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For the rationalized Planck units, see rationalized Planck units.

In particle physics and physical cosmology, Planck units are a set of units of measurement defined exclusively in terms of five universal physical constants, in such a manner that these five physical constants take on the numerical value of 1 when expressed in terms of these units.

Originally proposed in 1899 by German physicist Max Planck, these units are also known as natural units because the origin of their definition comes only from properties of nature and not from any human construct (e.g. luminous intensity (cd), luminous flux (lm), and equivalent dose (Sv)) nor any quality of earth or universe (e.g. standard gravity, standard atmosphere, and Hubble constant) nor any quality of a given substance (e.g. melting point of water, density of water, and specific heat capacity of water). Planck units are only one system of several systems of natural units, but Planck units are not based on properties of any prototype object or particle (e.g. elementary charge, electron rest mass, and proton rest mass) (that would be arbitrarily chosen), but rather on only the properties of free space (e.g. Planck speed is speed of light, Planck angular momentum is reduced Planck constant, Planck impedance is impedance of free space, Planck entropy is Boltzmann constant, all are properties of free space). Planck units have significance for theoretical physics since they simplify several recurring algebraic expressions of physical law by nondimensionalization. They are relevant in research on unified theories such as quantum gravity.

The term Planck scale refers to the magnitudes of space, time, energy and other units, below which (or beyond which) the predictions of the Standard Model, quantum field theory and general relativity are no longer reconcilable, and quantum effects of gravity are expected to dominate. This region may be characterized by energies around 5.52×10⁸ J (or 3.44×10²⁷ eV) or 1.96×10⁹ J (or 1.22×10²⁸ eV) (the Planck energy), time intervals around 1.91×10⁻⁴³ s or 5.39×10⁻⁴⁴ s (the Planck time) and lengths around 5.73×10⁻³⁵ m or 1.62×10⁻³⁵ m (the Planck length). At the Planck scale, current models are not expected to be a useful guide to the cosmos, and physicists have no scientific model to suggest how the physical universe behaves. The best known example is represented by the conditions in the first 10⁻⁴³ seconds of our universe after the Big Bang, approximately 13.8 billion years ago.

There are two versions of Planck units, Lorentz–Heaviside version (also called "rationalized") and Gaussian version (also called "non-rationalized").

The five universal constants that Planck units, by definition, normalize to 1 are:

• the speed of light in vacuum, c, (also known as Planck speed)
• the gravitational constant, G,
  • G for the Gaussian version, 4πG for the Lorentz–Heaviside version
• the reduced Planck constant, ħ, (also known as Planck action)
• the vacuum permittivity, ε₀ (also known as Planck permittivity)
  • ε₀ for the Lorentz–Heaviside version, 4πε₀ for the Gaussian version
• the Boltzmann constant, k_B (also known as Planck heat capacity)

Each of these constants can be associated with a fundamental physical theory or concept: c with special relativity, G with general relativity, ħ with quantum mechanics, ε₀ with electromagnetism, and k_B with the notion of temperature/energy (statistical mechanics and thermodynamics).

Introduction

Any system of measurement may be assigned a mutually independent set of base quantities and associated base units, from which all other quantities and units may be derived. In the International System of Units, for example, the SI base quantities include length with the associated unit of the metre. In the system of Planck units, a similar set of base quantities may be selected, and the Planck base unit of length is then known simply as the Planck length, the base unit of time is the Planck time, and so on. These units are derived from the five dimensional universal physical constants of Table 1, in such a manner that these constants are eliminated from fundamental selected equations of physical law when physical quantities are expressed in terms of Planck units. For example, Newton's law of universal gravitation,

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

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 are dimensionally consistent and equally valid in any system of units, but the second equation, with G missing, is relating only dimensionless quantities since any ratio of two like-dimensioned quantities is a dimensionless quantity. If, by a shorthand convention, it is understood that all physical quantities are expressed in terms of 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 (without G) is valid only if F, m₁, m₂, and r are the dimensionless numerical values of these quantities measured in terms of Planck units. This is why Planck units or any other use of natural units should be employed with care. Referring to G = c = 1, Paul S. Wesson wrote that, "Mathematically it is an acceptable trick which saves labour. Physically it represents a loss of information and can lead to confusion."

Definition

Table 1: Dimensional universal physical constants normalized with Planck units

Constant	Symbol	Dimension	Value (SI units)
Speed of light in vacuum	c	L T−1	299792458 m⋅s−1‍ (exact by definition of metre)
Gravitational constant	G (1 for the Gaussian version, ⁠1/4π⁠ for the Lorentz–Heaviside version)	L3 M−1 T−2	6.67430(15)×10−11 m3⋅kg−1⋅s−2‍
Reduced Planck constant	ħ = ⁠h/2π⁠ where h is the Planck constant	L2 M T−1	1.054571817...×10−34 J⋅s‍ (exact by definition of the kilogram since 20 May 2019)
Vacuum permittivity	ε0 (1 for the Lorentz–Heaviside version, ⁠1/4π⁠ for the Gaussian version)	L−3 M−1 T2 Q2	8.8541878188(14)×10−12 F⋅m−1‍ (exact by definitions of ampere and metre until 20 May 2019)
Boltzmann constant	kB	L2 M T−2 Θ−1	1.380649×10−23 J⋅K−1‍ (exact by definition of the kelvin since 20 May 2019)

Key: L = length, M = mass, T = time, Q = charge, Θ = temperature.

As can be seen above, the gravitational attractive force of two bodies of 1 Planck mass each, set apart by 1 Planck length is 1 Planck force in Gaussian version, or ⁠1/4π⁠ Planck force in Lorentz–Heaviside version. Likewise, the distance traveled by light during 1 Planck time is 1 Planck length. To determine, in terms of SI or another existing system of units, the quantitative values of the five base Planck units, those two equations and three others must be satisfied:

$l_{\text{P}}=c\ t_{\text{P}}$

$F_{\text{P}}={\frac {l_{\text{P}}m_{\text{P}}}{t_{\text{P}}^{2}}}=4\pi G\ {\frac {m_{\text{P}}^{2}}{l_{\text{P}}^{2}}}$ (Lorentz–Heaviside version)

$F_{\text{P}}={\frac {l_{\text{P}}m_{\text{P}}}{t_{\text{P}}^{2}}}=G\ {\frac {m_{\text{P}}^{2}}{l_{\text{P}}^{2}}}$ (Gaussian version)

$E_{\text{P}}={\frac {l_{\text{P}}^{2}m_{\text{P}}}{t_{\text{P}}^{2}}}=\hbar \ {\frac {1}{t_{\text{P}}}}$

$C_{\text{P}}={\frac {t_{\text{P}}^{2}q_{\text{P}}^{2}}{l_{\text{P}}^{2}m_{\text{P}}}}=\epsilon _{0}\ l_{\text{P}}$ (Lorentz–Heaviside version)

$C_{\text{P}}={\frac {t_{\text{P}}^{2}q_{\text{P}}^{2}}{l_{\text{P}}^{2}m_{\text{P}}}}=4\pi \epsilon _{0}\ l_{\text{P}}$ (Gaussian version)

$E_{\text{P}}={\frac {l_{\text{P}}^{2}m_{\text{P}}}{t_{\text{P}}^{2}}}=k_{\text{B}}\ T_{\text{P}}$

Solving the five equations above for the five unknowns results in a unique set of values for the five base Planck units:

Table 2: Base Planck units

Quantity	Expression		Approximate SI equivalent		Name
	Lorentz–Heaviside version	Gaussian version	Lorentz–Heaviside version	Gaussian version
Length (L)	$l_{\text{P}}={\sqrt {\frac {4\pi \hbar G}{c^{3}}}}$	$l_{\text{P}}={\sqrt {\frac {\hbar G}{c^{3}}}}$	5.72938×10−35 m	1.61623×10−35 m	Planck length
Mass (M)	$m_{\text{P}}={\sqrt {\frac {\hbar c}{4\pi G}}}$	$m_{\text{P}}={\sqrt {\frac {\hbar c}{G}}}$	6.13971×10−9 kg	2.17647×10−8 kg	Planck mass
Time (T)	$t_{\text{P}}={\sqrt {\frac {4\pi \hbar G}{c^{5}}}}$	$t_{\text{P}}={\sqrt {\frac {\hbar G}{c^{5}}}}$	1.91112×10−43 s	5.39116×10−44 s	Planck time
Charge (Q)	$q_{\text{P}}={\sqrt {\hbar c\epsilon _{0}}}$	$q_{\text{P}}={\sqrt {4\pi \hbar c\epsilon _{0}}}$	5.29082×10−19 C	1.87555×10−18 C	Planck charge
Temperature (Θ)	$T_{\text{P}}={\sqrt {\frac {\hbar c^{5}}{4\pi G{k_{\text{B}}}^{2}}}}$	$T_{\text{P}}={\sqrt {\frac {\hbar c^{5}}{G{k_{\text{B}}}^{2}}}}$	3.99674×1031 K	1.41681×1032 K	Planck temperature

Table 2 clearly defines Planck units in terms of the fundamental constants. Yet relative to other units of measurement such as SI, the values of the Planck units, other than the Planck charge, are only known approximately. This is due to uncertainty in the value of the gravitational constant G as measured relative to SI unit definitions.

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 length of the path travelled by light in vacuum during a time interval of ⁠1/299792458⁠ 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 the reduced Planck constant ħ has been determined experimentally to 12 parts per billion, while that of G has been determined experimentally to no better than 1 part in 21300 (or 47000 parts per billion). G appears in the definition of almost every Planck unit in Tables 2 and 3, but not all. 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 43500, or 23000 parts per billion.)

After 20 May 2019, h (and thus $\hbar ={\frac {h}{2\pi }}$) is exact, k_B is also exact, but since G is still not exact, the values of l_P, m_P, t_P, and T_P are also not exact. Besides, μ₀ (and thus $\epsilon _{0}={\frac {1}{c^{2}\mu _{0}}}$) is no longer exact (only e is exact), thus q_P is also not exact.

Derived units

In any system of measurement, units for many physical quantities can be derived from base units. Table 3 offers a 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 (since they are usually fundamental lower or upper bounds) and there are large uncertainties in their values.

