Wikipedia

# Planck units

In particle physics and physical cosmology, Planck units are a set of units of measurement defined exclusively in terms of five universal physical constants, in such a manner that these five physical constants take on the numerical value of 1 when expressed in terms of these units.

Originally proposed in 1899 by German physicist Max Planck, these units are also known as natural units because the origin of their definition comes only from properties of nature and not from any human construct. Planck units are only one system of several systems of natural units, but Planck units are not based on properties of any prototype object or particle (that would be arbitrarily chosen), but rather on only the properties of free space. They are relevant in research on unified theories such as quantum gravity.

The term Planck scale refers to the magnitudes of space, time, energy and other units, below which (or beyond which) the predictions of the Standard Model, quantum field theory and general relativity are no longer reconcilable, and quantum effects of gravity are expected to dominate. This region may be characterized by energies around 1.22×1019 GeV (the energy-equivalent of the Planck mass), time intervals around 5.39×10−44 s (the Planck time) and lengths around 1.62×10−35 m (the Planck length). At the Planck scale, current models are not expected to be a useful guide to the cosmos, and physicists have no scientific model to suggest how the physical universe behaves. The best known 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 five universal constants that Planck units, by definition, normalize to 1 are:

Each of these constants can be associated with a fundamental physical theory or concept: c with special relativity, G with general relativity, ħ with quantum mechanics, kB with thermodynamics, and ε0 with electromagnetism.

## 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 may be selected, and the Planck base unit of length is then known simply as the Planck length, the base unit of time is the Planck time, and so on. These units are derived from the five dimensional universal physical constants of Table 1, in such a manner that these constants are eliminated from fundamental selected equations of physical law when physical quantities are expressed in terms of Planck units. For example, Newton's law of universal gravitation,

{\displaystyle {\begin{aligned}F&=G{\frac {m_{1}m_{2}}{r^{2}}}\\\\&=\left({\frac {F_{\text{P}}l_{\text{P}}^{2}}{m_{\text{P}}^{2}}}\right){\frac {m_{1}m_{2}}{r^{2}}}\\\end{aligned}}}

can be expressed as:

${\displaystyle {\frac {F}{F_{\text{P}}}}={\frac {\left({\dfrac {m_{1}}{m_{\text{P}}}}\right)\left({\dfrac {m_{2}}{m_{\text{P}}}}\right)}{\left({\dfrac {r}{l_{\text{P}}}}\right)^{2}}}.}$

Both equations are dimensionally consistent and equally valid in any system of units, but the second equation, with G missing, 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 all physical quantities are expressed in terms of Planck units, the ratios above may be expressed simply with the symbols of physical quantity, without being scaled explicitly by their corresponding unit:

${\displaystyle F={\frac {m_{1}m_{2}}{r^{2}}}\ .}$

This last equation (without G) is valid only if F, m1, m2, and r are the dimensionless numerical values of these quantities measured in terms of Planck units. This is why Planck units or any other use of natural units should be employed with care. Referring to G = c = 1, 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."[1]

## Definition

Table 1: Dimensional universal physical constants normalized with Planck units
Constant Symbol Dimension in SI Quantities Value (SI units)
Speed of light in vacuum c L T−1 299792458 m/s [2]

(exact by definition)
Gravitational constant G L3 M−1 T−2 6.67430(15)×10−11 m3⋅kg−1⋅s−2 [3]
Reduced Planck constant ħ = h/2π

where h is the Planck constant
L2 M T−1 1.054571817...×10−34 J⋅s [4]

(defined as 6.62607015×10−34 J⋅s/ exactly)
Boltzmann constant k B L2 M T−2 Θ−1 1.380649×10−23 J⋅K−1 [5]

(exact by definition)
Coulomb constant ke = 1/4πε0

where ε0 is the permittivity of free space
L3 M T−2 Q−2 8.9875517923(14)×109 kg⋅m3⋅s−4⋅A−2 [6]

Key: L = length, M = mass, T = time, Q = electric charge, Θ = temperature.

A property of Planck units is that in order to obtain the value of any of the physical constants above it is enough to replace the dimensions of the constant with the corresponding Planck units. For example, the gravitational constant (G) has as dimensions L3 M−1 T−2. By replacing each dimension with the value of each corresponding Planck unit one obtains the value of (1 lP)3 × (1 mP)−1 × (1 tP)−2 = (1.616255×10−35 m)3 × (2.176435×10−8 kg)−1 × (5.391247×10−44 s)−2 = 6.674...×10−11 m3 kg−1 s−2 (which is the value of G).

This is the consequence of the fact that the system is internally coherent. For example, the gravitational attractive force of two bodies of 1 Planck mass each, set apart by 1 Planck length is 1 coherent Planck unit of force. Likewise, the distance traveled by light during 1 Planck time is 1 Planck length.

