A geometrized unit system or geometrodynamic unit system is a system of

natural units In physics, natural unit systems are measurement systems for which selected physical constants have been set to 1 through nondimensionalization of physical units. For example, the speed of light may be set to 1, and it may then be omitted, equa ...

in which the base

physical units A unit of measurement, or unit of measure, is a definite magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same kind of quantity. Any other quantity of that kind can ...

are chosen so that the

speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant exactly equal to ). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time i ...

in vacuum, ''c'', and the

gravitational constant The gravitational constant is an empirical physical constant involved in the calculation of gravitational effects in Sir Isaac Newton's law of universal gravitation and in Albert Einstein's general relativity, theory of general relativity. It ...

, ''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 Constant or The Constant may refer to: Mathematics * Constant (mathematics), a non-varying value * Mathematical constant, a special number that arises naturally in mathematics, such as or Other concepts * Control variable or scientific const ...

, to unity, for example

Stoney units In physics, the Stoney units form a system of units named after the Irish physicist George Johnstone Stoney, who first proposed them in 1874 (but published only in 1881). They are the earliest example of natural units, i.e., a coherent set of unit ...

and

Planck units In particle physics and physical cosmology, Planck units are a system of units of measurement defined exclusively in terms of four universal physical constants: ''Speed of light, c'', ''Gravitational constant, G'', ''Reduced Planck constant, ħ ...

. This system is useful in

physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...

, especially in the special and general theories of relativity. All

physical quantities A physical quantity (or simply quantity) is a property of a material or system that can be quantified by measurement. A physical quantity can be expressed as a ''value'', which is the algebraic multiplication of a '' numerical value'' and a '' ...

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 A black hole is a massive, compact astronomical object so dense that its gravity prevents anything from escaping, even light. Albert Einstein's theory of general relativity predicts that a sufficiently compact mass will form a black hole. Th ...

with mass ''m'' becomes . 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 Particle physics or high-energy physics is the study of Elementary particle, fundamental particles and fundamental interaction, forces that constitute matter and radiation. The field also studies combinations of elementary particles up to the s ...

and

cosmology Cosmology () is a branch of physics and metaphysics dealing with the nature of the universe, the cosmos. The term ''cosmology'' was first used in English in 1656 in Thomas Blount's ''Glossographia'', with the meaning of "a speaking of the wo ...

, in which instead. This introduces an additional factor of 8π into Newton's law of universal gravitation but simplifies the

Einstein field equations In the General relativity, general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of Matter#In general relativity and cosmology, matter within it. ...

, the

Einstein–Hilbert action The Einstein–Hilbert action in general relativity is the action that yields the Einstein field equations through the stationary-action principle. With the metric signature, the gravitational part of the action is given as :S = \int R \sqrt ...

, the

Friedmann equations The Friedmann equations, also known as the Friedmann–Lemaître (FL) equations, are a set of equations in physical cosmology that govern cosmic expansion in homogeneous and isotropic models of the universe within the context of general relativi ...

and the Newtonian

Poisson equation Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with th ...

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 John Archibald Wheeler (July 9, 1911April 13, 2008) was an American theoretical physicist. He was largely responsible for reviving interest in general relativity in the United States after World War II. Wheeler also worked with Niels Bohr to e ...

with the

c

, the

G

, and

Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative thermal energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin (K) and the ...

k_b

all set to 1. Some authors refer to these units as geometrodynamic units. In geometric units, every time interval is interpreted as the distance travelled by light during that given time interval. That is, one

second The second (symbol: s) is a unit of time derived from the division of the day first into 24 hours, then to 60 minutes, and finally to 60 seconds each (24 × 60 × 60 = 86400). The current and formal definition in the International System of U ...

is interpreted as one

light-second The light-second is a unit of length useful in astronomy, telecommunications Telecommunication, often used in its plural form or abbreviated as telecom, is the transmission of information over a distance using electronic means, typically t ...

, so time has the geometric units of

length Length is a measure of distance. In the International System of Quantities, length is a quantity with Dimension (physical quantity), dimension distance. In most systems of measurement a Base unit (measurement), base unit for length is chosen, ...

. This is dimensionally consistent with the notion that, according to the

kinematical In physics, kinematics studies the geometrical aspects of motion of physical objects independent of forces that set them in motion. Constrained motion such as linked machine parts are also described as kinematics. Kinematics is concerned with s ...

laws of

special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between Spacetime, space and time. In Albert Einstein's 1905 paper, Annus Mirabilis papers#Special relativity, "On the Ele ...

, time and distance are on an equal footing.

Energy Energy () is the physical quantity, quantitative physical property, property that is transferred to a physical body, body or to a physical system, recognizable in the performance of Work (thermodynamics), work and in the form of heat and l ...

and

momentum In Newtonian mechanics, momentum (: momenta or momentums; more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. ...

are interpreted as components of the

four-momentum In special relativity, four-momentum (also called momentum–energy or momenergy) is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum i ...

vector, and

mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...

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''². For example, the

Sun The Sun is the star at the centre of the Solar System. It is a massive, nearly perfect sphere of hot plasma, heated to incandescence by nuclear fusion reactions in its core, radiating the energy from its surface mainly as visible light a ...

's mass of in SI units is equivalent to . This is half the Schwarzschild radius of a one solar mass

. 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 /s and candela (1/683 /sr is a dimensionless ratio of two dimensionless ratios such as ratio of two volumes g⋅m²/s³= and ratio of two areas ²/m²= r while mole is only a dimensionless

Avogadro number The Avogadro constant, commonly denoted or , is an SI defining constant with an exact value of when expressed in reciprocal moles. It defines the ratio of the number of constituent particles to the amount of substance in a sample, where th ...

of entities such as atoms or particles. The

vacuum permittivity Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric const ...

and

are ''ε''₀ and ''k''_B.

References

* ''See Appendix F''

External links

Conversion factors for energy equivalents
{{DEFAULTSORT:Geometrized Unit System General relativity Systems of units Natural units