting from the tangent spaces, we introduce the operators and
vectors as $q$-variables,
$z_{t}=(\imath\mathtt{A}^{0},\mathtt{A}^{i}\hat{q}_{i})$ and
$D_{t}=(\imath\Delta_{0},\mathbf{\Delta}_{i}\hat{q}^{i})$. The position of the
indices is important here. $\mathtt{A}$ vectors are the fields;
$\mathtt{A}\mathtt{A}$ is the vector potential and $\mathtt{A}_{0}=\phi$
represents the scalar case. There are two more i.e.
$z_{c}=(\imath\mathtt{A}_{0},\mathtt{A}_{i}\hat{q}^{i})$ and
$D_{c}=(\imath\Delta^{0},\mathbf{\Delta}^{i}\hat{q}_{i})$.
We show that Maxwell’s equations are accessible with augmented quaternions.
The given content confirms the above definitions of the $q$-vectors. We
emphasize that the quaternion mapping creates two separate spaces: the tangent
and the cotangent $q$-vectors. Basically, the measures of these two spaces are
connected by the new matrix elements, not by the Minkowski metric. Our
calculations show that the law of direct and crossed multiplication between
$q$-vectors is given by
$H^{\mu\nu}=\left(\begin{array}[]{cccc}1&\imath&\imath&\imath\\\
-\imath&-1&1&-1\\\ -\imath&-1&-1&1\\\ -\imath&1&-1&-1\\\ \end{array}\right)$,
where

$\displaystyle
H^{\mu\nu}=\eta^{\mu\nu}+\left(\begin{array}[]{cccc}0&\imath&\imath&\imath\\\
-\imath&0&1&-1\\\ -\imath&-1&0&1\\\ -\imath&1&-1&0\\\
\end{array}\right):=\eta^{\mu\nu}+H^{\mu\nu}_{d},$

$\eta^{\mu\nu}=\mathrm{diag}(1,-1,-1,-1)$ is the Minkowski metric. With the
help of the matrix $H^{\mu\nu}$ we define the inverse matrix

$\displaystyle\tilde{H}^{\mu\nu}=(H^{\mu\nu})^{T}=\left(\begin{array}[]{cccc}1&-\imath&-\imath&-\imath\\\
\imath&-1&-1&1\\\ \imath&1&-1&-1\\\ \imath&-1&1&-1\\\
\end{array}\right)=\eta^{\mu\nu}+\tilde{H}^{\mu\nu}_{d},$

We also introduce a new tensor i.e. $n_{k}^{\ ij}$ that follows from the
behavior of the quaternions and has the following properties: it is
antisymmetric with respect to the exchange of indices, it can participate in
the raising and lowering of indices, it is spatial, but without counting, and
contains a temporal index. The $n_{k}^{\ ij}$-symbol is used when its indices
are all on the same row. When all indices are in a row, it is similar to the
Levi-Civita symbol. We will see later where the temporal index is located.
Since the product of the quaternions is a quaternion, the new q-variable in
the quaternoin space can be named as $\mathsf{I}$, so we obtain
$D_{t}\triangleright\
z_{c}=(\imath\mathsf{I}_{0},\mathsf{I}_{i}\hat{q}^{i})\equiv\mathsf{I}_{c}$.
Considering the indices, we obtain

$\mathsf{I}_{c}=(\imath\mathsf{I}_{0},\mathsf{I})$

(using the upper indices), our calculations show that

$\displaystyle\mathsf{I}_{c}$ $\displaystyle=(\mathtt{A}_{\nu,\mu}\
\eta^{\mu\nu},\mathtt{A}_{\nu,\mu}\ H_{d}^{\mu\nu})$

based on this result and Eq.I, we arrive at

$\displaystyle\imath\mathsf{I}_{0}$ $\displaystyle=\mathtt{A}_{\nu,\mu}\
\eta^{\mu\nu}$ (15)

and

$\displaystyle\mathsf{I}_{k}$ $\displaystyle=\mathtt{A}_{\nu,\mu}\
H_{d}^{\mu\nu}$

are available with the following definition:

$\displaystyle\mathsf{I}_{k}$
$\displaystyle=\imath\mathcal{E}_{0k}+\frac{1}{2}n^{\ \
ij}_{k}\mathcal{E}_{ij}$ $\displaystyle\equiv\imath\mathcal{E}_{0k}+D_{k},$

where $\mathcal{E}_{\mu\nu}$ is given by Eq.I. The equation of motion is the
directed action of the covariant derivative on the corresponding obtained
quaternion variables and vice versa. Due to the incompleteness of the
derivative effects, in order to obtain the equation of motion, we must define
a new commutator as follows: $[\ ,\ ]_{\triangleright}:\
\mathsf{Q}\times\mathsf{Q}\rightarrow\mathsf{Q}$ which is the same as
$[\mathtt{B},\mathtt{C}]_{\triangleright}=(\eta^{\mu\nu}[\mathtt{B}_{\mu},\mathtt{C}_{\nu}]_{\ast},[\mathtt{B},\mathtt{C}]^{\vee}_{\ast}+[\mathtt{B}\mathtt{B},\mathtt{C}\mathtt{C}]^{\wedge}_{\ast})$.
Due to the electrodynamics in Minkowski spacetime, the commutator of the
motion has the following form (under the matrix $\tilde{H}$):

$\displaystyle[(\imath
g^{0\zeta}\Delta_{\zeta},g^{s\zeta}\mathbf{\Delta}_{\zeta}),\mathsf{I}_{c}]^{\tilde{H}}_{\triangleright}$
$\displaystyle=(g^{\mu\zeta}\mathsf{I}_{\nu;\zeta}\eta^{\mu\nu},\mathsf{I}_{\nu;\mu}\
\tilde{H}_{d}^{\mu\nu})=\mathsf{J}_{c},$

$\mathsf{J}_{c}=(\imath\mathsf{J}_{0},\mathsf{J}_{i}\hat{q}^{i})$ is the
quaternion form of current density. So the equation of motion expands to,

$\displaystyle g^{\mu\zeta}\mathsf{I}_{\nu;\zeta}\eta^{\mu\nu}$
$\displaystyle=\imath\mathsf{j}_{0},$ $\displaystyle\mathsf{I}_{\nu;\mu}\
\tilde{H}_{d}^{\mu\nu}$ $\displaystyle=\mathsf{j}_{k}$ (16)

For any $\mathtt{B}$ without upper indices, the equality
$\Delta\ast\mathtt{B}\equiv[\Delta,\mathtt{B}]_{\ast}$ holds, and

$\displaystyle\Delta_{\nu}\ast\mathtt{B}_{\mu}=\mathtt{B}_{\mu,\nu}\equiv\mathtt{B}_{\mu;\nu}$
(17)

If $\mathtt{B}$ carries no upper indices, then
$\Delta_{\nu}\ast\mathtt{B}_{;\mu}=\mathtt{B}_{;\mu;\nu}$ and
$\Delta_{\mu}\ast g_{\kappa\lambda}=0$, $\Delta_{\mu}\ast
g^{\kappa\lambda}\neq 0$. Therefore, equations 8 and II based on equation 17
lead to the following equation:

$\displaystyle g^{0\zeta}\mathsf{I}_{0\ ;\zeta}-\imath
g^{s\zeta}\mathcal{E}_{0s;\zeta}-g^{s\zeta}D_{s;\zeta}=\imath\mathsf{J}_{0},$
(18) $\displaystyle-\imath
g^{0\zeta}(\imath\mathcal{E}_{0k;\zeta}+D_{k;\zeta})-n_{ks}^{\ \ \
l}g^{s\zeta}(\imath\mathcal{E}_{0l;\zeta}+D_{l;\zeta})$ (19)
$\displaystyle+\imath g^{k\zeta}\mathsf{I}_{0\ ;\zeta}=\mathsf{J}_{k},$ (20)

and finally, we arrive at

$\displaystyle-\imath g^{0\zeta}\mathsf{I}_{0\
;\zeta}-g^{s\zeta}\mathcal{E}_{0s;\zeta}$ $\displaystyle=\mathsf{J}_{0},$
$\displaystyle g^{s\zeta}D_{s;\zeta}$ $\displaystyle=0,$ $\displaystyle\imath
g^{k\zeta}\mathsf{I}_{0\ ;\zeta}+g^{0\zeta}\mathcal{E}_{0k;\zeta}-n_{ks}^{\ \
\ l}g^{s\zeta}D_{l;\zeta}$ $\displaystyle=\mathsf{J}_{k},$ $\displaystyle
g^{0\zeta}D_{k;\zeta}+n_{ks}^{\ \ \ l}g^{s\zeta}\mathcal{E}_{0l;\zeta}$
$\displaystyle=0$ (21)

