Planck units

Inparticle physicsandphysical cosmology,Planck unitsare asystem of units of measurementdefined exclusively in terms of four universalphysical constants:c,G,ħ,andk_B(described further below). Expressing one of these physical constants in terms of Planck units yields a numerical value of1.They are a system ofnatural units,defined using fundamental properties ofnature(specifically, properties offree space) rather than properties of a chosenprototype object.Originally proposed in 1899 by German physicistMax Planck,they are relevant in research on unified theories such asquantum gravity.

The termPlanck scalerefers to quantities of space, time, energy and other units that are similar in magnitude to corresponding Planck units. This region may be characterized by particleenergiesof around10¹⁹GeVor10⁹J,timeintervals of around5×10⁻⁴⁴sandlengthsof around10⁻³⁵m(approximately the energy-equivalent of the Planck mass, the Planck time and the Planck length, respectively). At the Planck scale, the predictions of theStandard Model,quantum field theoryandgeneral relativityare not expected to apply, andquantum effects of gravityare expected to dominate. One example is represented by the conditions in thefirst 10⁻⁴³secondsof our universe after theBig Bang,approximately 13.8 billion years ago.

The fouruniversal constantsthat, by definition, have a numeric value 1 when expressed in these units are:

• c,thespeed of lightin vacuum,
• G,thegravitational constant,
• ħ,thereduced Planck constant,and
• k_B,theBoltzmann constant.

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 associatedbase units,from which all other quantities and units may be derived. In theInternational System of Units,for example, theSI base quantitiesinclude length with the associated unit of themetre.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,

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

can be expressed as:

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

Both equations aredimensionally consistentand equally valid inanysystem of quantities, but the second equation, withGabsent, is relating onlydimensionless quantitiessince 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:

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

This last equation (withoutG) is valid withF′,m₁′,m₂′, andr′being the dimensionless ratio quantitiescorresponding tothe standard quantities, written e.g.F′≘ForF′=F/F_P,but not as a direct equality of quantities. This may seem to be "setting the constantsc,G,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"G=c= 1",Paul S. Wessonwrote 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 ofnatural unitswas introduced in 1874, whenGeorge Johnstone Stoney,noting that electric charge is quantized, derived units of length, time, and mass, now namedStoney unitsin his honor. Stoney chose his units so thatG,c,and theelectron chargeewould be numerically equal to 1.In 1899, one year before the advent of quantum theory,Max Planckintroduced what became later 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 ofaction,now usually known as the Planck constant, which appeared in theWien approximationforblack-body radiation.Planck underlined the universality of the new unit system, writing:

... die Möglichkeit gegeben ist, 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 nothwendig behalten und welche daher als »natürliche Maasseinheiten« 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$G$,$h$,$c$,and$k_{\rm {B}}$to arrive at natural units forlength,time,mass,andtemperature.His definitions differ from the modern ones by a factor of${\sqrt {2\pi }}$,because the modern definitions use$\hbar$rather than$h$.

Table 1: Modern values for Planck's original choice of quantities

Name	Dimension	Expression	Value (SIunits)
Planck length	length(L)	$l_{\text{P}}={\sqrt {\frac {\hbar G}{c^{3}}}}$	1.616255(18)×10−35m‍
Planck mass	mass(M)	$m_{\text{P}}={\sqrt {\frac {\hbar c}{G}}}$	2.176434(24)×10−8kg‍
Planck time	time(T)	$t_{\text{P}}={\sqrt {\frac {\hbar G}{c^{5}}}}$	5.391247(60)×10−44s‍
Planck temperature	temperature(Θ)	$T_{\text{P}}={\sqrt {\frac {\hbar c^{5}}{Gk_{\text{B}}^{2}}}}$	1.416784(16)×1032K‍

Unlike the case with theInternational 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 theCoulomb constant$k_{\text{e}}$or thevacuum permittivity$\epsilon _{0}$is normalized to 1. Thus, depending on the author's choice, this charge unit is given by $$q_{\text{P}}={\sqrt {4\pi \epsilon _{0}\hbar c}}\approx 1.875546\times 10^{-18}{\text{ C}}\approx 11.7\ e$$ for$k_{\text{e}}=1$,or $$q_{\text{P}}={\sqrt {\epsilon _{0}\hbar c}}\approx 5.290818\times 10^{-19}{\text{ C}}\approx 3.3\ e$$ for$\varepsilon _{0}=1$.Some of these tabulations also replace mass with energy when doing so.

