Planck units

In physics, Planck units are one of several systems of units of measurement called natural units because they normalize certain fundamental physical constants to 1. Planck units are designed so that the constants in Table 1 are normalized to 1. Planck units elegantly simplify many recurring algebraic expressions in theoretical physics, and are commonly employed in research on unified theories such as quantum gravity. The name "Planck units" honors Max Planck, who first proposed these units in 1899.

Table 1

Constant	Symbol	Dimension	Related Theory
speed of light in vacuum	${c}\$	L T⁻¹	Special relativity
Gravitational constant	${G}\$	L³ M⁻¹ T⁻²	General relativity; Newtonian gravitation
Dirac's constant or "reduced Planck's constant"	$\hbar ={\frac {h}{2\pi }}$ where ${h}\$ is Planck's constant	L² M T⁻¹	Quantum physics
Coulomb force constant	${\frac {1}{4\pi \epsilon _{0}}}$ where ${\epsilon _{0}}\$ is the permittivity of free space	L³ M T⁻² Q⁻²	Electrostatics
Boltzmann constant	${k}\$	L² M T⁻²Θ⁻¹	Thermodynamics; statistical mechanics

Base Planck units

Like all systems of measurement, Planck units feature "base units." In the SI system, the base unit of length is the meter; the same is true of the centimeter in the cgs system. The Planck base unit of length is the Planck length, as shown in Table 2. Any other unit of length can be defined as some multiple (or fraction) of the Planck length, the meter, or the centimeter.

Setting to unity the five fundamental constants in Table 1 defines base units of length, mass, time, charge, and temperature shown in Table 2. These base units are derived from the algebraic combinations of the dimensions of the fundamental constants, shown in the column "Expressions." These combinations are designed so that all dimensions cancel but one, that of the unit being defined. Like all systems of natural units, Planck units are an instance of dimensional analysis.

Table 2

Name	Dimension	Expressions	Approximate SI equivalent^[1]	Other equivalent
Planck length	Length (L)	$l_{P}={\sqrt {\frac {\hbar G}{c^{3}}}}$	1.616252 × 10⁻³⁵ m
Planck mass	Mass (M)	$m_{P}={\sqrt {\frac {\hbar c}{G}}}$	2.17645 × 10⁻⁸ kg	1.311 × 10¹⁹ u
Planck time	Time (T)	$t_{P}={\frac {l_{P}}{c}}={\frac {\hbar }{m_{P}c^{2}}}={\sqrt {\frac {\hbar G}{c^{5}}}}$	5.39121 × 10⁻⁴⁴ s
Planck charge	Electric charge (Q)	$q_{P}=m_{P}2\pi {\sqrt {G\epsilon _{0}}}={\sqrt {\hbar c4\pi \epsilon _{0}}}$	1.8755459 × 10⁻¹⁸ C	11.70624 e
Planck temperature	Temperature (Θ)	$T_{P}={\frac {m_{P}c^{2}}{k}}={\sqrt {\frac {\hbar c^{5}}{Gk^{2}}}}$	1.41679 × 10³² K

Derived Planck units

Table 3 shows how a variety of other physical units can be derived from the base units.