From the equations of motion, it follows that $g^{s\zeta}D_{s;\zeta}=0$, which
gives a Bianchi identity (without time index): $\frac{1}{2}n_{s}^{\
ij}g^{s\zeta}\mathcal{E}_{ij;\zeta}\rightarrow\frac{1}{2}\varepsilon^{sij}\mathcal{E}_{ij;s}=0$.
Moreover, the fourth equation is $g^{0\zeta}D_{k;\zeta}+n_{ks}^{\ \ \
l}g^{s\zeta}\mathcal{E}_{0l;\zeta}=\frac{1}{2}n_{k}^{\ ij}\mathcal{E}_{ij}^{\
\ ;0}+n_{k}^{\ \ sl}\mathcal{E}_{0l;s}=0$ which gives another Bianchi
identity: $\frac{1}{2}n_{k0}^{\ \ \ ij}\mathcal{E}_{ij}^{\ \ \ ;0}+n_{k}^{\
s0l}\mathcal{E}_{0l;s}=\frac{1}{2}\varepsilon^{k0ij}\mathcal{E}_{ij;0}+\varepsilon^{ks0l}\mathcal{E}_{0l;s}=0$.
Since $n_{ks}^{\ \ \
l}g^{s\zeta}D_{l;\zeta}=\frac{1}{2}\varepsilon_{ksl}g^{s\zeta}\varepsilon^{lij}\mathcal{E}_{ij;\zeta}=g^{s\zeta}\mathcal{E}_{ks;\zeta}$,
equation II is thus

$\displaystyle-g^{s\zeta}\mathcal{E}_{0s;\zeta}=\mathsf{J}_{0}+\imath
g^{0\zeta}\mathsf{I}_{0\ ;\zeta},$ $\displaystyle
g^{0\zeta}\mathcal{E}_{0k;\zeta}-g^{s\zeta}\mathcal{E}_{ks;\zeta}=\mathsf{J}_{k}-\imath
g^{k\zeta}\mathsf{I}_{0\ ;\zeta}$

to summarize,

$\displaystyle g^{\mu\zeta}\mathcal{E}_{\mu\nu;\zeta}$
$\displaystyle=\mathtt{J}_{\nu},$ (22)

where

$\displaystyle\mathtt{J}_{\nu}=\mathsf{J}_{\nu}+\imath\eta_{\nu\mu}g^{\mu\zeta}\mathsf{I}_{0\
;\zeta}.$ (23)

The above relations are simplified as follows:

$\displaystyle\mathsf{I}_{0}^{\ ;0}-\mathsf{I}_{s}^{\
;s}=\imath\mathsf{j}_{0}$ $\displaystyle-\imath\mathsf{I}_{k}^{\
;0}+\imath\mathsf{I}_{0}^{\ ;k}-n_{ki}^{\ \ j}\mathsf{I}_{j}^{\
;i}=\mathsf{j}_{k}$

From Eqs.11 or 12, we get the relation $n_{ki}^{\ \ j}\mathsf{I}_{j}^{\
;i;k}=\frac{e}{2}n_{ki}^{\ \ j}\hat{R}^{ik}\ast\mathsf{I}_{j}$ and
$\imath(\mathsf{I}_{k}^{\ ;k;0}-\mathsf{I}_{k}^{\ ;0;k})=\imath
e\hat{R}^{k0}\ast\mathsf{I}_{k}$. If we substitute these points, we obtain

$\displaystyle\mathsf{j}_{0}^{\ ;0}+\mathsf{j}_{k}^{\ ;k}$
$\displaystyle=-\imath\eta_{\mu\nu}g^{\mu\zeta}\mathsf{I}_{0;\zeta}^{\ \
;\nu}+\imath e\hat{R}^{k0}\ast\mathsf{I}_{k}-\frac{e}{2}n_{ki}^{\ \
j}\hat{R}^{ik}\ast\mathsf{I}_{j}$
$\displaystyle=-\imath\eta_{\mu\nu}g^{\mu\zeta}\mathsf{I}_{0;\zeta}^{\ \
;\nu}$ $\displaystyle+\frac{\imath e}{2}(n_{k}^{\
ij}\hat{R}^{k0}\ast\mathcal{E}_{ij}-n_{ki}^{\ \
j}\hat{R}^{ik}\ast\mathcal{E}_{0j})$
$\displaystyle-\frac{e}{4}(4\hat{R}^{k0}\ast\mathcal{E}_{0k}+n_{ki}^{\ \
j}n_{j}^{\ mn}\hat{R}^{ik}\ast\mathcal{E}_{mn})$

However, it can be seen that $n_{ki}^{\ \ j}n_{j}^{\
mn}\hat{R}^{ik}\ast\mathcal{E}_{mn}=\varepsilon_{kij}\varepsilon^{jmn}\hat{R}^{ik}\ast\mathcal{E}_{mn}=2\hat{R}^{nm}\ast\mathcal{E}_{mn}=2\hat{R}_{\nu\mu}\ast\mathcal{E}^{\mu\nu}-4\hat{R}_{0\mu}\ast\mathcal{E}^{\mu
0}$. Moreover, the second and third terms are zero because the equations of
motion $n_{k}^{\ ij}\hat{R}^{0k}\ast\mathcal{E}_{ij}+n_{ki}^{\ \
j}\hat{R}^{ik}\ast\mathcal{E}_{0j}=0$. This result is independent of the
physical conditions and has nothing to do with the sources. Substituting the
results, we get

