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{{short description|Unit system used in the physics of relativity}}
A '''geometrized unit system'''<ref name=MisnerThorneWheeler>{{cite book |last1=Misner |first1=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 used as defining constants.

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

This system is used in [[physics]], especially in the [[special theory of relativity|special]] and [[general theory of relativity|general theories of relativity]], which focus on [[physical quantity|physical quantities]] that are identified with [[Dimensional analysis|dynamic quantities]] such as time, length, mass, dimensionless quantities, area, energy, momentum, path curvatures and sectional curvatures.

Many equations in relativistic physics appear simpler when expressed in geometrized 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''{{sub|s}} = 2''m''}}. For this reason, many books and papers on relativistic physics use geometrized units. An alternative "rationalized" system of geometrized units is often used in [[particle physics]] and [[physical cosmology|cosmology]], in which {{nowrap|4π''G''}} or {{nowrap|8π''G''}} are used instead. This makes equations such as the [[Einstein field equations]], the [[Einstein–Hilbert action]], the [[Friedmann equations]] and the Newtonian [[Poisson equation]] seem simpler and more natural.

== Definition ==
Geometrized units were defined in the book [[Gravitation (book)|''Gravitation'']] by [[Charles W. Misner|Misner]], [[Kip S. Thorne|Thorne]], and [[John Archibald Wheeler|Wheeler]] such that the [[speed of light]] ''c'', the [[gravitational constant]] ''G'', and [[Boltzmann constant]] ''k''{{sub|B}} are 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 |article-number=084052 |bibcode=2021PhRvD.103h4052L |s2cid=235581301 }}</ref>

In geometrized 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 geometrized 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 [[invariant mass]] is the magnitude of this vector, so in geometrized 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''2. 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 combinations of the SI base units, based on the constants ''c'', ''G'', ''ε''0 ([[vacuum permittivity]]) and ''k''B ([[Boltzmann constant]]).

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

Geometrized unit system - Revision history

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