# Planck units

The Planck units form a system of natural units for the physical quantities .

They are calculated directly as products and quotients of the fundamental natural constants from:

• Gravitational constant ${\ displaystyle \ textstyle G}$
• Speed ​​of Light ${\ displaystyle \ textstyle c}$
• reduced Planck's quantum of action ${\ displaystyle \ textstyle \ hbar}$
• Boltzmann's constant ${\ displaystyle \ textstyle k _ {\ mathrm {B}}}$
• Coulomb's constant (with the electrical permittivity of the vacuum ).${\ displaystyle k _ {\ mathrm {C}} = {\ frac {1} {4 \ pi \ textstyle \ varepsilon _ {0}}}}$${\ displaystyle \ textstyle \ varepsilon _ {0}}$

Expressed in Planck units, these natural constants (or certain conventional multiples of these) therefore all have the numerical value 1. In this system of units, many calculations are numerically simpler. The Planck units are named after Max Planck , who noticed in 1899 that with his discovery of the quantum of action, enough fundamental natural constants were now known to define universal units for length, time, mass and temperature.

The importance of the Planck units is, on the one hand, that the Planck units mark minimum limits (e.g. for length and time ) up to which we can differentiate between cause and effect . This means that beyond this limit the previously known physical laws are no longer applicable, e.g. B. in the theoretical elucidation of the processes shortly after the Big Bang (see Planck scale ).

On the other hand, as Planck put it, the Planck units “independently of special bodies or substances retain their meaning for all times and for all, including extraterrestrial and extraterrestrial cultures, and [...] can therefore be called“ natural units of measurement ” ", D. In other words , our natural laws are universally applicable, understandable and communicable in the cosmos down to the Planck units .

## Definitions

### Basic sizes

The Planck units result from a simple dimension analysis . They result as mathematical expressions of the dimension of a length, time or mass, which only contain products and quotients of suitable powers of , and . If the electrical permittivity of the vacuum and the Boltzmann constant are also used, a Planck charge and a Planck temperature can also be determined as further basic quantities. The Planck charge fulfills the condition that the force of gravity between two Planck masses and the electromagnetic force between two Planck charges are equally strong: . ${\ displaystyle G}$${\ displaystyle c}$${\ displaystyle \ hbar}$ ${\ displaystyle \ varepsilon _ {0}}$ ${\ displaystyle k _ {\ mathrm {B}}}$${\ displaystyle m _ {\ mathrm {P}} ^ {2} G / l _ {\ mathrm {P}} ^ {2} = q _ {\ mathrm {P}} ^ {2} / 4 \ pi \ varepsilon _ { 0} l _ {\ mathrm {P}} ^ {2}}$

Surname size dimension term Value in SI units Value in other units
Planck mass Dimensions M. ${\ displaystyle m _ {\ mathrm {P}} = {\ sqrt {\ frac {\ hbar \, c} {G}}}}$ 2.176 434 (24) · 10 −8 kg 1.311 · 10 19 u , 1.221 · 10 19 GeV / c 2
Planck length length L. ${\ displaystyle l _ {\ mathrm {P}} = {\ sqrt {\ frac {\ hbar \, G} {c ^ {3}}}}}$ 1.616 255 (18) · 10 −35 m 3.054 · 10 −25 a 0
Planck time time T ${\ displaystyle \! \, t _ {\ mathrm {P}} = {\ frac {l _ {\ mathrm {P}}} {c}}}$ 5.391 247 (60) · 10 −44 s
Planck charge charge IT ${\ displaystyle q _ {\ mathrm {P}} = {\ sqrt {\ hbar \, c \, 4 \, \ pi \, \ varepsilon _ {0}}}}$ 1.876 · 10 −18 C 11.71 e
Planck temperature temperature Θ ${\ displaystyle \! \, T _ {\ mathrm {P}} = {\ frac {m _ {\ mathrm {P}} \, c ^ {2}} {k _ {\ mathrm {B}}}}}$ 1.416 784 (16) * 10 32 K

The symbols mean:

Instead of sometimes setting to one, the unit of mass is the reduced Planck mass: ${\ displaystyle \, G}$${\ displaystyle \, 8 \ pi G}$

${\ displaystyle {\ overline {m _ {\ mathrm {P}}}} = {\ sqrt {\ frac {\ hbar c} {8 \ pi G}}} \ approx 4 {,} 340 \, \ mu \ mathrm {g}}$.