To determine, in terms of SI or another existing system of units, the quantitative values of the five base Planck units, those two equations and three others must be satisfied:

${\displaystyle l_{\text{P}}=c\ t_{\text{P}}}$
${\displaystyle F_{\text{P}}={\frac {m_{\text{P}}l_{\text{P}}}{t_{\text{P}}^{2}}}=G\ {\frac {m_{\text{P}}^{2}}{l_{\text{P}}^{2}}}}$
${\displaystyle E_{\text{P}}={\frac {m_{\text{P}}l_{\text{P}}^{2}}{t_{\text{P}}^{2}}}=\hbar \ {\frac {1}{t_{\text{P}}}}}$
${\displaystyle E_{\text{P}}={\frac {m_{\text{P}}l_{\text{P}}^{2}}{t_{\text{P}}^{2}}}=k_{\text{B}}\ T_{\text{P}}.}$
${\displaystyle F_{\text{P}}={\frac {m_{\text{P}}l_{\text{P}}}{t_{\text{P}}^{2}}}={\frac {1}{4\pi \varepsilon _{0}}}\ {\frac {q_{\text{P}}^{2}}{l_{\text{P}}^{2}}}}$

Solving the five equations above for the five unknowns results in a unique set of values for the five base Planck units:

Table 2: Base Planck units
Name Dimension Expression Value (SI units)
Planck length Length (L) ${\displaystyle l_{\text{P}}={\sqrt {\frac {\hbar G}{c^{3}}}}}$ 1.616255(18)×10−35 m [7]
Planck mass Mass (M) ${\displaystyle m_{\text{P}}={\sqrt {\frac {\hbar c}{G}}}}$ 2.176435(24)×10−8 kg [8]
Planck time Time (T) ${\displaystyle t_{\text{P}}={\frac {l_{\text{P}}}{c}}={\frac {\hbar }{m_{\text{P}}c^{2}}}={\sqrt {\frac {\hbar G}{c^{5}}}}}$ 5.391247(60)×10−44 s [9]
Planck temperature Temperature (Θ) ${\displaystyle T_{\text{P}}={\frac {m_{\text{P}}c^{2}}{k_{\text{B}}}}={\sqrt {\frac {\hbar c^{5}}{Gk_{\text{B}}^{2}}}}}$ 1.416785(16)×1032 K [10]
Planck charge Electric charge (Q) ${\displaystyle q_{\text{P}}={\sqrt {\frac {\hbar c}{k_{\text{e}}}}}={\sqrt {4\pi \varepsilon _{0}\hbar c}}={\frac {e}{\sqrt {\alpha }}}}$ 1.875545956(41)×10−18 C [11] [4] [2]

Table 2 clearly defines Planck units in terms of the fundamental constants. Yet relative to other units of measurement such as SI, the values of the Planck units are only known approximately. This is due to uncertainty in the values of the gravitational constant G and ε0 in SI units.

The values of c, h, e and kB in SI units are exact due to the definition of the second, metre, kilogram and kelvin in terms of these constants, and contribute no uncertainty to the values of the Planck units expressed in terms of SI units. The vacuum permittivity ε0 has a relative uncertainty of 1.5×10−10.[11] The numerical value of G has been determined experimentally to a relative uncertainty of 2.2×10−5.[3] G appears in the definition of every Planck unit other than for charge in Tables 2 and 3. Hence the uncertainty in the values of the Table 2 and 3 SI equivalents of the Planck units derives almost entirely from uncertainty in the value of G. (The propagation of the error in G is a function of the exponent of G in the algebraic expression for a unit. Since that exponent is ±1/2 for every base unit other than Planck charge, the relative uncertainty of each base unit is about one half that of G.)

## Derived units

In any system of measurement, units for many physical quantities can be derived from base units. Table 3 offers a sample of derived Planck units, some of which in fact 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.

Table 4: Coherent derived units of Planck units
Derived unit of Expression Approximate SI equivalent
area (L2) ${\displaystyle l_{\text{P}}^{2}={\frac {\hbar G}{c^{3}}}}$ 2.6121×10−70 m2
volume (L3) ${\displaystyle l_{\text{P}}^{3}=\left({\frac {\hbar G}{c^{3}}}\right)^{\frac {3}{2}}={\sqrt {\frac {(\hbar G)^{3}}{c^{9}}}}}$ 4.2217×10−105 m3
momentum (LMT−1) ${\displaystyle m_{\text{P}}c={\frac {\hbar }{l_{\text{P}}}}={\sqrt {\frac {\hbar c^{3}}{G}}}}$ 6.5249 kg⋅m/s
energy (L2MT−2) ${\displaystyle E_{\text{P}}=m_{\text{P}}c^{2}={\frac {\hbar }{t_{\text{P}}}}={\sqrt {\frac {\hbar c^{5}}{G}}}}$ 1.9561×109 J
force (LMT−2) ${\displaystyle F_{\text{P}}={\frac {E_{\text{P}}}{l_{\text{P}}}}={\frac {\hbar }{l_{\text{P}}t_{\text{P}}}}={\frac {c^{4}}{G}}}$ 1.2103×1044 N
density (L−3M) ${\displaystyle \rho _{\text{P}}={\frac {m_{\text{P}}}{l_{\text{P}}^{3}}}={\frac {\hbar t_{\text{P}}}{l_{\text{P}}^{5}}}={\frac {c^{5}}{\hbar G^{2}}}}$ 5.1550×1096 kg/m3
acceleration (LT−2) ${\displaystyle a_{\text{P}}={\frac {c}{t_{\text{P}}}}={\sqrt {\frac {c^{7}}{\hbar G}}}}$ 5.5608×1051 m/s2
frequency (T−1) ${\displaystyle f_{p}={\frac {c}{l_{\text{P}}}}={\sqrt {\frac {c^{5}}{\hbar G}}}}$ 1.8549×1043 Hz

Most Planck units 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 fact, 1 Planck unit is often the largest or smallest value of a physical quantity that makes sense within present-day theories of physics. For example, our understanding of the Big Bang begins with the Planck epoch, when the universe was 1 Planck time old and 1 Planck length in diameter. Understanding the universe when it was less than 1 Planck time old requires a theory of quantum gravity that would incorporate quantum effects into general relativity. Such a theory does not yet exist.

