Speed for gamma equal to 2

Problem

At what speed must a clock travel for the Lorentz factor to be $\gamma = 2$, i.e. to dilate time by a factor of $2$?

Solution

Definition of the Lorentz factor. Recall that

$\gamma = \frac{1}{\sqrt{1-\beta^2}}, \qquad \beta = \frac{v}{c}.$

Inversion. Set $\gamma = 2$ and isolate $\beta$:

$\sqrt{1-\beta^2} = \frac{1}{\gamma} = \frac{1}{2} \quad\Rightarrow\quad 1-\beta^2 = \frac{1}{4} \quad\Rightarrow\quad \beta^2 = \frac{3}{4}.$

So

$\beta = \frac{\sqrt{3}}{2} \approx 0{,}866.$

Result.

$\ev{v = \frac{\sqrt{3}}{2}\,c \approx 0{,}87\,c}$

A speed of about $87\%$ that of light is needed just to double time: time dilation only becomes noticeable at enormous speeds.

Links

Topics: Relatività ristretta Concepts: Dilatazione dei tempi Skills: Impostazione simbolica