Geometrized unit system
Encyclopedia
A geometrized unit system or geometric unit system is a system of natural units
in which the base physical units are chosen so that the speed of light in a vacuum, c, and the gravitational constant
, G, are set equal to unity.
The geometrized unit system is not a completely defined or unique system: latitude is left to also set other constants to unity. We may, for example, also set Boltzmann's constant, kB, and the Coulomb constant, ke, to unity.
Setting the reduced Planck constant, ħ, to unity yields 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 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 r = 2m. 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 cosmology
, in which 8πG = 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.
Practical measurements and computations are usually done in SI
units, but conversions are generally quite straightforward.
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.
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*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:
Conversion from kg,s,C,K into m:
Conversion from m,s,C,K into kg:
Conversion from m,kg,C,K into s
Conversion from m,kg,s,K into C
Conversion from m,kg,s,C into K
All these units represents nothing else than length along dimension, thus revealing SI as redundant in comparison to geometrized units.
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.
Path curvature is the reciprocal of the magnitude of the 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 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 field tensor
.
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 potential
four-vector and the electromagnetic current
four-vector.
Physical quantities such as mass
and electric charge
which can be identified with the 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.
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.
This table can be augmented to include temperature, as indicated above, as well as further derived physical quantities such as various moments
.
Natural units
In physics, natural units are physical units of measurement based only on universal physical constants. For example the elementary charge e is a natural unit of electric charge, or the speed of light c is a natural unit of speed...
in which the base physical units are chosen so that the speed of light in a vacuum, c, and the gravitational constant
Gravitational constant
The gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass. It appears in Newton's law of universal gravitation and in Einstein's theory of general relativity. It is also known as the universal...
, G, are set equal to unity.
The geometrized unit system is not a completely defined or unique system: latitude is left to also set other constants to unity. We may, for example, also set Boltzmann's constant, kB, and the Coulomb constant, ke, to unity.
Setting the reduced Planck constant, ħ, to unity yields Planck units
Planck units
In physics, Planck units are physical units of measurement defined exclusively in terms of five universal physical constants listed below, in such a manner that these five physical constants take on the numerical value of 1 when expressed in terms of these units. Planck units elegantly simplify...
.
This system is useful in physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, especially in the special and general theories of relativity. All physical quantities
Physical quantity
A physical quantity is a physical property of a phenomenon, body, or substance, that can be quantified by measurement.-Definition of a physical quantity:Formally, the International Vocabulary of Metrology, 3rd edition defines quantity as:...
are identified with geometric quantities such as areas, lengths, dimensionless numbers, path curvatures, or sectional curvatures.
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
Schwarzschild radius
The Schwarzschild radius is the distance from the center of an object such that, if all the mass of the object were compressed within that sphere, the escape speed from the surface would equal the speed of light...
of a nonrotating uncharged black hole
Black hole
A black hole is a region of spacetime from which nothing, not even light, can escape. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon that...
with mass m becomes simply r = 2m. Therefore, many books and papers on relativistic physics use geometric units exclusively. An alternative system of geometrized units is often used in particle physics
Particle physics
Particle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...
and cosmology
Physical cosmology
Physical cosmology, as a branch of astronomy, is the study of the largest-scale structures and dynamics of the universe and is concerned with fundamental questions about its formation and evolution. For most of human history, it was a branch of metaphysics and religion...
, in which 8πG = 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
Einstein-Hilbert action
The Einstein–Hilbert action in general relativity is the action that yields the Einstein's field equations through the principle of least action...
, the Friedmann equations
Friedmann equations
The Friedmann equations are a set of equations in physical cosmology that govern the expansion of space in homogeneous and isotropic models of the universe within the context of general relativity...
and the Newtonian Poisson equation by removing the corresponding factor.
Practical measurements and computations are usually done in SI
Si
Si, si, or SI may refer to :- Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...
units, but conversions are generally quite straightforward.
Definition
In geometric units, every time interval is interpreted as the distance travelled by light during that given time interval. That is, one secondSecond
The second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock....
is interpreted as one light-second
Light-second
A light-second is a unit of length useful in astronomy, telecommunications and relativistic physics. It is defined as the distance that light travels in free space in one second, and is equal to exactly 299,792,458 metres...
, so time has the geometric units of length
Length
In geometric measurements, length most commonly refers to the longest dimension of an object.In certain contexts, the term "length" is reserved for a certain dimension of an object along which the length is measured. For example it is possible to cut a length of a wire which is shorter than wire...