Table 3: Derived Planck units

Name	Dimension	Expression		Approximate SI equivalent
		Lorentz–Heaviside version	Gaussian version	Lorentz–Heaviside version	Gaussian version
Linear/translational mechanical properties
Planck area	area (L2)	$A_{\text{P}}=l_{\text{P}}^{2}={\frac {4\pi \hbar G}{c^{3}}}$	$A_{\text{P}}=l_{\text{P}}^{2}={\frac {\hbar G}{c^{3}}}$	3.28258×10−69 m2	2.61220×10−70 m2
Planck volume	volume (L3)	$V_{\text{P}}=l_{\text{P}}^{3}={\sqrt {\frac {64\pi ^{3}\hbar ^{3}G^{3}}{c^{9}}}}$	$V_{\text{P}}=l_{\text{P}}^{3}={\sqrt {\frac {\hbar ^{3}G^{3}}{c^{9}}}}$	1.88072×10−103 m3	4.22191×10−105 m3
Planck wavenumber	wavenumber (L−1)	$N_{\text{P}}={\frac {1}{l_{\text{P}}}}={\sqrt {\frac {c^{3}}{4\pi \hbar G}}}$	$N_{\text{P}}={\frac {1}{l_{\text{P}}}}={\sqrt {\frac {c^{3}}{\hbar G}}}$	1.74539×1034 m−1	6.18724×1034 m−1
Planck density	density (L−3M)	$d_{\text{P}}={\frac {m_{\text{P}}}{V_{\text{P}}}}={\frac {\hbar t_{\text{P}}}{l_{\text{P}}^{5}}}={\frac {c^{5}}{16\pi ^{2}\hbar G^{2}}}$	$d_{\text{P}}={\frac {m_{\text{P}}}{V_{\text{P}}}}={\frac {\hbar t_{\text{P}}}{l_{\text{P}}^{5}}}={\frac {c^{5}}{\hbar G^{2}}}$	3.26456×1094 kg/m3	5.15518×1096 kg/m3
Planck specific volume	specific volume (L3M−1)	$\beta _{\text{P}}={\frac {1}{d_{\text{P}}}}={\frac {16\pi ^{2}\hbar G^{2}}{c^{5}}}$	$\beta _{\text{P}}={\frac {1}{d_{\text{P}}}}={\frac {\hbar G^{2}}{c^{5}}}$	3.06320×10−95 m3/kg	1.93980×10−97 m3/kg
Planck frequency	frequency (T−1)	$f_{\text{P}}={\frac {1}{t_{\text{P}}}}={\sqrt {\frac {c^{5}}{4\pi \hbar G}}}$	$f_{\text{P}}={\frac {1}{t_{\text{P}}}}={\sqrt {\frac {c^{5}}{\hbar G}}}$	5.23254×1042 Hz	1.85489×1043 Hz
Planck speed	speed (LT−1)	$v_{\text{P}}={\frac {l_{\text{P}}}{t_{\text{P}}}}=c$		2.99792×108 m/s
Planck acceleration	acceleration (LT−2)	$a_{\text{P}}={\frac {v_{\text{P}}}{t_{\text{P}}}}={\sqrt {\frac {c^{7}}{4\pi \hbar G}}}$	$a_{\text{P}}={\frac {v_{\text{P}}}{t_{\text{P}}}}={\sqrt {\frac {c^{7}}{\hbar G}}}$	1.56868×1051 m/s2	5.56082×1051 m/s2
Planck jerk	jerk (LT−3)	${\mathcal {J}}_{\text{P}}={\frac {a_{\text{P}}}{t_{\text{P}}}}={\frac {c^{6}}{4\pi \hbar G}}$	${\mathcal {J}}_{\text{P}}={\frac {a_{\text{P}}}{t_{\text{P}}}}={\frac {c^{6}}{\hbar G}}$	8.20817×1093 m/s3	1.03147×1095 m/s3
Planck snap	snap (LT−4)	${\mathcal {N}}_{\text{P}}={\frac {{\mathcal {J}}_{\text{P}}}{t_{\text{P}}}}={\sqrt {\frac {c^{17}}{64\pi ^{3}\hbar ^{3}G^{3}}}}$	${\mathcal {N}}_{\text{P}}={\frac {{\mathcal {J}}_{\text{P}}}{t_{\text{P}}}}={\sqrt {\frac {c^{17}}{\hbar ^{3}G^{3}}}}$	4.29496×10136 m/s4	1.91326×10138 m/s4
Planck crackle	crackle (LT−5)	${\mathcal {K}}_{\text{P}}={\frac {{\mathcal {N}}_{\text{P}}}{t_{\text{P}}}}={\frac {c^{11}}{16\pi ^{2}\hbar ^{2}G^{2}}}$	${\mathcal {K}}_{\text{P}}={\frac {{\mathcal {N}}_{\text{P}}}{t_{\text{P}}}}={\frac {c^{11}}{\hbar ^{2}G^{2}}}$	2.24736×10179 m/s5	3.54889×10181 m/s5
Planck pop	pop (LT−6)	${\mathcal {P}}_{\text{P}}={\frac {{\mathcal {K}}_{\text{P}}}{t_{\text{P}}}}={\sqrt {\frac {c^{27}}{1024\pi ^{5}\hbar ^{5}G^{5}}}}$	${\mathcal {P}}_{\text{P}}={\frac {{\mathcal {K}}_{\text{P}}}{t_{\text{P}}}}={\sqrt {\frac {c^{27}}{\hbar ^{5}G^{5}}}}$	1.17594×10222 m/s6	6.58279×10224 m/s6
Planck momentum	momentum (LMT−1)	$p_{\text{P}}=m_{\text{P}}v_{\text{P}}={\frac {\hbar }{l_{\text{P}}}}={\sqrt {\frac {\hbar c^{3}}{4\pi G}}}$	$p_{\text{P}}=m_{\text{P}}v_{\text{P}}={\frac {\hbar }{l_{\text{P}}}}={\sqrt {\frac {\hbar c^{3}}{G}}}$	1.84064 N⋅s	6.52489 N⋅s
Planck force	force (LMT−2)	$F_{\text{P}}=m_{\text{P}}a_{\text{P}}={\frac {p_{\text{P}}}{t_{\text{P}}}}={\frac {c^{4}}{4\pi G}}$	$F_{\text{P}}=m_{\text{P}}a_{\text{P}}={\frac {p_{\text{P}}}{t_{\text{P}}}}={\frac {c^{4}}{G}}$	9.63122×1042 N	1.21029×1044 N
Planck energy	energy (L2MT−2)	$E_{\text{P}}=m_{\text{P}}v_{\text{P}}^{2}={\frac {\hbar }{t_{\text{P}}}}={\sqrt {\frac {\hbar c^{5}}{4\pi G}}}$	$E_{\text{P}}=m_{\text{P}}v_{\text{P}}^{2}={\frac {\hbar }{t_{\text{P}}}}={\sqrt {\frac {\hbar c^{5}}{G}}}$	5.51809×108 J	1.95611×109 J
Planck power	power (L2MT−3)	$P_{\text{P}}={\frac {E_{\text{P}}}{t_{\text{P}}}}={\frac {\hbar }{t_{\text{P}}^{2}}}={\frac {c^{5}}{4\pi G}}$	$P_{\text{P}}={\frac {E_{\text{P}}}{t_{\text{P}}}}={\frac {\hbar }{t_{\text{P}}^{2}}}={\frac {c^{5}}{G}}$	2.88737×1051 W	3.62837×1052 W
Planck specific energy	specific energy (L2T−2)	$h_{\text{P}}={\frac {E_{\text{P}}}{m_{\text{P}}}}=c^{2}$		8.98755×1016 J/kg
Planck energy density	energy density (L−1MT−2)	$u_{\text{P}}={\frac {E_{\text{P}}}{V_{\text{P}}}}={\frac {c^{7}}{16\pi ^{2}\hbar G^{2}}}$	$u_{\text{P}}={\frac {E_{\text{P}}}{V_{\text{P}}}}={\frac {c^{7}}{\hbar G^{2}}}$	2.93404×10111 J/m3	4.63325×10113 J/m3
Planck intensity	intensity (MT−3)	${\mathcal {I}}_{\text{P}}={\frac {P_{\text{P}}}{A_{\text{P}}}}={\frac {c^{8}}{16\pi ^{2}\hbar G^{2}}}$	${\mathcal {I}}_{\text{P}}={\frac {P_{\text{P}}}{A_{\text{P}}}}={\frac {c^{8}}{\hbar G^{2}}}$	8.79603×10119 W/m2	1.38901×10122 W/m2
Planck action	action (L2MT−1)	${\mathcal {S}}_{\text{P}}=l_{\text{P}}p_{\text{P}}=E_{\text{P}}t_{\text{P}}=\hbar$		1.05457×10−34 J⋅s
Planck gravitational field	gravitational field (LT−2)	$g_{\text{P}}={\frac {F_{\text{P}}}{m_{\text{P}}}}={\sqrt {\frac {c^{7}}{4\pi \hbar G}}}$	$g_{\text{P}}={\frac {F_{\text{P}}}{m_{\text{P}}}}={\sqrt {\frac {c^{7}}{\hbar G}}}$	1.56868×1051 m/s2	5.56082×1051 m/s2
Planck gravitational potential	gravitational potential (L2T−2)	$\Delta _{\text{P}}={\frac {E_{\text{P}}}{m_{\text{P}}}}=c^{2}$		8.98755×1016 J/kg
Angular/rotational mechanical properties
Planck angle	angle (dimensionless)	$\theta _{\text{P}}={\frac {l_{\text{P}}}{l_{\text{P}}}}=1$		1.00000 rad
Planck angular speed	angular speed (T−1)	$\omega _{\text{P}}={\frac {\theta _{\text{P}}}{t_{\text{P}}}}={\sqrt {\frac {c^{5}}{4\pi \hbar G}}}$	$\omega _{\text{P}}={\frac {\theta _{\text{P}}}{t_{\text{P}}}}={\sqrt {\frac {c^{5}}{\hbar G}}}$	5.23254×1042 rad/s	1.85489×1043 rad/s
Planck angular acceleration	angular acceleration (T−2)	$\alpha _{\text{P}}={\frac {\omega _{\text{P}}}{t_{\text{P}}}}={\frac {c^{5}}{4\pi \hbar G}}$	$\alpha _{\text{P}}={\frac {\omega _{\text{P}}}{t_{\text{P}}}}={\frac {c^{5}}{\hbar G}}$	2.73795×1085 rad/s2	3.44061×1086 rad/s2
Planck angular jerk	angular jerk (T−3)	$\zeta _{\text{P}}={\frac {\alpha _{\text{P}}}{t_{\text{P}}}}={\sqrt {\frac {c^{15}}{64\pi ^{3}\hbar ^{3}G^{3}}}}$	$\zeta _{\text{P}}={\frac {\alpha _{\text{P}}}{t_{\text{P}}}}={\sqrt {\frac {c^{15}}{\hbar ^{3}G^{3}}}}$	1.43265×10128 rad/s3	6.38195×10129 rad/s3
Planck rotational inertia	rotational inertia (L2M)	$I_{\text{P}}=m_{\text{P}}l_{\text{P}}^{2}={\sqrt {\frac {4\pi \hbar ^{3}G}{c^{5}}}}$	$I_{\text{P}}=m_{\text{P}}l_{\text{P}}^{2}={\sqrt {\frac {\hbar ^{3}G}{c^{5}}}}$	2.01544×10−77 kg⋅m2	5.68546×10−78 kg⋅m2
Planck angular momentum	angular momentum (L2MT−1)	$\Lambda _{\text{P}}=I_{\text{P}}\omega _{\text{P}}=\hbar$		1.05457×10−34 J⋅s
Planck torque	torque (L2MT−2)	$\tau _{\text{P}}=I_{\text{P}}\alpha _{\text{P}}=F_{\text{P}}l_{\text{P}}={\frac {\Lambda _{\text{P}}}{t_{\text{P}}}}={\sqrt {\frac {\hbar c^{5}}{4\pi G}}}$	$\tau _{\text{P}}=I_{\text{P}}\alpha _{\text{P}}=F_{\text{P}}l_{\text{P}}={\frac {\Lambda _{\text{P}}}{t_{\text{P}}}}={\sqrt {\frac {\hbar c^{5}}{G}}}$	5.51809×108 N⋅m	1.95611×109 N⋅m
Planck specific angular momentum	specific angular momentum (L2T−1)	$\pi _{\text{P}}={\frac {\Lambda _{\text{P}}}{m_{\text{P}}}}={\sqrt {\frac {4\pi \hbar G}{c}}}$	$\pi _{\text{P}}={\frac {\Lambda _{\text{P}}}{m_{\text{P}}}}={\sqrt {\frac {\hbar G}{c}}}$	1.71763×10−26 m2/s	4.84533×10−27 m2/s
Planck solid angle	solid angle (dimensionless)	$\Omega _{\text{P}}=\theta _{\text{P}}^{2}={\frac {l_{\text{P}}^{2}}{l_{\text{P}}^{2}}}=1$		1.00000 sr
Planck radiant intensity	radiant intensity (L2MT−3)	$\iota _{\text{P}}={\frac {P_{\text{P}}}{\Omega _{\text{P}}}}={\frac {c^{5}}{4\pi G}}$	$\iota _{\text{P}}={\frac {P_{\text{P}}}{\Omega _{\text{P}}}}={\frac {c^{5}}{G}}$	2.88737×1051 W/sr	3.62837×1052 W/sr
Planck radiance	radiance (MT−3)	${\mathcal {L}}_{\text{P}}={\frac {P_{\text{P}}}{A_{\text{P}}\Omega _{\text{P}}}}={\frac {c^{8}}{16\pi ^{2}\hbar G^{2}}}$	${\mathcal {L}}_{\text{P}}={\frac {P_{\text{P}}}{A_{\text{P}}\Omega _{\text{P}}}}={\frac {c^{8}}{\hbar G^{2}}}$	8.79603×10119 W/sr⋅m2	1.38901×10122 W/sr⋅m2
Hydromechanical properties
Planck pressure	pressure (L−1MT−2)	$\Pi _{\text{P}}={\frac {F_{\text{P}}}{A_{\text{P}}}}={\frac {\hbar }{l_{\text{P}}^{3}t_{\text{P}}}}={\frac {c^{7}}{16\pi ^{2}\hbar G^{2}}}$	$\Pi _{\text{P}}={\frac {F_{\text{P}}}{A_{\text{P}}}}={\frac {\hbar }{l_{\text{P}}^{3}t_{\text{P}}}}={\frac {c^{7}}{\hbar G^{2}}}$	2.93404×10111 Pa	4.63325×10113 Pa
Planck surface tension	surface tension (MT−2)	$\sigma _{\text{P}}={\frac {F_{\text{P}}}{l_{\text{P}}}}={\sqrt {\frac {c^{11}}{64\pi ^{3}\hbar G^{3}}}}$	$\sigma _{\text{P}}={\frac {F_{\text{P}}}{l_{\text{P}}}}={\sqrt {\frac {c^{11}}{\hbar G^{3}}}}$	1.68102×1077 N/m	7.48839×1078 N/m
Planck volumetric flow rate	volumetric flow rate (L3T−1)	$Q_{\text{P}}={\frac {V_{\text{P}}}{t_{\text{P}}}}=l_{\text{P}}^{2}v_{\text{P}}={\frac {4\pi \hbar G}{c^{2}}}$	$Q_{\text{P}}={\frac {V_{\text{P}}}{t_{\text{P}}}}=l_{\text{P}}^{2}v_{\text{P}}={\frac {\hbar G}{c^{2}}}$	9.84093×10−61 m3/s	7.83116×10−62 m3/s
Planck mass flow rate	mass flow rate (MT−1)	$M_{\text{P}}={\frac {m_{\text{P}}}{t_{\text{P}}}}={\frac {c^{3}}{4\pi G}}$	$M_{\text{P}}={\frac {m_{\text{P}}}{t_{\text{P}}}}={\frac {c^{3}}{G}}$	3.21263×1034 kg/s	4.03711×1035 kg/s
Planck mass flux	mass flux (L−2MT−1)	$J_{\text{P}}={\frac {M_{\text{P}}}{A_{\text{P}}}}={\frac {c^{6}}{16\pi ^{2}\hbar G^{2}}}$	$J_{\text{P}}={\frac {M_{\text{P}}}{A_{\text{P}}}}={\frac {c^{6}}{\hbar G^{2}}}$	9.78690×10102 kg/s/m2	1.54549×10105 kg/s/m2
Planck stiffness	stiffness (MT−2)	$K_{\text{P}}={\frac {F_{\text{P}}}{l_{\text{P}}}}={\sqrt {\frac {c^{11}}{64\pi ^{3}\hbar G^{3}}}}$	$K_{\text{P}}={\frac {F_{\text{P}}}{l_{\text{P}}}}={\sqrt {\frac {c^{11}}{\hbar G^{3}}}}$	1.68102×1077 N/m	7.48839×1078 N/m