In SI units, the values ofc,h,eandk_Bare exact and the values ofε₀andGin SI units respectively have relative uncertainties of1.6×10⁻¹⁰‍and2.2×10⁻⁵.‍Hence, the uncertainties in the SI values of the Planck units derive almost entirely from uncertainty in the SI value ofG.

Compared toStoney units,Planck base units are all larger by a factor${\textstyle {\sqrt {{1}/{\alpha }}}\approx 11.7}$,where$\alpha$is thefine-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.

Table 2: Coherent derived units of Planck units

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

Some Planck units, such as of time and length, are manyorders of magnitudetoo 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 theBig Bangdoes not extend to thePlanck epoch,i.e., when the universe was less than one Planck time old. Describing the universe during the Planck epoch requires a theory ofquantum gravitythat would incorporate quantum effects intogeneral relativity.Such a theory does not yet exist.

Several quantities are not "extreme" in magnitude, such as the Planck mass, which is about22 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 littleanthropocentricarbitrariness, but do still involve some arbitrary choices in terms of the defining constants. Unlike themetreandsecond,which exist asbase unitsin theSIsystem for historical reasons, thePlanck lengthand Planck time are conceptually linked at a fundamental physical level. Consequently, natural units help physicists to reframe questions.Frank Wilczekputs 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 1/13quintillion.

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 iscomparing apples with oranges,becausemassandelectric chargeareincommensurablequantities. Rather, the disparity of magnitude of force is a manifestation of the fact that thecharge on the protonsis approximately theunit chargebut themass of the protonsis far less than the unit mass.

Planck scale

Inparticle physicsandphysical cosmology,the Planck scale is anenergy scalearound1.22×10²⁸eV(the Planck energy, corresponding to theenergy equivalentof the Planck mass,2.17645×10⁻⁸kg) at whichquantum effectsofgravitybecome significant. At this scale, present descriptions and theories of sub-atomic particle interactions in terms ofquantum field theorybreak down and become inadequate, due to the impact of the apparentnon-renormalizabilityof 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 otherfundamental 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 anapproximate lower limitat 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,gravityis problematic, and cannot be integrated withquantum mechanicsat 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 ofquantum gravityis necessary. Approaches to this problem includestring theoryandM-theory,loop quantum gravity,noncommutative geometry,andcausal set theory.

In cosmology

Main article:Chronology of the universe

Further information:Time-variation of fundamental constants

InBig Bang cosmology,thePlanck epochorPlanck erais the earliest stage of theBig Bang,before thetime passedwas equal to the Planck time,t_P,or approximately 10⁻⁴³seconds.There is no currently available physical theory to describe such short times, and it is not clear in what sense the concept oftimeis meaningful for values smaller than the Planck time. It is generally assumed thatquantum effects of gravitydominate physical interactions at this time scale. At this scale, theunified forceof theStandard Modelis assumed to beunified with gravitation.Immeasurably hot and dense, the state of the Planck epoch was succeeded by thegrand unification epoch,where gravitation is separated from the unified force of the Standard Model, in turn followed by theinflationary epoch,which ended after about 10⁻³²seconds (or about 10¹¹t_P).

Table 3 lists properties of the observable universe today expressed in Planck units.

Table 3: Today's universe in Planck units

Property of present-dayobservable universe	Approximate number of Planck units	Equivalents
Age	8.08 × 1060tP	4.35 × 1017s or 1.38 × 1010years
Diameter	5.4 × 1061lP	8.7 × 1026m or 9.2 × 1010light-years
Mass	approx. 1060mP	3 × 1052kg or 1.5 × 1022solar masses(only counting stars) 1080protons(sometimes known as theEddington number)
Density	1.8 × 10−123mP⋅lP−3	9.9 × 10−27kg⋅m−3
Temperature	1.9 × 10−32TP	2.725 K temperature of thecosmic microwave background radiation
Cosmological constant	≈ 10−122l−2 P	≈ 10−52m−2
Hubble constant	≈ 10−61t−1 P	≈ 10−18s−1≈ 102(km/s)/Mpc

After the measurement of the cosmological constant (Λ) in 1998, estimated at 10⁻¹²²in Planck units, it was noted that this is suggestively close to the reciprocal of theage of the universe(T) squared. Barrow and Shaw proposed a modified theory in whichΛis a field evolving in such a way that its value remains Λ ~T⁻²throughout the history of the universe.