An exception to the general pattern of Planck units being "extreme" in magnitude is the Planck mass, which is about 22 micrograms: very large compared to subatomic particles, but well within the mass range of living things.

## History

The concept of natural units was introduced in 1881, when George Johnstone Stoney, noting that electric charge is quantized, derived units of length, time, and mass, now named Stoney units in his honor, by normalizing G, c, and the electron charge, e, to 1.

In 1899 (one year before the advent of quantum theory), Max Planck introduced what became later known as the Planck constant.[12][13] At the end of the paper, Planck proposed, as a consequence of his discovery, the base units later named in his honor. The Planck units are based on the quantum of action, now usually known as the Planck constant. Planck called the constant b in his paper, though h (or the closely related ħ) is now common. However, at that time it was part of Wien's radiation law, which Planck thought to be correct. Planck underlined the universality of the new unit system, writing:

... die Möglichkeit gegebenist, Einheiten für Länge, Masse, Zeit und Temperatur aufzustellen, welche, unabhängig von speciellen Körpern oder Substanzen, ihre Bedeutung für alle Zeiten und für alle, auch außerirdische und außermenschliche Culturen notwendig behalten und welche daher als »natürliche Maßeinheiten« bezeichnet werden können.

... it is possible to set up units for length, mass, time and temperature, which are independent of special bodies or substances, necessarily retaining their meaning for all times and for all civilizations, including extraterrestrial and non-human ones, which can be called "natural units of measure".

Planck considered only the units based on the universal constants G, ħ, c, and kB to arrive at natural units for length, time, mass, and temperature.[13] Planck did not adopt any electromagnetic units. Planck's paper also gave numerical values for the base units that were close to modern values.

The original base units proposed by Planck in 1899 differed by a factor of ${\displaystyle {\sqrt {2\pi }}}$ from the Planck units in use today.[12][13] This is due to the use of the reduced Planck constant (${\displaystyle \hbar }$) in the modern units, which did not appear in the original proposal.

Original Planck units
Name Dimension Expression Value in SI units Value in modern Planck units
Original Planck length Length (L) ${\displaystyle {\sqrt {\frac {hG}{c^{3}}}}}$ 4.05135×10−35 m ${\displaystyle {\sqrt {2\pi }}\times l_{\text{P}}}$
Original Planck mass Mass (M) ${\displaystyle {\sqrt {\frac {hc}{G}}}}$ 5.45551×10−8 kg ${\displaystyle {\sqrt {2\pi }}\times m_{\text{P}}}$
Original Planck time Time (T) ${\displaystyle {\sqrt {\frac {hG}{c^{5}}}}}$ 1.35138×10−43 s ${\displaystyle {\sqrt {2\pi }}\times t_{\text{P}}}$
Original Planck temperature Temperature (Θ) ${\displaystyle {\sqrt {\frac {hc^{5}}{Gk_{\text{B}}}}}}$ 3.55135×1032 K ${\displaystyle {\sqrt {2\pi }}\times T_{\text{P}}}$

## 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:

We see that the question [posed] is not, "Why is gravity so feeble?" but rather, "Why is the proton's mass so small?" For in natural (Planck) units, the strength of gravity simply is what it is, a primary quantity, while the proton's mass is the tiny number [1/(13 quintillion)].[14]

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. From the point of view of Planck units, this is comparing apples to oranges, because mass and electric charge are incommensurable quantities. Rather, the disparity of magnitude of force is a manifestation of the fact that the charge on the protons is approximately the unit charge but the mass of the protons is far less than the unit mass.

## Planck scale

In particle physics and physical cosmology, the Planck scale is an energy scale around 1.22 × 1019 GeV (the Planck energy, corresponding to the mass–energy equivalence of the Planck mass, 2.17645 × 10−8 kg) at which quantum effects of gravity become strong. 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 is 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 where the effects of quantum gravity can no longer be ignored in other fundamental interactions, and where current calculations and approaches begin to break down, and a means to take account of its impact is required.[15][16]

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 required. Other approaches to this problem include string theory and M-theory, loop quantum gravity, noncommutative geometry, scale relativity, causal set theory and P-adic quantum mechanics.[17]

### 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.[18] 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 1010 tP).[19]

Relative to the Planck epoch, the observable universe today looks extreme when expressed in Planck units, as in this set of approximations:[20][21]

Table 5: Today's universe in Planck units.
Property of

present-day Observable Universe
Approximate number

of Planck units
Equivalents
Age 8.08 × 1060 tP 4.35 × 1017 s, or 13.8 × 109 years
Diameter 5.4 × 1061 lP 8.7 × 1026 m or 9.2 × 1010 light-years
Mass approx. 1060 mP 3 × 1052 kg or 1.5 × 1022 solar masses (only counting stars)

1080 protons (sometimes known as the Eddington number)
Density 1.8 × 10−123 ρP 9.9 × 10−27 kg m−3
Temperature 1.9 × 10−32 TP 2.725 K

temperature of the cosmic microwave background radiation
Cosmological constant 5.6 × 10−122 t −2

P
1.9 × 10−35 s−2
Hubble constant 1.18 × 10−61 t −1

P
2.2 × 10−18 s−1 or 67.8 (km/s)/Mpc

The recurrence of large numbers close or related to 1060 in the above table is a coincidence that intrigues some theorists. It is an example of the kind of large numbers coincidence that led theorists such as Eddington and Dirac to develop alternative physical hypotheses (e.g. a variable speed of light or Dirac varying-G hypothesis).[22] After the measurement of the cosmological constant in 1998, estimated at 10−122 in Planck units, it was noted that this is suggestively close to the reciprocal of the age of the universe squared.[23] Barrow and Shaw (2011) proposed a modified theory in which Λ is a field evolving in such a way that its value remains Λ ~ T−2 throughout the history of the universe.[24]