Planck flexibility	flexibility (M−1T2)	$x_{\text{P}}={\frac {1}{K_{\text{P}}}}={\sqrt {\frac {64\pi ^{3}\hbar G^{3}}{c^{11}}}}$	$x_{\text{P}}={\frac {1}{K_{\text{P}}}}={\sqrt {\frac {\hbar G^{3}}{c^{11}}}}$	5.94876×10−78 m/N	1.33540×10−79 m/N
Planck rotational stiffness	rotational stiffness (L2MT−2)	$O_{\text{P}}={\frac {\tau _{\text{P}}}{\theta _{\text{P}}}}={\sqrt {\frac {\hbar c^{5}}{4\pi G}}}$	$O_{\text{P}}={\frac {\tau _{\text{P}}}{\theta _{\text{P}}}}={\sqrt {\frac {\hbar c^{5}}{G}}}$	5.51809×108 N⋅m/rad	1.95611×109 N⋅m/rad
Planck rotational flexibility	rotational flexibility (L−2M−1T2)	$y_{\text{P}}={\frac {1}{O_{\text{P}}}}={\sqrt {\frac {4\pi G}{\hbar c^{5}}}}$	$y_{\text{P}}={\frac {1}{O_{\text{P}}}}={\sqrt {\frac {G}{\hbar c^{5}}}}$	1.81222×10−9 rad/N⋅m	5.11218×10−10 rad/N⋅m
Planck ultimate tensile strength	ultimate tensile strength (L−1MT−2)	$\Sigma _{\text{P}}={\frac {F_{\text{P}}}{A_{\text{P}}}}={\frac {c^{7}}{16\pi ^{2}\hbar G^{2}}}$	$\Sigma _{\text{P}}={\frac {F_{\text{P}}}{A_{\text{P}}}}={\frac {c^{7}}{\hbar G^{2}}}$	2.93404×10111 Pa	4.63325×10113 Pa
Planck indentation hardness	indentation hardness (L−1MT−2)	${\mathcal {H}}_{\text{P}}={\frac {F_{\text{P}}}{A_{\text{P}}}}={\frac {c^{7}}{16\pi ^{2}\hbar G^{2}}}$	${\mathcal {H}}_{\text{P}}={\frac {F_{\text{P}}}{A_{\text{P}}}}={\frac {c^{7}}{\hbar G^{2}}}$	2.93404×10111 Pa	4.63325×10113 Pa
Planck absolute hardness	absolute hardness (M)	${\mathcal {G}}_{\text{P}}={\frac {F_{\text{P}}}{g_{\text{P}}}}={\sqrt {\frac {\hbar c}{4\pi G}}}$	${\mathcal {G}}_{\text{P}}={\frac {F_{\text{P}}}{g_{\text{P}}}}={\sqrt {\frac {\hbar c}{G}}}$	6.13971×10−9 N⋅s/m2	2.17647×10−8 N⋅s/m2
Planck viscosity	viscosity (L−1MT−1)	$\eta _{\text{P}}=P_{\text{P}}t_{\text{P}}={\sqrt {\frac {c^{9}}{64\pi ^{3}\hbar G^{3}}}}$	$\eta _{\text{P}}=P_{\text{P}}t_{\text{P}}={\sqrt {\frac {c^{9}}{\hbar G^{3}}}}$	5.60729×1068 Pa⋅s	2.49786×1070 Pa⋅s
Planck kinematic viscosity	kinematic viscosity (L2T−1)	$\nu _{\text{P}}={\frac {A_{\text{P}}}{t_{\text{P}}}}={\frac {\eta _{\text{P}}}{d_{\text{P}}}}={\sqrt {\frac {4\pi \hbar G}{c}}}$	$\nu _{\text{P}}={\frac {A_{\text{P}}}{t_{\text{P}}}}={\frac {\eta _{\text{P}}}{d_{\text{P}}}}={\sqrt {\frac {\hbar G}{c}}}$	1.71763×10−26 m2/s	4.84533×10−27 m2/s
Planck toughness	toughness (L−1MT−2)	${\mathcal {T}}_{\text{P}}={\frac {E_{\text{P}}}{V_{\text{P}}}}={\frac {c^{7}}{16\pi ^{2}\hbar G^{2}}}$	${\mathcal {T}}_{\text{P}}={\frac {E_{\text{P}}}{V_{\text{P}}}}={\frac {c^{7}}{\hbar G^{2}}}$	2.93404×10111 J/m3	4.63325×10113 J/m3
Electromagnetic properties
Planck current	current (T−1Q)	$i_{\text{P}}={\frac {q_{\text{P}}}{t_{\text{P}}}}={\sqrt {\frac {c^{6}\epsilon _{0}}{4\pi G}}}$	$i_{\text{P}}={\frac {q_{\text{P}}}{t_{\text{P}}}}={\sqrt {\frac {4\pi c^{6}\epsilon _{0}}{G}}}$	2.76844×1024 A	3.47893×1025 A
Planck voltage	voltage (L2MT−2Q−1)	$U_{\text{P}}={\frac {E_{\text{P}}}{q_{\text{P}}}}={\frac {P_{\text{P}}}{i_{\text{P}}}}={\sqrt {\frac {c^{4}}{4\pi G\epsilon _{0}}}}$		1.04296×1027 V
Planck impedance	resistance (L2MT−1Q−2)	$Z_{\text{P}}={\frac {U_{\text{P}}}{i_{\text{P}}}}={\frac {\hbar }{q_{\text{P}}^{2}}}={\frac {1}{c\epsilon _{0}}}=c\mu _{0}={\sqrt {\frac {\mu _{0}}{\epsilon _{0}}}}=Z_{0}={\frac {1}{Y_{0}}}$	$Z_{\text{P}}={\frac {U_{\text{P}}}{i_{\text{P}}}}={\frac {\hbar }{q_{\text{P}}^{2}}}={\frac {1}{4\pi c\epsilon _{0}}}={\frac {c\mu _{0}}{4\pi }}={\sqrt {\frac {\mu _{0}}{16\pi ^{2}\epsilon _{0}}}}={\frac {Z_{0}}{4\pi }}={\frac {1}{4\pi Y_{0}}}$	376.730 Ω	29.9792 Ω
Planck admittance	conductance (L−2M−1TQ2)	$Y_{\text{P}}={\frac {1}{Z_{\text{P}}}}=c\epsilon _{0}={\frac {1}{c\mu _{0}}}={\sqrt {\frac {\epsilon _{0}}{\mu _{0}}}}=Y_{0}={\frac {1}{Z_{0}}}$	$Y_{\text{P}}={\frac {1}{Z_{\text{P}}}}=4\pi c\epsilon _{0}={\frac {4\pi }{c\mu _{0}}}={\sqrt {\frac {16\pi ^{2}\epsilon _{0}}{\mu _{0}}}}=4\pi Y_{0}={\frac {4\pi }{Z_{0}}}$	2.65442×10−3 S	3.33564×10−2 S
Planck capacitance	capacitance (L−2M−1T2Q2)	$C_{\text{P}}={\frac {q_{\text{P}}}{U_{\text{P}}}}={\frac {t_{\text{P}}q_{\text{P}}^{2}}{\hbar }}={\sqrt {\frac {4\pi \hbar G\epsilon _{0}^{2}}{c^{3}}}}$	$C_{\text{P}}={\frac {q_{\text{P}}}{U_{\text{P}}}}={\frac {t_{\text{P}}q_{\text{P}}^{2}}{\hbar }}={\sqrt {\frac {16\pi ^{2}\hbar G\epsilon _{0}^{2}}{c^{3}}}}$	5.07290×10−46 F	1.79830×10−45 F
Planck inductance	inductance (L2MQ−2)	$L_{\text{P}}={\frac {E_{\text{P}}}{i_{\text{P}}}}={\frac {m_{\text{P}}l_{\text{P}}^{2}}{q_{\text{P}}^{2}}}={\sqrt {\frac {4\pi \hbar G}{c^{7}\epsilon _{0}^{2}}}}$	$L_{\text{P}}={\frac {E_{\text{P}}}{i_{\text{P}}}}={\frac {m_{\text{P}}l_{\text{P}}^{2}}{q_{\text{P}}^{2}}}={\sqrt {\frac {\hbar G}{16\pi ^{2}c^{7}\epsilon _{0}^{2}}}}$	7.19975×10−41 H	1.61623×10−42 H
Planck electrical resistivity	electrical resistivity (L3MT−1Q−2)	$r_{\text{P}}=Z_{\text{P}}l_{\text{P}}={\sqrt {\frac {4\pi \hbar G}{c^{5}\epsilon _{0}^{2}}}}$	$r_{\text{P}}=Z_{\text{P}}l_{\text{P}}={\sqrt {\frac {\hbar G}{16\pi ^{2}c^{5}\epsilon _{0}^{2}}}}$	2.15843×10−32 Ω⋅m	4.84533×10−34 Ω⋅m
Planck electrical conductivity	electrical conductivity (L−3M−1TQ2)	$\kappa _{\text{P}}={\frac {1}{r_{\text{P}}}}={\sqrt {\frac {c^{5}\epsilon _{0}^{2}}{4\pi \hbar G}}}$	$\kappa _{\text{P}}={\frac {1}{r_{\text{P}}}}={\sqrt {\frac {16\pi ^{2}c^{5}\epsilon _{0}^{2}}{\hbar G}}}$	4.63299×1031 S/m	2.06384×1033 S/m
Planck charge-to-mass ratio	charge-to-mass ratio (M−1Q)	$\xi _{\text{P}}={\frac {q_{\text{P}}}{m_{\text{P}}}}={\sqrt {4\pi G\epsilon _{0}}}={\sqrt {\frac {G}{k_{e}}}}$		8.61738×10−11 C/kg
Planck mass-to-charge ratio	mass-to-charge ratio (MQ−1)	$\varsigma _{\text{P}}={\frac {1}{\xi _{\text{P}}}}={\frac {m_{\text{P}}}{q_{\text{P}}}}={\sqrt {\frac {1}{4\pi G\epsilon _{0}}}}={\sqrt {\frac {k_{e}}{G}}}$		1.16045×1010 kg/C
Planck charge density	charge density (L−3Q)	$\rho _{\text{P}}={\frac {q_{\text{P}}}{V_{\text{P}}}}={\sqrt {\frac {c^{10}\epsilon _{0}}{64\pi ^{3}\hbar ^{2}G^{3}}}}$	$\rho _{\text{P}}={\frac {q_{\text{P}}}{V_{\text{P}}}}={\sqrt {\frac {4\pi c^{10}\epsilon _{0}}{\hbar ^{2}G^{3}}}}$	2.81319×1084 C/m3	4.44242×1086 C/m3
Planck current density	current density (L−2T−1Q)	$j_{\text{P}}={\frac {i_{\text{P}}}{A_{\text{P}}}}=\rho _{\text{P}}v_{\text{P}}={\sqrt {\frac {c^{12}\epsilon _{0}}{64\pi ^{3}\hbar ^{2}G^{3}}}}$	$j_{\text{P}}={\frac {i_{\text{P}}}{A_{\text{P}}}}=\rho _{\text{P}}v_{\text{P}}={\sqrt {\frac {4\pi c^{12}\epsilon _{0}}{\hbar ^{2}G^{3}}}}$	8.43374×1092 A/m2	1.33180×1095 A/m2
Planck magnetic charge	magnetic charge (LT−1Q)	$b_{\text{P}}=q_{\text{P}}v_{\text{P}}={\sqrt {\hbar c^{3}\epsilon _{0}}}$	$b_{\text{P}}=q_{\text{P}}v_{\text{P}}={\sqrt {4\pi \hbar c^{3}\epsilon _{0}}}$	1.58634×10−10 A⋅m	5.62274×10−10 A⋅m
Planck magnetic current	magnetic current (L2MT−2Q−1)	$k_{\text{P}}=U_{\text{P}}={\sqrt {\frac {c^{4}}{4\pi G\epsilon _{0}}}}$		1.04296×1027 V
Planck magnetic current density	magnetic current density (MT−2Q−1)	$\delta _{\text{P}}={\frac {k_{\text{P}}}{A_{\text{P}}}}={\sqrt {\frac {c^{10}}{64\pi ^{3}\hbar ^{2}G^{3}\epsilon _{0}}}}$	$\delta _{\text{P}}={\frac {k_{\text{P}}}{A_{\text{P}}}}={\sqrt {\frac {c^{10}}{4\pi \hbar ^{2}G^{3}\epsilon _{0}}}}$	3.17725×1095 V/m2	3.99264×1096 V/m2
Planck electric field intensity	electric field intensity (LMT−2Q−1)	$e_{\text{P}}={\frac {F_{\text{P}}}{q_{\text{P}}}}={\sqrt {\frac {c^{7}}{16\pi ^{2}\hbar G^{2}\epsilon _{0}}}}$	$e_{\text{P}}={\frac {F_{\text{P}}}{q_{\text{P}}}}={\sqrt {\frac {c^{7}}{4\pi \hbar G^{2}\epsilon _{0}}}}$	1.82037×1061 V/m	6.45303×1061 V/m
Planck magnetic field intensity	magnetic field intensity (L−1T−1Q)	$H_{\text{P}}={\frac {i_{\text{P}}}{l_{\text{P}}}}={\sqrt {\frac {c^{9}\epsilon _{0}}{16\pi ^{2}\hbar G^{2}}}}$	$H_{\text{P}}={\frac {i_{\text{P}}}{l_{\text{P}}}}={\sqrt {\frac {4\pi c^{9}\epsilon _{0}}{\hbar G^{2}}}}$	4.83201×1058 A/m	2.15250×1060 A/m
Planck electric induction	electric induction (L−2Q)	$D_{\text{P}}={\frac {q_{\text{P}}}{l_{\text{P}}^{2}}}={\sqrt {\frac {c^{7}\epsilon _{0}}{16\pi ^{2}\hbar G^{2}}}}$	$D_{\text{P}}={\frac {q_{\text{P}}}{l_{\text{P}}^{2}}}={\sqrt {\frac {4\pi c^{7}\epsilon _{0}}{\hbar G^{2}}}}$	1.61179×1050 C/m2	7.17996×1051 C/m2
Planck magnetic induction	magnetic induction (MT−1Q−1)	$B_{\text{P}}={\frac {F_{\text{P}}}{l_{\text{P}}i_{\text{P}}}}={\sqrt {\frac {c^{5}}{16\pi ^{2}\hbar G^{2}\epsilon _{0}}}}$	$B_{\text{P}}={\frac {F_{\text{P}}}{l_{\text{P}}i_{\text{P}}}}={\sqrt {\frac {c^{5}}{4\pi \hbar G^{2}\epsilon _{0}}}}$	6.07208×1052 T	2.15250×1053 T
Planck electric potential	electric potential (L2MT−2Q−1)	$\phi _{\text{P}}={\frac {E_{\text{P}}}{q_{\text{P}}}}=U_{\text{P}}={\sqrt {\frac {c^{4}}{4\pi G\epsilon _{0}}}}$		1.04296×1027 V
Planck magnetic potential	magnetic potential (LMT−1Q−1)	$\psi _{\text{P}}={\frac {F_{\text{P}}}{i_{\text{P}}}}=B_{\text{P}}l_{\text{P}}={\frac {U_{\text{P}}}{v_{\text{P}}}}={\sqrt {\frac {c^{2}}{4\pi G\epsilon _{0}}}}$		3.47887×1018 T⋅m
Planck electromotive force	electromotive force (L2MT−2Q−1)	${\mathcal {E}}_{\text{P}}={\frac {E_{\text{P}}}{q_{\text{P}}}}={\sqrt {\frac {c^{4}}{4\pi G\epsilon _{0}}}}$		1.04296×1027 V
Planck magnetomotive force	magnetomotive force (T−1Q)	${\mathcal {F}}_{\text{P}}=i_{\text{P}}={\sqrt {\frac {c^{6}\epsilon _{0}}{4\pi G}}}$	${\mathcal {F}}_{\text{P}}=i_{\text{P}}={\sqrt {\frac {4\pi c^{6}\epsilon _{0}}{G}}}$	2.76844×1024 A	3.47893×1025 A
Planck permittivity	permittivity (L−3M−1T2Q2)	$\epsilon _{\text{P}}={\frac {C_{\text{P}}}{l_{\text{P}}}}=\epsilon _{0}$	$\epsilon _{\text{P}}={\frac {C_{\text{P}}}{l_{\text{P}}}}=4\pi \epsilon _{0}$	8.85419×10−12 F/m	1.11265×10−10 F/m
Planck permeability	permeability (LMQ−2)	$\mu _{\text{P}}={\frac {L_{\text{P}}}{l_{\text{P}}}}={\frac {1}{c^{2}\epsilon _{0}}}=\mu _{0}$	$\mu _{\text{P}}={\frac {L_{\text{P}}}{l_{\text{P}}}}={\frac {1}{4\pi c^{2}\epsilon _{0}}}={\frac {\mu _{0}}{4\pi }}$	1.25664×10−6 H/m	1.00000×10−7 H/m
