\documentclass[14pt]{article}
\usepackage{ pgfplots, tikz}
\usepackage{amsmath}
\usetikzlibrary{decorations.text, decorations.markings}
\usetikzlibrary{intersections}
\usetikzlibrary{arrows}  
\pgfdeclarelayer{bg}    % declare background layer
\usetikzlibrary{shapes.geometric,calc}
\pgfplotsset{
	%every tick label/.append style={scale=0.5},
	every axis label/.append style={font=\small},
	compat=newest,
}
\usepackage[%
	a4paper,%
	footskip=1cm,%
	headsep=0.3cm,% 
	top=2cm, %поле сверху
	bottom=2cm, %поле снизу
	left=2cm, %поле ліворуч
	right=2cm, %поле праворуч
]{geometry}

\begin{document}
Let's consider a only one spatial coordinate $x$. Metrics look:
\begin{equation}
	 ds^2 = g_{00}dx^0dx^0  + 2g_{01}dx^0dx^1 + g_{11}dx^1dx^1
\end{equation}

Tangential vector (4-velocity) to worldline: 
\begin{equation}
	v^{\nu} = \frac{dx^0}{d\lambda}\vec{e}_0 + \frac{dx^i}{d\lambda}\vec{e}_1
\end{equation} 
$\lambda$ here is the parameter of an particle worldline.

\begin{center}
    \begin{tikzpicture}[every node/.style={font=\footnotesize},scale = 1.5]
	\pgfmathsetmacro{\angle}{80}
	\pgfmathsetmacro{\xmax}{4}
	\pgfmathsetmacro{\ymax}{8}
	\pgfmathsetmacro{\gtx}{cos(\angle)}
	 %---------------------- particle coordinates ----------------------------            
	\pgfmathsetmacro{\xparticle}{3}
	\pgfmathsetmacro{\yparticle}{4}
	\def\particle{(\xparticle,\yparticle)}
    %---------------------- functions definitions ----------------------------	
	\pgfmathsetmacro{\lightforward}{1/(\gtx + sqrt(\gtx^2 + 1))}
	\pgfmathsetmacro{\lightbackward}{1/(\gtx - sqrt(\gtx^2 + 1))}

	\pgfmathdeclarefunction{x}{2}{\pgfmathparse{#1+#2*\gtx}} % Real x coord of canvas
	\pgfmathdeclarefunction{y}{2}{\pgfmathparse{#2*sin(\angle)}} % Real y coord of canvas
	\def\point(#1,#2){({x(#1,#2)},{y(#1,#2)})}
	\pgfmathdeclarefunction{len}{2}{\pgfmathparse{sqrt(#1^2 + #2^2 + 2*#1*#2*\gtx)}}
	\pgfmathdeclarefunction{projection}{2}{\pgfmathparse{#2 + #1*\gtx}}
	
	% --------------------------- axis drawing ------------------------------
    \draw[-latex'] (0,0) coordinate (O) node [below left] {$x^0$} -- \point(0,\ymax) node[above] {$x^0$};
    \draw[-latex'] (O) -- \point(\xmax,0) node[right] {$x^i$};
    %---------------------------- grid --------------------------------------
    \foreach \i in {1,...,\xmax} {
        \draw[gray!50, ultra thin] \point(\i,0) -- \point(\i,\ymax);
    }
    
    \foreach \i in {1,...,\ymax} {
        \draw[gray!50, ultra thin] \point(0,\i) -- \point(\xmax,\i) node [right, black] {$x^0 = \mathrm{const}$} ;
    }
    %----------------------------- ort --------------------------------------
        %\draw[-latex, blue] (0,0) -- (1,0) node[below] {$\vec{e}_i$};
        \draw[-latex, blue] (0,0) -- \point(0,1) node[left] {$u^{\mu}$};

	% ------------------Drawing v^{\nu} vector ------------------------------
    \draw [name path = velocity, blue, thick, -latex'](O) -- node [right] {$v^{\nu}$} \expandafter\point\particle coordinate (PE);
    
    % --------------------------- main --------------------------------------

%    \foreach \x [count=\i from 0] in {0,0.1,0.2,...,1}{
%    \draw [densely dashed, red] \point(\x*\xparticle,\x*\yparticle) coordinate (V\i)  --  \point(0,{projection(\x*\xparticle,\x*\yparticle)}) coordinate (A\i) \ifnum\i=10 node [sloped, pos=0.5, above=-2pt] {\tiny simultaneity} \fi;
%    \fill [red] (V\i) circle (0.05);
    
%   	\draw [gray] \point(\x*\xparticle,\x*\yparticle) -- \point(0,{\x*\yparticle-\x*\lightforward*\xparticle});
%   	\draw [gray] \point(\x*\xparticle,\x*\yparticle) -- node [above,sloped] {\tiny light} \point(0,{\x*\yparticle-\x*\lightbackward*\xparticle}) \ifnum\i=10 node [left, black] {\tiny Here the observer sees the particle at the distanse $d\ell$}\fi;
    	
%    }
   	\draw [gray] \point(\xparticle,\yparticle) -- node [above,sloped] {\tiny light} \point(0,{\yparticle-\lightforward*\xparticle});
   	\draw [gray] \point(\xparticle,\yparticle) -- node [above,sloped] {\tiny light} \point(0,{\yparticle-\lightbackward*\xparticle});
    \draw [densely dashed, red] \point(\xparticle,\yparticle) coordinate (V)  --  \point(0,{projection(\xparticle,\yparticle)}) coordinate (A10) node [sloped, pos=0.5, above=-2pt] {\tiny simultaneity};
    \fill [red] (V) circle (0.05);
   	
    %------------------ particle coordinates nodes --------------------------- 	
    \node [left] at \point(0,\yparticle) {$x^0 + dx^0$};
    \node [below] at \point(\xparticle,0) {$x^i + dx^i$};
    \ifnum\angle=90\relax\else\node [left] at (A10) {$\left( x^0 + dx^0\right)  + \frac{g_{0i}}{g_{00}}dx^0$};\fi
    
    \end{tikzpicture}
\end{center}
    
\begin{align}
	u^{\mu} &=  \left\lbrace \frac{1}{\sqrt{g_{00}}},0,0,0\right\rbrace \\
	v^{\nu} &=  \left\lbrace \frac{dx^0}{d\lambda},\frac{dx^i}{d\lambda}\right\rbrace 
\end{align} 
   
\begin{equation}
	\mathbf{g}(u^{\mu},v^{\nu}) = g_{00}u^0v^0 +   g_{01}u^0v^i  = \sqrt{g_{00}}v^0 + \frac{g_{01}}{\sqrt{g_{00}}} v^i = \sqrt{g_{00}}\left(v^0 + \frac{g_{01}}{g_{00}} v^i \right) 
\end{equation} 

\begin{equation}
	\mathbf{g}(u^{\mu},v^{\nu}) d\lambda = \sqrt{g_{00}}\left(dx^0 + \frac{g_{01}}{g_{00}} dx^i \right) 
\end{equation} 

The velocity $v$ of particle is measured in terms \emph{of the proper time of an observer} \[d\tau_\mathrm{sync} = \mathbf{g}(u^{\mu},v^{\nu}) d\lambda,\]% 
as determined by his clocks, \emph{but the clocks should be synchronized along the trajectory of the particle}. 
   
\end{document}