Planck units

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Planck units are units of measurement named after the German physicist Max Planck, who first proposed them in 1899. They are an example of natural units, i.e. units of measurement designed so that certain fundamental physical constants are normalized to 1. In Planck units, the constants thus normalized are:

the gravitational constant, G;
Dirac's constant or reduced Planck's constant, $\hbar$ ;
the speed of light in a vacuum, c;
the Coulomb force constant, $\frac{1}{4 \pi \epsilon_0} \text{;}$
Boltzmann's constant, k_B (or simply k).

Each of these constants can be associated with at least one fundamental physical theory: c with special relativity, G with general relativity and Newtonian gravity, $\hbar$ with quantum physics, ε₀ with electrostatics, and k with statistical mechanics and thermodynamics. Planck units have profound significance for theoretical physics since they elegantly simplify several recurring algebraic expressions of physical law. They are particularly relevant in research on unified theories such as quantum gravity.

1 Base Planck units
2 Derived Planck units
3 Planck units simplify the key equations of physics
4 Alternative normalizations
5 Uncertainties in values
6 Discussion
- 6.1 History
- 6.2 Planck units and the invariant scaling of nature
7 See also
8 Footnotes
9 References
10 External links

[edit] Base Planck units

All systems of measurement feature "base units". (In the SI system, for example, the base unit of length is the meter.) In the system of Planck units, the Planck base unit of length is known simply as the Planck length, the base unit of time is the Planck time, etc. These units are derived from the five fundamental physical constants in Table 1, which are arranged in Table 2 so as to cancel out the unwanted dimensions, leaving only the dimension appropriate to each unit. (Like all systems of natural units, Planck units are an instance of dimensional analysis.)

Table 1: Fundamental physical constants

Constant	Symbol	Dimension	Value in SI units with uncertainties^[1]
Speed of light in vacuum	$c$	L T ⁻¹	299 792 458 m/s
Gravitational constant	$G$	L³ M⁻¹ T ⁻²	6.674 28(67) × 10⁻¹¹ m³ kg⁻¹ s⁻²
Dirac's constant or "reduced Planck's constant"	$\hbar=\frac{h}{2 \pi}$ where $h$ is Planck's constant	L² M T ⁻¹	1.054 571 628(53) × 10⁻³⁴ J s
Coulomb force constant	$\frac{1}{4 \pi \epsilon_0}$ where $ε 0$ is the permittivity of free space	L³ M T ⁻² Q⁻²	8 987 551 787.368 1764 N m² C⁻²
Boltzmann constant	$k$	L² M T ⁻² Θ⁻¹	1.380 6504(24) × 10⁻²³ J K⁻¹

Key: L = length, T = time, M = mass, Q = electric charge, Θ = temperature. The values given without uncertainties are exact due to the definitions of the metre and the ampere.

Table 2: Base Planck units

Name	Dimension	Expressions	SI equivalent with uncertainties^[1]	Other equivalent
Planck length	Length (L)	$l_\text{P} = \sqrt{\frac{\hbar G}{c^3}}$	1.616 252(81) × 10⁻³⁵ m
Planck mass	Mass (M)	$m_\text{P} = \sqrt{\frac{\hbar c}{G}}$	2.176 44(11) × 10⁻⁸ kg	1.220 862(61) × 10¹⁹ Gev/c²
Planck time	Time (T)	$t_\text{P} = \frac{l_\text{P}}{c} = \frac{\hbar}{m_\text{P}c^2} = \sqrt{\frac{\hbar G}{c^5}}$	5.391 24(27) × 10⁻⁴⁴ s
Planck charge	Electric charge (Q)	$q_\text{P} = m_\text{P} 2 \pi \sqrt{G \epsilon_0} = \sqrt{\hbar c 4 \pi \epsilon_0}$	1.875 545 870(47) × 10⁻¹⁸ C	11.706 237 6398(40) e
Planck temperature	Temperature (Θ)	$T_\text{P} = \frac{m_\text{P} c^2}{k} = \sqrt{\frac{\hbar c^5}{G k^2}}$	1.416 785(71) × 10³² K

By setting to unity the five fundamental constants in Table 1, the base units of length, mass, time, charge, and temperature shown in Table 2 also acquire the value unity. This may be expressed in non-dimensional terms as follows:

$\text{If }G = c = \hbar = \frac{1}{4 \pi \epsilon_0} = k = 1 \text{, then } l_\text{P} = m_\text{P} = t_\text{P} = q_\text{P} = T_\text{P} = 1\text{.}$

[edit] Derived Planck units

In any system of measurement, units for many physical quantities can be derived from base units. Table 3 offers a random 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 (see Discussion and Uncertainties in values below).