$\displaystyle\mathsf{j}_{0}^{\ ;0}+\mathsf{j}_{k}^{\
;k}+\imath\eta_{\mu\nu}g^{\mu\zeta}\mathsf{I}_{0;\zeta}^{\ \
;\nu}=-\frac{e}{2}\hat{R}^{\nu\mu}\ast\mathcal{E}_{\mu\nu}$ (24)

Now we derive equation 22 and from the fact that

$\displaystyle g^{\nu\kappa}\Delta_{\kappa}\ast
g^{\mu\zeta}\mathcal{E}_{\mu\nu;\zeta}$
$\displaystyle=g^{\nu\kappa}\Delta_{\kappa}\ast\mathtt{J}_{\nu},$

we have:
$g^{\mu\zeta}\mathcal{E}_{\mu\nu;\zeta}=g_{\nu\beta}\mathcal{E}^{\alpha\beta}_{\
\ ;\alpha}$ On the other hand, $g_{\nu\beta}\mathcal{E}^{\alpha\beta}_{\ \
;\alpha}$ and the action of the derivative operator on the $\nu$ set is
proportional to the low index set. From this follows:
$g^{\nu\kappa}\Delta_{\kappa}\ast g_{\nu\beta}\mathcal{E}^{\alpha\beta}_{\ \
;\alpha}=\mathcal{E}^{\alpha\beta}_{\ \ \ ;\alpha;\beta}$. The same logic
applies to the
$g^{\nu\kappa}\Delta_{\kappa}\ast\mathtt{J}_{\nu}=\mathtt{J}_{\nu;\kappa}g^{\kappa\nu}$.
With these interpretations of equation I, we can write:
$\mathcal{E}^{\mu\nu}_{\ \ ;\mu;\nu}=-\mathcal{E}^{\nu\mu}_{\ \
;\nu;\mu}+e\hat{R}_{\mu\nu}\ast\mathcal{E}^{\mu\nu}$, and finally, we have

$\displaystyle\mathcal{E}^{\mu\nu}_{\ \
;\mu;\nu}=\frac{e}{2}\hat{R}_{\mu\nu}\ast\mathcal{E}^{\mu\nu}$ (25)

According to equations 24, 25, and the above, the continuity equation is given
as follows:

$\displaystyle\mathsf{j}_{0}^{\ ;0}+\mathsf{j}_{k}^{\
;k}+\imath\eta_{\mu\nu}g^{\mu\zeta}\mathsf{I}_{0;\zeta}^{\ \
;\nu}=\frac{e}{2}\hat{R}^{\mu\nu}\ast\mathcal{E}_{\mu\nu},$

But, $\mathcal{E}_{\mu\nu}^{\ \ ;\mu;\nu}=A_{\nu;\mu}^{\ \
;\mu;\nu}-A_{\mu;\nu}^{\ \ ;\mu;\nu}=e\hat{R}^{\mu\nu}A_{\nu;\mu}$ and based
on the relation in I, we reach $\mathcal{E}_{\mu\nu}^{\ \
;\mu;\nu}\rightarrow-eR_{\mu}^{\ \lambda\mu\nu}A_{\lambda;\nu}-eR_{\nu}^{\
\lambda\mu\nu}A_{\mu;\lambda}=-eR_{\nu}^{\
\lambda\nu\mu}\mathcal{E}_{\mu\lambda}=0$, which means that the condition
$\mathtt{J}_{\nu}^{\ ;\nu}=0$ as continuity equation is in agreement with
reference lobo . Inserting equation I, equation 22 becomes

$\displaystyle g^{\mu\zeta}(\Delta_{\mu}\ast A_{\nu}-\Delta_{\nu}\ast
A_{\mu})_{;\zeta}$ $\displaystyle=\mathtt{J}_{\nu},$

similar to the result of arbabb , or

$\displaystyle g^{\mu\zeta}\Delta_{\zeta}\ast\Delta_{\mu}\ast
A_{\nu}-g^{\mu\zeta}\Delta_{\zeta}\ast\Delta_{\nu}\ast A_{\mu}$
$\displaystyle=\mathtt{J}_{\nu},$

but
$g^{\mu\zeta}A_{\mu;\nu;\zeta}=g^{\mu\zeta}(\Delta_{\nu}\ast\Delta_{\zeta}+e\hat{R}_{\nu\zeta})\ast
A_{\mu}$, which gives us

