Worldlines

Coming from general relativity.
Given a relativistic spacetime, we assume the following:

Postulate 1. The worldline γ of a massive particle satisfies:
(i) $g_{\gamma (\lambda )}(v_{\gamma ,\gamma (\lambda )},v_{\gamma ,\gamma (\lambda )})>0$, and
(ii) $g_{\gamma (\lambda )}(T,v_{\gamma ,\gamma (\lambda )})>0$.

Postulate 2. The worldline γ of a massless particle satisfies:
(i) $g_{\gamma (\lambda )}(v_{\gamma ,\gamma (\lambda )},v_{\gamma ,\gamma (\lambda )})=0$, and
(ii) $g_{\gamma (\lambda )}(T,v_{\gamma ,\gamma (\lambda )})>0$.

Postulate 1 tells us that: (i) A massive particle’s worldline lies inside the cone structure, and (ii) it is future-directed.

The only difference with postulate 2 is that the worldline of a massless particle lies on the surface of the future cone.
It is at this point that we can identify the surface of the cone as the trajectory of light, as light is a massless particle.

\begin{document} 
\begin{tikzpicture}[scale=1.4][domain=0:4] 
 \draw [thick, blue, rotate around={-30:(1.415,1.3)}](1,2) arc (180:0:0.5cm and 0.1cm);
            \draw[red, ultra thick,->, rotate around={-12.5:(1.415,1.3)}] (1.415,1.3) -- (1.415,2.3); 
            \draw [thick,blue,rotate around={-30:(1.415,1.3)}](1,2) arc (180:360:0.5cm and 0.1cm) -- (1.415,1.3) -- cycle;
            \node[circle, fill, inner sep=1.5pt, label={left:\Large{$p$}}] at (1.415,1.3) {};]
            \node at (2,2.4) {\color{red}\Large{$v_{\gamma,p}$}};
            %
            \draw [thick, blue, rotate around={-10:(0.56,3.8)}] (-1,5) arc (180:0:1cm and 0.1cm);
            \draw[red, ultra thick,->, rotate around={-10:(0.56,3.8)}] (0.56,3.8) -- (0.56,5.5);
            \draw [thick,blue,rotate around={-10:(0.56,3.8)}] (-1,5) arc (180:360:1cm and 0.1cm) -- (0.56,3.8) -- cycle;
            \node[circle, fill, inner sep=1.5pt, label={left:\Large{$q$}}] at (0.56,3.8) {};
            \node at (1.2,5.7) {\color{red}\Large{$v_{\gamma,q}$}};
            %
            \draw[thick] (0,0) .. controls (4,2) and (-2,3.5) .. (2,5);
            \node at (-0.2,-0.2) {\Large{$\gamma$}};
            %
            \draw[thick] (8,0) .. controls (12,2) and (6,3.5) .. (10,5);
            \node at (7.8,-0.2) {\Large{$\gamma$}};
            %
            \draw [thick, blue, rotate around={-42:(9.415,1.3)}](9,2) arc (180:0:0.5cm and 0.1cm);
            \draw[red, ultra thick,->, rotate around={-12.5:(9.415,1.3)}] (9.415,1.3) -- (9.415,2.3); 
            \draw [thick,blue,rotate around={-42:(9.415,1.3)}](9,2) arc (180:360:0.5cm and 0.1cm) -- (9.415,1.3) -- cycle;
            \node[circle, fill, inner sep=1.5pt, label={left:\Large{$p$}}] at (9.415,1.3) {};
            \node at (10,2.4) {\color{red}\Large{$v_{\gamma,p}$}};
            %
            \draw [thick, blue, rotate around={10:(8.56,3.8)}] (7,5) arc (180:0:1cm and 0.1cm);
            \draw[red, ultra thick,->, rotate around={-10:(8.56,3.8)}] (8.56,3.8) -- (8.56,5.5);
            \draw [thick,blue,rotate around={10:(8.56,3.8)}] (7,5) arc (180:360:1cm and 0.1cm) -- (8.56,3.8) -- cycle;
            \node[circle, fill, inner sep=1.5pt, label={left:\Large{$q$}}] at (8.56,3.8) {};
            \node at (9.2,5.7) {\color{red}\Large{$v_{\gamma,q}$}};
\end{tikzpicture} 
\end{document}

World lines in spacetime of a massive particle (left) and a massless particle (right). It is important to remember that the cones and velocity vectors live in the tangent space to the point, not on the manifold itself, which the above picture might lead you to believe.