### Other uses

The Planck length is related to Planck energy by the uncertainty principle. At this scale, the concepts of size and distance break down, as quantum indeterminacy becomes virtually absolute. Because the Schwarzschild radius of a black hole is roughly equal to the Compton wavelength at the Planck scale, a photon with sufficient energy to probe this realm would yield no information whatsoever. Any photon energetic enough to precisely measure a Planck-sized object could actually create a particle of that dimension, but it would be massive enough to immediately become a black hole (a.k.a. Planck particle), thus completely distorting that region of space, and swallowing the photon. This is the most extreme example possible of the uncertainty principle, and explains why only a quantum gravity theory reconciling general relativity with quantum mechanics will allow us to understand the dynamics of space-time at this scale. Planck scale dynamics are important for cosmology because by tracing the evolution of the cosmos back to the very beginning, at some very early stage the universe should have been so hot that processes involving energies as high as the Planck energy (corresponding to distances as short as the Planck length) may have occurred. This period is therefore called the Planck era or Planck epoch.

### Analysis of the units

#### Planck density

The Planck density is a very large unit, about equivalent to 1023 solar masses squeezed into the space of a single atomic nucleus. The Planck density is thought to be the upper limit of density.

#### Planck energy

Most Planck units are extremely small, as in the case of Planck length or Planck time, or extremely large, as in the case of Planck temperature or Planck acceleration. For comparison, the Planck energy is approximately equal to the energy stored in an automobile gas tank (57.2 L of gasoline at 34.2 MJ/L of chemical energy). The ultra-high-energy cosmic ray observed in 1991 had a measured energy of about 50 joules, equivalent to about 2.5×10−8 EP.[25] Theoretically, the highest energy photon carries about 1 EP of energy (see Ultra-high-energy gamma ray), after which it becomes indistinguishable from a Planck particle carrying the same energy.

#### Planck force

The Planck force is the derived unit of force resulting from the definition of the base Planck units for time, length, and mass. It is equal to the natural unit of momentum divided by the natural unit of time.

${\displaystyle F_{\text{P}}={\frac {m_{\text{P}}c}{t_{\text{P}}}}={\frac {c^{4}}{G}}=1.210295\times 10^{44}{\mbox{ N.}}}$

The Planck force is associated[26] with the equivalence of gravitational potential energy and electromagnetic energy: the gravitational attractive force of two bodies of 1 Planck mass each, set apart by 1 Planck length is 1 Planck force; equivalently, the electrostatic attractive/repulsive force of two Planck charges set apart by 1 Planck length is 1 Planck force.

The Planck force appears in the Einstein field equations, describing the properties of a gravitational field surrounding any given mass:

${\displaystyle G_{\mu \nu }=8\pi {\frac {G}{c^{4}}}T_{\mu \nu }}$

where ${\displaystyle G_{\mu \nu }}$ is the Einstein tensor and ${\displaystyle T_{\mu \nu }}$ is the energy–momentum tensor. The Planck force thus describes how much or how easily space-time is curved by a given amount of mass-energy.

Since 1993, various authors (De Sabbata & Sivaram, Massa, Kostro & Lange, Gibbons, Schiller) have argued that the Planck force is the maximum force value that can be observed in nature. This limit property is valid both for gravitational force and for any other type of force.

#### Planck momentum

This plot of kinetic energy versus momentum has a place for most moving objects encountered in everyday life. It shows objects with the same kinetic energy (horizontally related) that carry different amounts of momentum, as well as how the speed of a low-mass object compares (by vertical extrapolation) to the speed after perfectly inelastic collision with a large object at rest. Highly sloped lines (rise/run=2) mark contours of constant mass, while lines of unit slope mark contours of constant speed. The plot further illustrates where lightspeed, Planck's constant, and kT figure in. (Note: the line labeled universe only tracks a mass estimate for the visible universe.)

The Planck momentum is equal to the Planck mass multiplied by the speed of light. Unlike most of the other Planck units, Planck momentum occurs on a human scale. By comparison, running with a five-pound object (108 × Planck mass) at an average running speed (10−8 × speed of light in a vacuum) would give the object Planck momentum. A 70 kg human moving at an average walking speed of 1.4 m/s (5.0 km/h; 3.1 mph) would have a momentum of about 15 ${\displaystyle m_{\text{P}}c}$. A baseball, which has mass ${\displaystyle m=}$ 0.145 kg, travelling at 45 m/s (160 km/h; 100 mph) would have a Planck momentum.

#### Planck temperature

The Planck temperature of 1 (unity), equal to 1.416785(16)×1032 K[10], is considered a fundamental limit of temperature.[27] An object with the temperature of 1.42×1032 kelvin (TP) would emit a black body radiation with a peak wavelength of 1.616×10−35 m (Planck length), where each photon and each individual collision would have the energy to create a Planck particle. There are no known physical models able to describe temperatures greater than or equal to TP.

