# On the quaternion transformation and field equations in curved space-time

B. C. Chanyal

###### Abstract

In this paper, we use four-dimensional quaternionic algebra to describing
space-time geometry in curvature form. The transformation relations of a
quaternionic variable are established with the help of basis transformations
of quaternion algebra. We deduced the quaternionic covariant derivative that
explains how the quaternion components vary with scalar and vector fields. The
quaternionic metric tensor and geodesic equation are also discussed to
describing the quaternionic line element in curved space-time. Moreover, we
discussed an expression for the Riemannian Christoffel curvature tensor in
terms of the quaternionic metric tensor. We have deduced the quaternionic
Einstein’s field-like equation which shows an equivalence between quaternionic
matter and geometry.

Keywords: quaternion, curvature space-time, Riemannian Christoffel curvature
tensor, Einstein field equation

PACS: 02.40.Hw, 02.40.Ky, 4.20.-q

Mathematics Subject Classification 2020: 12Hxx, 35Q76, 83C10

Department of Physics, G. B. Pant University of Agriculture & Technology
Pantnagar-263145 (Uttarakhand) India. Email<EMAIL_ADDRESS>
##  1 Introduction

The theory of the physical world has two fundamental components: matter and
its interactions. The fundamental forces or interactions can be described as a
field in which particles interact with one another. These fundamental
interactions broke down into four fundamental forces of nature [1]: the
gravitational force due to the attraction between massive bodies, the
electromagnetic force due to the force between electrically charged particles,
the weak force responsible for the radioactive decay of atoms, and the strong
force that binding fundamental particles together. Gravitational force is the
weakest force experiences as attractive. Because it is a very long-range
force, the dominant force on the macroscopic scale is the reason for the
formation, shape, and orbit of the astronomical bodies. This force is
responsible for an object in space to move in a circular orbit. In Newtonian
physics, the concept of absolute space and time is important for the dynamics
of an object but they are not relative to each other. Further, Albert Einstein
developed the theory of relativity, therefore, the traditional Newtonian
concept of absolute space and time has been replaced by the concept of space-
time. Time is considered as the fourth dimension in the four-spaces theory of
special and general relativity. The theory which explains the gravitation is
named ‘General Theory of Relativity’ (GTR) [2]. In GTR, gravity does not act
as a force but significance the space-time curvature where the source of the
curved space-time is energy and matter. The Einstein field equations also
elaborate that the mass and energy are responsible for the curvature of space-
time, and the geometry of space-time is responsible for the motion of matter.
Since, the description of space-time geometry should be non-Euclidean
geometry, such that the space-time path becomes curved in presence of the
gravitational field. In this case, the path of motion of particles in such a
field or non-Euclidean space-time shows curvilinear instead of straight.
Objects in nature have basic properties that are independent of the manual
selection of any specific coordinate system. The basic assumption of physics
is that the laws of nature are stated by equations, and should be valid for
all frames of reference.

The theory of relativity required the ‘covariant form’ that is independent of
the coordinate system. The covariant form remains the same in all co-ordinate
systems whether the reference frame will be changed or not. The laws of
physics must be expressed by an equation that remains covariant for all
coordinate systems. The tensor has such properties that show covariant form in
all coordinate systems. If all the equations, related to the general theory of
relativity, are written in the tensor form, then it will transform easily from
one reference frame to another. Therefore, in mathematics, the tensor calculus
is the best tool to explain the theory of curvature as ordinary calculus.
Besides, for a flat space-time, the Pythagoras theorem is valid perfectly but
for a curved space-time, this theorem no longer holds and uses a ‘metric’ to
measure the distance. Moreover, in GTR, the metric tensor behaves as the
gravitational potential of Newtonian gravitation. The Riemannian Christoffel’s
curvature tensor captures the notion of parallel transport. It gives the idea
about the connections of the parallel transported vector on a curve space-
time. Further, Riemannian curvature is required for all the changes in the
tangent vectors when we transport them around a curved space. The Riemann-
Christoffel tensor is the only tensor that can be constructed from the second
derivative of the metric tensor [3]. The contraction of the Riemann tensor
gives Ricci curvature and scalar curvature. Ricci curvature explains the
concept in which mass converges and diverges in time corresponding to the part
of the curvature of space-time. Scalar curvature gives the single real number
which represents the quantity of Riemannian manifold volume of geodesic ball
differs from the normal ball in Euclidean space. The equation of motion of
free particles is recognized by geodesic equation followed the analogous to
Newton’s equation of motion which clarifies the equation for the acceleration
of particles. The source of the curvature of space-time is generalized by the
energy and momentum of the field expressed as an energy-momentum tensor.

The study of algebraic structure in curvature space-time has an important role
to describe the natural world. In mathematical algebraic structure, there are
four types of division algebra [4], i.e., the real algebra, complex algebra,
quaternion algebra (Hamilton algebra)[5], and octonion algebra (Cayley
algebra)[6][7][8]. Real numbers are the numbers, which are used normally
without any imaginary number. The numbers written as the mixture of a real
number and the imaginary number are renowned by complex numbers. Quaternions
are the extension of a complex number with non-commutative property, used to
label the rotations in three-dimension. Cayley algebra is the extension of the
quaternions with non-commutative and non-associative as well. There are many
applications of quaternionic algebra in various branches of theoretical and
computational physics [9][10][11][12][13]. In electrodynamics, Maxwell’s
equations in the presence of magnetic monopoles [14] and the classical wave
equations of motion [15] have been constructed in terms of quaternionic
algebra. The quaternionic form of quantum electrodynamics has been discussed
[16] [17]. Chanyal [18][19] proposed the quaternionic covariant theory of
four-dimensional particle dyons in the form of relativistic quantum mechanics
and also focused on the quantized Dirac-Maxwell equations for dyons. Recently,
in magneto-hydrodynamics, the quaternionic dual fields equations for dyonic
cold plasma have been analyzed [20]. Further, it has been developed Dirac-
Maxwell, Bernoulli, and Navier Stokes like equations for dyonic fluid-plasma
in the generalized quaternionic field [21]. Currently, a new approach to
Dirac’s relativistic field equation for rotating free particle has been
investigated in quaternionic from [22]. Beyond the quaternionic algebra, many
authors [23][24][25][26][27][28][29][30][31][32][33][34][35] have studied the
role of higher dimensional hypercomplex division algebras in various fields of
modern physics. In GTR, Edmonds [36] discussed the quaternion wave equation in
curved space-time by using the curvilinear coordinate system in relativistic
quantum theory. The quaternionic form of curvature quantum theory fills the
gap between quantum and gravity theory. In the same way, Weng [37] studied the
electromagnetic and gravitational field equations in complex curved space with
the help of quaternions and octonions. Keeping in view the Riemannian space
and its connection with quaternions, we discussed the Einstein field-like
equation in quaternionic curvilinear form. Starting with quaternionic basis
transformation from one frame to another, we define the transformation of
scalar and vector field components of a quaternion curvature variable. From
the quaternionic covariant derivative, we expressed the Christoffel symbol,
quaternionic metric tensor, and quaternionic geodesic equation. We also
formulated the quaternionic Riemannian tensor that keeps track of how much
scalar and vector components of quaternion changed if we propagate parallel
along with a small parallelogram. Interestingly, if quaternionic Riemannian
Christoffel curvature is zero then the quaternionic curved space-time is
converted into flat space-time. Further, the quaternionic Ricci tensor is an
important contraction of the quaternionic Riemannian Christoffel tensor which
explains the changes in four space-time when an object parallels transport
along a geodesic. We have discussed the quaternionic form of the Einstein
field-like-equation in a compact form which shows that the quaternionic value
of matter-energy is equivalent to the quaternionic geometry.

##  2 Preliminaries

A tensor is the generalized form of a vector, defined by simply the
arrangement of numbers, or functions that transform according to certain rules
under a change of coordinates [38] [39]. According to Einstein summation
convention, if any same index appears twice in a term then that index stands
for the sum of all terms at the complete range of values. For example, we can
write the two forms of a vector $\mathbf{u}$ as

$\displaystyle\mathbf{u}\,=\,\,\sum_{i}u^{i}e_{i}\,\,,$
$\displaystyle\,\,\,\,\,\,\,\,\,\mathbf{u}\,=\,\,\sum_{j}u_{j}e^{j}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\forall\,\,(i,\,j=\,\,1,2,3)\,,$
(2.1)

where $i$ and $j$ are the repeated indices or summation indices while
$u^{i}\,\text{and\,}u_{j}$ are the vector components in contravariant and
covariant form, respectively. The scalar product of two vectors can be
represented in terms of indices with their components, i.e.,

$\displaystyle\mathbf{a}\cdot\mathbf{b}\,\,=\,\,$
$\displaystyle\left(\sum_{i}a_{i}e_{i}\right)\cdot\left(\sum_{j}b_{j}e_{j}\right)\,=\,\,\sum_{ij}a_{i}b_{j}\left(e_{i}\cdot
e_{j}\right)\,=\,\,\sum_{ij}a_{i}b_{j}\delta_{ij}\,=\,\,\sum_{i}a_{i}b_{i}\,,$
(2.2)

where $\delta_{ij}$ is the Kronecker delta symbol defined as
$\delta_{ij}=\,\,1,\,\text{for}\,(i=\,j)$ and
$\delta_{ij}=\,0,\,\text{for}\,(i\neq\,j)$. Also, the vector product becomes

$\displaystyle\left[\mathbf{a}\times\mathbf{b}\right]_{i}\,\,=\,\,$
$\displaystyle\sum_{jk}\epsilon_{ijk}a_{j}b_{k}\,.$ (2.3)

Here $\epsilon_{ijk}$ is the Levi-Civita symbol with three indices having
value $\epsilon_{ijk}=\,+1$ for cyclic permutation, $\epsilon_{ijk}=\,-1$ for
non-cyclic permutation, and $\epsilon_{ijk}=\,0$ for any two repeated indices.
The tensors recognized by their order viz. vectors are first-order tensors,
dyadics are second-order tensors, triadics are third-order tensors and
tetradics are fourth-order tensors. By adding any two tensors of the same rank
it gives the tensor of the same order, i.e.,

$\displaystyle\mathbf{A}_{st}^{r}\,\,=\,\,$
$\displaystyle\mathbf{B}_{st}^{r}+\mathbf{C}_{st}^{r}\,,$ (2.4)

where $\mathbf{A}_{st}^{r},\,\mathbf{B}_{st}^{r}$ and $\mathbf{C}_{st}^{r}$
are the tensors of same rank, $i.e.$ three-rank tensor. The product of two
tensors is given by

