One Second in Lightspeed Ship

Suppose the relative motion of two observers A and B is along their common $x$ -axes. A is static, B is in a starship moving at speed $u$ relative to A. Inside the starship, some object begins to move on the $x$ -axis.

A observes the moving object and records its coordinates $x$ , $y$ , $z$ at any time $t$ . B observes the same moving object, within his system, he records coordinates $x^{'}$ , $y^{'}$ , $z^{'}$ at any time $t^{'}$ .

Denote the speed of light as $c$ . According to the relativity of Einstien,

\begin{matrix} x^{'} & = & \frac{x - u t}{\sqrt{1 - u^{2} / c^{2}}} \\ y^{'} & = & y \\ z^{'} & = & z \\ t^{'} & = & \frac{t - u x / c^{2}}{\sqrt{1 - u^{2} / c^{2}}} \end{matrix}

In the last equation above, note as $u \leq c$ , $x / t \leq c$ ,

\begin{matrix} u x / t & \leq & c^{2} \\ u x / c^{2} & \leq & t \\ t - u x / c^{2} & \geq & 0 \end{matrix}

therefore

t^{'} \geq 0 .

One may verify that given B’s observations, A’s system “at rest” would record

\begin{matrix} x & = & \frac{x^{'} + u t^{'}}{\sqrt{1 - u^{2} / c^{2}}} \\ y & = & y^{'} \\ z & = & z^{'} \\ t & = & \frac{t^{'} + u x^{'} / c^{2}}{\sqrt{1 - u^{2} / c^{2}}} \end{matrix}

From the last equation, if $u$ is close to the lightspeed, the denominator can be infinitely small, as long as $t^{'} > 0$ , $t$ can be infinitely large. Hence why they say “one second in a lightspeed starship is a lifetime on earth.”

Actually one may show when one moves at variable speed, the above equations still hold.