Planck Epoch
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Planck Epoch

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 resistance 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 or (the Planck energy), time intervals around or (the Planck time) and lengths around or (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:

Each of these constants can be associated with a fundamental physical theory or concept: c with special relativity, G with general relativity, ? with quantum mechanics, ?0 with electromagnetism, and kB 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,

{\displaystyle {\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:

${\displaystyle {\frac {F}{F_{\text{P}}}}={\frac {\left({\dfrac {m_{1}}{m_{\text{P}}}}\right)\left({\dfrac {m_{2}}{m_{\text{P}}}}\right)}{\left({\dfrac {r}{l_{\text{P}}}}\right)^{2}}}.}$

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:

${\displaystyle F={\frac {m_{1}m_{2}}{r^{2}}}\ .}$

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 , 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."[1]

## Definition

Table 1: Dimensional universal physical constants normalized with Planck units
Constant Symbol Dimension Value (SI units)[2]
Speed of light in vacuum c L T-1 [3]
(exact by definition of metre)
Gravitational constant G
(1 for the Gaussian version, for the Lorentz-Heaviside version)
L3 M-1 T-2 [4]
Reduced Planck constant ? =
where h is the Planck constant
L2 M T-1 [5]
(exact by definition of the kilogram since 20 May 2019)
Vacuum permittivity ?0
(1 for the Lorentz-Heaviside version, for the Gaussian version)
[6]
(exact by definitions of ampere and metre until 20 May 2019)
Boltzmann constant kB L2 M T-2 ?-1 [7]
(exact by definition of the kelvin since 20 May 2019)

Key: L = length, M = mass, T = time, Q = charge, ? = 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 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:

${\displaystyle l_{\text{P}}=c\ t_{\text{P}}}$
${\displaystyle F_{\text{P}}={\frac {l_{\text{P}}m_{\text{P}}}{t_{\text{P}}^{2}}}=4\pi G\ {\frac {m_{\text{P}}^{2}}{l_{\text{P}}^{2}}}}$ (Lorentz-Heaviside version)
${\displaystyle F_{\text{P}}={\frac {l_{\text{P}}m_{\text{P}}}{t_{\text{P}}^{2}}}=G\ {\frac {m_{\text{P}}^{2}}{l_{\text{P}}^{2}}}}$ (Gaussian version)
${\displaystyle E_{\text{P}}={\frac {l_{\text{P}}^{2}m_{\text{P}}}{t_{\text{P}}^{2}}}=\hbar \ {\frac {1}{t_{\text{P}}}}}$
${\displaystyle C_{\text{P}}={\frac {t_{\text{P}}^{2}q_{\text{P}}^{2}}{l_{\text{P}}^{2}m_{\text{P}}}}=\epsilon _{0}\ l_{\text{P}}}$ (Lorentz-Heaviside version)
${\displaystyle C_{\text{P}}={\frac {t_{\text{P}}^{2}q_{\text{P}}^{2}}{l_{\text{P}}^{2}m_{\text{P}}}}=4\pi \epsilon _{0}\ l_{\text{P}}}$ (Gaussian version)
${\displaystyle E_{\text{P}}={\frac {l_{\text{P}}^{2}m_{\text{P}}}{t_{\text{P}}^{2}}}=k_{\text{B}}\ T_{\text{P}}}$

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) ${\displaystyle l_{\text{P}}={\sqrt {\frac {4\pi \hbar G}{c^{3}}}}}$ ${\displaystyle l_{\text{P}}={\sqrt {\frac {\hbar G}{c^{3}}}}}$ m m Planck length
Mass (M) ${\displaystyle m_{\text{P}}={\sqrt {\frac {\hbar c}{4\pi G}}}}$ ${\displaystyle m_{\text{P}}={\sqrt {\frac {\hbar c}{G}}}}$ kg kg Planck mass
Time (T) ${\displaystyle t_{\text{P}}={\sqrt {\frac {4\pi \hbar G}{c^{5}}}}}$ ${\displaystyle t_{\text{P}}={\sqrt {\frac {\hbar G}{c^{5}}}}}$ s s Planck time
Charge (Q) ${\displaystyle q_{\text{P}}={\sqrt {\hbar c\epsilon _{0}}}}$ ${\displaystyle q_{\text{P}}={\sqrt {4\pi \hbar c\epsilon _{0}}}}$ C C Planck charge
Temperature (?) ${\displaystyle T_{\text{P}}={\sqrt {\frac {\hbar c^{5}}{4\pi G{k_{\text{B}}}^{2}}}}}$ ${\displaystyle T_{\text{P}}={\sqrt {\frac {\hbar c^{5}}{G{k_{\text{B}}}^{2}}}}}$ K 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 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 ?0, due to the definition of ampere which sets the vacuum permeability ?0 to and the fact that ?0?0 = . The numerical value of the reduced Planck constant ? has been determined experimentally to 12 parts per billion, while that of G has been determined experimentally to no better than 1 part in (or parts per billion).[2]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 ± 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 , or parts per billion.)