$\displaystyle\mathrm{B}_{s}^{r}\,\mathbf{C}_{uv}^{t}\,\,=\,\,$
$\displaystyle\mathcal{A}_{suv}^{rt}\,.$ (2.5)

In the above representation, $\mathrm{B}_{s}^{r}$ is tensor of rank two having
indices $r$ and $s$; $\mathbf{C}_{uv}^{t}$ is a tensor of rank three having
indices $t,u\text{ and }v,$ while $\mathcal{A}_{suv}^{rt}$ is a tensor of rank
five. If a magnitude or scalar quantity $\phi$ transform from one reference
frame to another, it remains invariant, such that [40]

$\displaystyle\phi^{{}^{\prime}}\,\,=\,\,$ $\displaystyle\phi\,.$ (2.6)

This invariant quantity may also be known as contravariant tensor of rank zero
or covariant tensor of rank zero. On the other hand, a vector quantity can be
transformed as

$\displaystyle
v^{{}^{\prime}r}\,:\longmapsto\,v^{s},\,\,\,\,\Longrightarrow\,\,\,\,v^{{}^{\prime}r}\,\,=\,\,$
$\displaystyle\frac{\partial x^{{}^{\prime}r}}{\partial x^{s}}v^{s}\,.$ (2.7)

Equation (2.7) signifies that the components of vector $v^{s}$ transformed to
components of vector $v^{{}^{\prime}r}$ when the coordinate $x^{s}$
transformed to $x^{{}^{\prime}r}.$ We can also define the contravariant tensor
components of rank two as

$\displaystyle\mathrm{A}^{{}^{\prime}ru}\,:\longmapsto\,\mathrm{A}^{st},\,\,\,\,\Longrightarrow\,\,\,\,\mathrm{A}^{{}^{\prime}ru}\,\,=\,\,$
$\displaystyle\frac{\partial x^{{}^{\prime}r}}{\partial x^{s}}\frac{\partial
x^{{}^{\prime}u}}{\partial x^{t}}\mathrm{A}^{st}\,.$ (2.8)

Similarly, we can transform the tensors for higher rank.

##  3 The quaternionic algebra

In mathematics, a complex number over the real algebra $\mathbb{R}$ having the
imaginary number $i$, is defined by the algebraic extension of a normal real
number. Any complex number $Z\in\mathbb{C}$ can be expressed in the form of
its basis $\left(1,i\right)$ as

$\displaystyle Z\,\,=$
$\displaystyle\,\,\xi_{1}+i\xi_{2},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\forall\,\left(\xi_{1},\,\xi_{2}\right)\in\,\mathbb{R}^{2}\,\,\text{and
}\,Z\in\,\,\mathbb{C}\,.$ (3.1)

If $Re\left(Z\right)=0$ then the complex number is called purely imaginary. In
the same way, the quaternion is a number system that extends the complex
number in a form of algebra having some properties over addition and
multiplication. W. R. Hamilton [5], extended the complex number in terms of
four-dimensional norm-division algebra (quaternion) over real algebra
$\mathbb{R}.$ The quaternions has four unit elements
$\left(e_{0},e_{1},e_{2},e_{3}\right)$ called basis elements, in which $e_{0}$
is the scalar unit and $e_{1},e_{2},e_{3}$ are the imaginary units. A
quaternion $\mathbb{H}\in\,\mathbb{Q}$ can be written as

$\displaystyle\mathbb{H}\,\,=$
$\displaystyle\,\,e_{0}w+e_{1}x+e_{2}y+e_{3}z\,=\,\,e_{0}w+\sum_{j=1}^{3}e_{j}r_{j}\,,$
(3.2)

where $w,x,y,z$ are the real numbers. The quaternion may also be composed of
the scalar and vector parts as

$\displaystyle\mathbb{H}\,\,=$
$\displaystyle\,\,S_{\mathbb{H}}+V_{\mathbb{H}}\,.$ (3.3)

Here $e_{0}w$ is the scalar part of quaternion denoted by $S_{\mathbb{H}}$ and
$\left(e_{1}x+e_{2}y+e_{3}z\right)$ is the vector part of quaternion denoted
by $V_{\mathbb{H}}.$ If scalar part is zero in equation (3.3), then

$\displaystyle\mathbb{H}\,:\rightarrow\,\,V_{\mathbb{H}}\,\,=\,\,$
$\displaystyle e_{1}x+e_{2}y+e_{3}z\,,$ (3.4)

is known as right quaternion or pure quaternion. Although any quaternion can
be seen as a vector in a four-dimensional vector space, it is usually referred
to as the pure quaternion as a vector. Conditionally, if the vector part is
zero then

$\displaystyle\mathbb{H}\,:\rightarrow\,\,S_{\mathbb{H}}\,\,=\,\,$
$\displaystyle e_{0}w\,,$ (3.5)

is known as scalar quaternion. The addition of two quaternions will be

$\displaystyle\mathbb{A+B\,\,=}$
$\displaystyle\,\,\left(e_{0}A_{0}+e_{1}A_{1}+e_{2}A_{2}+e_{3}A_{3}\right)+\left(e_{0}B_{0}+e_{1}B_{1}+e_{2}B_{2}+e_{3}B_{3}\right)$
$\displaystyle\mathbb{=}$
$\displaystyle\,\,e_{0}\left(A_{0}+B_{0}\right)+e_{1}\left(A_{1}+B_{1}\right)+e_{2}\left(A_{2}+B_{2}\right)+e_{3}\left(A_{3}+B_{3}\right)$
$\displaystyle=$
$\displaystyle\,\,e_{0}\left(H_{0}\right)+e_{j}\left(H_{j}\right)\,\,\,\,\,\forall\,\,\,(j=1,2,3)\,,$
(3.6)

where $H_{0}\sim(A_{0}+B_{0})$ is the scalar part of quaternion while
$H_{j}\sim(A_{j}+B_{j})$ is the vector part of quaternion. Therefore, the
equation (3.6) shows the closure property concerning quaternionic addition.
The quaternionic algebra also satisfies the associative and commutative
properties of addition. Further, the multiplication properties of two
quaternions can be expressed by

$\displaystyle\mathbb{A\circ B}\,\,=\,\,$
$\displaystyle\left(e_{0}A_{0}+e_{1}A_{1}+e_{2}A_{2}+e_{3}A_{3}\right)\circ\left(e_{0}B_{0}+e_{1}B_{1}+e_{2}B_{2}+e_{3}B_{3}\right)$
$\displaystyle=\,\,$ $\displaystyle
e_{0}P_{0}+e_{1}P_{1}+e_{2}P_{2}+e_{3}P_{3}\,\,=\,\,\mathbb{P}\in\,\mathbb{Q}\,,$
(3.7)

where ${}^{\prime}\circ^{\prime}$ is a symbol used for the quaternionic
multiplication and the components of $\mathbb{P}$ given in equation (3.7) are
written by

$\displaystyle P_{0}:=\,\,$
$\displaystyle\left(A_{0}B_{0}-A_{1}B_{1}-A_{2}B_{2}-A_{3}B_{3}\right)\,\,\,\,\,\,\,\,\,\text{(coefficient
of $e_{0}$)}$ $\displaystyle P_{1}:=\,\,$
$\displaystyle\left(A_{0}B_{1}+A_{1}B_{0}+A_{2}B_{3}-A_{3}B_{2}\right)\,\,\,\,\,\,\,\,\,(\text{coefficient
of $e_{1}$})$ $\displaystyle P_{2}:=\,\,$
$\displaystyle\left(A_{0}B_{2}+A_{2}B_{0}+A_{3}B_{1}-A_{1}B_{3}\right)\,\,\,\,\,\,\,\,\,(\text{coefficient
of $e_{2}$})$ (3.8) $\displaystyle P_{3}:=\,\,$
$\displaystyle\left(A_{0}B_{3}+A_{3}B_{0}+A_{1}B_{2}-A_{2}B_{1}\right)\,\,\,\,\,\,\,\,\,\,(\text{coefficient
of $e_{3}$)}.$

Here the quaternionic multiplication follows the given rules for basis
$\left(e_{0},e_{1},e_{2},e_{3}\right),$ i.e.,

$\displaystyle e_{0}^{2}\,\,=$
$\displaystyle\,\,1\,,\,\,\,\,\,e_{i}^{2}\,\,=\,\,-1\,,\,\,\,e_{0}e_{i}\,\,=\,\,e_{i}e_{0}=\,\,e_{i}\,,$
$\displaystyle e_{i}e_{j}\,\,=$
$\displaystyle\,\,-\delta_{ij}e_{0}+\epsilon_{ijk}e_{k}\,,\,\,\,\,\,\forall\,\,\,(i,j,k=\,\,1,2,3)\,.$
(3.9)

All indices in Levi-Civita symbol $\epsilon_{ijk}$ are antisymmetric and
satisfy $e_{i}\times e_{j}=\,\,\sum_{k}\epsilon_{ijk}e_{k}.$ Thus, equation
(3.7) can be written in compact form in terms of ordinary dot and cross
product as,

$\displaystyle\mathbb{A\circ B\,\,=}$
$\displaystyle\,\,e_{0}\left(A_{0}B_{0}-\overrightarrow{A}\cdot\overrightarrow{B}\right)+e_{j}\left[A_{0}\overrightarrow{B}+B_{0}\overrightarrow{A}+\left(\overrightarrow{A}\times\overrightarrow{B}\right)_{j}\right]\,.$
(3.10)

In order to check the non-commutative property, we may write

$\displaystyle\mathbb{B\circ A}\,\,=$
$\displaystyle\,\,\left(e_{0}B_{0}+e_{1}B_{1}+e_{2}B_{2}+e_{3}B_{3}\right)\circ\left(e_{0}A_{0}+e_{1}A_{1}+e_{2}A_{2}+e_{3}A_{3}\right)$
$\displaystyle\mathbb{=}$
$\displaystyle\,\,e_{0}\left(B_{0}A_{0}-B_{1}A_{1}-B_{2}A_{2}-B_{3}A_{3}\right)+e_{1}\left(B_{0}A_{1}+B_{1}A_{0}+B_{2}A_{3}-B_{3}A_{2}\right)$
$\displaystyle+e_{2}\left(B_{0}A_{2}+B_{2}A_{0}+B_{3}A_{1}-B_{1}A_{3}\right)+e_{3}\left(B_{0}A_{3}+B_{3}A_{0}+B_{1}A_{2}-B_{2}A_{1}\right)$
$\displaystyle=$
$\displaystyle\,\,e_{0}\left(B_{0}A_{0}-\overrightarrow{B}\cdot\overrightarrow{A}\right)+e_{j}\left[B_{0}\overrightarrow{A}+A_{0}\overrightarrow{B}+\left(\overrightarrow{B}\times\overrightarrow{A}\right)_{j}\right]\,.$
$\displaystyle\neq$ $\displaystyle\,\,\mathbb{A\circ B}\,.$ (3.11)