$\displaystyle g^{\mu\zeta}\Delta_{\zeta}\ast\Delta_{\mu}\ast
A_{\nu}-eg^{\mu\zeta}\hat{R}_{\nu\zeta}\ast A_{\mu}$
$\displaystyle=\mathtt{J}_{\nu}+g^{\mu\zeta}\Delta_{\nu}\ast\Delta_{\zeta}\ast
A_{\mu}.$

Since $g^{\mu\zeta}\Delta_{\nu}\ast\Delta_{\zeta}\ast
A_{\mu}=g^{0\zeta}A_{0;\zeta;\nu}+g^{s\zeta}A_{s;\zeta;\nu}$ and the given
condition that $A^{\kappa}_{\
;\kappa}=0=g^{\kappa\lambda}A_{\lambda;\kappa}=g^{0\kappa}A_{0;\kappa}+g^{s\kappa}A_{s;\kappa}$,
we can derive that $g^{s\zeta}A_{s;\zeta}=-g^{0\zeta}A_{0;\zeta}$. We obtain
the following equation parker ; parke ; lindgren ; arbab ; cameron :

$\displaystyle g^{\mu\zeta}\Delta_{\zeta}\ast\Delta_{\mu}\ast
A_{\nu}-eg^{\mu\zeta}R^{\alpha}_{\ \mu\nu\zeta}A_{\alpha}$
$\displaystyle=\mathtt{J}_{\nu},$

is, by substitution of the Ricci tensor, equivalent to

$\displaystyle g^{\mu\zeta}\Delta_{\zeta}\ast\Delta_{\mu}\ast
A_{\nu}-e\mathrm{R}^{\ \ \alpha}_{\nu}A_{\alpha}$
$\displaystyle=\mathtt{J}_{\nu},$ (26)

which is

$\displaystyle g^{\mu\zeta}A_{\nu;\mu;\zeta}-e\mathrm{R}^{\ \
\alpha}_{\nu}A_{\alpha}$ $\displaystyle=\mathtt{J}_{\nu}.$

To reach the final goal (Eq.26), the relation
$\mathtt{A}_{s;s}-\mathtt{A}_{0;0}=\mathtt{A}_{s}^{\
;\zeta}g_{s\zeta}-\mathtt{A}_{0}^{\ ;\zeta}g_{0\zeta}$ must be established,
and this is a precondition (due to assumption 4). If the precondition is
satisfied, the continuity equation is $\mathsf{j}_{0}^{\ ;0}+\mathsf{j}_{k}^{\
;k}=0$.
To determine the position of the time index in the new tensor $n_{k}^{\ ij}$,
it is sufficient to define:

$\displaystyle
G^{\mu\nu}=\frac{1}{2}\varepsilon^{\mu\nu\alpha\beta}\mathcal{E}_{\alpha\beta},$

so that $G^{\mu\nu}_{\ \ ;\nu}=0$. For the case $\mu=0$ the second equation of
motion in Eq.II is now valid:

$\displaystyle G^{0k}_{\ \
;k}=\frac{1}{2}\varepsilon^{0sij}\mathcal{E}_{ij;s},$ (27)

and for the case $\mu\neq 0$ we get the fourth equation of motion in Eq.II

$\displaystyle G^{l\zeta}_{\ \
;\zeta}=\frac{1}{2}\varepsilon^{l0ij}\mathcal{E}_{ij;0}+\varepsilon^{li0j}\mathcal{E}_{0j;i}=0$
(28)

which gives the position of the time index in the comparison.

##  III Geometry

In this section, the gauge fields are replaced by the Christoffel connections
$\mathsf{A}$ to $\Gamma$. This change does not affect the definition of the
covariant derivative operator. Using the same method of the previous section
and multiplying the four components of the Riemann’s curvature tensor into two
two-component subspaces, we define the quaternion variables as
$z_{c}=(\imath\hat{\Gamma}_{0},\hat{\Gamma}_{i}\hat{q}^{i})$ and
$D_{c}=(\imath\Delta^{0},\mathbf{\Delta}^{i}\hat{q}_{i})$. It is clear that
$\hat{\Gamma}^{0}=g^{0\zeta}\hat{\Gamma}_{\zeta}$, and this is a general case.
If we set the above points and emphasize the position of the indicators, we
can write

$\displaystyle\mathsf{I}_{c}$
$\displaystyle=(\hat{\Gamma}_{\nu,\mu}\eta^{\mu\nu},\hat{\Gamma}_{\nu,\mu}\tilde{H}_{d}^{\mu\nu})$

With the help of these preliminary remarks, and similar to the relation in 15,
we arrive at the definitions

