Geometric units

Difficulty level:   ★ ★ ★

Geometric units are units of measurement in which Newton’s gravitational constant and the speed of light are 1: G=c=1. Quantities are hence expressed in units of length, which we will take to be metres. This may seem a pain at first, but it is very convenient when working with equations in relativity because you can omit any G and c terms which makes expressions shorter. Given any expression, it is straightforward to return it to ordinary SI (Système international) units which use metres, kilograms, seconds etc. Related are Planck units in which additionally the reduced Planck constant \hbar=1.

Quantity SI units ⇒ Conversion factor ⇒ Geometric units
Length m 1 m
Mass kg G/c2 m
Time s c m
Speed, velocity m/s 1/c 1
Acceleration m/s2 1/c2 m-1
Momentum kg m/s G/c3 m
Angular momentum kg m2/s G/c3 m2
Force N (kg m/s2) G/c4 1
Energy J (kg m2/s2) G/c4 m
Energy density J/m3 (kg/m/s2) G/c4 m-2
Power kg m2/s3 G/c5 1
Pressure, stress kg/m/s2 G/c4 m-2

“N” means newtons, “J” means joules, and units of “1” means dimensionless. To convert from SI units to geometric, multiply by the listed conversion factor. To convert from geometric units to SI, divide by it.

Examples:

  1. 2km. In geometrised units, unchanged, so 2km.
  2. mass of Sun is ≈ 2×1030 kg. Multiply by G/c^2 to get ≈ 1500m, which can be conveniently done in Wolfram Alpha. So the mass of the Sun is 1.5km!
  3. Suppose we want to go the other way, starting from geometric units and wanting to express in “normal” units. Time of 1.3×1026 metres. So we multiply by the inverse of the conversion factor, so that’s 1/c. Answer 4.4×1017 seconds or 14 billion years, which is the age of the universe as crudely estimated from the Hubble constant.

References

2 thoughts on “Geometric units”

  1. What about charge? I am actually facing issues in converting Q/M from geometric units to natural units. It will be great if you can help me with this…

    thanks

    1. hi Ashish, it seems this comes down to one’s choice of units for electromagnetism. If we take Gaussian units, then to convert from Coulombs, multiply by 1/\sqrt{4\pi\epsilon_0}. Hence 1 Coulomb is \approx 3\times 10^9 statC (\textrm{cm}^{3/2}\textrm{g}^{1/2}\textrm{s}^{-1}) or \approx 95000 \textrm{m}^{3/2}\textrm{kg}^{1/2}s^{-1}. Now convert to geometric units by multiplying by \sqrt{G}/c^2: Wald’s table has this one. The result is 8.6\times 10^{-18} metres, which is on the order of the “radius of a proton”. In particular, charge has units of length. Now, you asked for charge per mass, and I get 1 Coulomb/kg \approx 1.16\times 10^{10}, a dimensionless quantity in these Gaussian-geometric units.

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