From equation (3.11) it is clear that the product of two quaternions is non-
commutative because the product of two vectors is always non-commutative, so
that
$\overrightarrow{A}\times\overrightarrow{B}=-\overrightarrow{B}\times\overrightarrow{A}.$
On the other hand, the multiplication of quaternions satisfy the associative
property i.e.,
$\left(\mathbb{A}\circ\mathbb{B}\right)\circ\mathbb{C}=\,\mathbb{A}\circ\left(\mathbb{B}\circ\mathbb{C}\right)$.
Further, the quaternionic conjugate of equation (3.2) can be expressed as

$\displaystyle\mathbb{H^{\ast}}\,\,=\,\,$ $\displaystyle
e_{0}w-\left(e_{1}x+e_{2}y+e_{3}z\right)\,,$ (3.12)

and the product of two quaternions is given by [41]

$\displaystyle\mathbb{A\cdot B\,\,=}\,\,$
$\displaystyle-\frac{1}{2}\left(\mathbb{A\circ B^{\ast}+\mathbb{B\circ
A^{\ast}}}\right)=\,\,-\frac{1}{2}\left(\mathbb{A^{\ast}\circ
B+\mathbb{B^{\ast}\circ A}}\right)\,.$ (3.13)

The norm of a quaternion $\mathbb{H}$ denoted by $|\mathbb{H}|$ can be
represented as

$\displaystyle|\mathbb{H}|\,\,=\,\,$
$\displaystyle\sqrt{\mathbb{H}\circ\mathbb{H^{\ast}}}=\,\,\sqrt{w^{2}+x^{2}+y^{2}+z^{2}}\,.$
(3.14)

Now, the inverse of a quaternion can be defined by

$\displaystyle\mathbb{H}^{-1}\,\,=\,\,$
$\displaystyle\frac{\mathbb{H^{\ast}}}{|\mathbb{H}|^{2}}\,\,\equiv\,\,\frac{e_{0}w-e_{1}x-e_{2}y-e_{3}z}{w^{2}+x^{2}+y^{2}+z^{2}}\,.$
(3.15)

Moreover, the quotient of two vectors is also known as quaternion which can be
represented by [42]

$\displaystyle\mathbb{\mathbb{H}\,\,=}$
$\displaystyle\,\,\frac{\alpha}{\beta}\,,$ (3.16)

where $\mathbf{\alpha}$ and $\beta$ are the two vectors. The another approach
on a “Tensor of a quaternion” $\left(\mathbf{T}_{\mathbb{H}}\right)$ is given
by Hamilton [42], which can be written by

$\displaystyle\left(\mathbf{T_{\mathbb{H}}}\right)^{\mathrm{2}}\,\,=\,\,$
$\displaystyle\mathbb{H\circ H^{\ast}}\,\,=\,\,w^{2}+x^{2}+y^{2}+z^{2},$
(3.17)

so that $T_{\mathbb{H}}=\sqrt{\left(w^{2}+x^{2}+y^{2}+z^{2}\right)}\,.$ The
tensor of a quaternion is similar to the norm of a quaternion, it will be
always positive or real number [43]. If the scalar part is zero then
$\mathbf{T}\left(\mathbf{V}_{\mathbb{H}}\right)\,=\,\sqrt{\left(x^{2}+y^{2}+z^{2}\right)}$
is known as the tensor of a vector quaternion and if vector part is zero then
$\mathbf{T\left(\mathbf{S_{\mathbb{H}}}\right)}\,=\,w$ is known as the tensor
of a scalar quaternion. Further, the ’versor’ of a quaternion which indicates
direction, is a unit quaternion known as a normalized quaternion
$(\mathbf{U_{\mathbb{H}}})$, so that

$\displaystyle\mathbf{U_{\mathbb{H}}\,\,=\,\,}$
$\displaystyle\frac{\mathbb{H}}{|\mathbb{H}|}\,\,\equiv\,\,\frac{e_{0}w+e_{1}x+e_{2}y+e_{3}z}{\sqrt{\left(w^{2}+x^{2}+y^{2}+z^{2}\right)}}\,.$
(3.18)

The scalar and vector parts of a quaternionic versor can be expressed by
$\mathbf{\mathbf{S}\left(\mathbf{U_{\mathbb{H}}}\right)\,\,=}$
$w\left(w^{2}+x^{2}+y^{2}+z^{2}\right)^{-1/2}$ and
$\mathbf{V\left(\mathbf{U_{\mathbb{H}}}\right)\,\,=\,\,}(e_{1}x+e_{2}y+e_{3}z)$
$\left(w^{2}+x^{2}+y^{2}+z^{2}\right)^{-1/2}$. A quaternion can also be
written in terms of tensor of a quaternion and versor of a quaternion as [42]

$\displaystyle\mathbb{H}\,\,=\,\,$
$\displaystyle\mathbf{T}_{\mathbb{H}}\mathbf{U}_{\mathbb{H}}=\,\,e_{0}w+e_{1}x+e_{2}y+e_{3}z\,.$
(3.19)

##  4 Transformation of quaternionic basis elements

If a system of coordinates rotates from its original frame of reference then
there will be a change in unit vectors to original unit vectors. This change
of unit vectors represents the transformation equations. Here, we introduce
the transformation equations for the 2-dimensional system in which the
equations are used for the transformation of coordinates from one plane to
another plane, i.e.,

$\displaystyle
x^{{}^{\prime}}\,\,=\,\,x\cos\phi+y\sin\phi\,,\,\,\,\,\,\,\,\,\,y^{{}^{\prime}}\,\,=\,\,$
$\displaystyle-x\sin\phi+y\cos\phi\,,$ (4.1)

which gives

$\displaystyle\frac{\partial x^{{}^{\prime}}}{\partial x}\,\,=\,\,$
$\displaystyle\cos\phi,\,\,\,\,\,\frac{\partial x^{{}^{\prime}}}{\partial
y}\,\,=\,\,\sin\phi,\,\,\,\,\,\frac{\partial y^{{}^{\prime}}}{\partial
x}\,\,=\,\,-\sin\phi,\,\,\,\,\,\frac{\partial y^{{}^{\prime}}}{\partial
y}\,\,=\,\,\cos\phi\,.$ (4.2)

Thus, the transformation of basis elements can be written in the matrix form

$\displaystyle\left(\begin{array}[]{c}e_{1}^{{}^{\prime}}\\\
e_{2}^{{}^{\prime}}\end{array}\right)\,\,:\longmapsto\,\,$
$\displaystyle\left(\begin{array}[]{cc}\cos\phi&\sin\phi\\\
-\sin\phi&\cos\phi\end{array}\right)\left(\begin{array}[]{c}e_{1}\\\
e_{2}\end{array}\right)\,,$ (4.9)

which leads to following linear transformation equations

$\displaystyle e_{1}^{{}^{\prime}}\,\longmapsto\,\,$
$\displaystyle\frac{\partial x^{{}^{\prime}}}{\partial x}e_{1}+\frac{\partial
x^{{}^{\prime}}}{\partial y}e_{2}\,,$ $\displaystyle
e_{2}^{{}^{\prime}}\,\longmapsto\,\,$ $\displaystyle\frac{\partial
y^{{}^{\prime}}}{\partial x}e_{1}+\frac{\partial y^{{}^{\prime}}}{\partial
y}e_{2}\,,$ (4.10)

where $e_{1}$ and $e_{2}$ are the unit vectors corresponding to
$x\,\text{and}\,y-$ axis in $XY-$ plane, while $e_{1}^{{}^{\prime}}$ and
$e_{2}^{{}^{\prime}}$ are the unit vectors corresponding to
$x^{{}^{\prime}}\,\text{and}\,y^{{}^{\prime}}-$ axis in
$X^{{}^{\prime}}Y^{{}^{\prime}}-$ plane, respectively. $\phi$ is the angle
between these two planes. Like 2-D transformation, we may extend the
transformation relations of basis elements for 3-D system. Thus, the 3-D
transformation matrix $\left(D\right)$ can be written by

$\displaystyle D\,\,=\,\,$
$\displaystyle\left(\begin{array}[]{ccc}\cos\psi\cos\phi-\sin\psi\sin\phi\cos\theta&-\cos\psi\sin\phi-\sin\psi\cos\theta\cos\phi&\sin\psi\sin\theta\\\
\sin\psi\cos\phi+\cos\psi\sin\phi\cos\theta&-\sin\psi\sin\phi+\cos\psi\cos\theta\cos\phi&-\cos\psi\sin\theta\\\
\sin\theta\sin\phi&\sin\theta\cos\phi&\cos\theta\end{array}\right)\,,$ (4.14)

where $\psi,\theta,\phi$ are the three independent parameters known as Euler
angles which represents the rotation of axis in new frame of reference with
respect to original axis. Let $\left(e_{1},e_{2},e_{3}\right)$ are pure
quaternionic unit vectors along $x,y,z$\- axis in $S-$ frame while
$\left(e_{1}^{{}^{\prime}},e_{2}^{{}^{\prime}},e_{3}^{{}^{\prime}}\right)$ are
unit vectors for $x^{{}^{\prime}},y^{{}^{\prime}},z^{{}^{\prime}}$\- axis in
$S^{{}^{\prime}}-$ frame, then the linear transformation relations for pure
quaternionic unit vectors become,

