Problem — Why no mass reaches c

Problem

Explain why no object with mass can reach the speed of light, referring to the expression for $\gamma$ and/or relativistic momentum.

Solution

The behaviour of $\gamma$. The Lorentz factor

$\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$

diverges as $v \to c$: as $v$ tends to $c$ the denominator $\sqrt{1 - v^2/c^2} \to 0$, so $\gamma \to +\infty$.

Momentum and energy. The relativistic momentum and energy of a body of mass $m$ are

$p = \gamma m v, \qquad E = \gamma m c^2$

Since $\gamma \to \infty$ as $v \to c$, both $p$ and $E$ diverge: bringing a mass up to $c$ would require infinite momentum and energy.

Conclusion.

$\ev{v = c \text{ requires } E \to \infty \text{ for } m > 0 \;\Rightarrow\; \text{unreachable}}$

No source can supply infinite energy, so an object with mass can only get closer and closer to $c$ without ever reaching it. Only massless particles (such as photons) travel at exactly $c$: for them $\gamma m$ stays finite in the form $E = pc$.

Links

Topics: Special relativity Concepts: Relativistic momentum Skills: Limiting-case and sci-fi-physics analysis