Planck electric dipole moment	electric dipole moment (LQ)	${\mathcal {Q}}_{\text{P}}=q_{\text{P}}l_{\text{P}}={\sqrt {\frac {4\pi \hbar ^{2}G\epsilon _{0}}{c^{2}}}}$		3.03131×10−53 C⋅m
Planck magnetic dipole moment	magnetic dipole moment (L2T−1Q)	${\mathcal {M}}_{\text{P}}={\frac {E_{\text{P}}}{b_{\text{P}}}}={\sqrt {4\pi \hbar ^{2}G\epsilon _{0}}}$		9.08764×10−45 J/T
Planck electric flux	electric flux (L3MT−2Q−1)	$\Phi _{\text{P}}=e_{\text{P}}A_{\text{P}}={\sqrt {\frac {\hbar c}{\epsilon _{0}}}}$	$\Phi _{\text{P}}=e_{\text{P}}A_{\text{P}}={\sqrt {\frac {\hbar c}{4\pi \epsilon _{0}}}}$	5.97550×10−8 V⋅m	1.68566×10−8 V⋅m
Planck magnetic flux	magnetic flux (L2MT−1Q−1)	$\Psi _{\text{P}}=B_{\text{P}}A_{\text{P}}={\sqrt {\frac {\hbar }{c\epsilon _{0}}}}$	$\Psi _{\text{P}}=B_{\text{P}}A_{\text{P}}={\sqrt {\frac {\hbar }{4\pi c\epsilon _{0}}}}$	1.99321×10−16 Wb	5.62275×10−17 Wb
Planck electric polarizability	electric polarizability (M−1T2Q2)	${\mathcal {Y}}_{\text{P}}={\frac {{\mathcal {Q}}_{\text{P}}}{e_{\text{P}}}}={\sqrt {\frac {64\pi ^{3}\hbar ^{3}G^{3}\epsilon _{0}^{2}}{c^{9}}}}$	${\mathcal {Y}}_{\text{P}}={\frac {{\mathcal {Q}}_{\text{P}}}{e_{\text{P}}}}={\sqrt {\frac {16\pi ^{2}\hbar ^{3}G^{3}\epsilon _{0}^{2}}{c^{9}}}}$	1.66522×10−114 C⋅m2/V	4.69750×10−115 C⋅m2/V
Planck electric polarization	electric polarization (L−3M−1T2Q2)	${\mathcal {O}}_{\text{P}}={\frac {{\mathcal {Y}}_{\text{P}}}{V_{\text{P}}}}={\frac {1}{\epsilon _{0}}}$	${\mathcal {O}}_{\text{P}}={\frac {{\mathcal {Y}}_{\text{P}}}{V_{\text{P}}}}={\frac {1}{4\pi \epsilon _{0}}}$	1.12941×1011 C/V⋅m	8.98755×109 C/V⋅m
Planck electric field gradient	electric field gradient (MT−2Q−1)	${\mathcal {Z}}_{\text{P}}={\frac {k_{\text{P}}}{A_{\text{P}}}}={\sqrt {\frac {c^{10}}{64\pi ^{3}\hbar ^{2}G^{3}\epsilon _{0}}}}$	${\mathcal {Z}}_{\text{P}}={\frac {k_{\text{P}}}{A_{\text{P}}}}={\sqrt {\frac {c^{10}}{4\pi \hbar ^{2}G^{3}\epsilon _{0}}}}$	3.17725×1095 V/m2	3.99264×1096 V/m2
Planck gyromagnetic ratio	gyromagnetic ratio (M−1Q)	$\Theta _{\text{P}}={\frac {\theta _{\text{P}}}{t_{\text{P}}B_{\text{P}}}}={\sqrt {4\pi G\epsilon _{0}}}={\sqrt {\frac {G}{k_{e}}}}$		8.61738×10−11 rad/s/T
Planck magnetogyric ratio	magnetogyric ratio (MQ−1)	$\Xi _{\text{P}}={\frac {1}{\Theta _{\text{P}}}}={\frac {t_{\text{P}}B_{\text{P}}}{\theta _{\text{P}}}}={\sqrt {\frac {1}{4\pi G\epsilon _{0}}}}={\sqrt {\frac {k_{e}}{G}}}$		1.16045×1010 s⋅T/rad
Planck magnetic reluctance	magnetic reluctance (L−2M−1Q2)	${\mathcal {R}}_{\text{P}}={\frac {{\mathcal {F}}_{\text{P}}}{\Psi _{\text{P}}}}={\sqrt {\frac {c^{7}\epsilon _{0}^{2}}{4\pi \hbar G}}}$	${\mathcal {R}}_{\text{P}}={\frac {{\mathcal {F}}_{\text{P}}}{\Psi _{\text{P}}}}={\sqrt {\frac {16\pi ^{2}c^{7}\epsilon _{0}^{2}}{\hbar G}}}$	1.38894×1040 H−1	6.18724×1041 H−1
Radioactive properties
Planck specific activity	specific activity (T−1)	${\mathcal {A}}_{\text{P}}={\frac {1}{t_{\text{P}}}}={\sqrt {\frac {c^{5}}{4\pi \hbar G}}}$	${\mathcal {A}}_{\text{P}}={\frac {1}{t_{\text{P}}}}={\sqrt {\frac {c^{5}}{\hbar G}}}$	5.23254×1042 Bq	1.85489×1043 Bq
Planck radiation exposure	radiation exposure (M−1Q)	$X_{\text{P}}={\frac {q_{\text{P}}}{m_{\text{P}}}}={\sqrt {4\pi G\epsilon _{0}}}={\sqrt {\frac {G}{k_{e}}}}$		8.61738×10−11 C/kg
Planck absorbed dose	absorbed dose (L2T−2)	${\mathcal {D}}_{\text{P}}={\frac {E_{\text{P}}}{m_{\text{P}}}}=c^{2}$		8.98755×1016 Gy
Planck absorbed dose rate	absorbed dose rate (L2T−3)	$W_{\text{P}}={\frac {{\mathcal {D}}_{\text{P}}}{t_{\text{P}}}}={\sqrt {\frac {c^{9}}{4\pi \hbar G}}}$	$W_{\text{P}}={\frac {{\mathcal {D}}_{\text{P}}}{t_{\text{P}}}}={\sqrt {\frac {c^{9}}{\hbar G}}}$	4.70278×1059 Gy/s	1.66709×1060 Gy/s
Thermodynamic properties
Planck thermal expansion coefficient	thermal expansion coefficient (Θ−1)	$\gamma _{\text{P}}={\frac {1}{T_{\text{P}}}}={\sqrt {\frac {4\pi G{k_{\text{B}}}^{2}}{\hbar c^{5}}}}$	$\gamma _{\text{P}}={\frac {1}{T_{\text{P}}}}={\sqrt {\frac {G{k_{\text{B}}}^{2}}{\hbar c^{5}}}}$	2.50204×10−32 K−1	7.05812×10−33 K−1
Planck heat capacity	heat capacity (L2MT−2Θ−1)	$\Gamma _{\text{P}}={\frac {E_{\text{P}}}{T_{\text{P}}}}=k_{\text{B}}$		1.38065×10−23 J/K
Planck specific heat capacity	specific heat capacity (L2T−2Θ−1)	$c_{\text{P}}={\frac {E_{\text{P}}}{m_{\text{P}}T_{\text{P}}}}={\frac {\Gamma _{\text{P}}}{m_{\text{P}}}}={\sqrt {\frac {4\pi Gk_{\text{B}}^{2}}{\hbar c}}}$	$c_{\text{P}}={\frac {E_{\text{P}}}{m_{\text{P}}T_{\text{P}}}}={\frac {\Gamma _{\text{P}}}{m_{\text{P}}}}={\sqrt {\frac {Gk_{\text{B}}^{2}}{\hbar c}}}$	2.24872×10−15 J/kg⋅K	6.34352×10−16 J/kg⋅K
Planck volumetric heat capacity	volumetric heat capacity (L−1MT−2Θ−1)	$s_{\text{P}}={\frac {E_{\text{P}}}{V_{\text{P}}T_{\text{P}}}}={\frac {\Gamma _{\text{P}}}{V_{\text{P}}}}=c_{\text{P}}d_{\text{P}}={\sqrt {\frac {c^{9}k_{\text{B}}^{2}}{64\pi ^{3}\hbar ^{3}G^{3}}}}$	$s_{\text{P}}={\frac {E_{\text{P}}}{V_{\text{P}}T_{\text{P}}}}={\frac {\Gamma _{\text{P}}}{V_{\text{P}}}}=c_{\text{P}}d_{\text{P}}={\sqrt {\frac {c^{9}k_{\text{B}}^{2}}{\hbar ^{3}G^{3}}}}$	7.34108×1079 J/m3⋅K	3.27020×1081 J/m3⋅K
Planck thermal resistance	thermal resistance (L−2M−1T3Θ)	$R_{\text{P}}={\frac {T_{\text{P}}}{P_{\text{P}}}}={\sqrt {\frac {4\pi \hbar G}{c^{5}k_{\text{B}}^{2}}}}$	$R_{\text{P}}={\frac {T_{\text{P}}}{P_{\text{P}}}}={\sqrt {\frac {\hbar G}{c^{5}k_{\text{B}}^{2}}}}$	1.38424×10−20 K/W	3.90486×10−21 K/W
Planck thermal conductance	thermal conductance (L2MT−3Θ−1)	$G_{\text{P}}={\frac {1}{R_{\text{P}}}}={\sqrt {\frac {c^{5}k_{\text{B}}^{2}}{4\pi \hbar G}}}$	$G_{\text{P}}={\frac {1}{R_{\text{P}}}}={\sqrt {\frac {c^{5}k_{\text{B}}^{2}}{\hbar G}}}$	7.22420×1019 W/K	2.56091×1020 W/K
Planck thermal resistivity	thermal resistivity (L−1M−1T3Θ)	$\chi _{\text{P}}=R_{\text{P}}l_{\text{P}}={\sqrt {\frac {16\pi ^{2}\hbar ^{2}G^{2}}{c^{8}k_{\text{B}}^{2}}}}$	$\chi _{\text{P}}=R_{\text{P}}l_{\text{P}}={\sqrt {\frac {\hbar ^{2}G^{2}}{c^{8}k_{\text{B}}^{2}}}}$	7.93096×10−55 m⋅K/W	6.31126×10−56 m⋅K/W
Planck thermal conductivity	thermal conductivity (LMT−3Θ−1)	$\lambda _{\text{P}}={\frac {P_{\text{P}}}{l_{\text{P}}T_{\text{P}}}}={\frac {1}{\chi _{\text{P}}}}={\sqrt {\frac {c^{8}k_{\text{B}}^{2}}{16\pi ^{2}\hbar ^{2}G^{2}}}}$	$\lambda _{\text{P}}={\frac {P_{\text{P}}}{l_{\text{P}}T_{\text{P}}}}={\frac {1}{\chi _{\text{P}}}}={\sqrt {\frac {c^{8}k_{\text{B}}^{2}}{\hbar ^{2}G^{2}}}}$	1.26088×1054 W/m⋅K	1.58447×1055 W/m⋅K
Planck thermal insulance	thermal insulance (M−1T3Θ)	$o_{\text{P}}=R_{\text{P}}A_{\text{P}}={\sqrt {\frac {64\pi ^{3}\hbar ^{3}G^{3}}{c^{11}k_{\text{B}}^{2}}}}$	$o_{\text{P}}=R_{\text{P}}A_{\text{P}}={\sqrt {\frac {\hbar ^{3}G^{3}}{c^{11}k_{\text{B}}^{2}}}}$	4.54402×10−89 m2⋅K/W	1.10201×10−90 m2⋅K/W
Planck thermal transmittance	thermal transmittance (MT−3Θ−1)	$w_{\text{P}}={\frac {1}{o_{\text{P}}}}={\sqrt {\frac {c^{11}k_{\text{B}}^{2}}{64\pi ^{3}\hbar ^{3}G^{3}}}}$	$w_{\text{P}}={\frac {1}{o_{\text{P}}}}={\sqrt {\frac {c^{11}k_{\text{B}}^{2}}{\hbar ^{3}G^{3}}}}$	2.20069×1088 W/m2⋅K	9.80335×1089 W/m2⋅K
Planck entropy	entropy (L2MT−2Θ−1)	$S_{\text{P}}={\frac {E_{\text{P}}}{T_{\text{P}}}}=k_{\text{B}}$		1.38065×10−23 J/K
Molar properties
Planck amount of substance	amount of substance (N)	$n_{\text{P}}={\frac {1}{N_{\text{A}}}}$		1.66054×10−24 mol
Planck molar mass	molar mass (MN−1)	$\upsilon _{\text{P}}={\frac {m_{\text{P}}}{n_{\text{P}}}}={\sqrt {\frac {\hbar cN_{\text{A}}^{2}}{4\pi G}}}$	$\upsilon _{\text{P}}={\frac {m_{\text{P}}}{n_{\text{P}}}}={\sqrt {\frac {\hbar cN_{\text{A}}^{2}}{G}}}$	3.69742×1015 kg/mol	1.31070×1016 kg/mol
Planck molar volume	molar volume (L3N−1)	${\mathcal {V}}_{\text{P}}={\frac {V_{\text{P}}}{n_{\text{P}}}}={\sqrt {\frac {64\pi ^{3}\hbar ^{3}G^{3}N_{\text{A}}^{2}}{c^{9}}}}$	${\mathcal {V}}_{\text{P}}={\frac {V_{\text{P}}}{n_{\text{P}}}}={\sqrt {\frac {\hbar ^{3}G^{3}N_{\text{A}}^{2}}{c^{9}}}}$	1.13259×10−79 m3/mol	2.54249×10−81 m3/mol
Planck molar heat capacity	molar heat capacity (L2MT−2Θ−1N−1)	${\mathcal {U}}_{\text{P}}={\frac {E_{\text{P}}}{n_{\text{P}}T_{\text{P}}}}={\frac {\Gamma _{\text{P}}}{n_{\text{P}}}}=k_{\text{B}}N_{\text{A}}=R$		8.31446 J/mol⋅K
Planck mass fraction	mass fraction (dimensionless)	${\mathcal {W}}_{\text{P}}={\frac {m_{\text{P}}}{m_{\text{P}}}}=1$		100.000 %
Planck volume fraction	volume fraction (dimensionless)	$\varphi _{\text{P}}={\frac {V_{\text{P}}}{V_{\text{P}}}}=1$		100.000 %
Planck molality	molality (M−1N)	${\mathcal {B}}_{\text{P}}={\frac {n_{\text{P}}}{m_{\text{P}}}}={\sqrt {\frac {4\pi G}{\hbar cN_{\text{A}}^{2}}}}$	${\mathcal {B}}_{\text{P}}={\frac {n_{\text{P}}}{m_{\text{P}}}}={\sqrt {\frac {G}{\hbar cN_{\text{A}}^{2}}}}$	2.70459×10−16 mol/kg	7.62951×10−17 mol/kg
Planck molarity	molarity (L−3N)	${\mathcal {C}}_{\text{P}}={\frac {n_{\text{P}}}{V_{\text{P}}}}={\sqrt {\frac {c^{9}}{64\pi ^{3}\hbar ^{3}G^{3}N_{\text{A}}^{2}}}}$	${\mathcal {C}}_{\text{P}}={\frac {n_{\text{P}}}{V_{\text{P}}}}={\sqrt {\frac {c^{9}}{\hbar ^{3}G^{3}N_{\text{A}}^{2}}}}$	8.82929×1078 mol/m3	3.93315×1080 mol/m3
Planck mole fraction	mole fraction (dimensionless)	${\mathcal {X}}_{\text{P}}={\frac {n_{\text{P}}}{n_{\text{P}}}}=1$		1.00000
Planck heat of formation	heat of formation (L2MT−2N−1)	$\nabla _{\text{P}}={\frac {E_{\text{P}}}{n_{\text{P}}}}={\sqrt {\frac {\hbar c^{5}N_{\text{A}}^{2}}{4\pi G}}}$	$\nabla _{\text{P}}={\frac {E_{\text{P}}}{n_{\text{P}}}}={\sqrt {\frac {\hbar c^{5}N_{\text{A}}^{2}}{G}}}$	3.32307×1032 J/mol	1.17800×1033 J/mol
Planck catalytic activity	catalytic activity (T−1N)	$z_{\text{P}}={\frac {n_{\text{P}}}{t_{\text{P}}}}={\sqrt {\frac {c^{5}}{4\pi \hbar GN_{\text{A}}^{2}}}}$	$z_{\text{P}}={\frac {n_{\text{P}}}{t_{\text{P}}}}={\sqrt {\frac {c^{5}}{\hbar GN_{\text{A}}^{2}}}}$	8.68884×1018 kat	3.08012×1019 kat

