In particle physics and physical cosmology, Planck units are a system of units of measurement defined exclusively in terms of four universal physical constants: c, G, ħ, and kB. Expressing one of these physical constants in terms of Planck units yields a numerical value of 1. They are a system of natural units, defined using fundamental properties of nature (specifically, properties of free space) rather than properties of a chosen prototype object. Originally proposed in 1899 by German physicist Max Planck, they are relevant in research on unified theories such as quantum gravity.
The term Planck scale refers to quantities of space, time, energy and other units that are similar in magnitude to corresponding Planck units. This region may be characterized by particle energies of around or , time intervals of around and lengths of around (approximately the energy-equivalent of the Planck mass, the Planck time and the Planck length, respectively). At the Planck scale, the predictions of the Standard Model, quantum field theory and general relativity are not expected to apply, and quantum effects of gravity are expected to dominate. One example is represented by the conditions in the first 10−43 seconds of our universe after the Big Bang, approximately 13.8 billion years ago.
The four universal constants that, by definition, have a numerical value 1 when expressed in these units are:
c, the speed of light in vacuum, G, the gravitational constant, ħ, the reduced Planck constant, and kB, the Boltzmann constant.
Variants of the basic idea of Planck units exist, such as alternate choices of normalization that give other numerical values to one or more of the four constants above.
Introduction Any system of measurement may be assigned a mutually independent set of base quantities and associated base units, from which all other quantities and units may be derived. In the International System of Units, for example, the SI base quantities include length with the associated unit of the metre. In the system of Planck units, a similar set of base quantities and associated units may be selected, in terms of which other quantities and coherent units may be expressed. The Planck unit of length has become known as the Planck length, and the Planck unit of time is known as the Planck time, but this nomenclature has not been established as extending to all quantities.
All Planck units are derived from the dimensional universal physical constants that define the system, and in a convention in which these units are omitted (i.e. treated as having the dimensionless value 1), these constants are then eliminated from equations of physics in which they appear. For example, Newton's law of universal gravitation,
can be expressed as:
Both equations are dimensionally consistent and equally valid in any system of units, but the second equation, with absent, is relating only dimensionless quantities since any ratio of two like-dimensioned quantities is a dimensionless quantity. If, by a shorthand convention, it is understood that each physical quantity is the corresponding ratio with a coherent Planck unit (or "expressed in Planck units"), the ratios above may be expressed simply with the symbols of physical quantity, without being scaled explicitly by their corresponding unit: This last equation (without ) is valid with , , , and being the dimensionless ratio quantities corresponding to the standard quantities, written e.g. or , but not as a direct equality of quantities. This may seem to be "setting the constants , , etc., to 1" if the correspondence of the quantities is thought of as equality. For this reason, Planck or other natural units should be employed with care. Referring to "", Paul S. Wesson wrote that, "Mathematically it is an acceptable trick which saves labour. Physically it represents a loss of information and can lead to confusion."
History and definition
The concept of natural units was introduced in 1874, when George Johnstone Stoney, noting that electric charge is quantized, derived units of length, time, and mass, later named Stoney units in his honor. Stoney chose his units so that G, c, and the electron charge e would be numerically equal to 1. In 1899, one year before the advent of quantum theory, Max Planck introduced what later became known as the Planck constant. pp. 478–80 contain the first appearance of the Planck base units, and of the Planck constant, which Planck denoted by b. a and f in this paper correspond to the k and G in this article. At the end of the paper, he proposed the base units that were later named in his honor. The Planck units are based on the quantum of action, usually called the Planck constant, which appeared in the Wien approximation for black-body radiation. Planck underlined the universality of the new unit system, writing:
Planck considered only the units based on the universal constants , , , and to arrive at natural units for length, time, mass, and temperature. His definitions differ from the modern ones by a factor of , because the modern definitions use rather than .
+Table 1: Modern values for Planck's original choice of quantities Name Dimension Expression Value (SI units) Planck length length (L)
Planck mass mass (M)
Planck time time (T)
Planck temperature temperature (Θ)
Unlike the case with the International System of Units, there is no official entity that establishes a definition of a Planck unit system. Some authors define the base Planck units to be those of mass, length and time, regarding an additional unit for temperature to be redundant. Other tabulations add, in addition to a unit for temperature, a unit for electric charge, so that either the Coulomb constant or the vacuum permittivity is normalized to 1. Thus, depending on the author's choice, this charge unit is given by
(where is the fine-structure constant) for , or
for . Some of these tabulations also replace mass with energy when doing so. The former matches the table above in the sense that two particles of this charge and one Planck mass experience balanced electrostatic and gravitational forces, whereas the latter matches with the rationalized Planck units. In SI units, the values of c, h, e and kB are defined and therefore exact while the values of ε0 and G in SI units respectively have relative uncertainties of and Hence, the uncertainties in the SI values of the Planck units derive almost entirely from uncertainty in the SI value of G.