Table 3

Name	Dimensions	Expression	Approximate SI equivalent
Planck area	Area (L²)	$l_{P}^{2}={\frac {\hbar G}{c^{3}}}$	2.61223 × 10^-70 m²
Planck momentum	Momentum (LMT⁻¹)	$m_{P}c={\frac {\hbar }{l_{P}}}={\sqrt {\frac {\hbar c^{3}}{G}}}$	6.52485 kg m/s
Planck energy	Energy (L²MT⁻²)	$E_{P}=m_{P}c^{2}={\frac {\hbar }{t_{P}}}={\sqrt {\frac {\hbar c^{5}}{G}}}$	1.9561 × 10⁹ J
Planck force	Force (LMT⁻²)	$F_{P}={\frac {E_{P}}{l_{P}}}={\frac {\hbar }{l_{P}t_{P}}}={\frac {c^{4}}{G}}$	1.21027 × 10⁴⁴ N
Planck power	Power (L²MT⁻³)	$P_{P}={\frac {E_{P}}{t_{P}}}={\frac {\hbar }{t_{P}^{2}}}={\frac {c^{5}}{G}}$	3.62831 × 10⁵² W
Planck density	Density (LM⁻³)	$\rho _{P}={\frac {m_{P}}{l_{P}^{3}}}={\frac {\hbar t_{P}}{l_{P}^{5}}}={\frac {c^{5}}{\hbar G^{2}}}$	5.15500 × 10⁹⁶ kg/m³
Planck angular frequency	Frequency (T⁻¹)	$\omega _{P}={\frac {1}{t_{P}}}={\sqrt {\frac {c^{5}}{\hbar G}}}$	1.85487 × 10⁴³ s⁻¹
Planck pressure	Pressure (LM⁻¹T⁻²)	$p_{P}={\frac {F_{P}}{l_{P}^{2}}}={\frac {\hbar }{l_{P}^{3}t_{P}}}={\frac {c^{7}}{\hbar G^{2}}}$	4.63309 × 10¹¹³ Pa
Planck current	Electric current (QT⁻¹)	$I_{P}={\frac {q_{P}}{t_{P}}}={\sqrt {\frac {c^{6}4\pi \epsilon _{0}}{G}}}$	3.4789 × 10²⁵ A
Planck voltage	Voltage (L²MT⁻²Q⁻¹)	$V_{P}={\frac {E_{P}}{q_{P}}}={\frac {\hbar }{t_{P}q_{P}}}={\sqrt {\frac {c^{4}}{G4\pi \epsilon _{0}}}}$	1.04295 × 10²⁷ V
Planck impedance	Resistance (L²MT⁻¹Q⁻²)	$Z_{P}={\frac {V_{P}}{I_{P}}}={\frac {\hbar }{q_{P}^{2}}}={\frac {1}{4\pi \epsilon _{0}c}}={\frac {Z_{0}}{4\pi }}$	29.9792458 Ω

Planck units simplify the key equations of physics

Table 4 shows how Planck units, by setting the numerical values of the five fundamental constants to unity, simplify many equations of physics and make them nondimensional.

Table 4

	Usual form	Nondimensionalized form
Newton's Law of universal gravitation	$F=-G{\frac {m_{1}m_{2}}{r^{2}}}$	$F=-{\frac {m_{1}m_{2}}{r^{2}}}$
Schrödinger's equation	$-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi (\mathbf {r} ,t)+V(\mathbf {r} )\psi (\mathbf {r} ,t)$ $=i\hbar {\frac {\partial \psi }{\partial t}}(\mathbf {r} ,t)$	$-{\frac {1}{2m}}\nabla ^{2}\psi (\mathbf {r} ,t)+V(\mathbf {r} )\psi (\mathbf {r} ,t)$ $=i{\frac {\partial \psi }{\partial t}}(\mathbf {r} ,t)$
Equation relating particle energy to the radian frequency ${\omega }\$ of the wave function	${E=\hbar \omega }\$	${E=\omega }\$
Einstein's mass/energy equation of special relativity	${E=mc^{2}}\$	${E=m}\$
Einstein's field equation for general relativity	${G_{\mu \nu }=8\pi {G \over c^{4}}T_{\mu \nu }}\$	${G_{\mu \nu }=8\pi T_{\mu \nu }}\$
Thermal energy per particle per degree of freedom	${E={\frac {1}{2}}kT}\$	${E={\frac {1}{2}}T}\$
Coulomb's law	$F={\frac {1}{4\pi \epsilon _{0}}}{\frac {q_{1}q_{2}}{r^{2}}}$	$F={\frac {q_{1}q_{2}}{r^{2}}}$
Maxwell's equations	$\nabla \cdot \mathbf {E} ={\frac {1}{\epsilon _{0}}}\rho$ $\nabla \cdot \mathbf {B} =0\$ $\nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}$ $\nabla \times \mathbf {B} =\mu _{0}\mathbf {J} +\mu _{0}\epsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}$	$\nabla \cdot \mathbf {E} =4\pi \rho \$ $\nabla \cdot \mathbf {B} =0\$ $\nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}$ $\nabla \times \mathbf {B} =4\pi \mathbf {J} +{\frac {\partial \mathbf {E} }{\partial t}}$