(Note: $k_{e}$ is the Coulomb constant, $\mu _{0}$ is the vacuum permeability, $Z_{0}$ is the impedance of free space, $Y_{0}$ is the admittance of free space, $R$ is the gas constant)

(Note: $N_{\text{A}}$ is the Avogadro constant, which is also normalized to 1 in (both two versions of) Planck units)

The charge, as other Planck units, was not originally defined by Planck. It is a unit of charge that is a natural addition to the other units of Planck, and is used in some publications. The elementary charge $e$, measured in terms of the Planck charge, is

$e={\sqrt {4\pi \alpha }}\cdot q_{\text{P}}\approx 0.302822121\cdot q_{\text{P}}\,$ (Lorentz–Heaviside version)

$e={\sqrt {\alpha }}\cdot q_{\text{P}}\approx 0.085424543\cdot q_{\text{P}}\,$ (Gaussian version)

where ${\alpha }$ is the fine-structure constant

$\alpha ={\frac {k_{e}e^{2}}{\hbar c}}\approx {\frac {1}{137.03599911}}$

$\alpha ={\frac {1}{4\pi }}\left({\frac {e}{q_{\text{P}}}}\right)^{2}$ (Lorentz–Heaviside version)

$\alpha =\left({\frac {e}{q_{\text{P}}}}\right)^{2}$ (Gaussian version)

The fine-structure constant $\alpha$ is also called the electromagnetic coupling constant, thus comparing with the gravitational coupling constant $\alpha _{G}$. The electron rest mass $m_{e}$ measured in terms of the Planck mass, is

$m_{e}={\sqrt {4\pi \alpha _{G}}}\cdot m_{\text{P}}\approx 1.48368\times 10^{-22}\cdot m_{\text{P}}\,$ (Lorentz–Heaviside version)

$m_{e}={\sqrt {\alpha _{G}}}\cdot m_{\text{P}}\approx 4.18539\times 10^{-23}\cdot m_{\text{P}}\,$ (Gaussian version)

where ${\alpha _{G}}$ is the gravitational coupling constant

$\alpha _{G}={\frac {Gm_{e}^{2}}{\hbar c}}\approx 1.7518\times 10^{-45}$

$\alpha _{G}={\frac {1}{4\pi }}\left({\frac {m_{e}}{m_{\text{P}}}}\right)^{2}$ (Lorentz–Heaviside version)

$\alpha _{G}=\left({\frac {m_{e}}{m_{\text{P}}}}\right)^{2}$ (Gaussian version)

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

• 1 Planck mass is about 6.14 μg (Lorentz–Heaviside version) or 21.8 μg (Gaussian version);
• 1 Planck momentum is about 1.84 N⋅s (Lorentz–Heaviside version) or 6.52 N⋅s (Gaussian version);
• 1 Planck energy is about 153 kW⋅h (Lorentz–Heaviside version) or 543 kW⋅h (Gaussian version);
• 1 Planck angle is 1 radian (both versions);
• 1 Planck solid angle is 1 steradian (both versions);
• 1 Planck charge is about 3.3 elementary charges (Lorentz–Heaviside version) or 11.7 elementary charges (Gaussian version);
• 1 Planck impedance is about 377 ohms (Lorentz–Heaviside version) or 30 ohms (Gaussian version);
• 1 Planck admittance is about 2.65 mS (Lorentz–Heaviside version) or 33.4 mS (Gaussian version);
• 1 Planck permeability is about 1.26 μH/m (Lorentz–Heaviside version) or 0.1 μH/m (Gaussian version);
• 1 Planck electric flux is about 59.8 mV⋅μm (Lorentz–Heaviside version) or 16.9 mV⋅μm (Gaussian version).

However, most Planck units are many orders of magnitude too large or too small to be of 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 according to our current understanding. For example:

• 1 Planck speed is the speed of light in a vacuum, the maximum possible physical speed in special relativity; 1 nano-Planck speed is about 1.079 km/h.
• Our understanding of the Big Bang begins with the Planck epoch, when the universe was 1 Planck time old and 1 Planck length in diameter, and had a Planck temperature of 1. At that moment, 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 that would incorporate quantum effects into general relativity. Such a theory does not yet exist.

In Planck units, we have:

$\alpha ={\frac {e^{2}}{4\pi }}$ (Lorentz–Heaviside version)

$\alpha =e^{2}$ (Gaussian version)

$\alpha _{G}={\frac {m_{e}^{2}}{4\pi }}$ (Lorentz–Heaviside version)

$\alpha _{G}=m_{e}^{2}$ (Gaussian version)

where

$\alpha$ is the fine-structure constant

$e$ is the elementary charge

$\alpha _{G}$ is the gravitational coupling constant

$m_{e}$ is the electron rest mass

Hence the specific charge of electron (${\frac {e}{m_{e}}}$) is ${\sqrt {\frac {\alpha }{\alpha _{G}}}}$ Planck specific charge, in both two versions of Planck units.

Significance

Planck units are free of anthropocentric arbitrariness. Some physicists argue that communication with extraterrestrial intelligence would have to employ such a system of units in order to be understood. Unlike the metre and second, which exist as base units in the SI system for historical reasons, the Planck length and Planck time are conceptually linked at a fundamental physical level.

Natural units help physicists to reframe questions. Frank Wilczek puts 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/(13 quintillion)].

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 is comparing apples to oranges, because mass and electric charge are incommensurable quantities. Rather, the disparity of magnitude of force is a manifestation of the fact that the charge on the protons is approximately the unit charge but the mass of the protons is far less than the unit mass.

Cosmology

Main article: Chronology of the Universe

In Big Bang cosmology, the Planck epoch or Planck era is the earliest stage of the Big Bang, before the time passed was equal to the Planck time, t_P, or approximately 10⁻⁴³ seconds. There is no currently available physical theory to describe such short times, and it is not clear in what sense the concept of time is meaningful for values smaller than the Planck time. It is generally assumed that quantum effects of gravity dominate physical interactions at this time scale. At this scale, the unified force of the Standard Model is assumed to be unified with gravitation. Immeasurably hot and dense, the state of the Planck epoch was succeeded by the grand unification epoch, where gravitation is separated from the unified force of the Standard Model, in turn followed by the inflationary epoch, which ended after about 10⁻³² seconds (or about 10¹⁰ t_P).

Relative to the Planck epoch, the observable universe today looks extreme when expressed in Planck units, as in this set of approximations:

Further information: Time-variation of physical constants and Dirac large numbers hypothesis

The recurrence of large numbers close or related to 10⁶⁰ in the above table is a coincidence that intrigues some theorists. It is an example of the kind of large numbers coincidence that led theorists such as Eddington and Dirac to develop alternative physical theories (e.g. a variable speed of light or Dirac varying-G theory). After the measurement of the cosmological constant in 1998, estimated at 10⁻¹²² in Planck units, it was noted that this is suggestively close to the reciprocal of the age of the universe squared. Barrow and Shaw (2011) proposed a modified theory in which Λ is a field evolving in such a way that its value remains Λ ~ T⁻² throughout the history of the universe.

