Planck units

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Planck units, first proposed by Max Planck in 1899 and sometimes called natural units, are a system of units for measuring time, length, mass, electric charge, and temperature. Given Planck units, the numerical values of the five dimensionful universal physical constants in the table below all become 1, resulting in a material simplification of the notation of physics and of many equations physics employs. Planck units are especially popular in general relativity and quantum gravity.


Constant Symbol Dimension Expressed in current dimensions
speed of light in vacuum <math>{ c } \ </math> L T-1 speed
Gravitational constant <math>{ G } \ </math> M-1L3T-2 (volume / mass) / time2
"reduced Planck's constant" or Dirac's constant <math>\hbar=\frac{h}{2 \pi}</math> where <math>{h} \ </math> is Planck's constant ML2T-1 (mass * area) / time
Coulomb force constant <math> \frac{1}{4 \pi \epsilon_0} </math> where <math>{ \epsilon_0 } \ </math> is the permittivity in vacuum Q-2 M L3 T-2 (mass * volume) / (charge2 * time2)
Boltzmann constant <math>{ k } \ </math> ML2T-2Θ-1 (mass * area) / (temperature * time2)



Contents

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Base Planck units

Normalizing to 1 the numerical values of the five fundamental constants above defines the following five base units for time, length, mass, charge, and temperature.

Name Dimension Expression Approximate SI equivalent
Planck time Time (T) <math>t_P = \frac{l_P}{c} = \frac{\hbar}{m_Pc^2} = \sqrt{\frac{\hbar G}{c^5}} </math> 5.39121 × 10-44 s
Planck length Length (L) <math> l_P = \sqrt{\frac{\hbar G}{c^3}}</math> 1.61624 × 10-35 m
Planck mass Mass (M) <math>m_P = \sqrt{\frac{\hbar c}{G}}</math> 2.17645 × 10-8 kg
Planck charge Electric charge (Q) <math>q_P = \sqrt{\hbar c 4 \pi \epsilon_0} </math> 1.8755459 × 10-18 C
Planck temperature Temperature (Θ) <math>T_P = \frac{m_P c^2}{k} = \sqrt{\frac{\hbar c^5}{G k^2}}</math> 1.41679 × 1032 K


Planck units are defined only up to a factor of 2,4, or 8 times π, as one could normalize to 1:

Planck units are too small or too large for practical use, unless rescaled by many orders of magnitude. The SI equivalents of these units also suffer from uncertainties in the measurement of some of the fundamental constants on which they are based, especially of the gravitational constant G, which has an uncertainty of 1 in 7000.

Planck did not define or propose the Planck charge. Rather, it is a natural extension of the definitions of the other Planck units. Note that the elementary charge e, measured in terms of the Planck charge, is:

<math>e = \sqrt{\alpha} \ q_P = 0.085424543 \ q_P \ </math>

where <math> {\alpha} \ </math> is the fine-structure constant

<math> \alpha =\left ( \frac{e}{q_P} \right )^2 = \frac{e^2}{\hbar c 4 \pi \epsilon_0} = \frac{1}{137.03599911} </math> .


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Derived Planck units

From the base Planck units we can define the following derived units.

Name Dimension Expression Approximate SI equivalent
Planck energy Energy (ML2T-2) <math>E_P = m_P c^2 = \sqrt{\frac{\hbar c^5}{G}} </math> 1.9561 × 109 J
Planck force Force (MLT-2) <math>F_P = \frac{E_P}{l_P} = \frac{c^4}{G} </math> 1.21027 × 1044 N
Planck power Power (ML2T-3) <math>P_P = \frac{E_P}{t_P} = \frac{c^5}{G} </math> 3.62831 × 1052 W
Planck density Density (ML-3) <math>\rho_P = \frac{m_P}{l_P^3} = \frac{c^5}{\hbar G^2} </math> 5.15500 × 1096 kg/m3
Planck angular frequency Frequency (T-1) <math>\omega_P = \frac{1}{t_P} = \sqrt{\frac{c^5}{\hbar G}} </math> 1.85487 × 1043 s-1
Planck pressure Pressure (ML-1T-2) <math>p_P = \frac{F_P}{l_P^2} =\frac{c^7}{\hbar G^2} </math> 4.63309 × 10113 Pa
Planck current Electric current (QT-1) <math>I_P = \frac{q_P}{t_P} = \sqrt{\frac{c^6 4 \pi \epsilon_0}{G}} </math> 3.4789 × 1025 A
Planck voltage Voltage (ML2T-2Q-1) <math>V_P = \frac{E_P}{q_P} = \sqrt{\frac{c^4}{G 4 \pi \epsilon_0} } </math> 1.04295 × 1027 V
Planck impedance Resistance (ML2T-1Q-2) <math>Z_P = \frac{V_P}{I_P} = \frac{1}{4 \pi \epsilon_0 c} = \frac{Z_0}{4 \pi} </math> 2.99792458 × 101 Ω
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Planck units simplify certain fundamental equations