Discussion

Natural units began with George Johnstone Stoney's 1881 proposal of a set of units, now named Stoney units in his honor, derived by normalizing G, c, and the electron charge e to 1.

Natural units can help physicists reframe questions. For instance, they led Frank Wilczek to observe:

...We see that the question 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)]...
— June 2001 Physics Today

Physicists sometimes humorously refer to Planck units as "God's units," as Planck units eliminate anthropocentric arbitrariness from the system of units. While the SI units the meter and the second are associated mainly for historical reasons, the Planck length and Planck time are conceptually linked at a fundamental physical level.

Max Planck first set out the units now named in his honor (giving numerical values for them, in terms of the metric system of his day, remarkably close to those in Table 2) in a paper presented to the Prussian Academy of Sciences in May 1899.^[2] At that time, quantum physics had not been invented and hence Planck's constant was unknown. That constant first appeared (under another name) in Planck's 1900 paper on black-body radiation, for which he was later awarded the Nobel prize.

Because Planck's 1899 paper did not include algebraic details, we are not sure just how Planck came to discover his units, although he did say why he thought them valuable:

...ihre Bedeutung für alle Zeiten und für alle, auch außerirdische und außermenschliche Kulturen notwendig behalten und welche daher als »natürliche Maßeinheiten« bezeichnet werden können...

...These necessarily retain their meaning for all times and for all civilizations, even extraterrestrial and non-human ones, and can therefore be designated as "natural units"...

Planck's units are relevant only to theoretical physics. With the possible exception of Planck momentum, Planck impedance, Planck mass, and Planck energy, Planck units are many orders of magnitude too large or too small to be of any empirical and practical use. In fact, 1 Planck unit often is the largest or smallest value of a physical quantity that makes sense given the current state of physical theory. For instance:

A Planck velocity of 1 equals the speed of light in a vacuum, the maximum possible velocity given special relativity;
At lengths and times of less than about 1 Planck unit, quantum theory as presently understood ceases to apply;
At a Planck temperature of 1, the four fundamental forces of modern physics unify, and all symmetries broken since the early Big Bang are restored;
At present, our understanding of the Big Bang begins when the universe reached about 1 Planck time and 1Planck length in age and size, at which time its Planck temperature was about 1.

Physical theory applicable on the scale of approximately 1 Planck unit of distance, time, density, or temperature, requires incorporating quantum effects into general relativity. Doing so would require a theory of quantum gravity which does not yet exist.

At present, the numerical value of the gravitational constant G cannot be determined experimentally to an accuracy of better than about 1 part in 7000. This proportional uncertainty far exceeds that of any other fundamental empirical constant in Table 1. Tables 2 and 3 reveal that G appears in the base definition of every Planck unit except velocity, charge, and impedance. Hence a proportion of the uncertainty in the numerical value of G propagates to the value of the SI equivalent to nearly every Planck unit. This proportion depends on the net exponent of G in the base definition. An exponent of ±1⁄2 means that the uncertainty of the value of the Planck unit, in terms of SI units, is half of the uncertainty of G. By contrast, the speed of light in SI units is no longer subject to measurement error, because the SI base unit of length, the meter, is now defined as some stipulated fraction of the distance light travels in 1 second. Hence the value of c is now an exact defined quantity.