Analysis of the units

Planck length

The Planck length, denotedℓ_P,is a unit oflengthdefined as: $$\ell _{\mathrm {P} }={\sqrt {\frac {\hbar G}{c^{3}}}}$$

It is equal to1.616255(18)×10⁻³⁵m‍(the two digits enclosed by parentheses are the estimatedstandard errorassociated with the reported numerical value) or about10⁻²⁰times the diameter of aproton.It can be motivated in various ways, such as considering a particle whosereduced Compton wavelengthis comparable to itsSchwarzschild radius,though whether those concepts are in fact simultaneously applicable is open to debate.(The same heuristic argument simultaneously motivates the Planck mass.)

The Planck length is a distance scale of interest in speculations about quantum gravity. TheBekenstein–Hawking entropy of a black holeis one-fourth the area of itsevent horizonin units of Planck length squared.Since the 1950s, it has been conjectured that quantum fluctuations of the spacetime metric might make the familiar notion of distance inapplicable below the Planck length.This is sometimes expressed by saying that "spacetime becomes afoam at the Planck scale".It is possible that the Planck length is the shortest physically measurable distance, since any attempt to investigate the possible existence of shorter distances, by performing higher-energy collisions, would result in black hole production. Higher-energy collisions, rather than splitting matter into finer pieces, would simply produce bigger black holes.

The strings ofstring theoryare modeled to be on the order of the Planck length.In theories withlarge extra dimensions,the Planck length calculated from the observed value of$G$can be smaller than the true, fundamental Planck length.

Planck time

The Planck timet_Pis thetimerequired forlightto travel a distance of 1 Planck length invacuum,which is a time interval of approximately5.39×10⁻⁴⁴s.No current physical theory can describe timescales shorter than the Planck time, such as the earliest events after the Big Bang.Some conjectures state that the structure of time need not remain smooth on intervals comparable to the Planck time.

Planck energy

The Planck energyE_Pis approximately equal to the energy released in the combustion of the fuel in an automobile fuel tank (57.2 L at 34.2 MJ/L of chemical energy). Theultra-high-energy cosmic rayobserved in 1991had a measured energy of about 50 J, equivalent to about2.5×10⁻⁸E_P.

Proposals for theories ofdoubly special relativityposit that, in addition to the speed of light, an energy scale is also invariant for all inertial observers. Typically, this energy scale is chosen to be the Planck energy.

Planck unit of force

The Planck unit of force may be thought of as the derived unit offorcein the Planck system if the Planck units of time, length, and mass are considered to be base units. $$F_{\text{P}}={\frac {m_{\text{P}}c}{t_{\text{P}}}}={\frac {c^{4}}{G}}\approx \mathrm {1.2103\times 10^{44}~N}$$

It is the gravitational attractive force of two bodies of 1 Planck mass each that are held 1 Planck length apart. One convention for the Planck charge is to choose it so that the electrostatic repulsion of two objects with Planck charge and mass that are held 1 Planck length apart balances the Newtonian attraction between them.

Some authors have argued that the Planck force is on the order of the maximum force that can occur between two bodies.However, the validity of these conjectures has been disputed.

Planck temperature

The Planck temperatureT_Pis1.416784(16)×10³²K.‍At this temperature, the wavelength of light emitted bythermal radiationreaches the Planck length. There are no known physical models able to describe temperatures greater thanT_P;a quantum theory of gravity would be required to model the extreme energies attained.Hypothetically, a system inthermal equilibriumat the Planck temperature might contain Planck-scale black holes, constantly being formed from thermal radiation and decaying via Hawking evaporation. Adding energy to such a system mightdecreaseits temperature by creating larger black holes, whose Hawking temperature is lower.