$\displaystyle\imath\mathsf{I}_{0}$
$\displaystyle=\hat{\Gamma}_{\nu,\mu}\eta^{\mu\nu}$

and

$\displaystyle\mathsf{I}_{k}=\hat{\Gamma}_{\nu,\mu}\tilde{H}_{d}^{\mu\nu},$

which are written in the short form

$\displaystyle\mathsf{I}_{k}$
$\displaystyle=\imath\mathcal{F}_{0k}+\frac{1}{2}n_{k}^{\ ij}\mathcal{F}_{ij}$
$\displaystyle=\imath\mathcal{F}_{0k}+\mathfrak{D}_{k}$

The new notation $\mathcal{F}_{\mu\nu}$ (Eq.I), is only to visually separate
the variable of this part from that of the previous part, i.e.,
$\mathcal{E}_{\mu\nu}$. Therefore, all the previous relationships and their
placements are maintained with this transformation. Thus, the relation in 18
will be the equation of motion for the new fields $\mathcal{F}_{\mu\nu}$. By
entering the quaternion current density and separating the main equations into
real and imaginary components, we have

$\displaystyle-\imath g^{0\zeta}\mathsf{I}_{0\
;\zeta}-g^{s\zeta}\mathcal{F}_{0s;\zeta}$ $\displaystyle=\mathsf{J}_{0},$
$\displaystyle g^{s\zeta}D_{s;\zeta}$ $\displaystyle=0,$ $\displaystyle\imath
g^{k\zeta}\mathsf{I}_{0\ ;\zeta}+g^{0\zeta}\mathcal{F}_{0k;\zeta}-n_{ks}^{\ \
\ l}g^{s\zeta}D_{l;\zeta}$ $\displaystyle=\mathsf{J}_{k},$ $\displaystyle
g^{0\zeta}D_{k;\zeta}+n_{ks}^{\ \ \ l}g^{s\zeta}\mathcal{F}_{0l;\zeta}$
$\displaystyle=0$ (29)

The relationship between the dynamics of the geometric tensor and the sources
is clear and obvious. Comparing the results from the previous section, it is
clear that the properties of the gauge field strength tensor apply to both
components of Riemann’s curvature tensor. These performances emphasize that
the Christoffel connections have the gauge property. Equations 27 and III give
the source less equations:

$\displaystyle\frac{1}{2}\varepsilon^{0kij}\hat{R}_{ij;k}=0.$

and from equation 28

$\displaystyle\frac{1}{2}\varepsilon^{l0ij}\hat{R}_{ij;0}+\varepsilon^{li0j}\hat{R}_{0j;i}=0$
(30)

Eq.30 represents the Bianchi identities and can be derived from
$\hat{G}^{\mu\nu}=\frac{1}{2}\varepsilon^{\mu\nu\alpha\beta}\hat{R}_{\alpha\beta}$.
For the Bianchi identities, the sub-components of the released space behave
independently. But in the second relation, which is independent of the source,
the behavior of the second Bianichi identity is intertwined with the basis of
the subcomponents. The remaining two relations of the basic equation of motion
(Eq.III) are as follows:

$\displaystyle g^{\mu\zeta}\hat{R}_{\mu\nu;\zeta}$
$\displaystyle=\mathtt{J}_{\nu},$ (31)

which $\mathtt{J}_{\nu}$ in Eq.31, still gives the current density as Eq.23.
together with $\hat{R}_{\mu\nu}^{\ \ ;\mu;\nu}=0$ and the condition
$\mathtt{J}_{\nu}^{\ ;\nu}=0$. Besides the gauge condition, the precondition

$\eta_{\nu\mu}g^{\mu\zeta}\mathsf{I}_{0\ ;\zeta}=0,$

is also conceivable.
A new tensor, $G^{\kappa\lambda}_{\ \ \ \mu\nu}$, can be introduced here:

$\displaystyle G^{\kappa\lambda}_{\ \ \
\mu\nu}=\frac{1}{4}\varepsilon^{\kappa\lambda\alpha\beta}\varepsilon_{\mu\nu\theta\pi}R^{\theta\pi}_{\
\ \ \alpha\beta},$

which gets its special form:

$\displaystyle G^{\kappa\lambda}_{\ \ \ k\nu}$
$\displaystyle=-\frac{1}{2}\delta^{\lambda}_{\nu}R_{\alpha\beta}^{\ \ \
\alpha\beta}+R_{\nu\alpha}^{\ \ \ \lambda\alpha}.$

On the other hand, if we use the equation of motion (Eq.III)

