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| {{Cleanup|section|date=July 2009}}
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| A '''geometrized unit system''' or '''geometric unit system''' is a system of [[natural units]] in which the base [[unit of measurement|physical units]] are chosen so that the [[speed of light in vacuum]], ''c'', and the [[gravitational constant]], ''G'', are set equal to unity.
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| :<math> c = 1 \ </math>
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| :<math> G = 1 \ </math>
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| The geometrized unit system is not a completely defined or unique system: latitude is left to also set other [[Physical constants|constants]] to unity. We may, for example, also set [[Boltzmann's constant]], ''k''<sub>B</sub>, and the [[Coulomb constant]], ''k''<sub>e</sub>, to unity.
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| :<math> k_\mathrm{B} = 1 \ </math>
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| :<math> k_\mathrm{e} = \frac{1}{4 \pi \epsilon_0} = 1 </math>
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| Setting the [[reduced Planck constant]], ħ, to unity yields [[Planck units]].
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| :<math> \hbar = 1 \ </math>
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| This system is useful in [[physics]], especially in the [[special theory of relativity|special]] and [[general theory of relativity|general theories of relativity]]. All [[physical quantity|physical quantities]] are identified with geometric quantities such as areas, lengths, dimensionless numbers, path curvatures, or sectional curvatures.
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| 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 {{nowrap|1=''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 [[physical cosmology|cosmology]], in which {{nowrap|1=8π''G'' = 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.
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| Practical measurements and computations are usually done in [[SI]] units, but conversions are generally quite straightforward.
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| ==Definition==
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| 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 [[kinematics|kinematical]] laws of [[special relativity]], time and distance are on an equal footing.
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| [[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''/''c''<sup>2</sup>. For example, the [[Sun]]'s mass of {{val|2.0|e=30|u=kg}} in SI units is equivalent to {{val|1.5|u=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.
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| The small numerical size of the few conversion factors reflects the fact that relativistic effects are only noticeable when large masses or high speeds are considered.
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| In the table below are listed all conversion factors that are useful to convert between all combinations of the 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⋅m<sup>2</sup>/s<sup>3</sup>] = [W] and ratio of two areas [m<sup>2</sup>/m<sup>2</sup>] = [sr], while mole is only a dimensionless [[Avogadro number]] of entities such as atoms or particles: | |
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| Conversion from kg, s, C, K into m:
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| *''G''/''c''<sup>2</sup> [m/kg]
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| *''c'' [m/s]
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| *(''G''/(4πε<sub>0</sub>))<sup>1/2</sup>/''c''<sup>2</sup> [m/C]
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| *''Gk''<sub>B</sub>/''c''<sup>4</sup> [m/K]
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| Conversion from m, s, C, K into kg:
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| *''c''<sup>2</sup>/''G'' [kg/m]
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| *''c''<sup>3</sup>/''G'' [kg/s]
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| *1/(''G'' 4πε<sub>0</sub>)<sup>1/2</sup> [kg/C]
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| *''k''<sub>B</sub>/''c''<sup>2</sup> [kg/K]
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| Conversion from m, kg, C, K into s
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| *1/''c'' [s/m]
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| *''G''/''c''<sup>3</sup> [s/kg]
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| *(''G''/(4πε<sub>0</sub>))<sup>1/2</sup>/''c''<sup>3</sup> [s/C]
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| *''Gk''<sub>B</sub>/''c''<sup>5</sup> [s/K]
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| Conversion from m, kg, s, K into C
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| *''c''<sup>2</sup>/(''G''/(4πε<sub>0</sub>))<sup>1/2</sup> [C/m]
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| *(''G'' 4πε<sub>0</sub>)<sup>1/2</sup> [C/kg]
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| *c<sup>3</sup>/(''G''/(4πε<sub>0</sub>))<sup>1/2</sup> [C/s]
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| *''k''<sub>B</sub>(''G'' 4πε<sub>0</sub>)<sup>1/2</sup>/''c''<sup>2</sup> [C/K]
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| Conversion from m, kg, s, C into K
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| *''c''<sup>4</sup>/(''Gk''<sub>B</sub>) [K/m]
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| *''c''<sup>2</sup>/''k''<sub>B</sub> [K/kg]
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| *''c''<sup>5</sup>/(''Gk''<sub>B</sub>) [K/s]
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| *''c''<sup>2</sup>/(''k''<sub>B</sub>(''G'' 4πε<sub>0</sub>)<sup>1/2</sup>) [K/C]
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| All these units represents nothing else than length along dimension, thus revealing SI as redundant in comparison to geometrized units.
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| ==Geometrical quantities==
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| 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.
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| ''Path curvature'' is the reciprocal of the magnitude of the [[geodesic curvature|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 [[observation|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 tensor|electromagnetic field tensor]].
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| 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 four-potential|electromagnetic potential]] four-vector and the [[four-current|electromagnetic current]] four-vector.
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| Physical quantities such as [[mass]] and [[electric charge]] which can be identified with the [[magnitude (mathematics)|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''.