## 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 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
Planck area area (L2) ${\displaystyle A_{\text{P}}=l_{\text{P}}^{2}={\frac {4\pi \hbar G}{c^{3}}}}$ ${\displaystyle A_{\text{P}}=l_{\text{P}}^{2}={\frac {\hbar G}{c^{3}}}}$ m2 m2
Planck volume volume (L3) ${\displaystyle V_{\text{P}}=l_{\text{P}}^{3}={\sqrt {\frac {64\pi ^{3}\hbar ^{3}G^{3}}{c^{9}}}}}$ ${\displaystyle V_{\text{P}}=l_{\text{P}}^{3}={\sqrt {\frac {\hbar ^{3}G^{3}}{c^{9}}}}}$ m3 m3
Planck wavenumber wavenumber (L-1) ${\displaystyle N_{\text{P}}={\frac {1}{l_{\text{P}}}}={\sqrt {\frac {c^{3}}{4\pi \hbar G}}}}$ ${\displaystyle N_{\text{P}}={\frac {1}{l_{\text{P}}}}={\sqrt {\frac {c^{3}}{\hbar G}}}}$ m-1 m-1
Planck density density (L-3M) ${\displaystyle d_{\text{P}}={\frac {m_{\text{P}}}{V_{\text{P}}}}={\frac {\hbar t_{\text{P}}}{l_{\text{P}}^{5}}}={\frac {c^{5}}{16\pi ^{2}\hbar G^{2}}}}$ ${\displaystyle d_{\text{P}}={\frac {m_{\text{P}}}{V_{\text{P}}}}={\frac {\hbar t_{\text{P}}}{l_{\text{P}}^{5}}}={\frac {c^{5}}{\hbar G^{2}}}}$ kg/m3 kg/m3
Planck specific volume specific volume (L3M-1) ${\displaystyle \beta _{\text{P}}={\frac {1}{d_{\text{P}}}}={\frac {16\pi ^{2}\hbar G^{2}}{c^{5}}}}$ ${\displaystyle \beta _{\text{P}}={\frac {1}{d_{\text{P}}}}={\frac {\hbar G^{2}}{c^{5}}}}$ m3/kg m3/kg
Planck frequency frequency (T-1) ${\displaystyle f_{\text{P}}={\frac {1}{t_{\text{P}}}}={\sqrt {\frac {c^{5}}{4\pi \hbar G}}}}$ ${\displaystyle f_{\text{P}}={\frac {1}{t_{\text{P}}}}={\sqrt {\frac {c^{5}}{\hbar G}}}}$ Hz Hz
Planck speed speed (LT-1) ${\displaystyle v_{\text{P}}={\frac {l_{\text{P}}}{t_{\text{P}}}}=c}$ m/s
Planck acceleration acceleration (LT-2) ${\displaystyle a_{\text{P}}={\frac {v_{\text{P}}}{t_{\text{P}}}}={\sqrt {\frac {c^{7}}{4\pi \hbar G}}}}$ ${\displaystyle a_{\text{P}}={\frac {v_{\text{P}}}{t_{\text{P}}}}={\sqrt {\frac {c^{7}}{\hbar G}}}}$ m/s2 m/s2
Planck jerk jerk (LT-3) ${\displaystyle {\mathcal {J}}_{\text{P}}={\frac {a_{\text{P}}}{t_{\text{P}}}}={\frac {c^{6}}{4\pi \hbar G}}}$ ${\displaystyle {\mathcal {J}}_{\text{P}}={\frac {a_{\text{P}}}{t_{\text{P}}}}={\frac {c^{6}}{\hbar G}}}$ m/s3 m/s3
Planck snap snap (LT-4) ${\displaystyle {\mathcal {N}}_{\text{P}}={\frac {{\mathcal {J}}_{\text{P}}}{t_{\text{P}}}}={\sqrt {\frac {c^{17}}{64\pi ^{3}\hbar ^{3}G^{3}}}}}$ ${\displaystyle {\mathcal {N}}_{\text{P}}={\frac {{\mathcal {J}}_{\text{P}}}{t_{\text{P}}}}={\sqrt {\frac {c^{17}}{\hbar ^{3}G^{3}}}}}$ m/s4 m/s4