$\displaystyle G^{\kappa\lambda}_{\ \ \ \mu\nu;\kappa}$
$\displaystyle=\frac{1}{4}\varepsilon^{\kappa\lambda\theta\pi}\varepsilon_{\mu\nu\alpha\beta}R^{\alpha\beta}_{\
\ \ \theta\pi;\kappa}=0$
$\displaystyle=\frac{1}{2}\varepsilon^{\alpha\lambda\theta\pi}\varepsilon_{\mu\nu\alpha\beta}\mathtt{J}^{\beta}_{\
\ \theta\pi}-\frac{1}{2}\delta_{\nu}^{\lambda}\ \mathcal{R}_{;\mu}$
$\displaystyle+R^{\lambda\alpha}_{\ \ \ \nu\alpha;\mu},$ (32)

$\mathcal{R}$ is the Ricci scalar. As far as we know

$\displaystyle\varepsilon^{\alpha\lambda\theta\pi}\varepsilon_{\mu\nu\alpha\beta}\mathtt{J}^{\beta}_{\theta\pi}$
$\displaystyle=-\delta^{\alpha}_{\alpha}\det{\left(\begin{array}[]{ccc}\delta^{\lambda}_{\mu}&\delta^{\lambda}_{\nu}&\delta^{\lambda}_{\beta}\\\
\delta^{\theta}_{\mu}&\delta^{\theta}_{\nu}&\delta^{\theta}_{\beta}\\\
\delta^{\pi}_{\mu}&\delta^{\pi}_{\nu}&\delta^{\pi}_{\beta}\end{array}\right)}\mathtt{J}^{\beta}_{\theta\pi}$
$\displaystyle=(4-1)!(-2\delta^{\lambda}_{\mu}\mathtt{J}^{\beta}_{\nu\beta}+2\delta^{\lambda}_{\nu}\mathtt{J}^{\beta}_{\mu\beta}-2\delta^{\lambda}_{\beta}\mathtt{J}^{\beta}_{\mu\nu})$
$\displaystyle:=\mathtt{S}^{\lambda}_{\mu\nu},$

so Eq.III becomes

$\displaystyle-\frac{1}{2}\delta^{\lambda}_{\nu}\
\mathcal{R}_{;\mu}+R^{\lambda\alpha}_{\ \ \
\nu\alpha;\mu}+\mathtt{S}^{\lambda}_{\mu\nu}=0,$ (33)

if one now sets
$\mathtt{S}^{\lambda}_{\mu\nu}:=\mathtt{T}^{\lambda}_{\nu;\mu}$ and that based
on the equation I, $-\frac{1}{2}\delta^{\lambda}_{\nu}\
\mathcal{R}_{;\mu}+R^{\lambda\alpha}_{\ \ \
\nu\alpha;\mu}=[\Delta_{\mu},\frac{1}{2}\delta^{\lambda}_{\nu}\
\mathcal{R}+R^{\lambda\alpha}_{\ \ \ \nu\alpha}]_{\ast}$, one finds a more
completely form of 33,

$\displaystyle-\frac{1}{2}\delta^{\lambda}_{\nu}\
\mathcal{R}+R^{\lambda\alpha}_{\ \ \
\nu\alpha}=-\mathtt{T}^{\lambda}_{\nu}+\mathtt{C}^{\lambda}_{\nu}$

certainly

$\displaystyle\mathtt{C}^{\lambda}_{\nu;\mu}=0$ (34)

is like the famous case $\mathtt{C}_{\lambda\nu}=-\Lambda g_{\lambda\nu}$. It
turns out that $\mathtt{C}_{\lambda\nu}=f_{\lambda\nu}(\mathbf{g})$ (for
example $(e^{\kappa\mathbf{g}})_{\lambda\nu}$, $\mathbf{g}$ is the matrix for
the metric). $\mathtt{C}_{\lambda\nu}$, is a geometrical quantity, not matter.
However, it can be considered as general background matter and can be absorbed
by the tensor $\mathtt{T}_{\lambda\nu}$ and become zero.
In the case of lower indices,

$\displaystyle-\frac{1}{2}g^{\lambda\beta}g_{\beta\nu}\
\mathcal{R}+g^{\lambda\beta}R^{\ \alpha}_{\beta\
\nu\alpha}=-g^{\lambda\beta}\mathtt{T}_{\beta\nu}+g^{\lambda\beta}\mathtt{C}_{\beta\nu}$
(35)

or

$\displaystyle\mathrm{R}_{\lambda\nu}-\frac{1}{2}g_{\lambda\nu}\
\mathcal{R}+\mathtt{C}_{\lambda\nu}=\mathtt{T}_{\lambda\nu}$ (36)

In the last equation, the characters of geometry and matter are separated. The
fact that in Einstein’s theory of gravitation Einstein’s term (in agreement
with the covariant derivative) is not the only way to choose
$\mathtt{C}_{\lambda\nu}$ is another result of this mechanism. We must also
consider some constants that indicate the spacetime constraints to solve the
equation of gravity. It is a question of finding the constant from equation
36, because it is not a unique quantity. For any kind of principal change in
the constant of equation 36 (constant in agreement with covariant derivative),
we naturally enter a new universe. Among the possible options for the constant
$\mathtt{C}_{\lambda\nu}$, equation 36 makes a perturbation theorem. What can
be obtained, however, is the non-uniqueness of Einstein’s famous theorem.

