The low-velocity limit

If time dilation is real, why do we never notice it? The answer lies in the behaviour of the Lorentz factor when velocities are much smaller than that of light. For $v \ll c$ we can use the first-order expansion $(1-\varepsilon)^{-1/2} \approx 1 + \varepsilon/2$, with $\varepsilon = v^2/c^2$:

$\gamma \approx 1 + \frac{v^2}{2c^2}$

The deviation of $\gamma$ from $1$ does not grow as $v$, but as $v^2/c^2$: it is therefore doubly small at ordinary velocities. This is why Galilean physics works perfectly well in everyday life — the relativistic correction is there, but it is drowned out by a tiny factor of $(v/c)^2$.

Example — The Earth in orbit

The Earth orbits the Sun at $v = 30\;\text{km/s}$, i.e. $v/c \approx 10^{-4}$. The Lorentz factor is $\gamma \approx 1 + 5\cdot 10^{-9}$: the difference between proper time and dilated time is of the order of five billionths. Irrelevant in everyday life, but perfectly measurable with atomic clocks.

GPS and relativity

GPS satellites orbit at $\sim 14\,000\;\text{km/h}$ and experience a time dilation of a few microseconds per day. If this effect is not corrected for (along with a second effect from general relativity, due to the different gravitational potential), GPS accumulates errors of around $10\;\text{km}$ per day! Relativity is not just theory: it is in your smartphone.

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Topics: Special relativity Concepts: Time dilation Skills: Analysis of limiting cases and sci-fi physics · Dimensional analysis

Related exercises: Problem — Gravitational redshift on Everest · Problem — GPS, quantitatively · Problem — Lorentz factor of the muon