With the definition of a correspondingly reduced Planck charge , the above remains. Maintain equality of forces. ${\ displaystyle {\ overline {q _ {\ mathrm {P}}}} = {\ sqrt {\ frac {\ hbar \, c \, \ varepsilon _ {0}} {2}}}}$

### Derived quantities

In addition to these five basic quantities, the following derived quantities are also used:

Surname size dimension term Value in SI units
Planck surface surface L 2 ${\ displaystyle l _ {\ mathrm {P}} ^ {2} = {\ frac {\ hbar G} {c ^ {3}}}}$ 2.612 · 10 −70 m 2
Planck volume volume L 3 ${\ displaystyle l _ {\ mathrm {P}} ^ {3} = {\ sqrt {\ frac {\ hbar G} {c ^ {3}}}} ^ {\, 3}}$ 4.222 · 10 −105 m 3
Planck energy energy ML 2 T −2 ${\ displaystyle E _ {\ mathrm {P}} = m _ {\ mathrm {P}} c ^ {2} = {\ frac {\ hbar} {t _ {\ mathrm {P}}}} = {\ sqrt {\ frac {\ hbar c ^ {5}} {G}}}}$ 1.956 · 10 9 J
= 1.2209 · 10 28 eV
= 543.4 kWh
Planck impulse pulse MLT −1 ${\ displaystyle m _ {\ mathrm {P}} c = {\ frac {\ hbar} {l _ {\ mathrm {P}}}} = {\ sqrt {\ frac {\ hbar c ^ {3}} {G} }}}$ 6.525 kg m s −1
Planck force force MLT −2 ${\ displaystyle F _ {\ mathrm {P}} = {\ frac {E _ {\ mathrm {P}}} {l _ {\ mathrm {P}}}} = {\ frac {\ hbar} {l _ {\ mathrm { P}} t _ {\ mathrm {P}}}} = {\ frac {c ^ {4}} {G}}}$ 1.210 x 10 44 N
Planck performance power ML 2 T −3 ${\ displaystyle P _ {\ mathrm {P}} = {\ frac {E _ {\ mathrm {P}}} {t _ {\ mathrm {P}}}} = {\ frac {\ hbar} {t _ {\ mathrm { P}} ^ {2}}} = {\ frac {c ^ {5}} {G}}}$ 3.628 · 10 52 W.
Planck density density ML −3 ${\ displaystyle \ rho _ {\ mathrm {P}} = {\ frac {m _ {\ mathrm {P}}} {l _ {\ mathrm {P}} ^ {3}}} = {\ frac {\ hbar t_ {\ mathrm {P}}} {l _ {\ mathrm {P}} ^ {5}}} = {\ frac {c ^ {5}} {\ hbar G ^ {2}}}}$ 5.155 · 10 96 kg · m −3
Planck angular frequency Angular frequency T −1 ${\ displaystyle \ omega _ {\ mathrm {P}} = {\ frac {1} {t _ {\ mathrm {P}}}} = {\ sqrt {\ frac {c ^ {5}} {\ hbar G} }}}$ 1.855 · 10 43 s −1
Planck print pressure ML −1 T −2 ${\ displaystyle p _ {\ mathrm {P}} = {\ frac {F _ {\ mathrm {P}}} {l _ {\ mathrm {P}} ^ {2}}} = {\ frac {\ hbar} {l_ {\ mathrm {P}} ^ {3} t _ {\ mathrm {P}}}} = {\ frac {c ^ {7}} {\ hbar G ^ {2}}}}$ 4.633 · 10 113 Pa
Planck current Electrical current QT −1 ${\ displaystyle I _ {\ mathrm {P}} = {\ frac {q _ {\ mathrm {P}}} {t _ {\ mathrm {P}}}} = {\ sqrt {\ frac {c ^ {6} 4 \ pi \ varepsilon _ {0}} {G}}}}$ 3.479 x 10 25 A
Planck tension Electric voltage ML 2 T −2 Q −1 ${\ displaystyle U _ {\ mathrm {P}} = {\ frac {E _ {\ mathrm {P}}} {q _ {\ mathrm {P}}}} = {\ frac {\ hbar} {t _ {\ mathrm { P}} q _ {\ mathrm {P}}}} = {\ sqrt {\ frac {c ^ {4}} {G4 \ pi \ varepsilon _ {0}}}}}$ 1.043 x 10 27 V.
Planck impedance resistance ML 2 T −1 Q −2 ${\ displaystyle Z _ {\ mathrm {P}} = {\ frac {U _ {\ mathrm {P}}} {I _ {\ mathrm {P}}}} = {\ frac {\ hbar} {q _ {\ mathrm { P}} ^ {2}}} = {\ frac {1} {4 \ pi \ varepsilon _ {0} c}} = {\ frac {Z_ {0}} {4 \ pi}}}$ 29.98 Ω
Planck acceleration acceleration LT −2 ${\ displaystyle g _ {\ mathrm {P}} = {\ frac {F _ {\ mathrm {P}}} {m _ {\ mathrm {P}}}} = {\ sqrt {\ frac {c ^ {7}} {\ hbar G}}}}$ 5.56 · 10 51 m · s −2
Planck magnetic field Magnetic flux density MQ −1 T −1 ${\ displaystyle B _ {\ mathrm {P}} = {\ sqrt {{\ frac {\ mu _ {0}} {4 \ pi}} p _ {\ mathrm {P}}}} = {\ sqrt {\ frac {c ^ {5}} {\ hbar G ^ {2} 4 \ pi \ varepsilon _ {0}}}}}$ 2.1526 · 10 53 T
Planck magnetic flux Magnetic river ML 2 T −1 Q −1 ${\ displaystyle \ phi _ {\ mathrm {P}} = {\ frac {E _ {\ mathrm {P}}} {I _ {\ mathrm {P}}}} = {\ sqrt {\ frac {\ hbar} { 4 \ pi \ varepsilon _ {0} c}}}}$ 5.6227 · 10 −17 Wb

