Geometrized unit system

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In physics, especially in the general theory of relativity, geometrized units or geometric units constitute a physical unit system in which all physical quantities are identified with geometric quantities such as areas, lengths, dimensionless numbers, path curvatures, or sectional curvatures. In this system, the base physical units are chosen so that the speed of light and the gravitational constant are set equal to unity:

These are only two constraints, leaving latitude to also set the Boltzmann constant and Coulomb force constant equal to unity:

<math> \frac{1}{4 \pi \epsilon_0} = 1 </math>

If Dirac's constant (also called the "reduced Planck's constant") is also set equal to unity,

we then obtain Planck units.

Many equations in relativistic physics appear far simpler when expressed in geometric units, because all appearances of G or c drop out. For example, the Schwarzschild radius of a nonrotating uncharged black hole with mass m becomes simply r = 2 m. Therefore, many books and papers on relativistic physics use geometric units exclusively. An alternative system of geometrized units is often used in particle physics and cosmology, in which <math>8\pi G=1</math> instead. This introduces an additional factor of 8π into Newton's law of universal gravitation but simplifies Einstein's equations, the Einstein-Hilbert action, the Friedmann equations and the Newtonian Poisson equation by removing the corresponding factor.

In order to make practical computations, it is often necessary to use SI units. Fortunately, there are simple rules for converting any equation expressed in geometric units into one expressed in SI units.

1 Definition
2 Geometrical quantities
3 See also
4 Reference

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Definition

In geometric units, every time interval is interpreted as the distance travelled by light during that given time interval. That is, one second is interpreted as one light second, so time has the geometric units of length. This is dimensionally consistent with the notion that, according to the kinematical laws of special relativity, time and distance are on an equal footing.

Energy and momentum are interpreted as components of the four-momentum vector, and mass is the magnitude of this vector, so in geometric units these must all have the dimension of length. We can convert a mass expressed in kilograms to the equivalent mass expressed in meters by multiplying by the conversion factor G/c². For example, the Sun's mass of 2.0×10³⁰ kg in SI units is equivalent to 1.5 km. This is half the Schwarzschild radius of a one solar mass black hole. All other conversion factors can be worked out by combining these two.

The most useful conversion factors are listed in the table below. To convert from SI units to geometrized units, multiply by the given factor; to go the other way, divide.

quantity	factor	numeric value
time	c	3.00 × 10⁸ m/s
mass	G/c²	7.43 × 10⁻²⁸ m/kg
momentum	G/c³	2.48 × 10⁻³⁶ s/kg
energy	G/c⁴	8.26 × 10⁻⁴⁵ m/J
power	G/c⁵	2.75 × 10⁻⁵⁵ s/J
temperature	G k/c⁴	1.14 × 10⁻⁶⁷ m/K

The small numerical size of the last few conversion factors reflects the fact that relativistic effects are only noticeable when large masses or high speeds are considered.

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Geometrical quantities

The components of curvature tensors such as the Einstein tensor have, in geometric units, the dimensions of sectional curvature. So do the components of the stress-energy tensor. Therefore the Einstein field equation is dimensionally consistent in these units.

Path curvature is the reciprocal of the magnitude of the curvature vector of a curve, so in geometric units it has the dimension of inverse length. Path curvature measures the rate at which a nongeodesic curve bends in spacetime, and if we interpret a timelike curve as the world line of some observer, then its path curvature can be interpreted as the magnitude of the acceleration experienced by that observer. Physical quantities which can be identified with path curvature include the components of the electromagnetic field tensor.

Any velocity can be interpreted as the slope of a curve; in geometric units, slopes are evidently dimensionless ratios. Physical quantities which can be identified with dimensionless ratios include the components of the electromagnetic potential four-vector and the electromagnetic current four-vector.

Physical quantities such as mass and electric charge which can be identified with the magnitude of a timelike vector have the geometric dimension of length. Physical quantities such as angular momentum which can be identified with the magnitude of a bivector have the geometric dimension of area.

Here is a table collecting some important physical quantities according to their dimensions in geometrized units. They are listed together with the appopriate conversion factor.

Physical quantity	SI units	Conversion Factor
Sectional Curvature
mass density	kg · m^-3	G/c²
momentum density	kg · m^-2 · s^-1	G/c³
energy density, pressure	kg · m^-1 · s^-2	G/c⁴
electric charge density	kg^1/2 · m^-3/2 · s^-1	G^1/2/c²
electric current flux	kg^1/2 · m^-1/2 · s^-2	G^1/2/c³
Path Curvature
acceleration	m · s^-2	1/c²
magnetic field (induction)	kg^1/2 · m^-3/2	G^1/2
electric field	kg^1/2 · m^-1/2 · s^-1	G^1/2/c
Dimensionless Ratio
velocity	m · s^-1	1/c
force	kg · m · s^-2	G/c⁴
power	kg · m² · s^-3	G/c⁵
electric potential	kg^1/2 · m^1/2 · s^-1	G^1/2/c²
electric current	kg^1/2 · m^3/2 · s^-2	G^1/2/c³
Length
time	s	c
mass	kg	G/c²
momentum	kg · m · s^-1	G/c³
energy, work	kg · m² · s^-2	G/c⁴
electric charge	kg^1/2 · m^3/2 · s^-1	G^1/2/c²
specific angular momentum	m² · s^-1	1/c
Area
angular momentum	kg · m² · s^-1	G/c³