#### Planck time

A Planck time unit is the time required for light to travel a distance of 1 Planck length in a vacuum, which is a time interval of approximately 5.39 × 10−44 s (Gaussian version) or 1.911 × 10−43 s (Lorentz–Heaviside version).[28] All scientific experiments and human experiences occur over time scales that are many orders of magnitude longer than the Planck time,[29] making any events happening at the Planck scale undetectable with current scientific technology. As of November 2016, the smallest time interval uncertainty in direct measurements was on the order of 850 zeptoseconds (8.50 × 10−19 seconds).[30]

## List of physical equations

Physical quantities that have different dimensions (such as time and length) cannot be equated even if they are numerically equal (1 second is not the same as 1 metre). In theoretical physics, however, this scruple can be set aside, by a process called nondimensionalization. Table 6 shows how the use of Planck units simplifies many fundamental equations of physics, because this gives each of the five fundamental constants, and products of them, a simple numeric value of 1. In the SI form, the units should be accounted for. In the nondimensionalized form, the units, which are now Planck units, need not be written if their use is understood.

Table 6: How Planck units simplify the key equations of physics
SI form Lorentz-Heaviside version Planck form Gaussian version form
Newton's law of universal gravitation ${\displaystyle F=G{\frac {m_{1}m_{2}}{r^{2}}}}$ ${\displaystyle F={\frac {m_{1}m_{2}}{4\pi r^{2}}}}$ ${\displaystyle F={\frac {m_{1}m_{2}}{r^{2}}}}$
Einstein field equations in general relativity ${\displaystyle {G_{\mu \nu }=8\pi {G \over c^{4}}T_{\mu \nu }}\ }$ ${\displaystyle {G_{\mu \nu }=2T_{\mu \nu }}\ }$ ${\displaystyle {G_{\mu \nu }=8\pi T_{\mu \nu }}\ }$
Mass–energy equivalence in special relativity ${\displaystyle {E=mc^{2}}\ }$ ${\displaystyle {E=m}\ }$
Energy–momentum relation ${\displaystyle E^{2}=m^{2}c^{4}+p^{2}c^{2}\;}$ ${\displaystyle E^{2}=m^{2}+p^{2}\;}$
Thermal energy per particle per degree of freedom ${\displaystyle {E={\tfrac {1}{2}}k_{\text{B}}T}\ }$ ${\displaystyle {E={\tfrac {1}{2}}T}\ }$
Boltzmann's entropy formula ${\displaystyle {S=k_{\text{B}}\ln \Omega }\ }$ ${\displaystyle {S=\ln \Omega }\ }$
Planck–Einstein relation for energy and angular frequency ${\displaystyle {E=\hbar \omega }\ }$ ${\displaystyle {E=\omega }\ }$
Planck's law (surface intensity per unit solid angle per unit angular frequency) for black body at temperature T. ${\displaystyle I(\omega ,T)={\frac {\hbar \omega ^{3}}{4\pi ^{3}c^{2}}}~{\frac {1}{e^{\frac {\hbar \omega }{k_{\text{B}}T}}-1}}}$ ${\displaystyle I(\omega ,T)={\frac {\omega ^{3}}{4\pi ^{3}}}~{\frac {1}{e^{\omega /T}-1}}}$
Stefan–Boltzmann constantσ defined ${\displaystyle \sigma ={\frac {\pi ^{2}k_{\text{B}}^{4}}{60\hbar ^{3}c^{2}}}}$ ${\displaystyle \sigma ={\frac {\pi ^{2}}{60}}}$
BekensteinHawking black hole entropy[31] ${\displaystyle S_{\text{BH}}={\frac {A_{\text{BH}}k_{\text{B}}c^{3}}{4G\hbar }}={\frac {4\pi Gk_{\text{B}}m_{\text{BH}}^{2}}{\hbar c}}}$ ${\displaystyle S_{\text{BH}}=\pi A_{\text{BH}}=m_{\text{BH}}^{2}}$ ${\displaystyle S_{\text{BH}}={\frac {A_{\text{BH}}}{4}}=4\pi m_{\text{BH}}^{2}}$
Schrödinger's equation ${\displaystyle -{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi (\mathbf {r} ,t)+V(\mathbf {r} ,t)\psi (\mathbf {r} ,t)=i\hbar {\frac {\partial \psi (\mathbf {r} ,t)}{\partial t}}}$ ${\displaystyle -{\frac {1}{2m}}\nabla ^{2}\psi (\mathbf {r} ,t)+V(\mathbf {r} ,t)\psi (\mathbf {r} ,t)=i{\frac {\partial \psi (\mathbf {r} ,t)}{\partial t}}}$
Hamiltonian form of Schrödinger's equation ${\displaystyle H\left|\psi _{t}\right\rangle =i\hbar {\frac {\partial }{\partial t}}\left|\psi _{t}\right\rangle }$ ${\displaystyle H\left|\psi _{t}\right\rangle =i{\frac {\partial }{\partial t}}\left|\psi _{t}\right\rangle }$
Covariant form of the Dirac equation ${\displaystyle \ (i\hbar \gamma ^{\mu }\partial _{\mu }-mc)\psi =0}$ ${\displaystyle \ (i\gamma ^{\mu }\partial _{\mu }-m)\psi =0}$
Unruh temperature ${\displaystyle T={\frac {\hbar a}{2\pi ck_{B}}}}$ ${\displaystyle T={\frac {a}{2\pi }}}$
Coulomb's law ${\displaystyle F={\frac {1}{4\pi \epsilon _{0}}}{\frac {q_{1}q_{2}}{r^{2}}}}$ ${\displaystyle F={\frac {q_{1}q_{2}}{4\pi r^{2}}}}$ ${\displaystyle F={\frac {q_{1}q_{2}}{r^{2}}}}$
Maxwell's equations ${\displaystyle \nabla \cdot \mathbf {E} ={\frac {1}{\epsilon _{0}}}\rho }$