Table 3: Derived Planck units

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 (L⁻³M)	$\rho_P = \frac{m_P}{l_P^3} = \frac{\hbar t_P}{l_P^5} = \frac{c^5}{\hbar G^2}$	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)	$V_P = \frac{E_P}{q_P} = \frac{\hbar}{t_P q_P} = \sqrt{\frac{c^4}{G 4 \pi \epsilon_0} }$	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

[edit] Planck units simplify the key equations of physics

Ordinarily, 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). However, in theoretical physics this scruple can be set aside in order to simplify calculations. The process by which this is done is called Nondimensionalization. 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: Nondimensionalized equations

	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 = m c^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} k T } \$	${ 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 \$ $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}$ $\nabla \times \mathbf{B} = 4 \pi \mathbf{J} + \frac{\partial \mathbf{E}} {\partial t}$

[edit] Alternative normalizations

As already stated in the introduction, 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 4π, and multiples of it such as 8π, are ubiquitous in formulas in theoretical physics because it is the surface area of the unit sphere in three dimensions. 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, for example, reflects the fact that the flux of electric field distributes uniformly on the surface of the sphere. If space had more dimensions, the factor corresponding to 4π would be different.

In any event, a fundamental choice that has to be made when designing a system of natural units is which, if any, instances of 4nπ appearing in the equations of physics are to be eliminated via normalization.

Setting ε₀ = 1.

Planck normalized to 1 the Coulomb force constant 1/(4πε₀) (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 would make Z_P equal to Z₀, but would also eliminate 4π from Maxwell's equations. On the other hand, the nondimensionalized form of Coulomb's law would now include a factor of 1/(4π).

Setting 4nπG = 1.

In 1899, general relativity lay some years in the future, so that Newton's law of universal gravitation was still seen as fundamental, rather than as a convenient approximation holding for "small" velocities and distances. Hence Planck normalized to 1 the gravitational constant G in Newton's law. In theories emerging after 1899, G is nearly always multiplied by 4π or multiples.

4πG appears in the:
- Gauss's law for gravity, Φ_g = −4πGM;
- Characteristic impedance of gravitational radiation in free space, Z₀ = 4πG/c. ^[2] The c in the denominator stems from the general relativity prediction that gravitational radiation propagates at the same speed 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 density replacing charge density, and with 1/(4πG) replacing 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 Planck mass is divided by $\sqrt{8\pi}\text{.}$
Setting 16πG = 1 would eliminate the constant k = c⁴/(16πG) from the Einstein-Hilbert action. The Einstein field equations with cosmological constant Λ becomes R_μν − Λg_μν = (Rg_μν − T_μν)/2.

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

Setting 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.

[edit] Uncertainties in values

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

Today the value of the speed of light c in SI units is not subject to measurement error, because the SI base unit of length, the metre, is now defined as the the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second. Hence the value of c is now exact by definition, and contributes no uncertainty to the SI equivalents of the Planck units. The same is true of the value of the vacuum permittivity ε₀, due to the definition of ampere which sets the vacuum permeability μ₀ to 4π × 10⁻⁷ H/m and the fact that μ₀ε₀ = 1/c². The numerical value of Dirac's constant ℏ has been determined experimentally to 50 parts per billion, while that of G has been determined experimentally to no better than 1 part in 10000.^[1] G appears in the definition of almost every Planck unit 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. This is indeed the case; according to CODATA, the experimental values of the SI equivalents of the base Planck units are known to about 1 part in 20,000.)

[edit] Discussion

Physicists sometimes humorously refer to Planck units as "God's units", as Planck units are free of arbitrary anthropocentricity. Thus, while the meter and second are incorporated in the SI system for historical reasons, the Planck length and Planck time are conceptually linked at a fundamental physical level.

Some Planck units are suitable for measuring quantities that are familiar from daily experience. For example:

1 Planck mass is about 22 micrograms;
1 Planck momentum is about 6.5 kg m/s;
1 Planck energy is about 500 kWh;
1 Planck charge is slightly more than 11 elementary charges;
1 Planck impedance is very nearly 30 ohms.

However, most Planck units are many orders of magnitude too large or too small to be of any empirical and practical use, so that Planck units as a system are really only relevant to theoretical physics. In fact, 1 Planck unit is often the largest or smallest value of a physical quantity that makes sense given the current state of physical theory. For example:

a speed of 1 Planck length per Planck time is the speed of light in a vacuum, the maximum possible velocity in special relativity;
our understanding of the Big Bang begins with the Planck Epoch, when the universe grew older and larger than about 1 Planck time and 1 Planck length, at which time it cooled below about 1 Planck temperature and quantum theory as presently understood becomes applicable. Understanding the universe when it was less than 1 Planck time old requires a theory of quantum gravity, incorporating quantum effects into general relativity. Such a theory does not yet exist;
at a Planck temperature of 1, all symmetries broken since the early Big Bang would be restored, and the four fundamental forces of contemporary physical theory would become one force.

