The criterion for self-consistently translational motion of reference frames in universal kinematics

Abstract

Universal kinematics as mathematical objects may be interesting for astrophysics, because there exists the hypothesis, that in the large scale of the Universe, physical laws (in particular, the laws of kinematics) may be different from the laws, acting in the neighborhood of our Solar System. The present paper is devoted to investigation of self-consistently translational motion of reference frames in abstract universal kinematics. In the case of self-consistently translational motion we can give the clear and unambiguous definition of displacement as well average and instantaneous speed of the reference frame. Hence the uniform rectilinear motion is the particular case of self-consistently translational motion. So, the investigation of self-consistently translational motion is technically necessary for definition of classes of inertially-related reference frames (being in the state of uniform rectilinear mutual motion) in universal kinematics. In the paper we establish the necessary and sufficient condition on coordinate transformation between reference frames of universal kinematics assuring self-consistently translational motion of one reference frame relatively to another:

Theorem. Let $\mathcal{F}$ be vector universal kinematics.
The reference frame $\mathfrak{m}\in\mathcal{L}k\left(\mathcal{F}\right)$ is self-
consistently translational relatively the reference frame
$\mathfrak{l}\in\mathcal{L}k\left(\mathcal{F}\right)$ if and only if, there exist the functions:
$\Phi:\mathbb{M}k\left(\mathfrak{m}\right)\longmapsto\mathbf{Tm}\left(\mathfrak{l}\right)$,
$\mathbf{F}:\mathbf{Tm}\left(\mathfrak{l}\right)\longmapsto\mathbf{Zk}\left(\mathfrak{l}\right)$,
$\mathbf{g}:\mathbf{Zk}\left(\mathfrak{m}\right)\longmapsto\mathbf{Zk}\left(\mathfrak{l}\right)$,
satisfying the following conditions:
1. For each $\mathrm{x}\in\mathbf{Zk}\left(\mathfrak{m}\right)$ the function
$\Phi_{(\mathrm{x})}(t):=\Phi\left(t,\mathrm{x}\right)$,
$t\in\mathbf{Tm}\left(\mathfrak{m}\right)$ is a bijection between
$\mathbf{Tm}\left(\mathfrak{m}\right)$ and $\mathbf{Tm}\left(\mathfrak{l}\right)$.
2. The function $\mathbf{g}$ is a bijection between $\mathbf{Zk}\left(\mathfrak{m}\right)$
and $\mathbf{Zk}\left(\mathfrak{l}\right)$.
3. For every $\mathrm{w}\in\mathbb{M}k\left(\mathfrak{m}\right)$
the following equality is performed:
\[
\left[\mathfrak{l}{\leftarrow}\mathfrak{m}\right]\mathrm{w}=
\left(\Phi\left(t_{\mathrm{w}},\mathrm{x}_{\mathrm{w}}\right),
\mathbf{F}\left(\Phi\left(t_{\mathrm{w}},\mathrm{x}_{\mathrm{w}}\right)\right)
+\mathbf{g}\left(\mathrm{x}_{\mathrm{w}}\right)
\right),
\]
where $t_{\mathrm{w}}=\mathsf{tm}\left(\mathrm{w}\right)$,
$\mathrm{x}_{\mathrm{w}}=\mathsf{bs}\left(\mathrm{w}\right)$..