We know that because of entanglement,
\(max(v_x)=c\)
Entanglement shares energy and acts like a drag force proportional to \(v_x^2\), that give rise to a terminal velocity, \(max(v_x)\). And
\(max(v_t)=c\)
since all dimensions are equivalent, swapping \(x\) for \(t\) is arbitrary.
Since, \(v_t\) and \(v_x\) are oscillating and orthogonal,
\(v_x=max(v_x)sin(\omega t)=c.sin(\omega t)\)
\(v_t=max(v_t)cos(\omega t)=c.cos(\omega t)\)
With the trigonometrical identity,
\(sin^2(\omega t)+cos^2(\omega t)=1\)
\(c^2sin^2(\omega t)+c^2cos^2(\omega t)=c^2\)
So,
\(v^2_x+v^2_t=c^2\) --- (*)
where \(c\) is the speed limit in both dimensions.
We have conservation of energy across two dimensions for a single particle. As \(v_x\) increases in a field, \(v_t\) decreases and we have time dilation.
Good night.