Usual form with dimensional conversion factors Nondimensionalized form
Newton's Law of universal gravitation <math> F = G \frac{m_1 m_2}{r^2} </math> <math> F = \frac{m_1 m_2}{r^2} </math>
Einstein's field equation for general relativity <math>{ G_{\mu \nu} = 8 \pi {G \over c^4} T_{\mu \nu}} \ </math> <math>{ G_{\mu \nu} = 8 \pi T_{\mu \nu} } \ </math>
Particle energy equals radian frequency <math> \omega </math> of wave function <math>{ E = \hbar \omega } \ </math> <math>{ E = \omega } \ </math>
Einstein's equivalence of mass and energy <math>{ E = m c^2} \ </math> <math>{ E = m } \ </math>
Thermal energy per particle per degree of freedom <math>{ E = \frac{1}{2} k T } \ </math> <math>{ E = \frac{1}{2} T } \ </math>
Coulomb's law <math> F = \frac{1}{4 \pi \epsilon_0} \frac{q_1 q_2}{r^2} </math> <math> F = \frac{q_1 q_2}{r^2} </math>
Maxwell's equations <math>\nabla \cdot \mathbf{E} = \frac{1}{\epsilon_0}\rho</math>

<math>\nabla \cdot \mathbf{B} = 0 \ </math>
<math>\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}</math>
<math>\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}} {\partial t}</math>

<math>\nabla \cdot \mathbf{E} = 4 \pi \rho \ </math>

<math>\nabla \cdot \mathbf{B} = 0 \ </math>
<math>\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}</math>
<math>\nabla \times \mathbf{B} = 4 \pi \mathbf{J} + \frac{\partial \mathbf{E}} {\partial t}</math>

N.B. Planck units normalize the Coulomb force constant (4πε0)-1 rather than the permittivity of free space ε0. Hence the factor 4π above.

Schrödinger's equation <math>

- \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) </math>

<math>

- \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) </math>


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Discussion

Physicists have been known to refer to Planck units in jest as "God's units". Planck units eliminate anthropocentric arbitrariness from the system of units. An extra-terrestrial intelligent life form would arguably use, or at least recognize, such units.

Natural units can help physicists reframe questions. In this regard, Frank Wilczek has written:

...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)]...(June 2001 Physics Today)

The strengths of gravity and of the electromagnetic force are both simply what they are. The electromagnetic force operates on electric charge, so it cannot be compared directly to gravity, which operates on mass. To note that gravity is an extremely weak force is, from the point of view of natural units, like comparing apples to oranges. 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, because the charge of the proton is approximately a natural unit of charge, while the mass of the proton is far less than the natural unit of mass.

At the "Planck scales" in length, time, density, or temperature, one must consider both the effects of quantum mechanics and general relativity. Unfortunately this requires a theory of quantum gravity which does not as yet exist.


The dimensionless fine structure constant can be seen as taking on the value it does because of the amount of charge, measured in natural units (Planck charge), that nature has assigned to electrons, protons, and other charged particles. Because the electromagnetic force between two particles is proportional to the product of the charges of each particle (each which would, in Planck units, be proportional to <math> \sqrt{\alpha} \ </math>), the strength of the electromagnetic force relative to other fundamental forces is proportional to <math> {\alpha} \ </math>.

The Planck impedance is the characteristic impedance of free space Z0 scaled down by 4π, meaning that in terms of Planck units, Z0 = 4πZP. The factor 4π stems from the fact that Planck units, like cgs units, normalize to 1 the Coulomb force constant (4πε0)-1 in Coulomb's law, rather than the permittivity of free space ε0. This choice, as well as that of normalizing the gravitational constant G--rather than 4πG, 8πG, or 16πG--to 1, is an arbitrary one, and one also perhaps less than ideal if the goal is to eliminate from the equations of physics those instances of π lacking evident geometric motivation.


