Geometrized unit system

A geometrized unit system^[1] or geometrodynamic unit system is a system of natural units in which the base physical units are chosen so that the speed of light in vacuum, c, and the gravitational constant, G, are set equal to unity.

c=1\

G=1\

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 constants, to unity, for example Stoney units and 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 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 r = 2m. 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 cosmology, in which 8πG = 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 by Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler with the speed of light, $c$ , the gravitational constant, $G$ , and Boltzmann constant, $k_{b}$ all set to $1.0$ .^[1]^: 36 Some authors refer to these units as geometrodynamic units.^[2]

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 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⋅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:

	m	kg	s	C	K
m	1	c²/G [kg/m]	1/c [s/m]	c²/(G/(ε₀))^1/2 [C/m]	c⁴/(Gk_B) [K/m]
kg	G/c² [m/kg]	1	G/c³ [s/kg]	(Gε₀)^1/2 [C/kg]	c²/k_B [K/kg]
s	c [m/s]	c³/G [kg/s]	1	c³/(G/(ε₀))^1/2 [C/s]	c⁵/(Gk_B) [K/s]
C	(G/(ε₀))^1/2/c² [m/C]	1/(Gε₀)^1/2 [kg/C]	(G/(ε₀))^1/2/c³ [s/C]	1	c²/(k_B(Gε₀)^1/2) [K/C]
K	Gk_B/c⁴ [m/K]	k_B/c² [kg/K]	Gk_B/c⁵ [s/K]	k_B(Gε₀)^1/2/c² [C/K]	1

References

^ ^a ^b Misner, Charles W.; Thorne, Kip S.; Wheeler, John Archibald (2008). Gravitation (27. printing ed.). New York, NY: Freeman. ISBN 978-0-7167-0344-0.
^ Lobo, Francisco S. N.; Rodrigues, Manuel E.; Silva, Marcos V. de S.; Simpson, Alex; Visser, Matt (2021). "Novel black-bounce spacetimes: Wormholes, regularity, energy conditions, and causal structure". Physical Review D. 103 (8): 084052. arXiv:2009.12057. Bibcode:2021PhRvD.103h4052L. doi:10.1103/PhysRevD.103.084052. S2CID 235581301.

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

External links

Conversion factors for energy equivalents

[MisnerThorneWheeler-1] Misner, Charles W.; Thorne, Kip S.; Wheeler, John Archibald (2008). Gravitation (27. printing ed.). New York, NY: Freeman. ISBN 978-0-7167-0344-0.

[2] Lobo, Francisco S. N.; Rodrigues, Manuel E.; Silva, Marcos V. de S.; Simpson, Alex; Visser, Matt (2021). "Novel black-bounce spacetimes: Wormholes, regularity, energy conditions, and causal structure". Physical Review D. 103 (8): 084052. arXiv:2009.12057. Bibcode:2021PhRvD.103h4052L. doi:10.1103/PhysRevD.103.084052. S2CID 235581301.

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