${\displaystyle \nabla \cdot \mathbf {B} =0\ }$

${\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}$

${\displaystyle \nabla \times \mathbf {B} ={\frac {1}{c^{2}}}\left({\frac {1}{\epsilon _{0}}}\mathbf {J} +{\frac {\partial \mathbf {E} }{\partial t}}\right)}$

${\displaystyle \nabla \cdot \mathbf {E} =\rho }$

${\displaystyle \nabla \cdot \mathbf {B} =0\ }$

${\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}$

${\displaystyle \nabla \times \mathbf {B} =\mathbf {J} +{\frac {\partial \mathbf {E} }{\partial t}}}$

${\displaystyle \nabla \cdot \mathbf {E} =4\pi \rho \ }$

${\displaystyle \nabla \cdot \mathbf {B} =0\ }$

${\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}$

${\displaystyle \nabla \times \mathbf {B} =4\pi \mathbf {J} +{\frac {\partial \mathbf {E} }{\partial t}}}$

Ideal gas law ${\displaystyle PV=nRT}$ ${\displaystyle PV=NT}$

## Alternative choices of normalization

As already stated above, Planck units are derived by "normalizing" the numerical values of certain fundamental constants to 1. These normalizations are neither the only ones possible nor necessarily the best. Moreover, the choice of what factors to normalize, among the factors appearing in the fundamental equations of physics, is not evident, and the values of the Planck units are sensitive to this choice.

The factor 4π is ubiquitous in theoretical physics because the surface area of a sphere of radius r is 4πr2. This, along with the concept of flux, are the basis for the inverse-square law, Gauss's law, and the divergence operator applied to flux density. For example, gravitational and electrostatic fields produced by point charges have spherical symmetry (Barrow 2002: 214–15). The 4πr2 appearing in the denominator of Coulomb's law in rationalized form, for example, follows from the flux of an electrostatic field being distributed uniformly on the surface of a sphere. Likewise for Newton's law of universal gravitation. (If space had more than three spatial dimensions, the factor 4π would have to be changed according to the geometry of the sphere in higher dimensions.)

Hence a substantial body of physical theory developed since Planck (1899) suggests normalizing not G but either 4πG (or 8πG or 16πG) to 1. Doing so would introduce a factor of 1/4π (or 1/8π or 1/16π) into the nondimensionalized form of the law of universal gravitation, consistent with the modern rationalized formulation of Coulomb's law in terms of the vacuum permittivity. In fact, alternative normalizations frequently preserve the factor of 1/4π in the nondimensionalized form of Coulomb's law as well, so that the nondimensionalized Maxwell's equations for electromagnetism and gravitoelectromagnetism both take the same form as those for electromagnetism in SI, which do not have any factors of 4π. When this is applied to electromagnetic constants, ε0, this unit system is called "rationalized" Lorentz–Heaviside units. When applied additionally to gravitation and Planck units, these are called rationalized Planck units[32] and are seen in high-energy physics.

The rationalized Planck units are defined so that ${\displaystyle c=4\pi G=\hbar =\varepsilon _{0}=k_{\text{B}}=1}$.

There are several possible alternative normalizations.

### Gravity

In 1899, Newton's law of universal gravitation was still seen as exact, rather than as a convenient approximation holding for "small" velocities and masses (the approximate nature of Newton's law was shown following the development of general relativity in 1915). Hence Planck normalized to 1 the gravitational constant G in Newton's law. In theories emerging after 1899, G nearly always appears in formulae multiplied by 4π or a small integer multiple thereof. Hence, a choice to be made when designing a system of natural units is which, if any, instances of 4π appearing in the equations of physics are to be eliminated via the normalization.

• Normalizing 4πG to 1:
• Setting 8πG = 1. This would eliminate 8πG from the Einstein field equations, Einstein–Hilbert action, and the Friedmann equations, for gravitation. Planck units modified so that 8πG = 1 are known as reduced Planck units, because the Planck mass is divided by 8π. Also, the Bekenstein–Hawking formula for the entropy of a black hole simplifies to SBH = (mBH)2/2 = 2πABH.
• Setting 16πG = 1. This would eliminate the constant c4/16πG from the Einstein–Hilbert action. The form of the Einstein field equations with cosmological constant Λ becomes Rμν + Λgμν = 1/2(Rgμν + Tμν).

### Electromagnetism

Planck units normalize to 1 the Coulomb force constant ke = 1/4πε0 (as does the cgs system of units). This sets the derived unit of impedance, ZP equal to Z0/4π, where Z0 is the characteristic impedance of free space.

${\displaystyle e={\sqrt {4\pi \alpha }}\cdot q_{{\text{P}}'}\approx 0.30282212\cdot q_{{\text{P}}'}}$
where ${\displaystyle {\alpha }}$ is the fine-structure constant. This convention is seen in high-energy physics.

### Temperature

Planck normalized to 1 the Boltzmann constant kB.

• Normalizing 1/2kB to 1:
• Removes the factor of 1/2 in the nondimensionalized equation for the thermal energy per particle per degree of freedom.
• Introduces a factor of 2 into the nondimensionalized form of Boltzmann's entropy formula.
• Does not affect the value of any of the base or derived Planck units listed in Tables 3 and 4.

