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: c, G, ħ, and k_B. 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 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 . 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 numeric value 1 when expressed in these units are:

Variants of the basic idea of Planck units exist, such as alternate choices of normalization that give other numeric 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, 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 quantities, 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, the ratios above may be expressed simply with the symbols of physical quantity, without being scaled explicitly by their corresponding unit:
This last equation 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. 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.

Name	Dimension	Expression	Value
Planck length	length
Planck mass	mass
Planck time	time
Planck temperature	temperature

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
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 of 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 k_B are exact and 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, where is the fine-structure constant.

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. As with the base units, their use is mostly confined to theoretical physics because most of them are too large or too small for empirical or practical use and there are large uncertainties in their values.

Derived unit of	Expression	Approximate SI equivalent
area
volume
momentum
energy
force
density
acceleration

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 quantities are not "extreme" in magnitude, such as the Planck mass, which is about 22 micrograms: very large in comparison with subatomic particles, and within the mass range of living organisms. Similarly, the related units of energy and of 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 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 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 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 lesser energy levels 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.