An increasingly common convention in the literatures on particle physics and cosmology is reduced Planck units, whereby 8πG = 1, which "reduce" the Planck mass by <math> \sqrt{8 \pi} \ </math>. These units remove the factor 8π from the Einstein field equation, Einstein-Hilbert action, Friedmann equations, and the Poisson equation for gravitation, at the expense of introducing such a factor into Newton's law of universal gravitation. The normalization 16πG = 1 sets the coefficient of R in the Einstein-Hilbert action to unity.


Yet another convention sets 4πG = 1 so that the dimensionful constants in the gravitoelectromagnetic (GEM) counterparts to Maxwell's equations are eliminated. The GEM equations take the same form as Maxwell's equations (and the Lorentz force equation) of electromagnetic interaction, with mass (or mass density) replacing charge (or charge density) and (4πG)-1 replacing the permittivity ε0. The GEM equations hold in weak gravitational fields or reasonably flat space-time. The normalization 4πG = 1 also sets the characteristic impedance of free space, Z0 = 4πG/c, to unity. Note that the velocity of propagation of gravitational is the same as that of electromagnetic radiation, namely c.

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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. A few such questions that are relevant here might be: 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 measureable difference?

Barrow (2002) wrote:

"[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."

According to Michael Duff (2004), and Duff, Okun, and Veneziano (2002) (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, one is actually counting tick marks on a given standard or is measuring the length relative to that given standard, a dimensionless value. The same holds 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 (atomic structures would change) but if all dimensionless physical quantities remained constant (this includes all possible ratios of identically dimensioned physical quantity), we could not tell if a dimensionful quantity, such as the speed of light, c, has changed. Indeed, the Tompkins concept becomes meaningless in our existence if a dimensionful 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 <math> \sqrt{8} </math> from the point-of-view of some unaffected "god-like" observer on the outside. But then the sizes of atoms (approximately the Bohr radius) are related to the Planck length by an unchanging dimensionless constant:

<math>a_0 = {{4\pi\epsilon_0\hbar^2}\over{m_e e^2}}= {{m_P}\over{m_e \alpha}} l_P </math>

Then atoms would be bigger (in one dimension) by <math> \sqrt{8} </math>, each of us would be taller by <math> \sqrt{8} </math>, and so would our meter sticks be taller (and wider and thicker) by a factor of <math> \sqrt{8} </math> and we would not know the difference. 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 <math> \sqrt{32} </math> (from the point-of-view of this unaffected "god-like" observer) because the Planck time has increased by <math> \sqrt{32} </math> 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. We would not notice any difference.

This in one sense contradicts George Gamow in Mr. Tompkins who suggests that if a dimensionful universal constant such as c changed, we would easily notice the difference; however, as noted, the disagreement is better thought of as the ambiguity in the phrase "changing a physical constant", when one does not specify whether one does so keeping all other dimensionless constants the same, or does so keeping all other dimensionful constants the same. The latter is a somewhat confusing possibility since most of our unit definitions are related to the outcomes of physical experiments which themselves depend on the constants, the only exception being the kilogram. Gamow does not address this subtlety; the thought experiments he conducts in his popular works assume the latter.

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Planck's discovery of the natural units

The Planck units, along with the numerical values of their cgs equivalents remarkably close to their present-day values, first appeared in a paper Max Planck presented to the Prussian Academy of Sciences in May 1899. At that time, quantum mechanics had yet to be invented. Nor had he discovered the theory of black-body radiation (first published December 1900) in which Planck's constant <math> {h} \ </math> made its first appearance, and for which he was later awarded the Nobel prize in 1918. Planck's 1899 paper does not clearly reveal how he could have devised his units before such things as <math> \hbar \ </math> and quantum physics were known. Planck (1899: 479) wrote:

...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"...

Today we define these units using Dirac's constant <math> \hbar \ </math>, and invoke quantum physics to motivate them.

In 1881, George Stoney proposed a different set of natural units based on G, c, and e, the charge of the electron; see Barrow (2002).

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See also

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References

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External links


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