Ranking of Lorentz factors (0,1c–0,99c)

Problem

Ranking of $\gamma$. Order from smallest to largest the Lorentz factors corresponding to $v=0{,}1\,c$, $0{,}5\,c$, $0{,}9\,c$, $0{,}99\,c$. Calculate their approximate numerical value.

Solution

Formula. $\gamma = 1/\sqrt{1-\beta^2}$ increases monotonically with $\beta$: the order of $\gamma$ follows the order of the velocities.

Numerical values.

$\gamma(0{,}1\,c) = \frac{1}{\sqrt{1-0{,}01}} = \frac{1}{\sqrt{0{,}99}} \approx 1{,}005,$

$\gamma(0{,}5\,c) = \frac{1}{\sqrt{1-0{,}25}} = \frac{1}{\sqrt{0{,}75}} \approx 1{,}15,$

$\gamma(0{,}9\,c) = \frac{1}{\sqrt{1-0{,}81}} = \frac{1}{\sqrt{0{,}19}} \approx 2{,}29,$

$\gamma(0{,}99\,c) = \frac{1}{\sqrt{1-0{,}9801}} = \frac{1}{\sqrt{0{,}0199}} \approx 7{,}09.$

Ordering.

$\ev{\gamma(0{,}1\,c) < \gamma(0{,}5\,c) < \gamma(0{,}9\,c) < \gamma(0{,}99\,c)}$

At $0{,}1\,c$ the effect is about $0{,}5\%$, almost negligible; at $0{,}99\,c$ the factor is already $\approx 7$: the curve “explodes” as it approaches $c$.

Links

Topics: Special relativity Concepts: Time dilation Skills: Ranking reasoning