The Planck unit for the angular momentum results from the product of Planck length and Planck momentum to the value . This is precisely the unit of angular momentum quantization known from quantum mechanics . ${\ displaystyle \ hbar}$

The Planck surface plays an important role in string theories and in considerations of the entropy of black holes in connection with the holographic principle . ${\ displaystyle l _ {\ mathrm {P}} ^ {2}}$

## history

At the end of the 19th century, during his investigations into the theory of black body radiation , for which he received the Nobel Prize in Physics two decades later , Planck discovered the last natural constant required to define the Planck units, the quantum of action that was later named after him. He recognized the possibility of defining a universally valid system of units and mentioned this in a lecture “About irreversible radiation processes”. The following quote gives an impression of the importance Planck accorded these units

"... the possibility is given to set up units [...] which, regardless of special bodies or substances, necessarily retain their meaning for all times and for all, including extraterrestrial and extraterrestrial cultures, and which are therefore called 'natural units of measurement' can."

- Max Planck

Although Planck devoted a chapter (§ 159. Natural units of measurement) to this system of units in his book “Theory of Thermal Radiation”, published in 1906, and took up this topic again later, it was not used in physics either. The disadvantages that the value of the gravitational constant was not (and still is) known precisely enough for use in a system of measurement, and that practical quantities - expressed in its units - have absurd numerical values, were not offset by any advantage, as in no physical theory simultaneously the quantum of action and the constant of gravity appeared.

It was only after initial work on the unification of quantum theory and gravity in the late 1930s that the later field of application of the Planck units emerged. When John Archibald Wheeler and Oskar Klein published in 1955 about the Planck length as the limit of the applicability of general relativity, Planck's proposal was almost forgotten. After the "rediscovery" of Planck's proposals for such a system of measurements, the name Planck units became common from 1957 .

However, the Planck units commonly used today differ from Planck's original units, as the development of quantum mechanics has shown that the natural unit is more practical than the one chosen by Planck . ${\ displaystyle \ hbar = {\ frac {h} {2 \ pi}}}$${\ displaystyle h}$

## Today's meaning

If equations containing the fundamental constants , and are formulated in Planck units, the constants can be omitted. This simplifies the equations considerably in certain disciplines of theoretical physics , for example in general relativity, in quantum field theories and in the various approaches to quantum gravity . ${\ displaystyle G}$${\ displaystyle c}$${\ displaystyle \ hbar}$

The Planck units also allow an alternative view of the fundamental forces of nature, the strength of which is described in the International System of Units (SI) by very different coupling constants . When using the Planck units, the situation is as follows: Between two particles that have exactly the Planck mass and the Planck charge, the gravitational force and the electromagnetic force would be exactly the same. The different strength of these forces in our world is the result of the fact that a proton or an electron has a charge of about 0.085 Planck charges, while their masses are 19 or 22 orders of magnitude smaller than the Planck mass. The question: "Why is gravity so weak?" Is therefore equivalent to the question: "Why do the elementary particles have such low masses?"

Various physicists and cosmologists are concerned with the question of whether we could notice it if dimensional physical constants were to change slightly, and what the world would look like if there were major changes. Such speculations are u. a. employed at the speed of light and the gravitational constant , the latter in the expansion theory of the earth since around 1900 . The nuclear physicist George Gamow says in his popular science book Mr. Tompkins in Wonderland that a change in would result in significant changes. ${\ displaystyle c}$ ${\ displaystyle G}$${\ displaystyle c}$