Table 4: Some common physical quantities

Quantities	In Lorentz–Heaviside version Planck units	In Gaussian version Planck units
Quantities of Earth or universe
Standard gravity ($g$)	6.25154×10−51 $g_{\text{P}}$	1.76353×10−51 $g_{\text{P}}$
Standard atmosphere ($atm$)	3.45343×10−108 $\Pi _{\text{P}}$	2.18691×10−109 $\Pi _{\text{P}}$
Speed of sound ($v_{s}$)	1.14479×10−6 $v_{\text{P}}$
Mean solar day	4.52091×1047 $t_{\text{P}}$	1.60262×1048 $t_{\text{P}}$
Equatorial radius of the Earth	1.11323×1041 $l_{\text{P}}$	3.94629×1041 $l_{\text{P}}$
Equatorial circumference of the Earth	6.99465×1041 $l_{\text{P}}$	2.47954×1042 $l_{\text{P}}$
Diameter of the observable universe	1.53594×1061 $l_{\text{P}}$	5.44477×1061 $l_{\text{P}}$
Volume of the Earth	1.89062×1055 $V_{\text{P}}$	6.70208×1055 $V_{\text{P}}$
Volume of the observable universe	6.98156×10114 $V_{\text{P}}$	2.47490×10115 $V_{\text{P}}$
Mass of the Earth	9.72717×1032 $m_{\text{P}}$	2.74398×1032 $m_{\text{P}}$
Mass of the observable universe	2.37796×1061 $m_{\text{P}}$	6.70811×1060 $m_{\text{P}}$
Mean density of Earth	1.68905×10−91 $d_{\text{P}}$	1.06960×10−93 $d_{\text{P}}$
Density of the universe	3.03257×10−121 $d_{\text{P}}$	1.92040×10−123 $d_{\text{P}}$
Age of the Earth	7.49657×1059 $t_{\text{P}}$	2.65747×1060 $t_{\text{P}}$
Age of the universe	2.27853×1060 $t_{\text{P}}$	8.07719×1060 $t_{\text{P}}$
Mean temperature of the Earth	7.18485×10−30 $T_{\text{P}}$	2.02681×10−30 $T_{\text{P}}$
Temperature of the universe	6.81806×10−32 $T_{\text{P}}$	1.92334×10−32 $T_{\text{P}}$
Hubble constant ($H_{0}$)	4.20446×10−61 $t_{\text{P}}^{-1}$	1.18605×10−61 $t_{\text{P}}^{-1}$
Cosmological constant ($\Lambda$)	3.62922×10−121 $l_{\text{P}}^{-2}$	2.88805×10−122 $l_{\text{P}}^{-2}$
vacuum energy density ($\rho _{\text{vacuum}}$)	1.82567×10−121 $\rho _{\text{P}}$	1.15612×10−123 $\rho _{\text{P}}$
Quantities of given substance
Melting point of water	6.83432×10−30 $T_{\text{P}}$	1.92793×10−30 $T_{\text{P}}$
Boiling point of water	9.33636×10−30 $T_{\text{P}}$	2.63374×10−30 $T_{\text{P}}$
Pressure of triple point of water	2.08469×10−109 $\Pi _{\text{P}}$	1.32015×10−111 $\Pi _{\text{P}}$
Temperature of triple point of water	6.83457×10−30 $T_{\text{P}}$	1.92800×10−30 $T_{\text{P}}$
Pressure of critical point of water	7.52001×10−105 $\Pi _{\text{P}}$	4.76210×10−107 $\Pi _{\text{P}}$
Temperature of critical point of water	1.61906×10−29 $T_{\text{P}}$	4.56728×10−30 $T_{\text{P}}$
Density of water	3.06320×10−92 $d_{\text{P}}$	1.93980×10−94 $d_{\text{P}}$
Specific heat capacity of water	1.86061×1018 $c_{\text{P}}$	6.59570×1018 $c_{\text{P}}$
Molar volume of ideal ($V_{m}$)	2.00522×1077 ${\mathcal {V}}_{\text{P}}$	8.93256×1078 ${\mathcal {V}}_{\text{P}}$
hydrogen ion activity of water	1.13259×10−83 ${\mathcal {C}}_{\text{P}}$	2.54249×10−85 ${\mathcal {C}}_{\text{P}}$
Properties of prototype object or particle
Elementary charge ($e$)	3.02822×10−1 $q_{\text{P}}$	8.54245×10−2 $q_{\text{P}}$
Electron rest mass ($m_{e}$)	1.48368×10−22 $m_{\text{P}}$	4.18539×10−23 $m_{\text{P}}$
Proton rest mass ($m_{p}$)	2.72427×10−19 $m_{\text{P}}$	7.68502×10−20 $m_{\text{P}}$
Neutron rest mass ($m_{n}$)	2.72802×10−19 $m_{\text{P}}$	7.69562×10−20 $m_{\text{P}}$
Atomic mass constant ($u$)	2.70459×10−19 $m_{\text{P}}$	7.62951×10−20 $m_{\text{P}}$
Charge-to-mass ratio of electron ($\xi _{e}$)	−2.04102×1021 $\xi _{\text{P}}$
Charge-to-mass ratio of proton ($\xi _{p}$)	1.11157×1018 $\xi _{\text{P}}$
Classical electron radius ($r_{e}$)	4.91840×1019 $l_{\text{P}}$	1.74353×1020 $l_{\text{P}}$
charge radius of proton	1.46860×1019 $l_{\text{P}}$	5.20606×1019 $l_{\text{P}}$
Compton wavelength of electron ($\lambda _{c}$)	2.71873×1028 $l_{\text{P}}$	9.63763×1028 $l_{\text{P}}$
Compton wavelength of proton ($\lambda _{cp}$)	2.30637×1019 $l_{\text{P}}$	8.17588×1019 $l_{\text{P}}$
Compton wavelength of neutron ($\lambda _{cn}$)	2.30320×1019 $l_{\text{P}}$	8.16463×1019 $l_{\text{P}}$
Electron magnetic moment ($\mu _{e}$)	−1.02169×1021 ${\mathcal {M}}_{\text{P}}$
Proton magnetic moment ($\mu _{p}$)	1.55223×1018 ${\mathcal {M}}_{\text{P}}$
Neutron magnetic moment ($\mu _{n}$)	−1.06324×1018 ${\mathcal {M}}_{\text{P}}$
Electric polarizability of proton	6.41245×1054 $V_{\text{P}}$	2.85653×1056 $V_{\text{P}}$
Magnetic polarizability of proton	1.03684×1054 $V_{\text{P}}$	4.61877×1055 $V_{\text{P}}$
gyromagnetic ratio of proton ($\gamma _{p}$)	3.10445×1018 $\Theta _{\text{P}}$
Quantities of given nuclear
covalent radius of hydrogen	5.41071×1023 $l_{\text{P}}$	1.91805×1024 $l_{\text{P}}$
Van der Waals radius of hydrogen	2.09447×1024 $l_{\text{P}}$	7.42469×1024 $l_{\text{P}}$
mass of the isotope 1H	2.72575×10−19 $m_{\text{P}}$	7.68921×10−20 $m_{\text{P}}$
1st ionization energy of hydrogen	3.94815×10−27 $E_{\text{P}}/n_{\text{P}}$	1.11375×10−27 $E_{\text{P}}/n_{\text{P}}$
excess energy of the isotope 1H	2.11635×10−24 $E_{\text{P}}$	5.97011×10−25 $E_{\text{P}}$
mean lifetime of neutron	4.61257×1045 $t_{\text{P}}$	1.63511×1046 $t_{\text{P}}$
half-life of tritium	2.03432×1051 $t_{\text{P}}$	7.21146×1051 $t_{\text{P}}$
half-life of beryllium-8	4.28739×1026 $t_{\text{P}}$	1.51984×1027 $t_{\text{P}}$
Physical constants which are not normalized to 1 in both versions of Planck units
Faraday constant ($F$)	3.02822×10−1 $q_{\text{P}}/n_{\text{P}}$	8.54245×10−2 $q_{\text{P}}/n_{\text{P}}$
Bohr radius ($a_{0}$)	9.23620×1023 $l_{\text{P}}$	3.27415×1024 $l_{\text{P}}$
Bohr magneton ($\mu _{B}$)	1.02051×1021 $E_{\text{P}}/B_{\text{P}}$
Magnetic flux quantum ($\varphi _{0}$)	10.3744 $\Psi _{\text{P}}$	36.7762 $\Psi _{\text{P}}$
Rydberg constant ($R_{\infty }$)	6.28727×10−28 $l_{\text{P}}^{-1}$	1.77361×10−28 $l_{\text{P}}^{-1}$
Josephson constant ($K_{J}$)	9.63913×10−2 $f_{\text{P}}/U_{\text{P}}$	2.71915×10−2 $f_{\text{P}}/U_{\text{P}}$
von Klitzing constant ($R_{K}$)	68.5180 $Z_{\text{P}}$	861.023 $Z_{\text{P}}$
Stefan–Boltzmann constant ($\sigma$)	1.64493×10−1 $P_{\text{P}}/l_{\text{P}}^{2}/T_{\text{P}}^{4}$

History

Natural units began in 1881, when George Johnstone Stoney, noting that electric charge is quantized, derived units of length, time, and mass, now named Stoney units in his honor, by normalizing G, c, ⁠1/4πε₀⁠, k_B, and the electron charge, e, to 1.

Already in 1899 (i.e. one year before the advent of quantum theory) Max Planck introduced what became later known as Planck's constant. At the end of the paper, Planck introduced, as a consequence of his discovery, the base units later named in his honor. The Planck units are based on the quantum of action, now usually known as Planck's constant. Planck called the constant b in his paper, though h (or ħ) is now common. However, at that time it was entering Wien's radiation law which Planck thought to be correct. Planck underlined the universality of the new unit system, writing:

...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 considered only the units based on the universal constants G, ħ, c, and k_B to arrive at natural units for length, time, mass, and temperature. Planck did not adopt any electromagnetic units. However, since the non-rationalized gravitational constant, G, is set to 1, a natural extension of Planck units to a unit of electric charge is to also set the non-rationalized Coulomb constant, k_e, to 1 as well (as well as the Stoney units). This is the non-rationalized Planck units (Planck units with the Gaussian version), which is more convenient but not rationalized, there is also a Planck system which is rationalized (Planck units with the Lorentz–Heaviside version), set 4πG and ε₀ (instead of G and k_e) to 1, which may be less convenient but is rationalized. Another convention is to use the elementary charge as the basic unit of electric charge in the Planck system. Such a system is convenient for black hole physics. The two conventions for unit charge differ by a factor of the square root of the fine-structure constant. Planck's paper also gave numerical values for the base units that were close to modern values.

List of physical equations

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). In theoretical physics, however, this scruple can be set aside, by a process called nondimensionalization. Table 6 shows how the use of Planck units simplifies many fundamental equations of physics, because this gives each of the five fundamental constants, and products of them, a simple numeric value of 1. In the SI form, the units should be accounted for. In the nondimensionalized form, the units, which are now Planck units, need not be written if their use is understood.