Planck crackle crackle (LT-5) ${\displaystyle {\mathcal {C}}_{\text{P}}={\frac {{\mathcal {S}}_{\text{P}}}{t_{\text{P}}}}={\frac {c^{11}}{16\pi ^{2}\hbar ^{2}G^{2}}}}$ ${\displaystyle {\mathcal {C}}_{\text{P}}={\frac {{\mathcal {S}}_{\text{P}}}{t_{\text{P}}}}={\frac {c^{11}}{\hbar ^{2}G^{2}}}}$ m/s5 m/s5
Planck pop pop (LT-6) ${\displaystyle {\mathcal {P}}_{\text{P}}={\frac {{\mathcal {C}}_{\text{P}}}{t_{\text{P}}}}={\sqrt {\frac {c^{27}}{1024\pi ^{5}\hbar ^{5}G^{5}}}}}$ ${\displaystyle {\mathcal {P}}_{\text{P}}={\frac {{\mathcal {C}}_{\text{P}}}{t_{\text{P}}}}={\sqrt {\frac {c^{27}}{\hbar ^{5}G^{5}}}}}$ m/s6 m/s6
Planck momentum momentum (LMT-1) ${\displaystyle p_{\text{P}}=m_{\text{P}}v_{\text{P}}={\frac {\hbar }{l_{\text{P}}}}={\sqrt {\frac {\hbar c^{3}}{4\pi G}}}}$ ${\displaystyle p_{\text{P}}=m_{\text{P}}v_{\text{P}}={\frac {\hbar }{l_{\text{P}}}}={\sqrt {\frac {\hbar c^{3}}{G}}}}$ N?s N?s
Planck force force (LMT-2) ${\displaystyle F_{\text{P}}=m_{\text{P}}a_{\text{P}}={\frac {p_{\text{P}}}{t_{\text{P}}}}={\frac {c^{4}}{4\pi G}}}$ ${\displaystyle F_{\text{P}}=m_{\text{P}}a_{\text{P}}={\frac {p_{\text{P}}}{t_{\text{P}}}}={\frac {c^{4}}{G}}}$ N N
Planck energy energy (L2MT-2) ${\displaystyle E_{\text{P}}=m_{\text{P}}v_{\text{P}}^{2}={\frac {\hbar }{t_{\text{P}}}}={\sqrt {\frac {\hbar c^{5}}{4\pi G}}}}$ ${\displaystyle E_{\text{P}}=m_{\text{P}}v_{\text{P}}^{2}={\frac {\hbar }{t_{\text{P}}}}={\sqrt {\frac {\hbar c^{5}}{G}}}}$ J J
Planck power power (L2MT-3) ${\displaystyle P_{\text{P}}={\frac {E_{\text{P}}}{t_{\text{P}}}}={\frac {\hbar }{t_{\text{P}}^{2}}}={\frac {c^{5}}{4\pi G}}}$ ${\displaystyle P_{\text{P}}={\frac {E_{\text{P}}}{t_{\text{P}}}}={\frac {\hbar }{t_{\text{P}}^{2}}}={\frac {c^{5}}{G}}}$ W W
Planck specific energy specific energy (L2T-2) ${\displaystyle h_{\text{P}}={\frac {E_{\text{P}}}{m_{\text{P}}}}=c^{2}}$ J/kg
Planck energy density energy density (L-1MT-2) ${\displaystyle u_{\text{P}}={\frac {E_{\text{P}}}{V_{\text{P}}}}={\frac {c^{7}}{16\pi ^{2}\hbar G^{2}}}}$ ${\displaystyle u_{\text{P}}={\frac {E_{\text{P}}}{V_{\text{P}}}}={\frac {c^{7}}{\hbar G^{2}}}}$ J/m3 J/m3
Planck intensity intensity (MT-3) ${\displaystyle {\mathcal {I}}_{\text{P}}={\frac {P_{\text{P}}}{A_{\text{P}}}}={\frac {c^{8}}{16\pi ^{2}\hbar G^{2}}}}$ ${\displaystyle {\mathcal {I}}_{\text{P}}={\frac {P_{\text{P}}}{A_{\text{P}}}}={\frac {c^{8}}{\hbar G^{2}}}}$ W/m2 W/m2