. This is dimensionally consistent with the notion that, according to the kinematical
Kinematics
Kinematics is the branch of classical mechanics that describes the motion of bodies and systems without consideration of the forces that cause the motion....
laws of special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
, time and distance are on an equal footing.
Energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
and momentum
Momentum
In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...
are interpreted as components of the four-momentum
Four-momentum
In special relativity, four-momentum is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is a four-vector in spacetime...
vector, and mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
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
Sun
The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...
's mass of 2.0×1030 kg in SI units is equivalent to 1.5 km. This is half the Schwarzschild radius
Schwarzschild radius
The Schwarzschild radius is the distance from the center of an object such that, if all the mass of the object were compressed within that sphere, the escape speed from the surface would equal the speed of light...
of a one solar mass black hole
Black hole
A black hole is a region of spacetime from which nothing, not even light, can escape. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon that...
. 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.
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*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:
Conversion from kg,s,C,K into m:
- G/c2 [m/kg]
- c [m/s]
- ((G/(4πε0))1/2)/c2 [m/C]
- (G kB)/c4 [m/K]
Conversion from m,s,C,K into kg:
- c2/G [kg/m]
- c3/G [kg/s]
- 1/(G4πε0)1/2 [kg/C]
- kB/c2 [kg/K]
Conversion from m,kg,C,K into s
- 1/c [s/m]
- G/c3 [s/kg]
- ((G/(4πε0))1/2)/c3 [s/C]
- (G kB)/c5 [s/K]
Conversion from m,kg,s,K into C
- c2/((G/(4πε0))1/2) [C/m]
- (G 4πε0)1/2 [C/kg]
- c3/((G/(4πε0))1/2) [C/s]
- (kB(G 4πε0)1/2)/c2 [C/K]
Conversion from m,kg,s,C into K
- c4/(G kB) [K/m]
- c2/kB [K/kg]
- c5/(G kB) [K/s]
- c2/(kB(G 4πε0)1/2) [K/C]
All these units represents nothing else than length along dimension, thus revealing SI as redundant in comparison to geometrized units.
Geometrical quantities
The components of curvature tensors such as the Einstein tensorEinstein tensor
In differential geometry, the Einstein tensor , named after Albert Einstein, is used to express the curvature of a Riemannian manifold...
have, in geometric units, the dimensions of sectional curvature
Sectional curvature
In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature K depends on a two-dimensional plane σp in the tangent space at p...
. So do the components of the stress-energy tensor
Stress-energy tensor
The stress–energy tensor is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields...
. Therefore the Einstein field equation is dimensionally consistent in these units.
Path curvature is the reciprocal of the magnitude of the curvature vector
Geodesic curvature
In Riemannian geometry, the geodesic curvature k_g of a curve lying on a submanifold of the ambient space measures how far the curve is from being a geodesic...
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
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
, and if we interpret a timelike curve as the world line
World line
In physics, the world line of an object is the unique path of that object as it travels through 4-dimensional spacetime. The concept of "world line" is distinguished from the concept of "orbit" or "trajectory" by the time dimension, and typically encompasses a large area of spacetime wherein...
of some observer
Observation
Observation is either an activity of a living being, such as a human, consisting of receiving knowledge of the outside world through the senses, or the recording of data using scientific instruments. The term may also refer to any data collected during this activity...
, then its path curvature can be interpreted as the magnitude of the acceleration
Acceleration
In physics, acceleration is the rate of change of velocity with time. In one dimension, acceleration is the rate at which something speeds up or slows down. However, since velocity is a vector, acceleration describes the rate of change of both the magnitude and the direction of velocity. ...
experienced by that observer. Physical quantities which can be identified with path curvature include the components of the electromagnetic field tensor
Electromagnetic tensor
The electromagnetic tensor or electromagnetic field tensor is a mathematical object that describes the electromagnetic field of a physical system in Maxwell's theory of electromagnetism...
.
Any velocity
Velocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...
can be interpreted as the slope
Slope
In mathematics, the slope or gradient of a line describes its steepness, incline, or grade. A higher slope value indicates a steeper incline....