Table 5: How Planck units simplify the key equations of physics

Physical equation	SI form	Lorentz–Heaviside version Planck form	Gaussian version Planck form
Only include $c$
Mass–energy equivalence in special relativity	${E=mc^{2}}\$	${E=m}\$
Energy–momentum relation	$E^{2}=m^{2}c^{4}+p^{2}c^{2}\;$	$E^{2}=m^{2}+p^{2}\;$
Include $c$ and $G$
Newton's law of universal gravitation	$F=-G{\frac {m_{1}m_{2}}{r^{2}}}$	$F=-{\frac {m_{1}m_{2}}{4\pi r^{2}}}$	$F=-{\frac {m_{1}m_{2}}{r^{2}}}$
Einstein field equations in general relativity	${G_{\mu \nu }={8\pi G \over c^{4}}T_{\mu \nu }}\$	${G_{\mu \nu }=2T_{\mu \nu }}\$	${G_{\mu \nu }=8\pi T_{\mu \nu }}\$
Einstein's constant κ defined	$\kappa ={\frac {8\pi G}{c^{2}}}$	$\kappa =2$	$\kappa =8\pi$
The formula of Schwarzschild radius	$r_{s}={\frac {2Gm}{c^{2}}}$	$r_{s}={\frac {m}{2\pi }}$	$r_{s}=2m$
Gauss's law for gravity	$\mathbf {g} \cdot d\mathbf {A} =-4\pi GM$ $\nabla \cdot \mathbf {g} =-4\pi G\rho$	$\mathbf {g} \cdot d\mathbf {A} =-M$ $\nabla \cdot \mathbf {g} =-\rho$	$\mathbf {g} \cdot d\mathbf {A} =-4\pi M$ $\nabla \cdot \mathbf {g} =-4\pi \rho$
Poisson's equation	${\nabla }^{2}\phi =4\pi G\rho$ $\phi (r)={\dfrac {-Gm}{r}}$	${\nabla }^{2}\phi =\rho$ $\phi (r)={\dfrac {-m}{4\pi r}}$	${\nabla }^{2}\phi =4\pi \rho$ $\phi (r)={\dfrac {-m}{r}}$
The characteristic impedance	$Z_{0}={\frac {4\pi G}{c}}$	$Z_{0}=1$	$Z_{0}=4\pi$
The characteristic admittance	$Y_{0}={\frac {c}{4\pi G}}$	$Y_{0}=1$	$Y_{0}={\frac {1}{4\pi }}$
GEM equations	$\nabla \cdot \mathbf {E_{g}} =-4\pi G\rho _{g}$ $\nabla \cdot \mathbf {D_{g}} =\rho _{g}f$ $\nabla \cdot \mathbf {B_{g}} =0\$ $\nabla \times \mathbf {E_{g}} =-{\frac {\partial \mathbf {B_{g}} }{\partial t}}$ $\nabla \times \mathbf {B_{g}} ={\frac {1}{c^{2}}}\left(-4\pi G\mathbf {J_{g}} +{\frac {\partial \mathbf {E_{g}} }{\partial t}}\right)$ $\nabla \times \mathbf {H_{g}} =\mathbf {J_{g}} f+{\frac {\partial \mathbf {D_{g}} }{\partial t}}$	$\nabla \cdot \mathbf {E_{g}} =\rho _{g}$ $\nabla \cdot \mathbf {D_{g}} =\rho _{g}f$ $\nabla \cdot \mathbf {B_{g}} =0\$ $\nabla \times \mathbf {E_{g}} =-{\frac {\partial \mathbf {B_{g}} }{\partial t}}$ $\nabla \times \mathbf {B_{g}} =\mathbf {J_{g}} +{\frac {\partial \mathbf {E_{g}} }{\partial t}}$ $\nabla \times \mathbf {H_{g}} =\mathbf {J_{g}} f+{\frac {\partial \mathbf {D_{g}} }{\partial t}}$	$\nabla \cdot \mathbf {E_{g}} =4\pi \rho _{g}\$ $\nabla \cdot \mathbf {D_{g}} =\rho _{g}f$ $\nabla \cdot \mathbf {B_{g}} =0\$ $\nabla \times \mathbf {E_{g}} =-{\frac {\partial \mathbf {B_{g}} }{\partial t}}$ $\nabla \times \mathbf {B_{g}} =4\pi \mathbf {J_{g}} +{\frac {\partial \mathbf {E_{g}} }{\partial t}}$ $\nabla \times \mathbf {H_{g}} =\mathbf {J_{g}} f+{\frac {\partial \mathbf {D_{g}} }{\partial t}}$
Include $c$, $G$, and $\hbar$
Planck–Einstein relation	${E=h\nu }\$ ${E=\hbar \omega }\$	${E=2\pi \nu }\$ ${E=\omega }\$
Heisenberg's uncertainty principle	$\Delta x\cdot \Delta p\geq {\frac {\hbar }{2}}$	$\Delta x\cdot \Delta p\geq {\frac {1}{2}}$
Energy of photon	$E=\hbar \omega =h\nu ={\frac {hc}{\lambda }}$ = ⁠ħc/ƛ⁠	$E=\omega =2\pi \nu ={\frac {2\pi }{\lambda }}$ = ⁠1/ƛ⁠
Momentum of photon	$p=\hbar k={\frac {h\nu }{c}}={\frac {h}{\lambda }}$ = ⁠ħ/ƛ⁠	$p=k=2\pi \nu ={\frac {2\pi }{\lambda }}$ = ⁠1/ƛ⁠
Wavelength and reduced wavelength of matter wave	$\lambda ={\frac {h}{mv}}={\frac {2\pi \hbar }{mv}}$ ƛ = ⁠ħ/mv⁠	$\lambda ={\frac {2\pi }{mv}}$ ƛ = ⁠1/mv⁠
The formula of Compton wavelength and reduced Compton wavelength	$\lambda ={\frac {h}{mc}}={\frac {2\pi \hbar }{mc}}$ ƛ = ⁠ħ/mc⁠	$\lambda ={\frac {2\pi }{m}}$ ƛ = ⁠1/m⁠
Schrödinger's equation	$-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi (\mathbf {r} ,t)+V(\mathbf {r} ,t)\psi (\mathbf {r} ,t)=i\hbar {\frac {\partial \psi (\mathbf {r} ,t)}{\partial t}}$	$-{\frac {1}{2m}}\nabla ^{2}\psi (\mathbf {r} ,t)+V(\mathbf {r} ,t)\psi (\mathbf {r} ,t)=i{\frac {\partial \psi (\mathbf {r} ,t)}{\partial t}}$
Schrödinger's equation	$i\hbar {\frac {\partial }{\partial t}}\psi =H\cdot \psi$	$i{\frac {\partial }{\partial t}}\psi =H\cdot \psi$
Hamiltonian form of Schrödinger's equation	$H\left|\psi _{t}\right\rangle =i\hbar {\frac {\partial }{\partial t}}\left|\psi _{t}\right\rangle$	$H\left|\psi _{t}\right\rangle =i{\frac {\partial }{\partial t}}\left|\psi _{t}\right\rangle$
Covariant form of the Dirac equation	$\ (i\hbar \gamma ^{\mu }\partial _{\mu }-mc)\psi =0$	$\ (i\gamma ^{\mu }\partial _{\mu }-m)\psi =0$
The main role in quantum gravity	$\Delta r_{s}\Delta r\geq {\frac {\hbar G}{c^{3}}}$	$\Delta r_{s}\Delta r\geq {\frac {1}{4\pi }}$	$\Delta r_{s}\Delta r\geq 1$
Include $c$, $G$, $\hbar$, and $\epsilon _{0}$
The vacuum permeability	$\mu _{0}={\frac {1}{\epsilon _{0}c^{2}}}$	$\mu _{0}=1$	$\mu _{0}=4\pi$
The impedance of free space	$Z_{0}={\frac {\mathbf {E} }{\mathbf {H} }}={\sqrt {\frac {\mu _{0}}{\epsilon _{0}}}}={\frac {1}{\epsilon _{0}c}}=\mu _{0}c$	$Z_{0}=1$	$Z_{0}=4\pi$
The admittance of free space	$Y_{0}={\frac {\mathbf {H} }{\mathbf {E} }}={\sqrt {\frac {\epsilon _{0}}{\mu _{0}}}}=\epsilon _{0}c={\frac {1}{\mu _{0}c}}$	$Y_{0}=1$	$Y_{0}={\frac {1}{4\pi }}$
The Coulomb constant	$k_{e}={\frac {1}{4\pi \epsilon _{0}}}$	$k_{e}={\frac {1}{4\pi }}$	$k_{e}=1$
Coulomb's law	$F=k_{e}{\frac {q_{1}q_{2}}{r^{2}}}={\frac {1}{4\pi \epsilon _{0}}}{\frac {q_{1}q_{2}}{r^{2}}}$	$F={\frac {q_{1}q_{2}}{4\pi r^{2}}}$	$F={\frac {q_{1}q_{2}}{r^{2}}}$
Coulomb's law for two stationary magnetic charge	$F=k_{m}{\frac {b_{1}b_{2}}{r^{2}}}={\frac {\mu _{0}}{4\pi }}{\frac {b_{1}b_{2}}{r^{2}}}$	$F={\frac {4\pi b_{1}b_{2}}{r^{2}}}$	$F={\frac {b_{1}b_{2}}{r^{2}}}$
Biot–Savart law	$\Delta B={\frac {\mu _{0}I}{4\pi }}{\frac {\Delta L}{r^{2}}}\sin \theta$	$\Delta B={\frac {I}{4\pi }}{\frac {\Delta L}{r^{2}}}\sin \theta$	$\Delta B=I{\frac {\Delta L}{r^{2}}}\sin \theta$
Biot–Savart law	$\mathbf {B} (\mathbf {r} )={\frac {\mu _{0}}{4\pi }}\int _{C}{\frac {I\,d{\boldsymbol {\ell }}\times \mathbf {r'} }{|\mathbf {r'} |^{3}}}$	$\mathbf {B} (\mathbf {r} )={\frac {1}{4\pi }}\int _{C}{\frac {I\,d{\boldsymbol {\ell }}\times \mathbf {r'} }{|\mathbf {r'} |^{3}}}$	$\mathbf {B} (\mathbf {r} )=\int _{C}{\frac {I\,d{\boldsymbol {\ell }}\times \mathbf {r'} }{|\mathbf {r'} |^{3}}}$
Equation of electric field intensity and electric induction and polarization	$\mathbf {D} =\epsilon _{0}\mathbf {E} +\mathbf {P}$	$\mathbf {D} =\mathbf {E} +\mathbf {P}$	$\mathbf {D} ={\frac {\mathbf {E} }{4\pi }}+\mathbf {P}$
Equation of magnetic field intensity and magnetic induction and magnetization	$\mathbf {H} ={\frac {\mathbf {B} }{\mu _{0}}}+\mathbf {M}$	$\mathbf {H} =\mathbf {B} +\mathbf {M}$	$\mathbf {H} ={\frac {\mathbf {B} }{4\pi }}+\mathbf {M}$
Maxwell's equations	$\nabla \cdot \mathbf {E} ={\frac {1}{\epsilon _{0}}}\rho$ $\nabla \cdot \mathbf {D} =\rho _{f}$ $\nabla \cdot \mathbf {B} =0\$ $\nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}$ $\nabla \times \mathbf {B} ={\frac {1}{c^{2}}}\left({\frac {1}{\epsilon _{0}}}\mathbf {J} +{\frac {\partial \mathbf {E} }{\partial t}}\right)$ $\nabla \times \mathbf {H} =\mathbf {J} _{f}+{\frac {\partial \mathbf {D} }{\partial t}}$	$\nabla \cdot \mathbf {E} =\rho$ $\nabla \cdot \mathbf {D} =\rho _{f}$ $\nabla \cdot \mathbf {B} =0\$ $\nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}$ $\nabla \times \mathbf {B} =\mathbf {J} +{\frac {\partial \mathbf {E} }{\partial t}}$ $\nabla \times \mathbf {H} =\mathbf {J} _{f}+{\frac {\partial \mathbf {D} }{\partial t}}$	$\nabla \cdot \mathbf {E} =4\pi \rho \$ $\nabla \cdot \mathbf {D} =\rho _{f}$ $\nabla \cdot \mathbf {B} =0\$ $\nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}$ $\nabla \times \mathbf {B} =4\pi \mathbf {J} +{\frac {\partial \mathbf {E} }{\partial t}}$ $\nabla \times \mathbf {H} =\mathbf {J} _{f}+{\frac {\partial \mathbf {D} }{\partial t}}$
Poynting vector	${\mathcal {S}}_{\text{g}}=-{\frac {c^{2}}{4\pi G}}\mathbf {E} _{\text{g}}\times 4\mathbf {B} _{\text{g}}$ ${\mathcal {S}}=c^{2}\epsilon _{0}\mathbf {E} \times \mathbf {B}$	${\mathcal {S}}_{\text{g}}=-\mathbf {E} _{\text{g}}\times 4\mathbf {B} _{\text{g}}$ ${\mathcal {S}}=\mathbf {E} \times \mathbf {B}$	${\mathcal {S}}_{\text{g}}=-{\frac {1}{4\pi }}\mathbf {E} _{\text{g}}\times 4\mathbf {B} _{\text{g}}$ ${\mathcal {S}}=-{\frac {1}{4\pi }}\mathbf {E} \times \mathbf {B}$
Josephson constant KJ defined	$K_{J}={\frac {e}{\pi \hbar }}$	$K_{J}={\sqrt {\frac {4\alpha }{\pi }}}$	$K_{J}={\frac {\sqrt {\alpha }}{\pi }}$
von Klitzing constant RK defined	$R_{K}={\frac {2\pi \hbar }{e^{2}}}$	$R_{K}={\frac {1}{2\alpha }}$	$R_{K}={\frac {2\pi }{\alpha }}$
The charge-to-mass ratio of electron	$\xi _{e}={\frac {e}{m_{e}}}={\sqrt {\frac {G\alpha }{k_{e}\alpha _{G}}}}={\sqrt {\frac {4\pi G\epsilon _{0}\alpha }{\alpha _{G}}}}$	$\xi _{e}={\sqrt {\frac {\alpha }{\alpha _{G}}}}$
The Bohr radius	$a_{0}={\frac {4\pi \epsilon _{0}\hbar ^{2}}{m_{\text{e}}e^{2}}}={\frac {\hbar }{m_{\text{e}}c\alpha }}$	$a_{0}={\frac {1}{\alpha {\sqrt {4\pi \alpha _{G}}}}}$	$a_{0}={\frac {1}{\alpha {\sqrt {\alpha _{G}}}}}$
The Bohr magneton	$\mu _{B}={\frac {e\hbar }{2m_{e}}}$	$\mu _{B}={\sqrt {\frac {\alpha }{4\alpha _{G}}}}$
Rydberg constant R∞ defined	$R_{\infty }={\frac {m_{\text{e}}e^{4}}{8\epsilon _{0}^{2}h^{3}c}}={\frac {\alpha ^{2}m_{\text{e}}c}{4\pi \hbar }}$	$R_{\infty }={\sqrt {\frac {\alpha ^{4}\alpha _{G}}{4\pi }}}$	$R_{\infty }={\frac {\sqrt {\alpha ^{4}\alpha _{G}}}{4\pi }}$
Include $c$, $G$, $\hbar$, $\epsilon _{0}$, and $k_{\text{B}}$
Ideal gas law	$PV=nRT=Nk_{\text{B}}T$	$PV=NT$
Equation of the root-mean-square speed	$v_{rms}={\sqrt {\frac {3RT}{M}}}={\sqrt {\frac {3k_{\text{B}}T}{m}}}$	$v_{rms}={\sqrt {\frac {3T}{m}}}$
Kinetic theory of gases	$\Sigma {\frac {1}{2}}mv^{2}={\frac {3}{2}}Nk_{\text{B}}T$	$\Sigma {\frac {1}{2}}mv^{2}={\frac {3}{2}}NT$
Unruh temperature	$T={\frac {\hbar a}{2\pi ck_{B}}}$	$T={\frac {a}{2\pi }}$
Thermal energy per particle per degree of freedom	${E={\tfrac {1}{2}}k_{\text{B}}T}\$	${E={\tfrac {1}{2}}T}\$
Boltzmann's entropy formula	${S=k_{\text{B}}\ln \Omega }\$	${S=\ln \Omega }\$
Stefan–Boltzmann constant σ defined	$\sigma ={\frac {\pi ^{2}k_{\text{B}}^{4}}{60\hbar ^{3}c^{2}}}$	$\sigma ={\frac {\pi ^{2}}{60}}$
Planck's law (surface intensity per unit solid angle per unit angular frequency) for black body at temperature T.	$I(\omega ,T)={\frac {\hbar \omega ^{3}}{4\pi ^{3}c^{2}}}~{\frac {1}{e^{\frac {\hbar \omega }{k_{\text{B}}T}}-1}}$	$I(\omega ,T)={\frac {\omega ^{3}}{4\pi ^{3}}}~{\frac {1}{e^{\omega /T}-1}}$
The formula of Unruh temperature	$T={\frac {\hbar a}{2\pi ck_{B}}}$	$T={\frac {a}{2\pi }}$
Hawking temperature of a black hole	$T_{H}={\frac {\hbar c^{3}}{8\pi GMk_{B}}}$	$T_{H}={\frac {1}{2M}}$	$T_{H}={\frac {1}{8\pi M}}$
Bekenstein–Hawking black hole entropy	$S_{\text{BH}}={\frac {A_{\text{BH}}k_{\text{B}}c^{3}}{4G\hbar }}={\frac {4\pi Gk_{\text{B}}m_{\text{BH}}^{2}}{\hbar c}}$	$S_{\text{BH}}=\pi A_{\text{BH}}=m_{\text{BH}}^{2}$	$S_{\text{BH}}={\frac {A_{\text{BH}}}{4}}=4\pi m_{\text{BH}}^{2}$

Note:

• For the elementary charge $e$:

$e={\sqrt {4\pi \alpha }}$ (Lorentz–Heaviside version)

$e={\sqrt {\alpha }}$ (Gaussian version)

where $\alpha$ is the fine-structure constant.