Planck action action (L2MT-1) ${\displaystyle {\mathcal {S}}_{\text{P}}=l_{\text{P}}p_{\text{P}}=E_{\text{P}}t_{\text{P}}=\hbar }$ J?s
Planck gravitational induction gravitational field (LT-2) ${\displaystyle g_{\text{P}}={\frac {F_{\text{P}}}{m_{\text{P}}}}={\sqrt {\frac {c^{7}}{4\pi \hbar G}}}}$ ${\displaystyle g_{\text{P}}={\frac {F_{\text{P}}}{m_{\text{P}}}}={\sqrt {\frac {c^{7}}{\hbar G}}}}$ m/s2 m/s2
Planck mass current mass current (MT-1)
Planck mass flux mass flux (L-2MT-1)
Planck angle angle (dimensionless) ${\displaystyle \theta _{\text{P}}={\frac {l_{\text{P}}}{l_{\text{P}}}}=1}$ rad
Planck angular speed angular speed (T-1) ${\displaystyle \omega _{\text{P}}={\frac {\theta _{\text{P}}}{t_{\text{P}}}}={\sqrt {\frac {c^{5}}{4\pi \hbar G}}}}$ ${\displaystyle \omega _{\text{P}}={\frac {\theta _{\text{P}}}{t_{\text{P}}}}={\sqrt {\frac {c^{5}}{\hbar G}}}}$ rad/s rad/s
Planck angular acceleration angular acceleration (T-2) ${\displaystyle \alpha _{\text{P}}={\frac {\omega _{\text{P}}}{t_{\text{P}}}}={\frac {c^{5}}{4\pi \hbar G}}}$ ${\displaystyle \alpha _{\text{P}}={\frac {\omega _{\text{P}}}{t_{\text{P}}}}={\frac {c^{5}}{\hbar G}}}$ rad/s2 rad/s2
Planck angular jerk angular jerk (T-3) ${\displaystyle \zeta _{\text{P}}={\frac {\alpha _{\text{P}}}{t_{\text{P}}}}={\sqrt {\frac {c^{15}}{64\pi ^{3}\hbar ^{3}G^{3}}}}}$ ${\displaystyle \zeta _{\text{P}}={\frac {\alpha _{\text{P}}}{t_{\text{P}}}}={\sqrt {\frac {c^{15}}{\hbar ^{3}G^{3}}}}}$ rad/s3 rad/s3
Planck rotational inertia rotational inertia (L2M) ${\displaystyle I_{\text{P}}=m_{\text{P}}l_{\text{P}}^{2}={\sqrt {\frac {4\pi \hbar ^{3}G}{c^{5}}}}}$ ${\displaystyle I_{\text{P}}=m_{\text{P}}l_{\text{P}}^{2}={\sqrt {\frac {\hbar ^{3}G}{c^{5}}}}}$ kg?m2 kg?m2
Planck angular momentum angular momentum (L2MT-1) ${\displaystyle \Lambda _{\text{P}}=I_{\text{P}}\omega _{\text{P}}=\hbar }$ J?s
Planck torque torque (L2MT-2) ${\displaystyle \tau _{\text{P}}=I_{\text{P}}\alpha _{\text{P}}=F_{\text{P}}l_{\text{P}}={\frac {\Lambda _{\text{P}}}{t_{\text{P}}}}={\sqrt {\frac {\hbar c^{5}}{4\pi G}}}}$ ${\displaystyle \tau _{\text{P}}=I_{\text{P}}\alpha _{\text{P}}=F_{\text{P}}l_{\text{P}}={\frac {\Lambda _{\text{P}}}{t_{\text{P}}}}={\sqrt {\frac {\hbar c^{5}}{G}}}}$ N?m N?m