$\displaystyle e_{1}^{{}^{\prime}}\,\longmapsto$
$\displaystyle\,\,\frac{\partial x^{{}^{\prime}}}{\partial
x}e_{1}+\frac{\partial x^{{}^{\prime}}}{\partial y}e_{2}+\frac{\partial
x^{{}^{\prime}}}{\partial z}e_{3}\,,$ $\displaystyle
e_{2}^{{}^{\prime}}\,\longmapsto$ $\displaystyle\,\,\frac{\partial
y^{{}^{\prime}}}{\partial x}e_{1}+\frac{\partial y^{{}^{\prime}}}{\partial
y}e_{2}+\frac{\partial y^{{}^{\prime}}}{\partial z}e_{3}\,,$ $\displaystyle
e_{3}^{{}^{\prime}}\,\longmapsto$ $\displaystyle\,\,\frac{\partial
z^{{}^{\prime}}}{\partial x}e_{1}+\frac{\partial z^{{}^{\prime}}}{\partial
y}e_{2}+\frac{\partial z^{{}^{\prime}}}{\partial z}e_{3}\,,$ (4.15)

where the derivatives become

$\displaystyle\frac{\partial x^{{}^{\prime}}}{\partial x}\,=\,\,$
$\displaystyle\cos\psi\cos\phi-\sin\psi\sin\phi\cos\theta,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{\partial
x^{{}^{\prime}}}{\partial
y}\,=\,\,-\cos\psi\sin\phi-\sin\psi\cos\theta\cos\phi\,,$
$\displaystyle\frac{\partial x^{{}^{\prime}}}{\partial z}\,=\,\,$
$\displaystyle\sin\psi\sin\theta,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{\partial
y^{{}^{\prime}}}{\partial
x}\,=\,\,\sin\psi\cos\phi+\cos\psi\sin\phi\cos\theta\,,$
$\displaystyle\frac{\partial y^{{}^{\prime}}}{\partial y}\,=\,\,$
$\displaystyle-\sin\psi\sin\phi+\cos\psi\cos\theta\cos\phi,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{\partial
y^{{}^{\prime}}}{\partial z}\,=\,\,-\cos\psi\sin\theta\,,$
$\displaystyle\frac{\partial z^{{}^{\prime}}}{\partial x}\,=\,\,$
$\displaystyle\sin\theta\sin\phi,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{\partial
z^{{}^{\prime}}}{\partial
y}\,=\,\,\sin\theta\cos\phi,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{\partial
z^{{}^{\prime}}}{\partial z}\,=\,\,\cos\theta\,.$

Interestingly, these transformation equations satisfy the properties of vector
algebra. The cross product of two vectors is always non-commutative for pure
quaternionic space. Moreover, the properties of pure quaternionic unit
elements $\left(e_{1},e_{2},e_{3}\right)$ corresponding to vector algebra
basis can be represented as

$\displaystyle e_{1}^{{}^{\prime}}\circ e_{1}^{{}^{\prime}}\,=\,\,$
$\displaystyle\left(\frac{\partial x^{{}^{\prime}}}{\partial
x}e_{1}+\frac{\partial x^{{}^{\prime}}}{\partial y}e_{2}+\frac{\partial
x^{{}^{\prime}}}{\partial z}e_{3}\right)\circ\left(\frac{\partial
x^{{}^{\prime}}}{\partial x}e_{1}+\frac{\partial x^{{}^{\prime}}}{\partial
y}e_{2}+\frac{\partial x^{{}^{\prime}}}{\partial z}e_{3}\right)$
$\displaystyle=\,\,$ $\displaystyle-\left(\frac{\partial
x^{{}^{\prime}}}{\partial x}\right)^{2}-\left(\frac{\partial
x^{{}^{\prime}}}{\partial y}\right)^{2}-\left(\frac{\partial
x^{{}^{\prime}}}{\partial z}\right)^{2}$ $\displaystyle=\,\,$
$\displaystyle-\left(\cos\psi\cos\phi-\sin\psi\sin\phi\cos\theta\right)^{2}-\left(-\cos\psi\sin\phi-\sin\psi\cos\theta\cos\phi\right)^{2}$
$\displaystyle-\left(\sin\psi\sin\theta\right)^{2}=\,\,-1\,.$ (4.16)

and

$\displaystyle e_{2}^{{}^{\prime}}\circ e_{2}^{{}^{\prime}}\,\,=\,\,$
$\displaystyle-1,\,\,\,\,\,\,\,e_{3}^{{}^{\prime}}\circ
e_{3}^{{}^{\prime}}\,\,=\,\,-1\,.$ (4.17)

On the other hand

$\displaystyle e_{1}^{{}^{\prime}}\circ e_{2}^{{}^{\prime}}\,\,=\,\,$
$\displaystyle\left(\frac{\partial x^{{}^{\prime}}}{\partial
x}e_{1}+\frac{\partial x^{{}^{\prime}}}{\partial y}e_{2}+\frac{\partial
x^{{}^{\prime}}}{\partial z}e_{3}\right)\circ\left(\frac{\partial
y^{{}^{\prime}}}{\partial x}e_{1}+\frac{\partial y^{{}^{\prime}}}{\partial
y}e_{2}+\frac{\partial y^{{}^{\prime}}}{\partial z}e_{3}\right)$
$\displaystyle=\,\,$ $\displaystyle-\frac{\partial x^{{}^{\prime}}}{\partial
x}\frac{\partial y^{{}^{\prime}}}{\partial x}-\frac{\partial
x^{{}^{\prime}}}{\partial y}\frac{\partial y^{{}^{\prime}}}{\partial
y}-\frac{\partial x^{{}^{\prime}}}{\partial z}\frac{\partial
y^{{}^{\prime}}}{\partial z}-\left(\frac{\partial x^{{}^{\prime}}}{\partial
z}\frac{\partial y^{{}^{\prime}}}{\partial y}-\frac{\partial
x^{{}^{\prime}}}{\partial y}\frac{\partial y^{{}^{\prime}}}{\partial
z}\right)e_{1}$ $\displaystyle-\,\,$ $\displaystyle\left(\frac{\partial
x^{{}^{\prime}}}{\partial x}\frac{\partial y^{{}^{\prime}}}{\partial
z}-\frac{\partial x^{{}^{\prime}}}{\partial z}\frac{\partial
y^{{}^{\prime}}}{\partial x}\right)e_{2}-\left(\frac{\partial
x^{{}^{\prime}}}{\partial y}\frac{\partial y^{{}^{\prime}}}{\partial
x}-\frac{\partial x^{{}^{\prime}}}{\partial x}\frac{\partial
y^{{}^{\prime}}}{\partial y}\right)e_{3}$ $\displaystyle=\,\,$
$\displaystyle-\left(\cos\psi\cos\phi-\sin\psi\sin\phi\cos\theta\right)\left(\sin\psi\cos\phi+\cos\psi\sin\phi\cos\theta\right)$
$\displaystyle-\,\,$
$\displaystyle\left(-\cos\psi\sin\phi-\sin\psi\cos\theta\cos\phi\right)\left(-\sin\psi\sin\phi+\cos\psi\cos\theta\cos\phi\right)$
$\displaystyle-\,\,$
$\displaystyle\left(\sin\psi\sin\theta\right)\left(-\cos\psi\sin\theta\right)-e_{1}[\left(\sin\psi\sin\theta\right)\left(-\sin\psi\sin\phi+\cos\psi\cos\theta\cos\phi\right)$
$\displaystyle+\,\,$
$\displaystyle\left(-\cos\psi\sin\phi-\sin\psi\cos\theta\cos\phi\right)\left(-\cos\psi\sin\theta\right)]$
$\displaystyle-\,\,$ $\displaystyle
e_{2}[\left(\cos\psi\cos\phi-\sin\psi\sin\phi\cos\theta\right)\left(-\cos\psi\sin\theta\right)$
$\displaystyle+\,\,$
$\displaystyle\left(\sin\psi\sin\theta\right)\left(\sin\psi\cos\phi+\cos\psi\sin\phi\cos\theta\right)]$
$\displaystyle-\,\,$ $\displaystyle
e_{3}[\left(-\cos\psi\sin\phi-\sin\psi\cos\theta\cos\phi\right)\left(\sin\psi\cos\phi+\cos\psi\sin\phi\cos\theta\right)$
$\displaystyle+\,\,$
$\displaystyle\left(\cos\psi\cos\phi-\sin\psi\sin\phi\cos\theta\right)\left(-\sin\psi\sin\phi+\cos\psi\cos\theta\cos\phi\right)]$
$\displaystyle=\,\,$ $\displaystyle e_{3}^{{}^{\prime}}\,,$ (4.18)

and

$\displaystyle e_{2}^{{}^{\prime}}\circ e_{1}^{{}^{\prime}}\,\,=\,\,$
$\displaystyle\left(\frac{\partial y^{{}^{\prime}}}{\partial
x}e_{1}+\frac{\partial y^{{}^{\prime}}}{\partial y}e_{2}+\frac{\partial
y^{{}^{\prime}}}{\partial z}e_{3}\right)\circ\left(\frac{\partial
x^{{}^{\prime}}}{\partial x}e_{1}+\frac{\partial x^{{}^{\prime}}}{\partial
y}e_{2}+\frac{\partial x^{{}^{\prime}}}{\partial z}e_{3}\right)$
$\displaystyle=\,\,$ $\displaystyle-\frac{\partial y^{{}^{\prime}}}{\partial
x}\frac{\partial x^{{}^{\prime}}}{\partial x}-\frac{\partial
y^{{}^{\prime}}}{\partial y}\frac{\partial x^{{}^{\prime}}}{\partial
y}-\frac{\partial y^{{}^{\prime}}}{\partial z}\frac{\partial
x^{{}^{\prime}}}{\partial z}+\left(\frac{\partial y^{{}^{\prime}}}{\partial
y}\frac{\partial x^{{}^{\prime}}}{\partial z}-\frac{\partial
y^{{}^{\prime}}}{\partial z}\frac{\partial x^{{}^{\prime}}}{\partial
y}\right)e_{1}$ $\displaystyle+\,\,$ $\displaystyle\left(\frac{\partial
y^{{}^{\prime}}}{\partial z}\frac{\partial x^{{}^{\prime}}}{\partial
x}-\frac{\partial y^{{}^{\prime}}}{\partial x}\frac{\partial
x^{{}^{\prime}}}{\partial z}\right)e_{2}+\left(\frac{\partial
y^{{}^{\prime}}}{\partial x}\frac{\partial x^{{}^{\prime}}}{\partial
y}-\frac{\partial y^{{}^{\prime}}}{\partial y}\frac{\partial
x^{{}^{\prime}}}{\partial x}\right)e_{3}$ $\displaystyle=\,\,$
$\displaystyle-\left(\sin\psi\cos\phi+\cos\psi\sin\phi\cos\theta\right)\left(\cos\psi\cos\phi-\sin\psi\sin\phi\cos\theta\right)$
$\displaystyle-\,\,$
$\displaystyle\left(-\sin\psi\sin\phi+\cos\psi\cos\theta\cos\phi\right)\left(-\cos\psi\sin\phi-\sin\psi\cos\theta\cos\phi\right)$
$\displaystyle-\,\,$
$\displaystyle\left(-\cos\psi\sin\theta\right)\left(\sin\psi\sin\theta\right)+e_{1}[\left(\sin\psi\sin\theta\right)\left(-\sin\psi\sin\phi+\cos\psi\cos\theta\cos\phi\right)$
$\displaystyle-\,\,$
$\displaystyle\left(-\cos\psi\sin\phi-\sin\psi\cos\theta\cos\phi\right)\left(-\cos\psi\sin\theta\right)]$
$\displaystyle+\,\,$ $\displaystyle
e_{2}[\left(\cos\psi\cos\phi-\sin\psi\sin\phi\cos\theta\right)\left(-\cos\psi\sin\theta\right)$
$\displaystyle-\,\,$
$\displaystyle\left(\sin\psi\sin\theta\right)\left(\sin\psi\cos\phi+\cos\psi\sin\phi\cos\theta\right)]$
$\displaystyle+\,\,$ $\displaystyle
e_{3}[\left(-\cos\psi\sin\phi-\sin\psi\cos\theta\cos\phi\right)\left(\sin\psi\cos\phi+\cos\psi\sin\phi\cos\theta\right)$
$\displaystyle-\,\,$
$\displaystyle\left(\cos\psi\cos\phi-\sin\psi\sin\phi\cos\theta\right)\left(-\sin\psi\sin\phi+\cos\psi\cos\theta\cos\phi\right)]$
$\displaystyle=\,\,\,$ $\displaystyle-e_{3}^{{}^{\prime}}\,.$ (4.19)

