Editing Geometrized unit system

{{short description|Unit system used in the physics of relativity}}
A '''geometrized unit system'''<ref name=MisnerThorneWheeler>{{Cite book |last=Misner |first=Charles W. |title=Gravitation |last2=Thorne |first2=Kip S. |last3=Wheeler |first3=John Archibald |date=2008 |publisher=Freeman |isbn=978-0-7167-0344-0 |edition=27. printing |location=New York, NY}}</ref> or '''geometrodynamic 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.

:<math> c = 1 \ </math>
:<math> G = 1 \ </math>

The geometrized unit system is not a completely defined system. Some systems are geometrized unit systems in the sense that they set these, in addition to other [[Physical constants|constants]], to unity, for example [[Stoney units]] and [[Planck units]].

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.

Many equations in relativistic physics appear simpler when expressed in geometric units, because all occurrences of ''G'' and of ''c'' drop out. For example, the [[Schwarzschild radius]] of a nonrotating uncharged [[black hole]] with mass ''m'' becomes {{nowrap|1=''r'' = 2''m''}}. For this reason, many books and papers on relativistic physics use geometric units. 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 the [[Einstein field equations]], the [[Einstein–Hilbert action]], the [[Friedmann equations]] and the Newtonian [[Poisson equation]] by removing the corresponding factor.

==Definition==
Geometrized units were defined in the book [[Gravitation (book)|''Gravitation'']] by [[Charles W. Misner]], [[Kip S. Thorne]], and [[John Archibald Wheeler]] with the [[speed of light]], <math>c</math>, the [[gravitational constant]], <math>G</math>, and [[Boltzmann constant]], <math>k_b</math> all set to 1.<ref name=MisnerThorneWheeler/>{{rp|36}} Some authors refer to these units as geometrodynamic units.<ref>{{Cite journal|arxiv=2009.12057|doi=10.1103/PhysRevD.103.084052|title=Novel black-bounce spacetimes: Wormholes, regularity, energy conditions, and causal structure|year=2021|last1=Lobo|first1=Francisco S. N.|last2=Rodrigues|first2=Manuel E.|last3=Silva|first3=Marcos V. de S.|last4=Simpson|first4=Alex|last5=Visser|first5=Matt|journal=Physical Review D|volume=103|issue=8|page=084052|bibcode=2021PhRvD.103h4052L|s2cid=235581301}}</ref>

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.

[[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.

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.

==Conversions==

Listed below are 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. The [[vacuum permittivity]] and [[Boltzmann constant]] are ''ε''<sub>0</sub> and ''k''<sub>B</sub>.

{| class="wikitable center"
|-
!
! m
! kg
! s
! C
! K
|-
! scope="col" | '''m'''
| 1
|''c''<sup>2</sup>/''G'' [kg/m]
|1/''c'' [s/m]
|''c''<sup>2</sup>/(''G''/''ε''<sub>0</sub>)<sup>1/2</sup> [C/m]
|''c''<sup>4</sup>/(''Gk''<sub>B</sub>) [K/m]
|-
! scope="col" | '''kg'''
|''G''/''c''<sup>2</sup> [m/kg]
| 1
|''G''/''c''<sup>3</sup> [s/kg]
|(''Gε''<sub>0</sub>)<sup>1/2</sup> [C/kg]
|''c''<sup>2</sup>/''k''<sub>B</sub> [K/kg]
|-
! scope="col" | '''s'''
|''c'' [m/s]
|''c''<sup>3</sup>/''G'' [kg/s]
| 1
|c<sup>3</sup>/(''G''/''ε''<sub>0</sub>)<sup>1/2</sup> [C/s]
|''c''<sup>5</sup>/(''Gk''<sub>B</sub>) [K/s]
|-
! scope="col" | '''C'''
|(''G''/''ε''<sub>0</sub>)<sup>1/2</sup>/''c''<sup>2</sup> [m/C]
|1/(''Gε''<sub>0</sub>)<sup>1/2</sup> [kg/C]
|(''G''/''ε''<sub>0</sub>)<sup>1/2</sup>/''c''<sup>3</sup> [s/C]
| 1
|''c''<sup>2</sup>/(''k''<sub>B</sub>(''Gε''<sub>0</sub>)<sup>1/2</sup>) [K/C]
|-
! scope="col" | '''K'''
|''Gk''<sub>B</sub>/''c''<sup>4</sup> [m/K]
|''k''<sub>B</sub>/''c''<sup>2</sup> [kg/K]
|''Gk''<sub>B</sub>/''c''<sup>5</sup> [s/K]
|''k''<sub>B</sub>(''Gε''<sub>0</sub>)<sup>1/2</sup>/''c''<sup>2</sup> [C/K]
| 1
|}

==References==
{{Reflist}}
* {{cite book | author=Wald, Robert M. |author-link=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''

==External links==
* [http://www.physics.nist.gov/cuu/Constants/energy.html Conversion factors for energy equivalents]

{{systems of measurement|sp=oxford}}
{{DEFAULTSORT:Geometrized Unit System}}

[[Category:General relativity]]
[[Category:Systems of units]]
[[Category:Natural units]]