## Planck units and the invariant scaling of nature

Some theorists (such as Dirac and Milne) have proposed cosmologies that conjecture that physical "constants" might actually change over time (e.g. a variable speed of light or Dirac varying-G theory). Such cosmologies have not gained mainstream acceptance and yet there is still considerable scientific interest in the possibility that physical "constants" might change, although such propositions introduce difficult questions. Perhaps the first question to address is: How would such a change make a noticeable operational difference in physical measurement or, more fundamentally, our perception of reality? If some particular physical constant had changed, how would we notice it, or how would physical reality be different? Which changed constants result in a meaningful and measurable difference in physical reality? If a physical constant that is not dimensionless, such as the speed of light, did in fact change, would we be able to notice it or measure it unambiguously? – a question examined by Michael Duff in his paper "Comment on time-variation of fundamental constants".[33][34]

George Gamow argued in his book Mr Tompkins in Wonderland that a sufficient change in a dimensionful physical constant, such as the speed of light in a vacuum, would result in obvious perceptible changes. But this idea is challenged:

[An] important lesson we learn from the way that pure numbers like α define the world is what it really means for worlds to be different. The pure number we call the fine structure constant and denote by α is a combination of the electron charge, e, the speed of light, c, and Planck's constant, h. At first we might be tempted to think that a world in which the speed of light was slower would be a different world. But this would be a mistake. If c, h, and e were all changed so that the values they have in metric (or any other) units were different when we looked them up in our tables of physical constants, but the value of α remained the same, this new world would be observationally indistinguishable from our world. The only thing that counts in the definition of worlds are the values of the dimensionless constants of Nature. If all masses were doubled in value [including the Planck mass mP ] you cannot tell because all the pure numbers defined by the ratios of any pair of masses are unchanged.

— Barrow 2002[20]

Referring to Duff's "Comment on time-variation of fundamental constants"[33] and Duff, Okun, and Veneziano's paper "Trialogue on the number of fundamental constants",[35] particularly the section entitled "The operationally indistinguishable world of Mr. Tompkins", if all physical quantities (masses and other properties of particles) were expressed in terms of Planck units, those quantities would be dimensionless numbers (mass divided by the Planck mass, length divided by the Planck length, etc.) and the only quantities that we ultimately measure in physical experiments or in our perception of reality are dimensionless numbers. When one commonly measures a length with a ruler or tape-measure, that person is actually counting tick marks on a given standard or is measuring the length relative to that given standard, which is a dimensionless value. It is no different for physical experiments, as all physical quantities are measured relative to some other like-dimensioned quantity.

We can notice a difference if some dimensionless physical quantity such as fine-structure constant, α, changes or the proton-to-electron mass ratio, mp/me, changes (atomic structures would change) but if all dimensionless physical quantities remained unchanged (this includes all possible ratios of identically dimensioned physical quantity), we cannot tell if a dimensionful quantity, such as the speed of light, c, has changed. And, indeed, the Tompkins concept becomes meaningless in our perception of reality if a dimensional quantity such as c has changed, even drastically.

If the speed of light c, were somehow suddenly cut in half and changed to 1/2c (but with the axiom that all dimensionless physical quantities remain the same), then the Planck length would increase by a factor of 22 from the point of view of some unaffected observer on the outside. Measured by "mortal" observers in terms of Planck units, the new speed of light would remain as 1 new Planck length per 1 new Planck time – which is no different from the old measurement. But, since by axiom, the size of atoms (approximately the Bohr radius) are related to the Planck length by an unchanging dimensionless constant of proportionality:

${\displaystyle a_{0}={\frac {4\pi \epsilon _{0}\hbar ^{2}}{m_{e}e^{2}}}={\frac {m_{\text{P}}}{m_{e}\alpha }}l_{\text{P}}.}$

Then atoms would be bigger (in one dimension) by 22, each of us would be taller by 22, and so would our metre sticks be taller (and wider and thicker) by a factor of 22. Our perception of distance and lengths relative to the Planck length is, by axiom, an unchanging dimensionless constant.

Our clocks would tick slower by a factor of 42 (from the point of view of this unaffected observer on the outside) because the Planck time has increased by 42 but we would not know the difference (our perception of durations of time relative to the Planck time is, by axiom, an unchanging dimensionless constant). This hypothetical unaffected observer on the outside might observe that light now propagates at half the speed that it previously did (as well as all other observed velocities) but it would still travel 299792458 of our new metres in the time elapsed by one of our new seconds (1/2c × 42 ÷ 22 continues to equal 299792458 m/s). We would not notice any difference.

This contradicts what George Gamow writes in his book Mr. Tompkins; there, Gamow suggests that if a dimension-dependent universal constant such as c changed significantly, we would easily notice the difference. The disagreement is better thought of as the ambiguity in the phrase "changing a physical constant"; what would happen depends on whether (1) all other dimensionless constants were kept the same, or whether (2) all other dimension-dependent constants are kept the same. The second choice is a somewhat confusing possibility, since most of our units of measurement are defined in relation to the outcomes of physical experiments, and the experimental results depend on the constants. Gamow does not address this subtlety; the thought experiments he conducts in his popular works assume the second choice for "changing a physical constant". And Duff or Barrow would point out that ascribing a change in measurable reality, i.e. α, to a specific dimensional component quantity, such as c, is unjustified. The very same operational difference in measurement or perceived reality could just as well be caused by a change in h or e if α is changed and no other dimensionless constants are changed. It is only the dimensionless physical constants that ultimately matter in the definition of worlds.[33][36]

This unvarying aspect of the Planck-relative scale, or that of any other system of natural units, leads many theorists to conclude that a hypothetical change in dimensionful physical constants can only be manifest as a change in dimensionless physical constants. One such dimensionless physical constant is the fine-structure constant. There are some experimental physicists who assert they have in fact measured a change in the fine structure constant[37] and this has intensified the debate about the measurement of physical constants. According to some theorists[38] there are some very special circumstances in which changes in the fine-structure constant can be measured as a change in dimensionful physical constants. Others however reject the possibility of measuring a change in dimensionful physical constants under any circumstance.[33] The difficulty or even the impossibility of measuring changes in dimensionful physical constants has led some theorists to debate with each other whether or not a dimensionful physical constant has any practical significance at all and that in turn leads to questions about which dimensionful physical constants are meaningful.[35]