Similarly, we obtain

$\displaystyle e_{2}^{{}^{\prime}}\circ e_{3}^{{}^{\prime}}\,\,=\,\,$
$\displaystyle e_{1}^{{}^{\prime}}\,,\,\,\,\,e_{3}^{{}^{\prime}}\circ
e_{2}^{{}^{\prime}}\,=\,\,-e_{1}^{{}^{\prime}}\,,$ $\displaystyle
e_{3}^{{}^{\prime}}\circ e_{1}^{{}^{\prime}}\,\,=\,\,$ $\displaystyle
e_{2}^{{}^{\prime}}\,,\,\,\,\,e_{1}^{{}^{\prime}}\circ
e_{3}^{{}^{\prime}}\,=\,\,-e_{2}^{{}^{\prime}}\,.$ (4.20)

Now, we easily can extend the pure-quaternion to quaternion by adding the
scalar unit element $e_{0}$. It should be noticed that the scalar unit element
becomes invariant under transformation given in equation (2.6) but it plays an
important role with unit elements $e_{1},e_{2},e_{3}$ in quaternionic
transformation. Myszkowski [44] has given the idea of 4-D transformation
similar to the 3-D transformation. Thus, we assume the transformation
equations for quaternionic basis elements as

$\displaystyle e_{0}^{{}^{\prime}}\,\longmapsto\,\,$ $\displaystyle e_{0}\,,$
$\displaystyle e_{1}^{{}^{\prime}}\,\longmapsto\,\,$
$\displaystyle\frac{\partial x^{{}^{\prime}}}{\partial t}e_{0}+\frac{\partial
x^{{}^{\prime}}}{\partial x}e_{1}+\frac{\partial x^{{}^{\prime}}}{\partial
y}e_{2}+\frac{\partial x^{{}^{\prime}}}{\partial z}e_{3}\,,$ $\displaystyle
e_{2}^{{}^{\prime}}\,\longmapsto\,\,$ $\displaystyle\frac{\partial
y^{{}^{\prime}}}{\partial t}e_{0}+\frac{\partial y^{{}^{\prime}}}{\partial
x}e_{1}+\frac{\partial y^{{}^{\prime}}}{\partial y}e_{2}+\frac{\partial
y^{{}^{\prime}}}{\partial z}e_{3}\,,$ $\displaystyle
e_{3}^{{}^{\prime}}\,\longmapsto\,\,$ $\displaystyle\frac{\partial
z^{{}^{\prime}}}{\partial t}e_{0}+\frac{\partial z^{{}^{\prime}}}{\partial
x}e_{1}+\frac{\partial z^{{}^{\prime}}}{\partial y}e_{2}+\frac{\partial
z^{{}^{\prime}}}{\partial z}e_{3}\,.$ (4.21)

Here, we focused on space-time structure for quaternionic transformation, the
quaternionic properties will also be satisfied as similar to the 3-D rotation.
Now, in the next section, we shall interpret these relations for Riemannian
geometry.

##  5 Quaternionic approach on Riemannian geometry

###  5.1 Quaternion transformation

In Riemannian geometry, the motion of object takes place in curvature space-
time in which 4-dimensional frame of reference rotate along the curve. The
structure of quaternionic space-time coordinates can be written as
$P^{\mu}\equiv(P^{0},P^{j})\,\simeq\,\,\left(P^{\xi},P^{1},P^{2},P^{3}\right)$.
The time coordinate may refer corresponding to $e_{0}$ i.e. $t\mapsto
P^{\xi},$ and the spatial coordinates may refer corresponding to $e_{j}$ i.e.
$(x,y,z)\mapsto\left(P^{1},P^{2},P^{3}\right)$ where $j=\,\,1,2,3$. Now, the
quaternionic elements may transformation as

$\displaystyle e_{0}^{{}^{\prime}}\,\,=\,\,$ $\displaystyle
e_{0},\,\,\,\,e_{i}^{{}^{\prime}}\,\,=\,\,\frac{\partial
P^{{}^{\prime}i}}{\partial
P^{\mu}}e_{\mu},\,\,\>\,\,\,\,\,\,\,\,\,\,\,\,\,\,\forall\,\,(i=\,\,1,2,3)\,.$
(5.1)

The transformation of scalar and vector field components of a quaternion
variable ($\mathbb{H}$) from $S$ to $S^{{}^{\prime}}$-frame can be expressed
as

$\displaystyle H^{\xi}\,\,=\,\,$ $\displaystyle
H^{{}^{\prime}\xi}+\frac{\partial P^{\xi}}{\partial
P^{{}^{\prime}i}}H^{{}^{\prime}i}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{(coefficient
of }e_{0}\text{)}\,,$ (5.2) $\displaystyle H^{j}\,\,=\,\,$
$\displaystyle\frac{\partial P^{j}}{\partial
P^{{}^{\prime}i}}H^{{}^{\prime}i}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{(coefficient
of }e_{j}\text{)}\,.$ (5.3)

In the above transformation relation (5.2), we assume that the effect of
$H^{{}^{\prime}\xi}$ on $H^{\xi}$ is linear because it shows the simple
transformation on flat space-time. So, we can neglect the component
$H^{{}^{\prime}\xi}$ to get the quaternionic transformation (Q-transformation)
in curved space-time. Therefore, we have

$\displaystyle\mathbb{H}^{{}^{\prime}}\,\,:\longmapsto\,\,\mathbb{H}\,\,=\,\,\,$
$\displaystyle\left(\frac{\partial P^{\xi}}{\partial
P^{{}^{\prime}i}}H^{{}^{\prime}i},\,\,\frac{\partial P^{j}}{\partial
P^{{}^{\prime}i}}H^{{}^{\prime}i}\right)\,.$ (5.4)

Equation (5.4) represents the Q-transformation in contravariant form, we can
also write the above transformation relations in quaternionic covariant form
as

$\displaystyle H_{\xi}\,\,=\,\,$ $\displaystyle\frac{\partial
P^{{}^{\prime}i}}{\partial
P^{\xi}}H_{i}^{{}^{\prime}}\,\,,\,\,\,\,\,\,H_{j}\,\,=\,\,\frac{\partial
P^{{}^{\prime}i}}{\partial P^{j}}H_{i}^{{}^{\prime}}\,\,.$ (5.5)

Now, we can write the transformation of quaternionic differential operator
$\square\simeq\\{\partial_{\phi},\,\partial_{j}\\}$ as

$\displaystyle\partial_{\phi}\,\,=\,\,$ $\displaystyle\frac{\partial
P^{\phi}}{\partial
P^{{}^{\prime}k}}\partial_{k}^{{}^{\prime}}\,,\,\,\,\,\,\,\,\,\,\,\partial_{j}\,\,=\,\,\frac{\partial
P^{j}}{\partial
P^{{}^{\prime}k}}\partial_{k}^{{}^{\prime}}\,,\,\,\,\,\,\,\,\,\,\,\,\,\forall\,(j,k=1,2,3)\,.$
(5.6)

Since the components of a quaternion can also represent in form of
coordinates, then we can write the transformation of quaternionic coordinates
[36],

$\displaystyle dP^{\xi}\,\,=\,\,$ $\displaystyle\frac{\partial
P^{\xi}}{\partial
P^{{}^{\prime}i}}\,dP^{{}^{\prime}i}\,,\,\,\,\,\,\,\,\,\,dP^{j}\,\,=\,\,\frac{\partial
P^{j}}{\partial
P^{{}^{\prime}i}}\,dP^{{}^{\prime}i},\,\,\,\,\,\,\,\forall\,\left(i,j=\,\,1,2,3\right)\,.$
(5.7)

###  5.2 Quaternionic covariant derivative

In the quaternionic form of curvature space-time, the scalar field derivative
of equation (5.5) can be written as

$\displaystyle H_{;\phi}^{\xi}\,\,=\,\,$ $\displaystyle
H_{,\phi}^{\xi}+\left\\{\begin{array}[]{c}\xi\\\
\phi\phi\end{array}\right\\}\,H^{\phi}\,,$ (5.10)

where the Christoffel symbol is denoted by $\left\\{\begin{array}[]{c}\xi\\\
\phi\phi\end{array}\right\\}\,=\,\frac{\partial}{\partial
P^{\phi}}\left(\frac{\partial P^{\xi}}{\partial P^{\phi}}\right)$. Similarly,
we have

$\displaystyle H_{\xi;\phi}\,\,=\,\,$ $\displaystyle
H_{\xi,\phi}-\left\\{\begin{array}[]{c}\phi\\\
\phi\xi\end{array}\right\\}\,H_{\phi}\,.$ (5.13)

Here the role of Christoffel symbol is very important for Riemannian space-
time. If $\left\\{\begin{array}[]{c}\xi\\\
\phi\phi\end{array}\right\\}\rightarrow\,0$, then the quaternionic curvature
covariant derivative transforms into the quaternionic usual partial derivative
which indicates the linear transformation. Correspondingly, applying
quaternionic derivative $\partial_{\phi}$ on $H^{j}$, we obtain

$\displaystyle H_{;\phi}^{j}\,\,=\,\,$ $\displaystyle
H_{,\phi}^{j}+\left\\{\begin{array}[]{c}j\\\
\phi\phi\end{array}\right\\}\,H^{\phi}\,.$

As such, we also can operate the quaternionic derivative $\partial_{k}$
($k=1,2,3)$ on given equation (5.5), so that