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| Here is a table collecting some important physical quantities according to their dimensions in geometrized units. They are listed together with the appropriate conversion factor for SI units.
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| {|class="wikitable"
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| |-
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| !Quantity!!SI dimension!!Geometrical dimension!!Multiplication factor
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| |-
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| |[[Length]]
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| |[''L'']
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| |[''L'']
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| |1
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| |-
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| |[[Time]]
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| |[''T'']
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| |[''L'']
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| |''c''
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| |-
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| |[[Mass]]
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| |[''M'']
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| |[''L'']
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| |''G'' ''c''<sup>−2</sup>
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| |-
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| |[[Velocity]]
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| |[''L'' ''T''<sup>−1</sup>]
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| |1
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| |''c''<sup>−1</sup>
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| |-
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| |[[Angular velocity]]
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| |[''T''<sup>−1</sup>]
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| |[''L''<sup>−1</sup>]
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| |''c''<sup>−1</sup>
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| |-
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| |[[Acceleration]]
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| |[''L'' ''T''<sup>−2</sup>]
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| |[''L''<sup>−1</sup>]
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| |''c''<sup>−2</sup>
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| |-
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| |[[Energy]]
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| |[''M'' ''L''<sup>2</sup> ''T''<sup>−2</sup>]
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| |[''L'']
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| |''G'' ''c''<sup>−4</sup>
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| |-
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| |[[Energy density]]
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| |[''M'' ''L''<sup>−1</sup> ''T''<sup>−2</sup>]
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| |[''L''<sup>−2</sup>]
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| |''G'' ''c''<sup>−4</sup>
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| |-
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| |[[Angular momentum]]
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| |[''M'' ''L''<sup>2</sup> ''T''<sup>−1</sup>]
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| |[''L''<sup>2</sup>]
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| |''G'' ''c''<sup>−3</sup>
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| |-
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| |[[Force (physics)|Force]]
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| |[''M'' ''L'' ''T''<sup>−2</sup>]
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| |1
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| |''G'' ''c''<sup>−4</sup>
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| |-
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| |[[Power (physics)|Power]]
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| |[''M'' ''L''<sup>2</sup> ''T''<sup>−3</sup>]
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| |1
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| |''G'' ''c''<sup>−5</sup>
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| |-
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| |[[Pressure]]
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| |[''M'' ''L''<sup>−1</sup> ''T''<sup>−2</sup>]
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| |[''L''<sup>−2</sup>]
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| |''G'' ''c''<sup>−4</sup>
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| |-
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| |[[Density]]
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| |[''M'' ''L''<sup>−3</sup>]
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| |[''L''<sup>−2</sup>]
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| |''G'' ''c''<sup>−2</sup>
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| |-
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| |[[Electric charge]]
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| |[''I'' ''T'']
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| |[''L'']
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| |''G''<sup>1/2</sup> ''c''<sup>−2</sup> (4πε<sub>0</sub>)<sup>−1/2</sup>
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| |-
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| |[[Electric potential]]
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| |[''M'' ''L''<sup>2</sup> ''T''<sup>−3</sup> ''I''<sup>−1</sup>]
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| |1
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| |''G''<sup>1/2</sup> ''c''<sup>−2</sup> (4πε<sub>0</sub>)<sup>1/2</sup>
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| |-
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| |[[Electric field]]
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| |[''M'' ''L'' ''T''<sup>−3</sup> ''I''<sup>−1</sup>]
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| |[''L''<sup>−1</sup>]
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| |''G''<sup>1/2</sup> ''c''<sup>−2</sup> (4πε<sub>0</sub>)<sup>1/2</sup>
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| |-
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| |[[Magnetic field]]
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| |[''M'' ''T''<sup>−2</sup> ''I''<sup>−1</sup>]
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| |[''L''<sup>−1</sup>]
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| |''G''<sup>1/2</sup> ''c''<sup>−1</sup> (4πε<sub>0</sub>)<sup>1/2</sup>
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| |-
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| |[[Potential]]
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| |[''M'' ''L'' ''T''<sup>−2</sup> ''I''<sup>−1</sup>]
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| |1
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| |''G''<sup>1/2</sup> ''c''<sup>−1</sup> (4πε<sub>0</sub>)<sup>1/2</sup>
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| |}
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| This table can be augmented to include temperature, as indicated above, as well as further derived physical quantities such as various [[moment (physics)|moments]].
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| ==References==
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| * {{cite book | author=Wald, Robert M. |authorlink=Robert Wald | title=[[General Relativity (book)|General Relativity]] | location=Chicago | publisher=University of Chicago Press | year = 1984 | isbn=0-226-87033-2}} ''See Appendix F''
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| ==External links==
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| * [http://www.physics.nist.gov/cuu/Constants/energy.html Conversion factors for energy equivalents]
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| {{systems of measurement}}
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| {{DEFAULTSORT:Geometrized Unit System}}
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| [[Category:General relativity]]
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| [[Category:Systems of units]]
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| [[Category:Natural units]]
| |
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