the classical gravitational theory as near as possible. Therefore the gravitational field equations, Eqs.(20)-(23), are able to be transformed into that in the three-dimensional vector space, $$\begin{aligned}
&& \blacktriangledown \cdot \textbf{b}^\prime = 0 ~,
\\
&& \partial_0 \textbf{b}^\prime + \blacktriangledown \times \textbf{g}^{\prime\prime} / v_0 = 0 ~,
\\
&& \blacktriangledown \cdot \textbf{g}^{\prime\prime} / v_0 = \mu_g s_0   ~,
\\
&& \blacktriangledown \times \textbf{b}^\prime - \partial_0 \textbf{g}^{\prime\prime} / v_0 = \mu_g \textbf{s}^\prime ~.\end{aligned}$$

Comparing Eq.(26) with Poisson equation for the Newton’s law of universal gravitation reveals that those two equations are the same formally. In case the gravitational strength component $\textbf{b}$ is weak and the velocity ratio $\textbf{v} / v_0$ is quite tiny, Eq.(26) will be reduced into the Newton’s law of universal gravitation in the classical gravitational theory.

\[sec:level2\]Electromagnetic field equations
---------------------------------------------

By means of the transformation of basis vectors of $\nabla$ in the quaternion operator, it is able to translate further the electromagnetic field equations, Eqs.(16)-(19), from the $\emph{S}$-quaternion space $\mathbb{H}_e$ into that in the three-dimensional vector space $(\emph{\textbf{j}}_1 ,~\emph{\textbf{j}}_2 ,~\emph{\textbf{j}}_3)$. In the electromagnetic field equations, the operator $\nabla$ is substituted by $\blacktriangledown$. And $\textbf{B}$, $\textbf{E}$, and $\textbf{S}$ are substituted by $\textbf{B}^\prime$, $\textbf{E}^\prime$, and $\textbf{S}^\prime$ respectively. Therefore the electromagnetic 