$\displaystyle H_{;k}^{\xi}\,\,=\,\,$ $\displaystyle
H_{,k}^{\xi}+\left\\{\begin{array}[]{c}\xi\\\
km\end{array}\right\\}\,H^{m}\,,$ (5.16) $\displaystyle H_{;k}^{j}\,\,=\,\,$
$\displaystyle H_{,k}^{j}+\left\\{\begin{array}[]{c}j\\\
km\end{array}\right\\}\,H^{m}\,,$ (5.19)

where $j,k,m=1,2,3.$ Equations (5.16) and (5.19) are represented scalar to
vector and vector to vector field transformations, respectively, for
generalized quaternionic fields [42]. It should be remarked that through
quaternionic derivative, the generalized covariant derivative gives
enlargement to the rank of quaternionic tensors which is important to provide
more information about the transformed coordinate. In more general the
quaternionic tensor derivatives are

$\displaystyle T_{\xi\xi;\phi}\,\,=\,\,$ $\displaystyle
T_{\xi\xi,\phi}-\,\left\\{\begin{array}[]{c}\eta\\\
\phi\xi\end{array}\right\\}T_{\xi\eta}-\,\left\\{\begin{array}[]{c}\eta\\\
\phi\xi\end{array}\right\\}T_{\eta\xi}\,,$ (5.24) $\displaystyle
T_{\xi\xi;k}\,\,=\,\,$ $\displaystyle
T_{\xi\xi,k}-\,\left\\{\begin{array}[]{c}l\\\ k\xi\end{array}\right\\}T_{\xi
l}-\,\left\\{\begin{array}[]{c}l\\\ k\xi\end{array}\right\\}T_{l\xi}\,,$
(5.29) $\displaystyle T_{\xi n;\phi}\,\,=\,\,$ $\displaystyle T_{\xi
n,\phi}-\,\left\\{\begin{array}[]{c}\eta\\\ \phi
n\end{array}\right\\}T_{\xi\eta}-\,\left\\{\begin{array}[]{c}\eta\\\
\phi\xi\end{array}\right\\}T_{\eta n}\,,$ (5.34) $\displaystyle T_{\xi
n;k}\,\,=\,\,$ $\displaystyle T_{\xi n,k}-\,\left\\{\begin{array}[]{c}l\\\
kn\end{array}\right\\}T_{\xi l}-\,\left\\{\begin{array}[]{c}l\\\
k\xi\end{array}\right\\}T_{ln}\,,$ (5.39) $\displaystyle
T_{j\xi;\phi}\,\,=\,\,$ $\displaystyle
T_{j\xi,\phi}-\,\left\\{\begin{array}[]{c}\eta\\\
\phi\xi\end{array}\right\\}T_{j\eta}-\,\left\\{\begin{array}[]{c}\eta\\\ \phi
j\end{array}\right\\}T_{\eta\xi}\,,$ (5.44) $\displaystyle
T_{j\xi;k}\,\,=\,\,$ $\displaystyle
T_{j\xi,k}-\,\left\\{\begin{array}[]{c}l\\\
k\xi\end{array}\right\\}T_{jl}-\,\left\\{\begin{array}[]{c}l\\\
kj\end{array}\right\\}T_{l\xi}\,,$ (5.49) $\displaystyle T_{jn;\phi}\,\,=\,\,$
$\displaystyle T_{jn,\phi}-\,\left\\{\begin{array}[]{c}\eta\\\ \phi
n\end{array}\right\\}T_{j\eta}-\,\left\\{\begin{array}[]{c}\eta\\\ \phi
j\end{array}\right\\}T_{\eta n}\,,$ (5.54) $\displaystyle T_{jn;k}\,\,=\,\,$
$\displaystyle T_{jn,k}-\,\left\\{\begin{array}[]{c}l\\\
kn\end{array}\right\\}T_{jl}-\,\left\\{\begin{array}[]{c}l\\\
kj\end{array}\right\\}T_{ln}\,,$ (5.59)

where the right-hand side of equations (5.24)-(5.59), the first term shows the
tensorial transformation of derivative of quaternionic tensor and the last two
terms show the Christoffel symbols which tell us how the geodesic path changes
from point to point.

###  5.3 Quaternionic metric tensor

To expressing the quaternionic metric tensor in terms of the Christoffel
symbol, we can use the covariant derivative of quaternionic tensor. Therefore,
one can write

$\displaystyle g_{\xi\xi;\phi}\,\,=\,\,$ $\displaystyle
g_{\xi\xi,\phi}-\,\left\\{\begin{array}[]{c}\eta\\\
\phi\xi\end{array}\right\\}g_{\xi\eta}-\,\left\\{\begin{array}[]{c}\eta\\\
\phi\xi\end{array}\right\\}g_{\eta\xi}\,.$ (5.64)

where the line element becomes
$\mathsf{ds}=\sqrt{-g_{\mu\nu}(x)\,dx^{\mu}dx^{\nu}.}$ Under the covariant
differentiation, the quaternionic metric tensor is constant i.e.,

$\displaystyle 0\,\,=\,\,$ $\displaystyle
g_{\xi\xi,\phi}-\,\left\\{\begin{array}[]{c}\eta\\\
\phi\xi\end{array}\right\\}g_{\xi\eta}-\,\left\\{\begin{array}[]{c}\eta\\\
\phi\xi\end{array}\right\\}g_{\eta\xi}$ (5.69) $\displaystyle
g_{\xi\xi,\phi}\,\,=\,\,$ $\displaystyle\left\\{\begin{array}[]{c}\eta\\\
\phi\xi\end{array}\right\\}g_{\xi\eta}+\,\left\\{\begin{array}[]{c}\eta\\\
\phi\xi\end{array}\right\\}g_{\eta\xi}\,.$ (5.74)

Here the metric tensor $g_{im}^{{}^{\prime}}=\,\,\frac{\partial
P^{\mu}}{\partial P^{{}^{\prime}i}}\,\frac{\partial P^{\nu}}{\partial
P^{{}^{\prime}m}}\,\Omega_{\mu\nu}$, where $\Omega_{\mu\nu}$ is the four-
dimensional Minkowski metric define by

$\displaystyle\Omega_{\mu\nu}\,\,=\,\,$
$\displaystyle\left(\begin{array}[]{cccc}-1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&0\\\
0&0&0&1\end{array}\right)\,,$

and the Christoffel symbol satisfy the following properties

$\displaystyle\left\\{\begin{array}[]{c}i\\\
kl\end{array}\right\\}\,\,=\,\,\left\\{\begin{array}[]{c}i\\\
lk\end{array}\right\\},\,\,\,\,\,\,\,\left\\{\begin{array}[]{c}m\\\
il\end{array}\right\\}\,g_{mk}\,\,=\,\,$
$\displaystyle\left\\{\begin{array}[]{c}k\\\ il\end{array}\right\\}\,.$ (5.83)

Now, the value of Christoffel symbol corresponding to quaternionic scalar
field can be written as,

$\displaystyle\left\\{\begin{array}[]{c}\eta\\\
\xi\xi\end{array}\right\\}\,\,=\,\,$
$\displaystyle\frac{1}{2}g^{\eta\phi}\,\left(g_{\phi\xi,\xi}+g_{\xi\phi,\xi}-g_{\xi\xi,\phi}\right)\,,$
(5.86)

where $g^{\eta\phi}$ is the inverse matrix of $g_{\eta\phi}$. Correspondingly,
in quaternionic pure vectorial field, the Christoffel symbol can be played by

$\displaystyle\left\\{\begin{array}[]{c}l\\\ jn\end{array}\right\\}\,\,=\,\,$
$\displaystyle\frac{1}{2}g^{lk}\,\left(g_{kj,n}+g_{nk,j}-g_{jn,k}\right)\,,$
(5.89)

and the other diversified quaternionic scalar and vector fields show a mixed
form of quaternionic Christoffel symbol in terms of the metric tensor, i.e.

$\displaystyle\left\\{\begin{array}[]{c}l\\\
\xi\xi\end{array}\right\\}\,\,=\,\,$
$\displaystyle\frac{1}{2}g^{lk}\,\left(g_{k\xi,\xi}+g_{\xi
k,\xi}-g_{\xi\xi,k}\right)\,,$ (5.92)
$\displaystyle\left\\{\begin{array}[]{c}\eta\\\ \xi
n\end{array}\right\\}\,\,=\,\,$
$\displaystyle\frac{1}{2}g^{\eta\phi}\,\left(g_{\phi\xi,n}+g_{n\phi,\xi}-g_{\xi
n,\phi}\right)\,,$ (5.95) $\displaystyle\left\\{\begin{array}[]{c}l\\\ \xi
n\end{array}\right\\}\,\,=\,\,$
$\displaystyle\frac{1}{2}g^{lk}\,\left(g_{k\xi,n}+g_{nk,\xi}-g_{\xi
n,k}\right)\,,$ (5.98) $\displaystyle\left\\{\begin{array}[]{c}\eta\\\
jn\end{array}\right\\}\,\,=\,\,$
$\displaystyle\frac{1}{2}g^{\eta\phi}\,\left(g_{\phi
j,n}+g_{n\phi,j}-g_{jn,\phi}\right)\,,$ (5.101)

where $(\xi,\eta,\phi)$ are used for generalized quaternionic scalar field
variables while $(i,j,k,l,m,n)$ are the generalized quaternionic vector field
variables.