Planck neither defined nor proposed the Planck charge. Its definition in Table 2 is a natural extension of the definitions of the other Planck units.^[3] Note that the elementary charge e, measured in terms of the Planck charge, is

e={\sqrt {\alpha }}\ q_{P}=0.085424543\ q_{P}\

where α is the dimensionless fine-structure constant

\alpha =\left({\frac {e}{q_{P}}}\right)^{2}={\frac {e^{2}}{\hbar c4\pi \epsilon _{0}}}={\frac {1}{137.03599907}}.

The numerical value of the fine-structure constant is a function of the charge, measured in Planck units, that nature has assigned to electrons, protons, and other charged particles. Because the electromagnetic force between two charged particles is proportional to the product of the charges of each particle (these charges in Planck units are proportional to α^1/2 ), the strength of the electromagnetic force relative to other fundamental forces is proportional to α.

Alternative normalizations

As already stated, 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 constants to normalize is not evident, and the values of the Planck units are sensitive to this choice.

The factor n4π, n=1,2, or 4, is ubiquitous in physical theory because it appears in the formulas for the surface area and volume of a sphere (Barrow 2002). Gravitational and electrostatic fields are sphere-like in that their strengths vary with distance but not direction. In any event, a fundamental choice that has to be made when designing a system of natural units is whether to eliminate particular instances of 4π via normalization.

Normalizing Boltzmann's constant k to 2. This would remove the factor of 2 in the nondimensionalized equation for the thermal energy per particle per degree of freedom, and would not affect the value of any base or derived unit other than the Planck temperature. But the Planck temperature would then not equal the temperature of the universe at the moment where our understanding of the Big Bang begins, as discussed in the preceding section.

Eliminating 4π from the equations for electromagnetism. Planck's choices of what to normalize were also a consequence of the state of physical theory in 1899. When he introduced the units now named after him, the understanding of electromagnetism was not what is today, so that Coulomb's law was seen as more fundamental than Maxwell's equations. Hence Planck normalized to 1 the Coulomb force constant (4πε₀)⁻¹ (as does the cgs system of units). This sets the Planck impedance, Z_P equal to Z₀/4π, where Z₀ is the characteristic impedance of free space. Normalizing the permittivity of free space ε₀ to 1 not only makes Z_P equal to Z₀ but also eliminates 4π from Maxwell's equations. On the other hand, the nondimensionalized form of Coulomb's law would now include a factor of (4π)⁻¹.

Eliminating n4π from the equations for general relativity and cosmology (see geometrized unit system). In 1899, general relativity lay some years in the future, so that Newton's law was still seen as fundamental, rather than as a convenient approximation holding for "small" velocities and distances. Hence Planck normalized to 1 the G in Newton's law of universal gravitation. In theories emerging after 1899, G is nearly always multiplied by n4π, n=1,2, or 4:

4πG appears in the:
- Gravitational form of Gauss's Law, $\Phi _{g}=-4\pi GM$ ;
- Characteristic impedance of gravitational radiation in free space, Z₀ = 4πG/c. The c in the denominator stems from the general relativity result that both gravitational radiation propagates at the same velocity as electromagnetic radiation;
- Gravitoelectromagnetic (GEM) equations, which hold in weak gravitational fields or reasonably flat space-time. These equations have the same form as Maxwell's equations (and the Lorentz force equation) of electromagnetism, with mass(mass density) replacing charge (charge density), and (4πG)⁻¹ replacing the permittivity ε₀.
8πG appears in the Einstein field equations, Einstein-Hilbert action, Friedmann equations, and the Poisson equation for gravitation. Planck units modified so that 8πG=1 are known as reduced Planck units because the factor (8π)^1/2 divides the Planck mass.
16πG. Normalizing this to 1 eliminates the constant $k={\frac {c^{4}}{16\pi G}}$ from the Einstein-Hilbert action. The Einstein field equations with cosmological constant Λ becomes $R_{ab}-\Lambda g_{ab}=(Rg_{ab}-T_{ab})/2.$

Hence a substantial body of physical theory discovered since Planck (1899) suggests normalizing to 1 not G but n4πG, n=1,2, or 4. However, doing so would introduce a factor of (n4π)⁻¹ into the nondimensionalized law of universal gravitation.

