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, and the speed of light c is a natural unit of
speed. A purely natural system of units has all of its units defined in this
way, and usually such that the numerical values of the selected physical
constants in terms of these units are exactly 1. These constants are then
typically omitted from mathematical expressions of physical laws, and while
this has the apparent advantage of simplicity, it may entail a loss of
clarity due to the loss of information for dimensional analysis.
Introduction Natural units are intended to elegantly
simplify particular algebraic expressions appearing in the laws of
physics or to normalize some chosen physical quantities that are properties
of universal elementary particles and are reasonably believed to be constant.
However, there is a choice of which quantities to set to unity in a natural
system of units, and quantities which are set to unity in one system may take
a different value or even be assumed to vary in another natural unit system.
Natural units are "natural" because the origin of their definition comes only
from properties of nature and not from any human construct. Planck units are
often, without qualification, called "natural units", although they
constitute only one of several systems of natural units, albeit the best known
such system. Planck units might be considered one of the most "natural"
systems in that the set of units is not based on properties of any prototype,
object, or particle but are solely derived from the properties of free
space. As with other systems of units, the base
units of a set of natural units will include definitions and values for
length, mass, time, temperature, and electric charge. Some physicists do not
recognize temperature as a fundamental physical quantity, since it expresses
the energy per degree of freedom of a particle, which can be expressed in
terms of energy. Virtually every system of natural units normalizes Boltzmann's
constant kB to 1, which can be thought of as simply a way of defining the unit
temperature. In the SI unit system, electric charge
is a separate fundamental dimension of physical quantity, but in natural unit
systems charge is expressed in terms of the mechanical units of mass, length,
and time, similarly to cgs. There are two common ways to relate charge to
mass, length, and time: In Lorentz–Heaviside units, Coulomb's law
is F = q1q2/(4πr2), and in Gaussian units, Coulomb's law is F = q1q2/r2.
Both possibilities are incorporated into different natural unit systems.
Notation and use Natural units are most commonly used by
setting the units to one. For example, many natural unit systems include the
equation c = 1 in the unit-system definition, where c is the speed of
light. If a velocity v is half the speed of light, then as v = c⁄2 and c = 1,
hence v = 1⁄2. The equation v = 1⁄2 means "the velocity v has the value
one-half when measured in Planck units", or "the velocity v is one-half the
Planck unit of velocity". The equation c = 1 can be plugged in
anywhere else. For example, Einstein's equation E = mc2 can be rewritten in
Planck units as E = m. This equation means "The energy of a particle,
measured in Planck units of energy, equals the mass of the particle,
measured in Planck units of mass." = Advantages and disadvantages =
Compared to SI or other unit systems, natural units have both advantages and
disadvantages: Simplified equations: By setting
constants to 1, equations containing those constants appear more compact and
in some cases may be simpler to understand. For example, the special
relativity equation E2 = p2c2 + m2c4 appears somewhat complicated, but the
natural units version, E2 = p2 + m2, appears simpler.
Physical interpretation: Natural unit systems automatically subsume
dimensional analysis. For example, in Planck units, the units are defined by
properties of quantum mechanics and gravity. Not coincidentally, the Planck
unit of length is approximately the distance at which quantum gravity
effects become important. Likewise, atomic units are based on the mass and
charge of an electron, and not coincidentally the atomic unit of length
is the Bohr radius describing the orbit of the electron in a hydrogen atom.
No prototypes: A prototype is a physical object that defines a unit, such as the
International Prototype Kilogram, a physical cylinder of metal whose mass is
by definition exactly one kilogram. A prototype definition always has
imperfect reproducibility between different places and between different
times, and it is an advantage of natural unit systems that they use no
prototypes. Less precise measurements: SI units are
designed to be used in precision measurements. For example, the second is
defined by an atomic transition frequency in cesium atoms, because this
transition frequency can be precisely reproduced with atomic clock technology.
Natural unit systems are generally not based on quantities that can be
precisely reproduced in a lab. Therefore, in order to retain the same
degree of precision, the fundamental constants used still have to be measured
in a laboratory in terms of physical objects that can be directly observed.
If this is not possible, then a quantity expressed in natural units can be less
precise than the same quantity expressed in SI units. For example, Planck units
use the gravitational constant G, which is measurable in a laboratory only to
four significant digits. Choosing constants to normalize
Out of the many physical constants, the designer of a system of natural unit
systems must choose a few of these constants to normalize. It is not
possible to normalize just any set of constants. For example, the mass of a
proton and the mass of an electron cannot both be normalized: if the mass
of an electron is defined to be 1, then the mass of a proton has to be
approximately 1836. In a less trivial example, the fine-structure constant, α
≈ 1/137, cannot be set to 1, at least not independently, because it is a
dimensionless number defined in terms of other quantities, some of which one may
want to set to unity as well. The fine-structure constant is related to
other fundamental constants through where ke is the Coulomb constant, e is
the elementary charge, ℏ is the reduced Planck constant, and c is the speed of
light. Electromagnetism units
In SI units, electric charge is expressed in coulombs, a separate unit
which is additional to the "mechanical" units, even though the traditional
definition of the ampere refers to some of these other units. In natural unit
systems, however, electric charge has units of [mass]1/2 [length]3/2 [time]−1.
There are two main natural unit systems for electromagnetism:
Lorentz–Heaviside units. Gaussian units.