###  5.4 Quaternionic geodesic equation

In general theory of relativity, geodesic equation gives the replacement of
linear space-time to curved space-time. In other words, a particle moving in a
curvature space-time is always follow the path of geodesic. The geodesic path
in curvature space-time structure can be described in terms of four-vector
form with the principle of least action, i.e. $dH^{\mu}=\,0,$ see ref.[3]. In
the pure quaternionic scalar field,

$\displaystyle
H_{;\phi}^{\xi}\,\,\equiv\,\,H_{,\phi}^{\xi}+\left\\{\begin{array}[]{c}\xi\\\
\phi\phi\end{array}\right\\}\,H^{\phi}\,\,=\,\,$ $\displaystyle 0$ (5.104)
$\displaystyle\Rightarrow\,dH^{\xi}+\left\\{\begin{array}[]{c}\xi\\\
\phi\phi\end{array}\right\\}\,dP^{\phi}\,H^{\phi}\,\,=\,\,$ $\displaystyle
0\,.$ (5.107)

Now, dividing equation (5.107) by $ds$ (e.g. a scalar parameter of motion as
proper time) and substituting $H^{\xi}\,=\,\frac{dP^{\xi}}{ds}$,
$H^{\phi}\,=\,\frac{dP^{\phi}}{ds}$, we get

$\displaystyle\frac{d^{2}P^{\xi}}{ds^{2}}+\left\\{\begin{array}[]{c}\xi\\\
\phi\phi\end{array}\right\\}\,\frac{dP^{\phi}}{ds}\frac{dP^{\phi}}{ds}\,\,=\,\,$
$\displaystyle 0\,.$ (5.110)

Similarly, for quaternionic vector fields, we get

$\displaystyle\frac{d^{2}P^{j}}{ds^{2}}+\left\\{\begin{array}[]{c}j\\\
km\end{array}\right\\}\,\frac{dP^{k}}{ds}\frac{dP^{m}}{ds}\,\,=\,\,$
$\displaystyle 0\,,$ (5.113)

and for a mixed fields,

$\displaystyle\frac{d^{2}P^{j}}{ds^{2}}+\left\\{\begin{array}[]{c}j\\\
\phi\phi\end{array}\right\\}\,\frac{dP^{\phi}}{ds}\frac{dP^{\phi}}{ds}\,\,=\,\,$
$\displaystyle 0\,,$ (5.116)
$\displaystyle\frac{d^{2}P^{\xi}}{ds^{2}}+\left\\{\begin{array}[]{c}\xi\\\
km\end{array}\right\\}\,\frac{dP^{k}}{ds}\frac{dP^{m}}{ds}\,\,=\,\,$
$\displaystyle 0\,.$ (5.119)

Equations (5.110)-(5.119) represent the non-linear equations called the
quaternionic geodesic equations that arise due to the effect of Christoffel
symbols. In quaternionic formulation, we also can emphasize that the space-
time curvature path is not only followed by vector components but also
followed by the scalar components and mixed components of a quaternion
variable. If the Christoffel symbol is zero, then the quaternionic geodesic
equations of motion lead to

$\displaystyle\frac{d^{2}P^{\mu}}{ds^{2}}\,\,=\,\,$ $\displaystyle
0\,,\,\,\,\,\,\,\,\forall\,(\mu=0,1,2,3)\,.$ (5.120)

This implies that the generalized quaternionic acceleration will be zero i.e.
the uniform velocity of the particle moving in a straight line (or non-
Riemannian space-time).

###  5.5 Quaternionic Riemannian Christoffel curvature tensor

The Riemannian Christoffel curvature tensor is a four-indices tensor, can be
obtained by subtraction of two covariant derivatives of quaternionic tensors
in which indices are interchanged, such that

$\displaystyle T_{\xi\xi;\phi}-T_{\xi\phi;\xi}\,\,=\,\,$
$\displaystyle\left(R_{\xi\xi\phi}^{\eta}\right)H_{\eta}\,,$ (5.121)

where

$R_{\xi\xi\phi}^{\eta}\,\,=\,\,\partial_{\xi}\left(\left\\{\begin{array}[]{c}\eta\\\
\phi\xi\end{array}\right\\}\right)-\partial_{\phi}\left(\left\\{\begin{array}[]{c}\eta\\\
\xi\xi\end{array}\right\\}\right)+\left\\{\begin{array}[]{c}\varphi\\\
\phi\xi\end{array}\right\\}\left\\{\begin{array}[]{c}\eta\\\
\varphi\xi\end{array}\right\\}-\left\\{\begin{array}[]{c}\varphi\\\
\xi\xi\end{array}\right\\}\left\\{\begin{array}[]{c}\eta\\\
\varphi\phi\end{array}\right\\}$ (5.122)

is the Riemannian Christoffel curvature tensor for purely quaternionic scalar
field. It is a four-indices tensor in curved space-time which describes the
curvature of manifolds. Further, we can obtain the Riemannian Christoffel
curvature tensor to other quaternionic components given in equation
(5.29)-(5.59), i.e.,

$\displaystyle T_{\xi\xi;k}-T_{\xi k;\xi}\,\,=\,\,$
$\displaystyle\left(R_{\xi\xi k}^{l}\right)H_{l}\,,$ (5.123) $\displaystyle
T_{\xi n;\phi}-T_{\xi\phi;n}\,\,=\,\,$ $\displaystyle\left(R_{\xi
n\phi}^{\eta}\right)H_{\eta}\,,$ (5.124) $\displaystyle T_{\xi n;k}-T_{\xi
k;n}\,\,=\,\,$ $\displaystyle\left(R_{\xi nk}^{l}\right)H_{l}\,,$ (5.125)
$\displaystyle T_{j\xi;\phi}-T_{j\phi;\xi}\,\,=\,\,$
$\displaystyle\left(R_{j\xi\phi}^{\eta}\right)H_{\eta}\,,$ (5.126)
$\displaystyle T_{j\xi;k}-T_{jk;\xi}\,\,=\,\,$ $\displaystyle\left(R_{j\xi
k}^{l}\right)H_{l}\,,$ (5.127) $\displaystyle
T_{jn;\phi}-T_{j\phi;n}\,\,=\,\,$
$\displaystyle\left(R_{jn\phi}^{\eta}\right)H_{\eta}\,,$ (5.128)
$\displaystyle T_{jn;k}-T_{jk;n}\,\,=\,\,$
$\displaystyle\left(R_{jnk}^{l}\right)H_{l}\,,$ (5.129)

where the Riemannian Christoffel curvature tensors lead to the following way
as

$\displaystyle R_{\xi\xi k}^{l}\,\,=\,\,$
$\displaystyle\partial_{\xi}\left(\left\\{\begin{array}[]{c}l\\\
k\xi\end{array}\right\\}\right)-\partial_{k}\left(\left\\{\begin{array}[]{c}l\\\
\xi\xi\end{array}\right\\}\right)+\left\\{\begin{array}[]{c}i\\\
k\xi\end{array}\right\\}\left\\{\begin{array}[]{c}l\\\
i\xi\end{array}\right\\}-\left\\{\begin{array}[]{c}i\\\
\xi\xi\end{array}\right\\}\left\\{\begin{array}[]{c}l\\\
ik\end{array}\right\\}\,,$ (5.142) $\displaystyle R_{\xi
n\phi}^{\eta}\,\,=\,\,$
$\displaystyle\partial_{n}\left(\left\\{\begin{array}[]{c}\eta\\\
\phi\xi\end{array}\right\\}\right)-\partial_{\phi}\left(\left\\{\begin{array}[]{c}\eta\\\
\xi n\end{array}\right\\}\right)+\left\\{\begin{array}[]{c}\varphi\\\
\phi\xi\end{array}\right\\}\left\\{\begin{array}[]{c}\eta\\\ \varphi
n\end{array}\right\\}-\left\\{\begin{array}[]{c}\varphi\\\
n\xi\end{array}\right\\}\left\\{\begin{array}[]{c}\eta\\\
\varphi\phi\end{array}\right\\}\,,$ (5.155) $\displaystyle R_{\xi
nk}^{l}\,\,=\,\,$
$\displaystyle\partial_{n}\left(\left\\{\begin{array}[]{c}l\\\
k\xi\end{array}\right\\}\right)-\partial_{k}\left(\left\\{\begin{array}[]{c}l\\\
\xi n\end{array}\right\\}\right)+\left\\{\begin{array}[]{c}i\\\
k\xi\end{array}\right\\}\left\\{\begin{array}[]{c}l\\\
in\end{array}\right\\}-\left\\{\begin{array}[]{c}i\\\
n\xi\end{array}\right\\}\left\\{\begin{array}[]{c}l\\\
ik\end{array}\right\\}\,,$ (5.168) $\displaystyle
R_{j\xi\phi}^{\eta}\,\,=\,\,$
$\displaystyle\partial_{\xi}\left(\left\\{\begin{array}[]{c}\eta\\\ \phi
j\end{array}\right\\}\right)-\partial_{\phi}\left(\left\\{\begin{array}[]{c}\eta\\\
j\xi\end{array}\right\\}\right)+\left\\{\begin{array}[]{c}\varphi\\\ \phi
j\end{array}\right\\}\left\\{\begin{array}[]{c}\eta\\\
\varphi\xi\end{array}\right\\}-\left\\{\begin{array}[]{c}\varphi\\\ \xi
j\end{array}\right\\}\left\\{\begin{array}[]{c}\eta\\\
\varphi\phi\end{array}\right\\}\,,$ (5.181) $\displaystyle R_{j\xi
k}^{l}\,\,=\,\,$
$\displaystyle\partial_{\xi}\left(\left\\{\begin{array}[]{c}l\\\
kj\end{array}\right\\}\right)-\partial_{k}\left(\left\\{\begin{array}[]{c}l\\\
j\xi\end{array}\right\\}\right)+\left\\{\begin{array}[]{c}i\\\
kj\end{array}\right\\}\left\\{\begin{array}[]{c}l\\\
i\xi\end{array}\right\\}-\left\\{\begin{array}[]{c}i\\\ \xi
j\end{array}\right\\}\left\\{\begin{array}[]{c}l\\\ ik\end{array}\right\\}\,,$
(5.194) $\displaystyle R_{jn\phi}^{\eta}\,\,=\,\,$
$\displaystyle\partial_{n}\left(\left\\{\begin{array}[]{c}\eta\\\ \phi
j\end{array}\right\\}\right)-\partial_{\phi}\left(\left\\{\begin{array}[]{c}\eta\\\
jn\end{array}\right\\}\right)+\left\\{\begin{array}[]{c}\varphi\\\ \phi
j\end{array}\right\\}\left\\{\begin{array}[]{c}\eta\\\ \varphi
n\end{array}\right\\}-\left\\{\begin{array}[]{c}\varphi\\\
nj\end{array}\right\\}\left\\{\begin{array}[]{c}\eta\\\
\varphi\phi\end{array}\right\\}\,,$ (5.207) $\displaystyle
R_{jnk}^{l}\,\,=\,\,$
$\displaystyle\partial_{n}\left(\left\\{\begin{array}[]{c}l\\\
kj\end{array}\right\\}\right)-\partial_{k}\left(\left\\{\begin{array}[]{c}l\\\
jn\end{array}\right\\}\right)+\left\\{\begin{array}[]{c}i\\\
kj\end{array}\right\\}\left\\{\begin{array}[]{c}l\\\
in\end{array}\right\\}-\left\\{\begin{array}[]{c}i\\\
nj\end{array}\right\\}\left\\{\begin{array}[]{c}l\\\
ik\end{array}\right\\}\,.$ (5.220)

Here, the quaternionic Riemannian tensor keeps track that how much scalar and
vector components of quaternion change when we propagate parallel along with a
small parallelogram. If the value of quaternionic Riemannian Christoffel
curvature is zero then the quaternionic curved space-time is converted into
flat space-time. On the other hand, the quaternionic Ricci tensor is an
important contraction of quaternionic Riemannian Christoffel tensor which
explains the volume changes when an object parallels transport along a
geodesic. In this case