Planck specific angular momentum specific angular momentum (L2T-1) ${\displaystyle \pi _{\text{P}}={\frac {\Lambda _{\text{P}}}{m_{\text{P}}}}={\sqrt {\frac {4\pi \hbar G}{c}}}}$ ${\displaystyle \pi _{\text{P}}={\frac {\Lambda _{\text{P}}}{m_{\text{P}}}}={\sqrt {\frac {\hbar G}{c}}}}$ m2/s m2/s
Planck solid angle solid angle (dimensionless) ${\displaystyle \Omega _{\text{P}}=\theta _{\text{P}}^{2}={\frac {l_{\text{P}}^{2}}{l_{\text{P}}^{2}}}=1}$ sr
Planck radiant intensity radiant intensity (L2MT-3) ${\displaystyle J_{\text{P}}={\frac {P_{\text{P}}}{\Omega _{\text{P}}}}={\frac {c^{5}}{4\pi G}}}$ ${\displaystyle J_{\text{P}}={\frac {P_{\text{P}}}{\Omega _{\text{P}}}}={\frac {c^{5}}{G}}}$ W/sr W/sr
Planck radiance radiance (MT-3) ${\displaystyle {\mathcal {L}}_{\text{P}}={\frac {P_{\text{P}}}{A_{\text{P}}\Omega _{\text{P}}}}={\frac {c^{8}}{16\pi ^{2}\hbar G^{2}}}}$ ${\displaystyle {\mathcal {L}}_{\text{P}}={\frac {P_{\text{P}}}{A_{\text{P}}\Omega _{\text{P}}}}={\frac {c^{8}}{\hbar G^{2}}}}$ W/sr?m2 W/sr?m2
Planck pressure pressure (L-1MT-2) ${\displaystyle \Pi _{\text{P}}={\frac {F_{\text{P}}}{A_{\text{P}}}}={\frac {\hbar }{l_{\text{P}}^{3}t_{\text{P}}}}={\frac {c^{7}}{16\pi ^{2}\hbar G^{2}}}}$ ${\displaystyle \Pi _{\text{P}}={\frac {F_{\text{P}}}{A_{\text{P}}}}={\frac {\hbar }{l_{\text{P}}^{3}t_{\text{P}}}}={\frac {c^{7}}{\hbar G^{2}}}}$ Pa Pa
Planck surface tension surface tension (MT-2) ${\displaystyle \sigma _{\text{P}}={\frac {F_{\text{P}}}{l_{\text{P}}}}={\sqrt {\frac {c^{11}}{64\pi ^{3}\hbar G^{3}}}}}$ ${\displaystyle \sigma _{\text{P}}={\frac {F_{\text{P}}}{l_{\text{P}}}}={\sqrt {\frac {c^{11}}{\hbar G^{3}}}}}$ N/m N/m
Planck volumetric flow rate volumetric flow rate (L3T-1) ${\displaystyle Q_{\text{P}}={\frac {V_{\text{P}}}{t_{\text{P}}}}=l_{\text{P}}^{2}v_{\text{P}}={\frac {4\pi \hbar G}{c^{2}}}}$ ${\displaystyle Q_{\text{P}}={\frac {V_{\text{P}}}{t_{\text{P}}}}=l_{\text{P}}^{2}v_{\text{P}}={\frac {\hbar G}{c^{2}}}}$ m3/s m3/s
Planck mass flow rate mass flow rate (MT-1) ${\displaystyle M_{\text{P}}={\frac {m_{\text{P}}}{t_{\text{P}}}}={\frac {c^{3}}{4\pi G}}}$ ${\displaystyle M_{\text{P}}={\frac {m_{\text{P}}}{t_{\text{P}}}}={\frac {c^{3}}{G}}}$ kg/s kg/s
Planck stiffness stiffness (MT-2) ${\displaystyle K_{\text{P}}={\frac {F_{\text{P}}}{l_{\text{P}}}}={\sqrt {\frac {c^{11}}{64\pi ^{3}\hbar G^{3}}}}}$ ${\displaystyle K_{\text{P}}={\frac {F_{\text{P}}}{l_{\text{P}}}}={\sqrt {\frac {c^{11}}{\hbar G^{3}}}}}$ N/m N/m