## Notes

1. General relativity predicts that gravitational radiation propagates at the same speed as electromagnetic radiation.

## References

### Citations

1. Wesson, P. S. (1980). "The application of dimensional analysis to cosmology". Space Science Reviews. 27 (2): 117. Bibcode:1980SSRv...27..109W. doi:10.1007/bf00212237.
2. ^ a b "2018 CODATA Value: speed of light in vacuum". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 20 May 2019.
3. ^ a b "2018 CODATA Value: Newtonian constant of gravitation". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 20 May 2019.
4. ^ a b "2018 CODATA Value: reduced Planck constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 28 August 2019.
5. "2018 CODATA Value: Boltzmann constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 20 May 2019.
6. Derived from ke = 1/(4πε0) – "2018 CODATA Value: vacuum electric permittivity". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 20 May 2019.
7. "2018 CODATA Value: Planck length". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 20 May 2019.
8. "2018 CODATA Value: Planck mass". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 20 May 2019.
9. "2018 CODATA Value: Planck time". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 20 May 2019.
10. ^ a b "2018 CODATA Value: Planck temperature". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 20 May 2019.
11. ^ a b "2018 CODATA Value: vacuum electric permittivity". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 20 May 2019.
12. ^ a b Planck (1899), p. 479.
13. ^ a b c Tomilin, K. A. (1999). Natural Systems of Units. To the Centenary Anniversary of the Planck System (PDF). Proceedings Of The XXII Workshop On High Energy Physics And Field Theory. pp. 287–296.
14. Wilczek, Frank (2001). "Scaling Mount Planck I: A View from the Bottom". Physics Today. 54 (6): 12–13. Bibcode:2001PhT....54f..12W. doi:10.1063/1.1387576.
15. Number Theory as the Ultimate Physical Theory, Igor V. Volovich, empslocal.ex.ac.uk/~mwatkins/zeta/volovich1.pdf, 10.1134/S2070046610010061
16. Staff. "Birth of the Universe". University of Oregon. Retrieved 24 September 2016. - discusses "Planck time" and "Planck era" at the very beginning of the Universe
17. Edward W. Kolb; Michael S. Turner (1994). The Early Universe. Basic Books. p. 447. ISBN 978-0-201-62674-2. Retrieved 10 April 2010.
18. ^ a b John D. Barrow, 2002. The Constants of Nature; From Alpha to Omega – The Numbers that Encode the Deepest Secrets of the Universe. Pantheon Books. ISBN 0-375-42221-8.
19. P.A.M. Dirac (1938). "A New Basis for Cosmology". Proceedings of the Royal Society A. 165 (921): 199–208. Bibcode:1938RSPSA.165..199D. doi:10.1098/rspa.1938.0053.
20. J.D. Barrow and F.J. Tipler, The Anthropic Cosmological Principle, Oxford UP, Oxford (1986), chapter 6.9.
21. Barrow, John D.; Shaw, Douglas J. (2011). "The value of the cosmological constant". General Relativity and Gravitation. 43 (10): 2555–2560. arXiv:1105.3105. Bibcode:2011GReGr..43.2555B. doi:10.1007/s10714-011-1199-1.
22. "HiRes - The High Resolution Fly's Eye Ultra High Energy Cosmic Ray Observatory". www.cosmic-ray.org. Retrieved 21 December 2016.
23. "Gravity and the Photon". HyperPhysics. Georgia State University. Retrieved 12 September 2012.
24. Nova: Absolute Hot
25. "First Second of the Big Bang". How The Universe Works 3. 2014. Discovery Science.
26. MacDonald, Fiona (14 November 2016). "Scientists have measured the smallest fragment of time ever". Science Alert. Retrieved 14 November 2016.
27. Also see Roger Penrose (1989) The Road to Reality. Oxford Univ. Press: 714-17. Knopf.
28. Sorkin, Rafael (1983). "Kaluza-Klein Monopole". Physical Review Letters. 51 (2): 87–90. Bibcode:1983PhRvL..51...87S. doi:10.1103/PhysRevLett.51.87.
29. ^ a b c d Michael Duff (2002). "Comment on time-variation of fundamental constants". arXiv:hep-th/0208093.
30. Michael Duff (2014). How fundamental are fundamental constants?. arXiv:1412.2040. doi:10.1080/00107514.2014.980093 (inactive 22 January 2020).
31. ^ a b Duff, Michael; Okun, Lev; Veneziano, Gabriele (2002). "Trialogue on the number of fundamental constants". Journal of High Energy Physics. 2002 (3): 023. arXiv:physics/0110060. Bibcode:2002JHEP...03..023D. doi:10.1088/1126-6708/2002/03/023.
32. Webb, J. K.; et al. (2001). "Further evidence for cosmological evolution of the fine structure constant". Phys. Rev. Lett. 87 (9): 884. arXiv:astro-ph/0012539v3. Bibcode:2001PhRvL..87i1301W. doi:10.1103/PhysRevLett.87.091301. PMID 11531558.
33. Davies, Paul C.; Davis, T. M.; Lineweaver, C. H. (2002). "Cosmology: Black Holes Constrain Varying Constants". Nature. 418 (6898): 602–3. Bibcode:2002Natur.418..602D. doi:10.1038/418602a. PMID 12167848.