Problem — Minkowski diagram of a rocket

Problem

The graph in the figure (Minkowski diagram) shows the world line of a rocket travelling at $v = 0{,}9c$ relative to an observer at rest. A light pulse also propagates from the origin. Determine from the graph the factor $\gamma$ corresponding to this speed and calculate the time dilation for a proper interval $\Delta t = 100$ ns.

Solution

Reading the graph. The rocket’s world line has a smaller slope than light’s ($x = ct$, the bisector): this confirms $v < c$. The rocket’s line has the inclination corresponding to $v = 0{,}9c$.

Lorentz factor. With $v = 0{,}9c$, $\dfrac{v^2}{c^2} = 0{,}81$:

$\gamma = \frac{1}{\sqrt{1 - 0{,}81}} = \frac{1}{\sqrt{0{,}19}} \approx 2{,}29$

$\ev{\gamma \approx 2{,}29}$

Time dilation. The proper interval $\Delta t = 100$ ns (measured aboard the rocket) appears dilated for the observer at rest:

$\Delta t' = \gamma\,\Delta t = 2{,}29 \times 100\;\text{ns}$

$\ev{\Delta t' \approx 229\;\text{ns}}$

For the observer at rest, between two “ticks” of the onboard clock, $229$ ns pass instead of $100$: the moving clock runs slow.

Links

Topics: Relatività ristretta Concepts: Dilatazione dei tempi Skills: Costruzione di diagrammi di Minkowski