##  IV Discussion

Here, we have used a quaternion mechanism to form new vectors. We have given
$q$-vectors as physical quantities-such as events in the manifold of
spacetime-corresponding to the position of the Lorentz indices and coupled to
the generators of quaternion algebra. By introducing the operator
$\hat{\Gamma}_{\mu}$ and the mapping $"\mathrm{ast}"$ ($\ast$-product), we
then succeeded in obtaining the theory of gauge fields in a general spacetime.
If we introduce the new antisymmetric tensor $n_{k}^{\ ij}$ and use the
equations of gauge theory motion, we obtain the Bianchi identities. We have
also shown that the Bianchi identities are physical relations, not
mathematical structures. As we proceeded, we replaced the gauge fields with
the Christoffel coefficients and followed the improved quaternion mechanism to
access the theory of gravity. Our calculations have shown that with this
substitution and with more degrees of freedom for Riemann’s curvature tensor,
the gravitation theory is still followed and that the Einstein constant term
is a necessity for the equation of motion. The results of gravity are much
higher than those of electrodynamic theory, and the reason for this difference
is Riemann’s curvature four-component tensor. As for the Bianchi identities in
gravity, we have concluded that the identities are part of the equations of
motion. Perhaps the most important result after the derivation of the equation
of gravity is the formalization of the constant term (as Einstein’s constant
term) in the Einstein equation. We have also established that the constant
term is exclusively geometric. Having decomposed the four-component Riemann’s
curvature tensor into one with two component subsections, another result of
this work is to emphasize the independent behavior of the two component
subsections of the curvature tensor. The result of the present work is a
commentary on parisi ; faisal  and many other references and matches the
physical data with the solutions of the Einstein equation, with many
corrections to the Einstein theory of gravity using the unprincipled
modification approach to the Einstein equation (modifications that do not
apply to equation 34). By separating the components of the Riemann’s curvature
tensor, we can also develop a new approach to quantum gravity. The conclusion
of this article is that any type of manipulation in Einstein’s equation is not
correct and must be compatible with the condition, “Covariant derivative is
equal to zero”.

##  V Conclusions

We demonstrate that, proving gravitational issues, such as dark matter,
Einstein’s equation can’t take any additional terms. Modifying Einstein’s
equation is possible only by vanishing covariant derivative terms.

##  VI Acknowledgments

The author would like to thank Shahrekord University for providing the
research grant.

##  VII Appendix

After defining the Christoffel coefficients $\Gamma^{\alpha}_{\kappa\beta}$,
according to assumption 3 and using the ”bra-ket” notation, we can write:

$\displaystyle\frac{1}{2}\mathbf{g}^{\mu\alpha}(\mathbf{g}_{\mu\beta,\kappa}$
$\displaystyle+\mathbf{g}_{\kappa\mu,\beta}-\mathbf{g}_{\beta\kappa,\mu})$
$\displaystyle=g^{\mu\alpha}(g_{\mu\beta,\kappa}+g_{\kappa\mu,\beta}-g_{\beta\kappa,\mu})e_{\alpha}e_{\mu}\times
e^{\mu}e^{\kappa}e^{\beta}$ $\displaystyle=\Gamma^{\alpha}_{\kappa\beta}\
e_{\alpha}e^{\kappa}e^{\beta}=e_{\alpha}\ (\Gamma^{\alpha}_{\kappa\beta}\
e^{\kappa})\ e^{\beta},$ (37)

this occurs when $\kappa$ is selected. Actually, to define a gamma operator in
the space of indices, we need to choose either $\kappa$ or $\beta$. Now, if we
choose the $\hat{\Gamma}_{\kappa}$ operator, then we do not have summation on
$\kappa$ while there is summation on $\alpha$ and $\beta$. In this way, we
will have

$\displaystyle\hat{\Gamma}_{\kappa}$
$\displaystyle=(\Gamma^{\alpha}_{\beta\kappa})\
|\beta><\alpha|:=\hat{\Gamma}_{\kappa}$ (38)

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