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 potential
Electromagnetic four-potential
The electromagnetic four-potential is a potential from which the electromagnetic field can be derived. It combines both the electric scalar potential and the magnetic vector potential into a single space-time four-vector. In a given reference frame, the first component is the scalar potential and...
four-vector and the electromagnetic current
Four-current
In special and general relativity, the four-current is the Lorentz covariant four-vector that replaces the electromagnetic current density, or indeed any conventional charge current density...
four-vector.
Physical quantities such as mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
and electric charge
Electric charge
Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...
which can be identified with the magnitude
Magnitude (mathematics)
The magnitude of an object in mathematics is its size: a property by which it can be compared as larger or smaller than other objects of the same kind; in technical terms, an ordering of the class of objects to which it belongs....
of a timelike vector have the geometric dimension of length. Physical quantities such as angular momentum
Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
which can be identified with the magnitude of a bivector
Bivector
In mathematics, a bivector or 2-vector is a quantity in geometric algebra or exterior algebra that generalises the idea of a vector. If a scalar is considered a zero dimensional quantity, and a vector is a one dimensional quantity, then a bivector can be thought of as two dimensional. Bivectors...
have the geometric dimension of area.
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.
| Quantity | SI dimension | Geometrical dimension | Multiplication factor |
|---|---|---|---|
| Length Length In geometric measurements, length most commonly refers to the longest dimension of an object.In certain contexts, the term "length" is reserved for a certain dimension of an object along which the length is measured. For example it is possible to cut a length of a wire which is shorter than wire... |
[L] | [L] | 1 |
| Time Time Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects.... |
[T] | [L] | c |
| Mass Mass Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:... |
[M] | [L] | G c −2 |
| Velocity Velocity In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ... |
[L T −1] | 1 | c −1 |
| Angular velocity Angular velocity In physics, the angular velocity is a vector quantity which specifies the angular speed of an object and the axis about which the object is rotating. The SI unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, revolutions per... |
[T −1] | [L −1] | c −1 |
| Acceleration Acceleration In physics, acceleration is the rate of change of velocity with time. In one dimension, acceleration is the rate at which something speeds up or slows down. However, since velocity is a vector, acceleration describes the rate of change of both the magnitude and the direction of velocity. ... |
[L T −2] | [L −1] | c −2 |
| Energy Energy In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems... |
[M L 2 T −2] | [L] | G c −4 |
| Energy density Energy density Energy density is a term used for the amount of energy stored in a given system or region of space per unit volume. Often only the useful or extractable energy is quantified, which is to say that chemically inaccessible energy such as rest mass energy is ignored... |
[M L −1 T −2] | [L −2] | G c −4 |
| Angular momentum Angular momentum In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system... |
[M L 2 T −1] | [L 2] | G c −3 |
| Force | [M L T −2] | 1 | G c −4 |
| Power Power (physics) In physics, power is the rate at which energy is transferred, used, or transformed. For example, the rate at which a light bulb transforms electrical energy into heat and light is measured in watts—the more wattage, the more power, or equivalently the more electrical energy is used per unit... |
[M L 2 T −3] | 1 | G c −5 |
| Pressure Pressure Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :... |
[M L −1 T −2] | [L −2] | G c −4 |
| Density Density The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight... |
[M L −3] | [L −2] | G c −2 |
| Electric charge Electric charge Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two... |
[I T] | [L] | G 1/2 c −2 (4πε0) −1/2 |
| Electric potential Electric potential In classical electromagnetism, the electric potential at a point within a defined space is equal to the electric potential energy at that location divided by the charge there... |
[M L 2 T −3 I −1] | 1 | G 1/2 c −2 (4πε0) 1/2 |
| Electric field Electric field In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding... |
[M L T −3 I −1] | [L −1] | G 1/2 c −2 (4πε0) 1/2 |
| Magnetic field Magnetic field A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;... |
[M T −2 I −1] | [L −1] | G 1/2 c −1 (4πε0) 1/2 |
| Potential Potential *In linguistics, the potential mood*The mathematical study of potentials is known as potential theory; it is the study of harmonic functions on manifolds... |
[M L T −2 I −1] | 1 | G 1/2 c −1 (4πε0) 1/2 |
This table can be augmented to include temperature, as indicated above, as well as further derived physical quantities such as various moments
Moment (physics)
In physics, the term moment can refer to many different concepts:*Moment of force is the tendency of a force to twist or rotate an object; see the article torque for details. This is an important, basic concept in engineering and physics. A moment is valued mathematically as the product of the...
.