Planck flexibility flexibility (M-1T2) ${\displaystyle x_{\text{P}}={\frac {1}{K_{\text{P}}}}={\sqrt {\frac {64\pi ^{3}\hbar G^{3}}{c^{11}}}}}$ ${\displaystyle x_{\text{P}}={\frac {1}{K_{\text{P}}}}={\sqrt {\frac {\hbar G^{3}}{c^{11}}}}}$ m/N m/N
Planck rotational stiffness rotational stiffness (L2MT-2) ${\displaystyle O_{\text{P}}={\frac {\tau _{\text{P}}}{\theta _{\text{P}}}}={\sqrt {\frac {\hbar c^{5}}{4\pi G}}}}$ ${\displaystyle O_{\text{P}}={\frac {\tau _{\text{P}}}{\theta _{\text{P}}}}={\sqrt {\frac {\hbar c^{5}}{G}}}}$ N?m/rad N?m/rad
Planck rotational flexibility rotational flexibility (L-2M-1T2) ${\displaystyle y_{\text{P}}={\frac {1}{O_{\text{P}}}}={\sqrt {\frac {4\pi G}{\hbar c^{5}}}}}$ ${\displaystyle y_{\text{P}}={\frac {1}{O_{\text{P}}}}={\sqrt {\frac {G}{\hbar c^{5}}}}}$ rad/N?m rad/N?m
Planck ultimate tensile strength ultimate tensile strength (L-1MT-2) ${\displaystyle \Sigma _{\text{P}}={\frac {F_{\text{P}}}{A_{\text{P}}}}={\frac {c^{7}}{16\pi ^{2}\hbar G^{2}}}}$ ${\displaystyle \Sigma _{\text{P}}={\frac {F_{\text{P}}}{A_{\text{P}}}}={\frac {c^{7}}{\hbar G^{2}}}}$ Pa Pa
Planck indentation hardness indentation hardness (L-1MT-2) ${\displaystyle {\mathcal {H}}_{\text{P}}={\frac {F_{\text{P}}}{A_{\text{P}}}}={\frac {c^{7}}{16\pi ^{2}\hbar G^{2}}}}$ ${\displaystyle {\mathcal {H}}_{\text{P}}={\frac {F_{\text{P}}}{A_{\text{P}}}}={\frac {c^{7}}{\hbar G^{2}}}}$ Pa Pa
Planck absolute hardness absolute hardness (M) ${\displaystyle {\mathcal {R}}_{\text{P}}={\frac {F_{\text{P}}}{g_{\text{P}}}}={\sqrt {\frac {\hbar c}{4\pi G}}}}$ ${\displaystyle {\mathcal {R}}_{\text{P}}={\frac {F_{\text{P}}}{g_{\text{P}}}}={\sqrt {\frac {\hbar c}{G}}}}$ N?s/m2 N?s/m2
Planck viscosity viscosity (L-1MT-1) ${\displaystyle \eta _{\text{P}}=P_{\text{P}}t_{\text{P}}={\sqrt {\frac {c^{9}}{64\pi ^{3}\hbar G^{3}}}}}$ ${\displaystyle \eta _{\text{P}}=P_{\text{P}}t_{\text{P}}={\sqrt {\frac {c^{9}}{\hbar G^{3}}}}}$ Pa?s Pa?s
Planck kinematic viscosity kinematic viscosity (L2T-1) ${\displaystyle \nu _{\text{P}}={\frac {A_{\text{P}}}{t_{\text{P}}}}={\frac {\eta _{\text{P}}}{d_{\text{P}}}}={\sqrt {\frac {4\pi \hbar G}{c}}}}$ ${\displaystyle \nu _{\text{P}}={\frac {A_{\text{P}}}{t_{\text{P}}}}={\frac {\eta _{\text{P}}}{d_{\text{P}}}}={\sqrt {\frac {\hbar G}{c}}}}$ m2/s m2/s
Planck toughness toughness (L-1MT-2) ${\displaystyle {\mathcal {T}}_{\text{P}}={\frac {E_{\text{P}}}{V_{\text{P}}}}={\frac {c^{7}}{16\pi ^{2}\hbar G^{2}}}}$ ${\displaystyle {\mathcal {T}}_{\text{P}}={\frac {E_{\text{P}}}{V_{\text{P}}}}={\frac {c^{7}}{\hbar G^{2}}}}$ J/m3 J/m3
