Reading the gamma(v) graph

Problem

Reading a graph: $\gamma(v)$. A graph shows $\gamma$ on the vertical axis and $v/c$ on the horizontal axis. Describe the shape of the curve: where it equals $1$, where it grows slowly, where it “explodes”. Indicate the asymptote and explain why, physically, infinite energy is needed to reach $v=c$.

Solution

Value at rest. For $v=0$ we have $\beta = 0$ and

$\gamma = \frac{1}{\sqrt{1-0}} = 1.$

The curve starts at $\gamma = 1$: at low speed the physics is (almost) Newtonian.

Slow growth at small velocities. For $\beta \ll 1$, $\gamma \approx 1 + \tfrac{1}{2}\beta^2$: the curve is nearly flat, growing like a very gentle parabola. Up to $v \approx 0{,}5\,c$ the factor is barely $\approx 1{,}15$. This is why we do not notice relativistic effects in everyday life.

“Explosion” near $c$. As $v$ approaches $c$, the denominator $\sqrt{1-\beta^2} \to 0$ and $\gamma \to \infty$: the curve rises very steeply. At $0{,}9\,c$ it is $\approx 2{,}3$, at $0{,}99\,c$ already $\approx 7$, at $0{,}999\,c$ over $22$.

Asymptote. The line $v/c = 1$ is a vertical asymptote: $\gamma$ diverges but is never reached at $v = c$.

$\ev{\gamma(0)=1;\ \ \text{vertical asymptote at } v/c=1}$

Why infinite energy. The relativistic kinetic energy is $K = (\gamma - 1)\,m c^2$. Since $\gamma \to \infty$ as $v \to c$, bringing a mass to $v = c$ would require infinite energy. No body with mass can therefore reach the speed of light: only massless particles (photons) travel at exactly $c$.

Links

Topics: Special relativity Concepts: Time dilation Skills: Limiting-case and science-fiction analysis