• For the electron rest mass $m_{e}$:

$m_{e}={\sqrt {4\pi \alpha _{G}}}$ (Lorentz–Heaviside version)

$m_{e}={\sqrt {\alpha _{G}}}$ (Gaussian version)

where $\alpha _{G}$ is the gravitational coupling constant.

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 4π is ubiquitous in theoretical physics because the surface area of a sphere of radius r is 4πr². This, along with the concept of flux, are the basis for the inverse-square law, Gauss's law, and the divergence operator applied to flux density. 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 in rationalized form, for example, follows from the flux of an electrostatic field being distributed uniformly on the surface of a sphere. Likewise for Newton's law of universal gravitation. (If space had more than three spatial dimensions, the factor 4π would have to be changed according to the geometry of the sphere in higher dimensions.)

Hence a substantial body of physical theory developed since Planck (1899) suggests normalizing not G but either 4πG (or 8πG or 16πG) to 1. Doing so would introduce a factor of ⁠1/4π⁠ (or ⁠1/8π⁠ or ⁠1/16π⁠) 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 of ⁠1/4π⁠ in the nondimensionalized form of Coulomb's law as well, so that the nondimensionalized Maxwell's equations for electromagnetism and gravitoelectromagnetism both 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" Lorentz–Heaviside units. When applied additionally to gravitation and Planck units, these are called rationalized Planck units and are seen in high-energy physics.

The rationalized Planck units are defined so that $c=4\pi G=\hbar =\epsilon _{0}=k_{\text{B}}=1$. These are the Planck units based on Lorentz–Heaviside units (instead of on the more conventional Gaussian units) as depicted above.

There are several possible alternative normalizations.

Gravity

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 of general relativity in 1915). Hence Planck normalized to 1 the gravitational constant G in Newton's law. In theories emerging after 1899, G nearly always appears in formulae multiplied by 4π or a small integer multiple thereof. Hence, a choice to be made when designing a system of natural units is which, if any, instances of 4π appearing in the equations of physics are to be eliminated via the normalization.

• Normalizing 4πG to 1: (like the Lorentz–Heaviside version Planck units)
  • Newton's law of universal gravitation has 4πr² remaining in the denominator (which is the surface area of the enclosing sphere at radius r).
  • Gauss's law for gravity becomes Φ_g = −M (rather than Φ_g = −4πM in Gaussian version Planck units).
  • Eliminates 4πG from the Poisson equation.
  • Eliminates 4πG in the gravitoelectromagnetic (GEM) equations, which hold in weak gravitational fields or locally 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.
  • Normalizes the characteristic impedance Z₀ of gravitational radiation in free space to 1. (Normally expressed as ⁠4πG/c⁠)
  • Eliminates 4πG from the Bekenstein–Hawking formula (for the entropy of a black hole in terms of its mass m_BH and the area of its event horizon A_BH) which is simplified to S_BH = πA_BH = (m_BH)².
  • In this case the electron rest mass, measured in terms of this rationalized Planck mass, is

$m_{e}={\sqrt {4\pi \alpha _{G}}}\cdot m_{\text{P}}\approx 1.48368\times 10^{-22}\cdot m_{\text{P}}\,$

where ${\alpha _{G}}\$ is the gravitational coupling constant. This convention is seen in high-energy physics.

• Setting 8πG = 1. This would eliminate 8πG from the Einstein field equations, Einstein–Hilbert action, and the Friedmann equations, for gravitation. Planck units modified so that 8πG = 1 are known as reduced Planck units, because the Planck mass is divided by √8π. Also, the Bekenstein–Hawking formula for the entropy of a black hole simplifies to S_BH = (m_BH)²/2 = 2πA_BH.
• Setting 16πG = 1. This would eliminate the constant ⁠c⁴/16πG⁠ from the Einstein–Hilbert action. The form of the Einstein field equations with cosmological constant Λ becomes R_μν + Λg_μν = ⁠1/2⁠(Rg_μν + T_μν).

Electromagnetism

In order to build natural units in electromagnetism one can use:

• Lorentz–Heaviside units (classified as a rationalized system of electromagnetism units).
• Gaussian units (classified as a non-rationalized system of electromagnetism units).

Of these, Lorentz–Heaviside is somewhat more common, mainly because Maxwell's equations are simpler in Lorentz–Heaviside units than they are in Gaussian units.

In the two unit systems, the Planck unit charge q_P is:

• q_P = √4παħc (Lorentz–Heaviside),
• q_P = √αħc (Gaussian)

where ħ is the reduced Planck constant, c is the speed of light, and α ≈ ⁠1/137.036⁠ is the fine-structure constant.

In a natural unit system where c = 1, Lorentz–Heaviside units can be derived from units by setting ε₀ = μ₀ = 1. Gaussian units can be derived from units by a more complicated set of transformations, such as multiplying all electric fields by (4πε₀)^−1⁄2, multiplying all magnetic susceptibilities by 4π, and so on.

Planck units normalize to 1 the Coulomb force constant k_e = ⁠1/4πε₀⁠ (as does the cgs system of units and the Gaussian 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: (as does the Lorentz–Heaviside units) (like the Lorentz–Heaviside version Planck units)
  • Sets the permeability of free space μ₀ = 1 (because c = 1).
  • Sets the unit impedance or unit resistance to the characteristic impedance of free space, Z_P = Z₀ (or sets the characteristic impedance of free space Z₀ to 1).
  • Eliminates 4π from the nondimensionalized form of Maxwell's equations.
  • Coulomb's law has 4πr² remaining in the denominator (which is the surface area of the enclosing sphere at radius r).
  • Equates the notions of flux density and field strength in free space (electric field intensity E and electric induction D, magnetic field intensity H and magnetic induction B)
  • In this case the elementary charge, measured in terms of this rationalized Planck charge, is

$e={\sqrt {4\pi \alpha }}\cdot q_{\text{P}}\approx 0.302822121\cdot q_{\text{P}}\,$

where ${\alpha }\$ is the fine-structure constant. This convention is seen in high-energy physics.

Temperature

Planck normalized to 1 the Boltzmann constant k_B.

• Normalizing ⁠1/2⁠k_B to 1:
  • Removes the factor of ⁠1/2⁠ in the nondimensionalized equation for the thermal energy per particle per degree of freedom.
  • Introduces a factor of 2 into the nondimensionalized form of Boltzmann's entropy formula.

Planck units and the invariant scaling of nature

Some theorists (such as Dirac and Milne) have proposed cosmologies that conjecture that physical "constants" might actually change over time (e.g. a variable speed of light or Dirac varying-G theory). Such cosmologies have not gained mainstream acceptance and yet there is still considerable scientific interest in the possibility that physical "constants" might change, although such propositions introduce difficult questions. Perhaps the first question to address is: How would such a change make a noticeable operational difference in physical measurement or, more fundamentally, our perception of reality? If some particular physical constant had changed, how would we notice it, or how would physical reality be different? Which changed constants result in a meaningful and measurable difference in physical reality? If a physical constant that is not dimensionless, such as the speed of light, did in fact change, would we be able to notice it or measure it unambiguously? – a question examined by Michael Duff in his paper "Comment on time-variation of fundamental constants".

George Gamow argued in his book Mr Tompkins in Wonderland that a sufficient change in a dimensionful physical constant, such as the speed of light in a vacuum, would result in obvious perceptible changes. But this idea is challenged:

[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 Duff's "Comment on time-variation of fundamental constants" and Duff, Okun, and Veneziano's paper "Trialogue on the number of fundamental constants", particularly the section entitled "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 quantity.

We can notice a difference if some dimensionless physical quantity such as fine-structure constant, α, changes or the proton-to-electron mass ratio, ⁠m_p/m_e⁠, changes (atomic structures would change) but if all dimensionless physical quantities remained unchanged (this includes all possible ratios of identically dimensioned physical quantity), we cannot tell if a dimensionful quantity, such as the speed of light, c, has changed. And, indeed, the Tompkins concept becomes meaningless in our perception of reality 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 ⁠1/2⁠c (but with the axiom that all dimensionless physical quantities remain the same), then the Planck length would increase by a factor of 2√2 from the point of view of some unaffected observer on the outside. Measured by "mortal" observers in terms of Planck units, the new speed of light would remain as 1 new Planck length per 1 new Planck time – which is no different from the old measurement. But, since by axiom, the size of atoms (approximately the Bohr radius) are related to the Planck length by an unchanging dimensionless constant of proportionality:

$a_{0}={\frac {4\pi \epsilon _{0}\hbar ^{2}}{m_{e}e^{2}}}={\frac {m_{\text{P}}}{m_{e}\alpha }}l_{\text{P}}.$

Then atoms would be bigger (in one dimension) by 2√2, each of us would be taller by 2√2, and so would our metre sticks be taller (and wider and thicker) by a factor of 2√2. 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 4√2 (from the point of view of this unaffected observer on the outside) because the Planck time has increased by 4√2 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 unaffected observer on the outside might observe that light now propagates at half the speed that it previously did (as well as all other observed velocities) but it would still travel 299792458 of our new metres in the time elapsed by one of our new seconds (⁠1/2⁠c × 4√2 ÷ 2√2 continues to equal 299792458 m/s). We would not notice any difference.

This contradicts what George Gamow writes in his book Mr. Tompkins; there, Gamow suggests that if a dimension-dependent universal constant such as c changed significantly, 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. Gamow does not address this subtlety; the thought experiments he conducts in his popular works assume the second choice for "changing a physical constant". And Duff or Barrow would point out that ascribing a change in measurable reality, i.e. α, to a specific dimensional component quantity, such as c, is unjustified. The very same operational difference in measurement or perceived reality could just as well be caused by a change in h or e if α is changed and no other dimensionless constants are changed. It is only the dimensionless physical constants that ultimately matter in the definition of worlds.

This unvarying aspect of the Planck-relative scale, or that of any other system of natural units, leads many theorists to conclude that a hypothetical change in dimensionful physical constants can only be manifest as a change in dimensionless physical constants. One such dimensionless physical constant is the fine-structure constant. There are some experimental physicists who assert they have in fact measured a change in the fine structure constant and this has intensified the debate about the measurement of physical constants. According to some theorists there are some very special circumstances in which changes in the fine-structure constant can be measured as a change in dimensionful physical constants. Others however reject the possibility of measuring a change in dimensionful physical constants under any circumstance. The difficulty or even the impossibility of measuring changes in dimensionful physical constants has led some theorists to debate with each other whether or not a dimensionful physical constant has any practical significance at all and that in turn leads to questions about which dimensionful physical constants are meaningful.

See also

• cGh physics
• Dimensional analysis
• Doubly special relativity
• Gaussian units
• Hartree atomic units
• Lorentz–Heaviside units
• Rationalized Planck units
• Natural units
• Planck scale
• Planck particle
• Stoney units
• Zero-point energy

References

Sources

• Barrow, John D. (2002). The Constants of Nature; From Alpha to Omega – The Numbers that Encode the Deepest Secrets of the Universe. New York: Pantheon Books. ISBN 978-0-375-42221-8. Easier.
• Barrow, John D.; Tipler, Frank J. (1986). The Anthropic Cosmological Principle. Oxford: Claredon Press. ISBN 978-0-19-851949-2. Harder.
• Penrose, Roger (2005). "Section 31.1". The Road to Reality. New York: Alfred A. Knopf. ISBN 978-0-679-45443-4. {{cite book}}: Unknown parameter |nopp= ignored (|no-pp= suggested) (help)
• Planck, Max (1899). "Über irreversible Strahlungsvorgänge". Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin (in German). 5: 440–480. 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.
• Tomilin, K. A. (1999). Natural Systems of Units: To the Centenary Anniversary of the Planck System (PDF). Proceedings Of The XXII Workshop On High Energy Physics And Field Theory. pp. 287–296. Archived from the original (PDF) on 17 June 2006.

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. As of 2011, those pages had been removed from the planck.org web site. Use the Wayback Machine to access pre-2011 versions of the website. Good discussion of why 8πG should be normalized to 1 when doing general relativity and quantum gravity. Many links.
• "Planck Era" and "Planck Time" (up to 10⁻⁴³ seconds after birth of Universe) (University of Oregon).
• Constants of nature: Quantum Space Theory offers a different set of Planck units and defines 31 physical constants in terms of them.
• See the Tool bag.