Natural units

Throughout the relativistic treatment it is convenient to use a somewhat unusual but extremely handy system of units: space is measured in light-seconds, time in seconds. In these units the speed of light is simply

$\ev{c = 1\ \text{(dimensionless)}}$

and the relativistic formulas lose a great many $c$‘s and $c^2$‘s that were only needed to sort out the units of measurement. This is not a trick: it is the choice of measuring distances with the same “ruler” we use to measure time, namely light.

Principle — Natural units of relativity

We define:

• light-second (ls) = distance travelled by light in $1\,\text{s}$ $\approx 3{.}00\cdot 10^8\,\text{m}\approx 300\,000\,\text{km}$;
• light-minute (lm) $\approx 1{.}80\cdot 10^{10}\,\text{m}$;
• light-hour (lh) $\approx 1{.}08\cdot 10^{12}\,\text{m}$;
• light-year (ly) $\approx 9{.}46\cdot 10^{15}\,\text{m}$.

A velocity, in these units, is a pure number (ls/s or ly/year) always between $0$ and $1$. Light travels at $1$; nothing else can.

Tip

In natural units, $v=0{.}5$ means “half the speed of light”. A velocity $v=2$ is impossible in principle: a material object cannot exceed $c=1$.

These units have a distinctly cosmic flavour: the universe, on an astronomical scale, is already naturally measured in light-times. The Moon is $\approx 1{.}3$ light-seconds from Earth; the Sun, $\approx 8{.}3$ light-minutes; Pluto, $\approx 5{.}5$ light-hours; the nearest star, Proxima Centauri, $\approx 4{.}24$ light-years; the centre of our galaxy, $\approx 26\,000$ light-years; the Andromeda galaxy, $\approx 2{.}5$ million light-years. It is as if the universe itself were suggesting: “stop converting to kilometres, use light-times”.

Why these particular units?

Measuring space in light-seconds means, at bottom, measuring distances using light as a ruler. This is already done with radar (terrestrial distances) and with laser reflectors on the Moon (Apollo programme). GPS works the same way: the satellite transmits the time at which it sent the signal, the receiver measures the delay, and from the relation $d = c\,\Delta t$ reconstructs the distance. So, in practice, in everyday life we already measure space in light-times; it is we who artificially convert back into metres because SI requires it.

Here are the invariant and the main formulas in both forms, with and without $c$:

	SI units	Natural units ($c=1$)
Invariant	$(\Delta\tau)^2 = (\Delta t)^2 - (\Delta s/c)^2$	$(\Delta\tau)^2 = (\Delta t)^2 - (\Delta s)^2$
Velocity	$0\leq v\leq c$	$0\leq \beta\leq 1$
Lorentz factor	$\gamma = 1/\sqrt{1-v^2/c^2}$	$\gamma = 1/\sqrt{1-\beta^2}$
Rest energy	$E_0 = mc^2$	$E_0 = m$
Energy–momentum	$E^2 = (pc)^2 + (mc^2)^2$	$E^2 = p^2 + m^2$

Tip

In natural units $\beta\equiv v/c$. When we see $v=0{.}9$ written without units, it should be read as $v=0{.}9\,c$.

Links

Topics: Special relativity Concepts: Space-time invariant Skills: Dimensional analysis Methods: Natural units

Related exercises: Spacelike or timelike · Problem — Particle or photon · Mr Rossi goes to the theatre