goganesyan said:
Have you considered that our observable 3 dimensions are actually 6 and the vector you talk about may not be what one thinks?
ps: mathematically don’t we know there are 10-11 dimensions? I’m just hoping one of you gurus put this in mathematical terms
Ok, now finally I must give the mathematical description, no matter whether you want it or not.
You start with four-vector components ##(x^{\mu})=(ct,\vec{x})## for time and space (the spacetime four-vector). The motion of a particle is described as a world line in this four-dimensional vector space. For massive particles this world line must be time-like, and thus you can choose proper time as the world-line parameter. This is the time measured by an ideal clock co-moving with the particle. It is defined by
$$\mathrm{d} \tau=\sqrt{\mathrm{d} t^2-\mathrm{d} \vec{x}^2/c^2}.$$
It is thus related to the coordinate time wrt. the inertial frame used to do the calulation by
$$\frac{\mathrm{d} \tau}{\mathrm{d} t}=\sqrt{1-(\mathrm{d}_t \vec{x})^2/c^2}=\sqrt{1-\vec{v}^2/c^2}=1/\gamma.$$
In order to have a covariant description one defines a four-vector
$$p^{\mu}=m \mathrm{d}_{\tau} x^{\mu}.$$
Since ##x^{\mu}## is a four-vector and ##\mathrm{\tau}## is a scalar, ##m## necessarily is a scalar too in order to have ##p^{\mu}## as a four-vector.
Expressing ##p^{\mu}## in terms of coordinate-time derivatives gives
$$(p^{\mu})=m \gamma \begin{pmatrix}c \\ \vec{v} \end{pmatrix}.$$
In an inertial frame, where ##|\vec{v}| \ll c## you have ##\gamma=1+\vec{v}^2/(2 c^2)+\mathcal{O}(v^4/c^4)## and thus
$$(p^{\mu}) \simeq \begin{pmatrix} m c +m v^2/(2 c)+\cdots \\ m \vec{v} + \cdots \end{pmatrix}.$$
This shows that
$$p^0=m c + E_{\text{kin}}/c,$$
and ##\vec{p}## takes the same form as in Newtonian physics with ##m## the usual mass known from Newtonian physics.
The relativistic connection between energy and momentum, where energy is defined such that it includes the socalled rest energy ##E_0=m c^2##, thus is
$$E=m \gamma c^2=\frac{m c^2}{\sqrt{1-v^2/c^2}}, \quad \vec{p}=m \gamma \vec{v}=\frac{m \vec{v}}{1-v^2/c^2}.$$
This shows that necessarily ##|v|<c## and to reach the limit ##v \rightarrow c## you need an infinite amount of energy.
There are no 6 dimensions nor 10-11 dimensions in standard relativistic theory. Also Minkowski space is a real 4D vector space. One should avoid textbooks using the ancient ##\mathrm{i} c t## convention, because it is quite confusing and also cannot be extended to noninertial reference frames in SR, let alone to GR, where you work with arbitrary spacetime coordinates anyway.
You find some introduction to special relativity in my (still unfinished) manuscript
https://itp.uni-frankfurt.de/~hees/pf-faq/srt.pdf
There you also find, how to covariantly formulate classical electrodynamics.