Compared to Stoney units, Planck base units are all larger by a factor .
Derived units In any system of measurement, units for many physical quantities can be derived from base units. Table 2 offers a sample of derived Planck units, some of which are seldom used.
+ Table 2: Coherent derived units of Planck units Derived unit of Expression Approximate SI equivalent area (L2)
volume (L3)
momentum (LMT−1)
energy (L2MT−2)
force (LMT−2)
density (L−3M)
acceleration (LT−2)
frequency (T−1)
Some Planck units, such as of time and length, are many orders of magnitude too large or too small to be of practical use, so that Planck units as a system are typically only relevant to theoretical physics. In some cases, a Planck unit may suggest a limit to a range of a physical quantity where present-day theories of physics apply. For example, our understanding of the Big Bang does not extend to the Planck epoch, i.e., when the universe was less than one Planck time old. Describing the universe during the Planck epoch requires a theory of quantum gravity that would explain both quantum effects and general relativity in their respective domains of applicability. Such a theory does not yet exist.
Several Planck units are not extreme in magnitude. For example, the Planck mass is about 22 micrograms, which is very large compared with subatomic particles, but lies within the mass range of living organisms. Similarly, the units of energy and momentum are in the range of some everyday phenomena.
Significance Planck units have little anthropocentric arbitrariness, but do still involve some arbitrary choices in terms of the defining constants. Unlike the metre and second, which exist as base units in the SI system for historical reasons, the Planck length and Planck time are conceptually linked at a fundamental physical level. Consequently, natural units help physicists to reframe questions. Frank Wilczek puts it succinctly:
While it is true that the electrostatic repulsive force between two protons (alone in free space) greatly exceeds the gravitational attractive force between the same two protons, this is not about the relative strengths of the two fundamental forces.
When Planck proposed his units, the goal was only that of establishing a universal ("natural") way of measuring objects, without giving any special meaning to quantities that measured one single unit. During the 1950s, multiple authors including Lev Landau and Oskar Klein argued that quantities on the order of the Planck scale indicated the limits of the validity of quantum field theory. John Archibald Wheeler proposed in 1955 that quantum fluctuations of spacetime become significant at the Planck scale, though at the time he was unaware of the Planck units.
Planck scale
In particle physics and physical cosmology, the Planck scale is an energy scale around (the Planck energy, corresponding to the energy equivalent of the Planck mass, ) at which quantum effects of gravity become significant. At this scale, present descriptions and theories of sub-atomic particle interactions in terms of quantum field theory break down and become inadequate, due to the impact of the apparent non-renormalizability of gravity within current theories.
Relationship to gravity At the Planck length scale, the strength of gravity is expected to become comparable with the other forces, and it has been theorized that all the fundamental forces are unified at that scale, but the exact mechanism of this unification remains unknown. The Planck scale is therefore the point at which the effects of quantum gravity can no longer be ignored in other fundamental interactions, where current calculations and approaches begin to break down, and a means to take account of its impact is necessary. On these grounds, it has been speculated that it may be an approximate lower limit at which a black hole could be formed by collapse.
While physicists have a fairly good understanding of the other fundamental interactions of forces on the quantum level, gravity is problematic, and cannot be integrated with quantum mechanics at very high energies using the usual framework of quantum field theory. At lower energies it is usually ignored, while for energies approaching or exceeding the Planck scale, a new theory of quantum gravity is necessary. Approaches to this problem include string theory and M-theory, loop quantum gravity, noncommutative geometry, and causal set theory.
In cosmology
In Big Bang cosmology, the Planck epoch or Planck era is the earliest stage of the Big Bang, before the time passed was equal to the Planck time, tP, or approximately 10−43 seconds. Discusses "Planck time" and "Planck era" at the very beginning of the Universe There is no currently available physical theory to describe such short times, and it is not clear in what sense the concept of time is meaningful for values smaller than the Planck time. It is generally assumed that quantum effects of gravity dominate physical interactions at this time scale. At this scale, the unified force of the Standard Model is assumed to be unified with gravitation. Immeasurably hot and dense, the state of the Planck epoch was succeeded by the grand unification epoch, where gravitation is separated from the unified force of the Standard Model, in turn followed by the inflationary epoch, which ended after about 10−32 seconds (or about 1011 tP).