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A geometrized unit system or geometric unit system is a system of natural units in which the base physical units are chosen so that the speed of light, c, and the gravitational constant, G, are set equal to unity.

The geometrized unit system is not a completely defined or unique system: latitude is left to also set other constants to unity. We may, for example, also set Boltzmann's constant, k, and the Coulomb force constant, k_C, to unity.

Setting Dirac's constant (also called the "reduced Planck's constant"), , to unity yields Planck units.

This system is useful in physics, especially in the special and general theories of relativity. All physical quantities are identified with geometric quantities such as areas, lengths, dimensionless numbers, path curvatures, or sectional curvatures.

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 = 2m. 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 8πG = 1 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.

Practical measurements and computations are usually done in SI units, however, conversions are generally quite straightforward.

Contents 1 Definition 2 SI base geometrical quantities 3 Various geometrical quantities 4 References 5 External links

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 metres 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
charge	(G kC)1/2/c2	8.62 × 10−18 m/C
momentum	G/c2	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.

Below are presented all conversion factors that covers all combinations between SI base units, and if not possible, between them and their unique elements, because ampere is a dimensionless ratio of two lengths such as [C/s], and candela (1/683 [W/sr]) is a dimensionless ratio of two dimensionless ratios such as ratio of two volumes [kg*m²/s³]=[W] and ratio of two areas [m²/m²]=[sr], while mole is only a dimensionless Avogadro number of entities such as atoms or particles:

Conversion from kg,s,C,K into m:

• G/c² [m/kg]
• c [m/s]
• ((G/(4*π*ε₀))^0.5)/c² [m/C]
• (G*k)/c⁴ [m/K]

Conversion from m,s,C,K into kg:

• c²/G [kg/m]
• c³/G [kg/s]
• 1/(G*4*π*ε₀)^0.5 [kg/C]
• k/c² [kg/K]

Conversion from m,kg,C,K into s

• 1/c [s/m]
• G/c³ [s/kg]
• ((G/(4*π*ε₀))^0.5)/c³ [s/C]
• (G*k)/c⁵ [s/K]

Conversion from m,kg,s,K into C

• c²/((G/(4*π*ε₀))^0.5) [C/m]
• (G*4*π*ε₀)^0.5 [C/kg]
• c³/((G/(4*π*ε₀))^0.5) [C/s]
• (k*(G*4*π*ε₀)^0.5)/c² [C/K]

Conversion from m,kg,s,C into K

• c⁴/(G*k) [K/m]
• c²/k [K/kg]
• c⁵/(G*k) [K/s]
• c²/(k*(G*4*π*ε₀)^0.5) [K/C]

All these units represents nothing else than length along dimension, thus revealing SI as redundant in comparison to geometrized units.

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 for both Gaussian cgs units and SI units

Physical quantity	Gaussian units	Gauss Conversion Factor	SI units	SI Conversion Factor
Sectional Curvature (1 / length2)
mass density	g·cm-3	G/c2	kg · m-3	G/c2
momentum density	g·cm-2·s-1	G/c2	kg · m-2 · s-1	G/c2
energy density, pressure	g·cm-1·s-2	G/c4	kg · m-1 · s-2	G/c4
electric charge density	g1/2·cm-3/2·s-1	G1/2/c2	C·m-3	(G/4 π ε0 ) 1/2/c2
electric current flux	g1/2 · cm-1/2 · s-2	G1/2/c2	C·m-2·s-1	(G/4 π ε01/2/c2
Path Curvature (1/length)
cm·s-2	1/c2	m·s-2	1/c2
magnetic field (induction)	g1/2·cm-3/2	G1/2	kg·s-1·C-1	( 4 π ε0 G ) 1/2 / c
electric field	g1/2·cm-1/2·s-1	G1/2/c	kg·m·s-2·C-1	( 4 π ε 0 G) 1/2 / c2
Dimensionless Ratio
velocity	cm·s-1	1/c	m·s-1	1/c
force	g·cm·s-2	G/c4	kg·m·s-2	G/c4
power	g·cm2·s-3	G/c5	kg·m2·s-3	G/c5
electric potential	g1/2·cm1/2·s-1	G1/2/c2	kg·m2·s-2·C-1	( 4 π ε 0 G ) 1/2 / c2
electric current	g1/2·cm3/2·s-2	G1/2/c2	C·s-1	( G / 4 π ε 0 ) 1/2/c2
Length
time	s	c	s	c
mass	g	G/c2	kg	G/c2
momentum	g·cm·s-1	G/c2	kg·m·s-1	G/c2
energy, work	g·cm2·s-2	G/c4	kg·m2·s-2	G/c4
electric charge	g1/2·m3/2·s-1	G1/2/c2	C	(G / 4 π ε0) 1/2 /c2
specific angular momentum	cm2·s-1	1/c	m2·s-1	1/c
Area
angular momentum	g·cm2·s-1	G/c2	kg·m2·s-1	G/c2

This table can be augmented to include temperature, as indicated above, as well as further derived physical quantities such as various moments.

• Wald, Robert M. (1984). General Relativity (book). Chicago: University of Chicago Press. ISBN 0-226-87033-2.  See Appendix F

• Conversion factors for energy equivalents

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