$\displaystyle g_{h\eta}R_{\xi n\phi}^{\eta}\,\,=\,\,$ $\displaystyle R_{h\xi
n\phi}$ $\displaystyle=\,\,$
$\displaystyle\partial_{n}\left(g_{h\eta}\left\\{\begin{array}[]{c}\eta\\\
\phi\xi\end{array}\right\\}\right)-\partial_{\phi}\left(g_{h\eta}\left\\{\begin{array}[]{c}\eta\\\
\xi n\end{array}\right\\}\right)+g_{h\eta}\left\\{\begin{array}[]{c}\varphi\\\
\phi\xi\end{array}\right\\}\left\\{\begin{array}[]{c}\eta\\\ \varphi
n\end{array}\right\\}$ (5.229) $\displaystyle-
g_{h\eta}\left\\{\begin{array}[]{c}\varphi\\\
n\xi\end{array}\right\\}\left\\{\begin{array}[]{c}\eta\\\
\varphi\phi\end{array}\right\\}\,,$ (5.234)

which can be simplified in terms of a metric tensor as

$\displaystyle R_{h\xi n\phi}\,\,=\,\,$
$\displaystyle\frac{1}{2}\left[\frac{\partial^{2}g_{h\phi}}{\partial
P^{n}\partial P^{\xi}}+\frac{\partial^{2}g_{\xi n}}{\partial P^{h}\partial
P^{\phi}}-\frac{\partial^{2}g_{\xi\phi}}{\partial P^{n}\partial
P^{h}}-\frac{\partial^{2}g_{nh}}{\partial P^{\phi}\partial P^{\xi}}\right]$
$\displaystyle+\,\,$ $\displaystyle g_{\eta
h}\left\\{\begin{array}[]{c}\varphi\\\
\phi\xi\end{array}\right\\}\left\\{\begin{array}[]{c}\eta\\\ \phi
n\end{array}\right\\}-g_{\eta h}\left\\{\begin{array}[]{c}\varphi\\\
n\xi\end{array}\right\\}\left\\{\begin{array}[]{c}\eta\\\
\varphi\phi\end{array}\right\\}\,.$ (5.243)

Now, contracting equation (5.243) with the metric tensor as $g^{hn}R_{h\xi
n\phi}=\,\,R_{\xi\phi}$, where $R_{\xi\phi}$ is a quaternionic form of Ricci
tensor in scalar field, so that

$\displaystyle R_{\xi\phi}\,\,=\,\,$
$\displaystyle\partial_{n}\left(\left\\{\begin{array}[]{c}n\\\
\phi\xi\end{array}\right\\}\right)-\partial_{\phi}\left(\left\\{\begin{array}[]{c}n\\\
\xi n\end{array}\right\\}\right)+\left\\{\begin{array}[]{c}\varphi\\\
\phi\xi\end{array}\right\\}\left\\{\begin{array}[]{c}n\\\ \varphi
n\end{array}\right\\}-\left\\{\begin{array}[]{c}\varphi\\\
n\xi\end{array}\right\\}\left\\{\begin{array}[]{c}n\\\
\varphi\phi\end{array}\right\\}\,.$ (5.256)

Similarly, to involving quaternionic vector field in addition to scalar field,
also in pure vector field the Ricci tensors become

$\displaystyle R_{\xi k}\,\,=\,\,$
$\displaystyle\partial_{n}\left(\left\\{\begin{array}[]{c}n\\\
k\xi\end{array}\right\\}\right)-\partial_{k}\left(\left\\{\begin{array}[]{c}n\\\
\xi n\end{array}\right\\}\right)+\left\\{\begin{array}[]{c}i\\\
k\xi\end{array}\right\\}\left\\{\begin{array}[]{c}n\\\
in\end{array}\right\\}-\left\\{\begin{array}[]{c}i\\\
n\xi\end{array}\right\\}\left\\{\begin{array}[]{c}n\\\
ik\end{array}\right\\}\,,$ (5.269) $\displaystyle R_{jk}\,\,=\,\,$
$\displaystyle\partial_{n}\left(\left\\{\begin{array}[]{c}n\\\
kj\end{array}\right\\}\right)-\partial_{k}\left(\left\\{\begin{array}[]{c}n\\\
jn\end{array}\right\\}\right)+\left\\{\begin{array}[]{c}i\\\
kj\end{array}\right\\}\left\\{\begin{array}[]{c}n\\\
in\end{array}\right\\}-\left\\{\begin{array}[]{c}i\\\
nj\end{array}\right\\}\left\\{\begin{array}[]{c}n\\\
ik\end{array}\right\\}\,.$ (5.282)

We should notice that the Riemannian tensor ($R_{h\xi n\phi}$) is symmetric to
first with third indices and second with fourth indices [3]. Thus the Ricci
tensor is also symmetric. The quaternionic form of Ricci tensor can further
contract with metric tensor and obtain scalar curvature tensor ($R$) of rank
zero, such that $g^{\xi\phi}R_{\xi\phi}=\,\,R\,.$

###  5.6 Quaternionic Einstein-field like equation

To express the quaternionic Einstein field like equation, let us start with
the Poisson equation as

$\displaystyle\nabla^{2}\Phi\,\,=\,\,$ $\displaystyle 4\pi G\rho\,,$ (5.283)

where $\Phi$ is the Newtonian gravitational potential, $G$ is gravitational
constant and $\rho$ is the mass density. Moreover, in metric tensor form [45],
we have $\nabla^{2}\Phi=-\frac{1}{2}\nabla^{2}g_{00}$. Thus

$\displaystyle\nabla^{2}g_{00}\,\,=\,\,$ $\displaystyle-8\pi G\rho\,.$ (5.284)

It should be noticed that in equation (5.284) we introduced the quaternionic
form of scalar metric tensor ($g_{\xi\xi}$) in place of Newtonian potential
($\Phi$) and quaternionic mass density ($T_{\xi\xi}$) in place of ($\rho$).
Generally, the quaternionic energy-momentum tensor can be written as

$\displaystyle T^{\mu\nu}\,\,\simeq\,\,$
$\displaystyle\left[\begin{array}[]{cccc}T^{\xi\xi}&T^{\xi 1}&T^{\xi 2}&T^{\xi
3}\\\ T^{1\xi}&T^{11}&T^{12}&T^{13}\\\ T^{2\xi}&T^{21}&T^{22}&T^{23}\\\
T^{3\xi}&T^{31}&T^{32}&T^{33}\end{array}\right]\,.$ (5.289)

Therefore, equation (5.284) leads to $\nabla^{2}g_{\xi\xi}\,=\,-8\pi
GT_{\xi\xi}.$ Since the quaternionic Ricci tensor can also be written in the
form of a double derivative of the quaternionic metric tensor. So, we get

$\displaystyle R_{\xi\xi}\,\,=\,\,$ $\displaystyle-8\pi GT_{\xi\xi}\,.$
(5.290)

Further, the rest components of quaternionic Riemannian Christoffel curvature
tensors can express as

$\displaystyle R_{\xi 1}\,\,=\,$ $\displaystyle-8\pi GT_{\xi
1}\,,\,\,\,\,\,\,\,\,R_{\xi 2}\,\,=\,-8\pi GT_{\xi 2}\,,\,\,\,\,\,\,\,\,R_{\xi
3}\,\,=\,-8\pi GT_{\xi 3}\,,$ $\displaystyle R_{11}\,\,=\,$
$\displaystyle-8\pi GT_{11}\,,\,\,\,\,\,\,\,R_{22}\,\,=\,-8\pi
GT_{22}\,,\,\,\,\,\,\,\,R_{33}\,\,=\,-8\pi GT_{33}\,,$ $\displaystyle
R_{12}\,\,=\,$ $\displaystyle-8\pi GT_{12}\,,\,\,\,\,\,\,\,R_{23}\,\,=\,-8\pi
GT_{23}\,,\,\,\,\,\,\,\,R_{31}\,\,=\,-8\pi GT_{31}\,,$ (5.291)

where the other components i.e.
$R_{1\xi},\,R_{2\xi},\,R_{3\xi},\,R_{21},\,R_{32},\,R_{13}$ are symmetrical to
$R_{\xi 1},\,R_{2\xi},\,R_{3\xi},$$\,R_{12},\,R_{23},\,R_{31}$, respectively.
Therefore, we may write a generalized quaternionic form of Einstein field-like
equation as $R_{\mu\nu}\,\,=\,\,-8\pi GT_{\mu\nu}\,.$ Since the quaternionic
energy-momentum tensor is conserved quantity but quaternionic Ricci tensor is
not, so it should be divergence-free. Thus, by using Bianchi identity
$R_{\alpha\mu\beta\gamma;\,\sigma}+R_{\alpha\mu\gamma\sigma;\,\beta}+R_{\alpha\mu\sigma\beta;\,\gamma}\,\,=\,\,0$,
the quaternionic Einstein field-like equation can be written as
$R_{\mu\nu}-\frac{1}{2}\,g_{\mu\nu}R\,\,=\,\,-8\pi GT_{\mu\nu}$. Here, we can
see that the right-hand part shows quaternionic energy and matter, while the
left-hand part shows the quaternionic geometry. Thus the quaternionic
formalism is perfectly fitted to describe a new approach in the dynamics of a
particle in curved space-time.

##  6 Conclusion

In Riemannian geometry, the curvature of space-time is directly connected to
the energy and momentum of whatever matter and radiation are present.
Generally, the space-time can be shown using simple experiments following the
free-fall trajectories of different test particles, the result of transporting
space-time vectors that can denote a particle’s velocity (time-like vectors)
will vary with the particle’s trajectory. The origin of the idea of this
space-time curvature is a mathematical concept. Thus the four-dimensional
curvature space-time has been visualized by a quaternionic algebra. The
transformation rules for two quaternionic frames have been discussed.
Interestingly, we discussed that the effect of curvature space-time arises due
to the quaternionic Christoffel symbol. Also, it is concluded that the curved
path of freely moving particles is effectively explained by the quaternionic
geodesic equation. The Christoffel symbol has been established in terms of the
derivative of quaternionic metric tensor while the gravitational potential in
tensorial form is represented by the metric tensor. It has been confirmed that
the volume element of an object obeying the geodesic path changes due to the
contraction of the quaternionic Riemannian tensor while the contraction of the
quaternionic Ricci tensor gives a scalar quantity representing the magnitude
of change in volume of the object. We also have been discussed the
quaternionic Einstein field-like equation for gravitation that relates the
geometry of quaternionic space-time with the distribution of matter within it.
The simplification of quaternionic formalism will be achieved by the
approximation as quaternionic flat space-time with a small deviation, which
leads to the linearized Einstein field equation. These quaternionic equations
may be used to study the phenomena such as gravitational waves.

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