Planck units and the invariant scaling of nature

Some theoreticians and experimentalists have conjectured that some physical "constants" might actually change over time, a proposition that introduces many difficult questions such as:

How would such a change make a noticeable operational difference in physical measurement or, more basically, our perception of reality?
If some physical constant had changed, would we even notice it?
How would physical reality be different?
Which changed constants would result in a meaningful and measurable difference?

John Barrow has spoken to these questions as follows:

[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 m_P] you cannot tell because all the pure numbers defined by the ratios of any pair of masses are unchanged.
— Barrow 2002

Referring to Michael Duff (Comment on time-variation of fundamental constants) and Duff, Okun, and Veneziano (Trialogue on the number of fundamental constants - 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 values.

We can notice a difference if some dimensionless physical quantity such as α or the proton/electron mass ratio changes; either change would alter atomic structures. But if all dimensionless physical quantities remained constant (this includes all possible ratios of identically dimensioned physical quantities), we could not tell if a dimensionful quantity, such as the speed of light, c, had changed. And, indeed, the Tompkins concept becomes meaningless in our existence 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 c⁄2, (but with all dimensionless physical quantities continuing to remain constant), then the Planck length would increase by a factor of √(8) from the point-of-view of some unaffected "god-like" observer on the outside. But then the size of atoms (approximately the Bohr radius) are related to the Planck length by an unchanging dimensionless constant:

a_{0}={{4\pi \epsilon _{0}\hbar ^{2}} \over {m_{e}e^{2}}}={{m_{P}} \over {m_{e}\alpha }}l_{P}

Then atoms would be bigger (in one dimension) by √(8), each of us would be taller by √(8), and so would our meter sticks be taller (and wider and thicker) by a factor of √(8) and we would not know the difference. Our perception of distance and lengths relative to the Planck length is logically an unchanging dimensionless constant.

Moreover, our clocks would tick slower by a factor of √(32) (from the point-of-view of this unaffected "god-like" observer) because the Planck time has increased by √(32) 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 god-like observer on the outside might observe that light now travels at half the speed that it used to (as well as all other observed velocities) but it would still travel 299792458 of our new meters in the time elapsed by one of our new seconds (c⁄2 √(32)⁄√(8) continues to equal 299792458 m/s). We would not notice any difference.

In one sense, this contradicts what George Gamow wrote in his book Mr. Tompkins; where he suggested that if a dimension-dependent universal constant such as c changed, 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. (The only exception is the kilogram.) Gamow does not address this subtlety; the thought experiments he conducts in his popular works tacitly assume that (2) defines "changing a physical constant."

Footnotes

^ Planck Length, NIST 2006 CODATA
^ Planck, M. (1899) 'Über irreversible Strahlungsvorgänge' (On irreversible radiative processes), Sitzungsberichte der Preußischen Akademie der Wissenschaften 5: 479.
^ Comment on time-variation of fundamental constants July 11, 2004

References

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.

External links

The NIST website(National Institute of Standards and Technology) is a convenient source of data on the commonly recognized constants, including Planck units.
Planck's original paper Planck, M., "Ueber irreversible Strahlungsvorgaenge", Sitzungsbericht Deutsche Akad. Wiss. Berlin, Math-Phys Tech. Kl., 440 (1899)
K.A. Tomilin: NATURAL SYSTEMS OF UNITS; To the Centenary Anniversary of the Planck System An overview/tutorial of various systems of natural units with some emphasis on Planck units and comparing all systems.

Template:Planckunits

[1] Planck Length, NIST 2006 CODATA

[2] Planck, M. (1899) 'Über irreversible Strahlungsvorgänge' (On irreversible radiative processes), Sitzungsberichte der Preußischen Akademie der Wissenschaften 5: 479.

[3] Comment on time-variation of fundamental constants July 11, 2004

[1]

[2]

[3]