Nondimensionalized equations

Physical quantities that have different dimensions (such as time and length) cannot be equated even if they are numerically equal (e.g., 1 second is not the same as 1 metre). In theoretical physics, however, this scruple may be set aside, by a process callednondimensionalization.The effective result is that many fundamental equations of physics, which often include some of the constants used to define Planck units, become equations where these constants are replaced by a 1.

Examples include theenergy–momentum relationE²= (mc²)²+ (pc)²,which becomesE²=m²+p²,and theDirac equation(iħγ^μ∂_μ−mc)ψ= 0,which becomes(iγ^μ∂_μ−m)ψ= 0.

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 4πis ubiquitous intheoretical physicsbecause in three-dimensional space, the surface area of asphereof radiusris 4πr².This, along with the concept offlux,are the basis for theinverse-square law,Gauss's law,and thedivergenceoperator applied toflux density.For example,gravitationalandelectrostatic fieldsproduced by point objects have spherical symmetry, and so the electric flux through a sphere of radiusraround a point charge will be distributed uniformly over that sphere. From this, it follows that a factor of 4πr²will appear in the denominator of Coulomb's law inrationalized form.(Both the numerical factor and the power of the dependence onrwould change if space were higher-dimensional; the correct expressions can be deduced from the geometry ofhigher-dimensional spheres.) Likewise for Newton's law of universal gravitation: a factor of 4πnaturally appears inPoisson's equationwhen relating the gravitational potential to the distribution of matter.

Hence a substantial body of physical theory developed since Planck's 1899 paper suggests normalizing notGbut 4πG(or 8πG) to 1. Doing so would introduce a factor of1/4π(or1/8π) 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 of1/4πin the nondimensionalized form of Coulomb's law as well, so that the nondimensionalized Maxwell's equations for electromagnetism andgravitoelectromagnetismboth 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".When applied additionally to gravitation and Planck units, these are calledrationalized Planck unitsand are seen in high-energy physics.

The rationalized Planck units are defined so thatc= 4πG=ħ=ε₀=k_B= 1.

There are several possible alternative normalizations.

Gravitational constant

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 ofgeneral relativityin 1915). Hence Planck normalized to 1 thegravitational constantGin Newton's law. In theories emerging after 1899,Gnearly always appears in formulae multiplied by 4πor a small integer multiple thereof. Hence, a choice to be made when designing a system of natural units is which, if any, instances of 4πappearing in the equations of physics are to be eliminated via the normalization.

• Normalizing 4πGto 1 (and therefore settingG=1/4π):
  • Gauss's law for gravitybecomesΦ_g= −M(rather thanΦ_g= −4πMin Planck units).
  • Eliminates 4πGfrom thePoisson equation.
  • Eliminates 4πGin thegravitoelectromagnetic(GEM) equations, which hold in weakgravitational fieldsorlocally flat spacetime.These equations have the same form as Maxwell's equations (and theLorentz forceequation) ofelectromagnetism,withmass densityreplacingcharge density,and with1/4πGreplacingε₀.
  • Normalizes thecharacteristic impedanceZ_gofgravitational radiationin free space to 1 (normally expressed as4πG/c).
  • Eliminates 4πGfrom the Bekenstein–Hawking formula (for theentropy of a black holein terms of its massm_BHand the area of itsevent horizonA_BH) which is simplified toS_BH=πA_BH= (m_BH)².
• Setting8πG= 1(and therefore settingG=1/8π). This would eliminate 8πGfrom theEinstein field equations,Einstein–Hilbert action,and theFriedmann equations,for gravitation. Planck units modified so that8πG= 1are known asreduced Planck units,because the Planck mass is divided by√8π.Also, the Bekenstein–Hawking formula for the entropy of a black hole simplifies toS_BH= (m_BH)²/2 = 2πA_BH.

See also

• cGh physics
• Dimensional analysis
• Doubly special relativity
• Trans-Planckian problem
• Zero-point energy

External links

• Value of the fundamental constants,including the Planck units, as reported by theNational Institute of Standards and Technology(NIST).
• The Planck scale: relativity meets quantum mechanics meets gravityfrom 'Einstein Light' at UNSW