Relative to the Planck Epoch, the universe today looks extreme when expressed in Planck units, as in this set of approximations (see for example):^[3] and^[4]

Table 5: Today's universe in Planck units

Feature of present-day universe	Approximate number of Planck units
Age	8.0 × 10⁶⁰ t_P (4.3 × 10¹⁷ seconds)
Diameter	5.4 x 10⁶¹ l_P (8.7 × 10²⁶ meters)
Mass	Roughly 10⁶⁰ m_P (3 × 10⁵² kilograms, only counting stars); 10⁸⁰ protons (sometimes known as the Eddington number)
Temperature	1.9 × 10⁻³² T_P (temperature of the cosmic microwave background radiation, 2.725 kelvins)

Natural units can help physicists reframe questions. An example of such reframing is the following passage by Frank Wilczek:

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 √N [where N is Gm_p²/ħc ≈ 3 × 10⁻³⁹].

– June 2001 Physics Today

[edit] History

Natural units began in 1881, when George Johnstone Stoney 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. (Stoney was also the first to hypothesize that electric charge is quantized and hence to see the fundamental character of e.) Max Planck first set out the base units (q_P excepted) later named in his honor, in a paper presented to the Prussian Academy of Sciences in May 1899.^[5]^[6]That paper also includes the first appearance of a constant named b, and later called h and named after him. The paper gave numerical values for the base units, in terms of the metric system of his day, that were remarkably close to those in Table 2. We are not sure just how Planck came to discover these units because his paper gave no algebraic details. But he did explain why he valued these units as follows:

...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"...

[edit] Planck units and the invariant scaling of nature

Dirac (1937), and others after him, 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

When measuring a length with a ruler or tape measure, one is actually counting tick marks on a given standard, i.e., measuring the length relative to that given standard; the result is a dimensionless value. It is no different for physical experiments, as all physical quantities are measured relative to some other like-dimensioned values. 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 we would measure when observing nature or conducting experiments would be dimensionless numbers. See Duff (2004) and section III.5 (by Duff alone) of Duff, Okun, and Veneziano (2002).

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:

$a_0 = {{4\pi\epsilon_0\hbar^2}\over{m_e e^2}}= {{m_P}\over{m_e \alpha}} l_P$

Atoms would then 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, axiomatically, an unchanging dimensionless constant.) This hypothetical god-like outside observer 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.

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 stems from the ambiguity in the phrase "changing a physical constant"; what would happen depends on whether we hold constant all other (1) dimensionless constants, or (2) dimension-dependent constants. (2) is a somewhat confusing alternative, 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 a "changing physical constant."

According to doubly special relativity (a recent and unconventional development of relativity theory) the Planck length is an invariant, minimum length in the same way that the speed of light is an invariant, maximum velocity.

[edit] See also

[edit] Footnotes

^ ^a ^b ^c Fundamental Physical Constants from NIST
^ arXiv:0710.1378v4
^ *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.
^ Frank J. Tipler, 1986. The Anthropic Cosmological Principle. Oxford University Press. Harder.
^ Planck (1899), p. 479.
^ *Tomilin, K. A., 1999, "Natural Systems of Units: To the Centenary Anniversary of the Planck System," 287-96.

[edit] 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. Easier.
-------- and Frank J. Tipler, 1986. The Anthropic Cosmological Principle. Oxford University Press. Harder.
Michael Duff, 2004, "Comment on time-variation of fundamental constants."
--------, L. B. Okun, and Gabriele Veneziano, 2002, "Trialogue on the number of fundamental constants," Journal of High Energy Physics.
Max Planck, 1899, "Über Irreversible Strahlungsvorgänge," ("On irreversible radiative processes") Sitzungsbericht Deutsche Akad. Wiss. Berlin, Math-Phys Tech. Kl 5: 440-80. Pp. 478-80 contain the first appearance of the Planck base units other than the Planck charge, and of Planck's constant, which Planck denoted by b. a and f in this paper correspond to k and G in this entry.
Roger Penrose, 2004. The Road to Reality. Alfred A. Knopf. Section 31.1.
Tomilin, K. A., 1999, "Natural Systems of Units: To the Centenary Anniversary of the Planck System," 287-96.

[edit] External links

Value of the fundamental constants, including the Planck base units, as reported by the National Institute of Standards and Technology (NIST).
Sections C-E of collection of resources bear on Planck units. Good discussion of why 8πG should be normalized to 1 when doing general relativity and quantum gravity. Many links.

v • d • e Planck's natural units

Base Planck units	Planck time · Planck length · Planck mass · Planck charge · Planck temperature

Derived Planck units	Planck energy · Planck force · Planck power · Planck density · Planck angular frequency · Planck pressure · Planck current · Planck voltage · Planck impedance · Planck momentum

v • d • e Systems of measurement

Metric systems	International System of Units · centimetre-gram-second · metre-tonne-second · Gravitational units

Natural units	Geometric systems · Planck · Stoney · "Schrödinger" · Atomic · Electronic · Quantum electrodynamical

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