Of these, Lorentz–Heaviside is somewhat more common, mainly because Maxwell's
equations are simpler in Lorentz-Heaviside units than they are in
Gaussian units. In the two unit systems, the elementary
charge e satisfies: where ℏ is the reduced Planck constant,
c is the speed of light, and α ≈ 1/137 is the fine-structure constant.
In a natural unit system where c = 1, Lorentz–Heaviside units can be derived
from SI units by setting ε0 = μ0 = 1. Gaussian units can be derived from SI
units by a more complicated set of transformations, such as multiplying all
electric fields by−1/2, multiplying all magnetic susceptibilities by 4π, and so
on. Systems of natural units
= Planck units = Planck units are defined by
where c is the speed of light, ℏ is the reduced Planck constant, G is the
gravitational constant, ke is the Coulomb constant, and kB is the
Boltzmann constant. Planck units are a system of natural
units that is not defined in terms of properties of any prototype, physical
object, or even elementary particle. They only refer to the basic structure
of the laws of physics: c and G are part of the structure of spacetime in general
relativity, and ℏ captures the relationship between energy and
frequency which is at the foundation of quantum mechanics. This makes Planck
units particularly useful and common in theories of quantum gravity, including
string theory. Planck units may be considered "more
natural" even than other natural unit systems discussed below, as Planck units
are not based on any arbitrarily chosen prototype object or particle. For
example, some other systems use the mass of an electron as a parameter to be
normalized. But the electron is just one of 16 known massive elementary
particles, all with different masses, and there is no compelling reason,
within fundamental physics, to emphasize the electron mass over some other
elementary particle's mass. = Stoney units =
Stoney units are defined by: where c is the speed of light, G is the
gravitational constant, ke is the Coulomb constant, e is the elementary
charge, and kB is the Boltzmann constant.
George Johnstone Stoney was the first physicist to introduce the concept of
natural units. He presented the idea in a lecture entitled "On the Physical
Units of Nature" delivered to the British Association in 1874. Stoney
units differ from Planck units by fixing the elementary charge at 1, instead of
Planck's constant. Stoney units are rarely used in modern
physics for calculations, but they are of historical interest.
= Atomic units = There are two types of atomic units,
closely related. Hartree atomic units:
Rydberg atomic units: Coulomb's constant is generally
expressed as These units are designed to simplify
atomic and molecular physics and chemistry, especially the hydrogen atom,
and are widely used in these fields. The Hartree units were first proposed by
Douglas Hartree, and are more common than the Rydberg units.
The units are designed especially to characterize the behavior of an electron
in the ground state of a hydrogen atom. For example, using the Hartree
convention, in the Bohr model of the hydrogen atom, an electron in the ground
state has orbital velocity = 1, orbital radius = 1, angular momentum = 1,
ionization energy = 1⁄2, etc. The unit of energy is called the Hartree
energy in the Hartree system and the Rydberg energy in the Rydberg system.
They differ by a factor of 2. The speed of light is relatively large in atomic
units, which comes from the fact that an electron in hydrogen tends to move much
slower than the speed of light. The gravitational constant is extremely
small in atomic units, which comes from the fact that the gravitational force
between two electrons is far weaker than the Coulomb force. The unit length, lA,
is the Bohr radius, a0. The values of c and e shown above imply
that e =1/2, as in Gaussian units, not Lorentz–Heaviside units. However,
hybrids of the Gaussian and Lorentz–Heaviside units are sometimes
used, leading to inconsistent conventions for magnetism-related units.
= Quantum chromodynamics units = The electron mass is replaced with that
of the proton. Strong units are "convenient for work in QCD and nuclear
physics, where quantum mechanics and relativity are omnipresent and the
proton is an object of central interest".
= "Natural units" = In particle physics and cosmology, the
phrase "natural units" generally means: where ℏ is the reduced Planck constant,
c is the speed of light, and kB is the Boltzmann constant.
Both Planck units and QCD units are this type of Natural units. Like the other
systems, the electromagnetism units can be based on either Lorentz–Heaviside
units or Gaussian units. The unit of charge is different in each.
Finally, one more unit is needed to construct a usable system of units that
includes energy and mass. Most commonly, electron-volt is used, despite the fact
that this is not a "natural" unit in the sense discussed above – it is defined by
a natural property, the elementary charge, and the anthropogenic unit of
electric potential, the volt. With the addition of eV, any quantity
can be expressed. For example, a distance of 1.0 cm can be expressed in
terms of eV, in natural units, as: = Geometrized units =
The geometrized unit system, used in general relativity, is not a completely
defined system. In this system, the base physical units are chosen so that the
speed of light and the gravitational constant are set equal to unity. Other
units may be treated however desired. Planck units and Stoney units are
examples of geometrized unit systems. = Summary table =
where: α is the fine-structure constant,2 ≈
0.007297, αG is the gravitational coupling
constant,2 ≈ 6955175200000000000♠1.752×10−45,
See also Notes and references
External links The NIST website is a convenient source
of data on the commonly recognized constants.
K.A. Tomilin: NATURAL SYSTEMS OF UNITS; To the Centenary Anniversary of the
Planck System A comparative overview/tutorial of various systems of
natural units having historical use. Pedagogic Aides to Quantum Field Theory
Click on the link for Chap. 2 to find an extensive, simplified introduction to
natural units.