Planck specific activity specific activity (T-1) ${\displaystyle {\mathcal {A}}_{\text{P}}={\frac {1}{t_{\text{P}}}}={\sqrt {\frac {c^{5}}{4\pi \hbar G}}}}$ ${\displaystyle {\mathcal {A}}_{\text{P}}={\frac {1}{t_{\text{P}}}}={\sqrt {\frac {c^{5}}{\hbar G}}}}$ Bq Bq
Planck radiation exposure radiation exposure (M-1Q) ${\displaystyle X_{\text{P}}={\frac {q_{\text{P}}}{m_{\text{P}}}}={\sqrt {4\pi G\epsilon _{0}}}={\sqrt {\frac {G}{k_{e}}}}}$ C/kg
Planck absorbed dose absorbed dose (L2T-2) ${\displaystyle {\mathcal {D}}_{\text{P}}={\frac {E_{\text{P}}}{m_{\text{P}}}}=c^{2}}$ Gy
Planck absorbed dose rate absorbed dose rate (L2T-3) ${\displaystyle W_{\text{P}}={\frac {{\mathcal {D}}_{\text{P}}}{t_{\text{P}}}}={\sqrt {\frac {c^{9}}{4\pi \hbar G}}}}$ ${\displaystyle W_{\text{P}}={\frac {{\mathcal {D}}_{\text{P}}}{t_{\text{P}}}}={\sqrt {\frac {c^{9}}{\hbar G}}}}$ Gy/s Gy/s
Planck current current (T-1Q) ${\displaystyle i_{\text{P}}={\frac {q_{\text{P}}}{t_{\text{P}}}}={\sqrt {\frac {c^{6}\epsilon _{0}}{4\pi G}}}}$ ${\displaystyle i_{\text{P}}={\frac {q_{\text{P}}}{t_{\text{P}}}}={\sqrt {\frac {4\pi c^{6}\epsilon _{0}}{G}}}}$ A A
Planck voltage voltage (L2MT-2Q-1) ${\displaystyle U_{\text{P}}={\frac {E_{\text{P}}}{q_{\text{P}}}}={\frac {P_{\text{P}}}{i_{\text{P}}}}={\sqrt {\frac {c^{4}}{4\pi G\epsilon _{0}}}}}$ V
Planck impedance resistance (L2MT-1Q-2) ${\displaystyle Z_{\text{P}}={\frac {U_{\text{P}}}{i_{\text{P}}}}={\frac {\hbar }{q_{\text{P}}^{2}}}={\frac {1}{c\epsilon _{0}}}=c\mu _{0}={\sqrt {\frac {\mu _{0}}{\epsilon _{0}}}}=Z_{0}={\frac {1}{Y_{0}}}}$ ${\displaystyle Z_{\text{P}}={\frac {U_{\text{P}}}{i_{\text{P}}}}={\frac {\hbar }{q_{\text{P}}^{2}}}={\frac {1}{4\pi c\epsilon _{0}}}={\frac {c\mu _{0}}{4\pi }}={\sqrt {\frac {\mu _{0}}{16\pi ^{2}\epsilon _{0}}}}={\frac {Z_{0}}{4\pi }}={\frac {1}{4\pi Y_{0}}}}$ ? ?
Planck admittance conductance (L-2M-1TQ2) ${\displaystyle Y_{\text{P}}={\frac {1}{Z_{\text{P}}}}=c\epsilon _{0}={\frac {1}{c\mu _{0}}}={\sqrt {\frac {\epsilon _{0}}{\mu _{0}}}}=Y_{0}={\frac {1}{Z_{0}}}}$ ${\displaystyle Y_{\text{P}}={\frac {1}{Z_{\text{P}}}}=4\pi c\epsilon _{0}={\frac {4\pi }{c\mu _{0}}}={\sqrt {\frac {16\pi ^{2}\epsilon _{0}}{\mu _{0}}}}=4\pi Y_{0}={\frac {4\pi }{Z_{0}}}}$ S S
Planck capacitance capacitance (L-2M-1T2Q2) ${\displaystyle C_{\text{P}}={\frac {q_{\text{P}}}{U_{\text{P}}}}={\frac {t_{\text{P}}q_{\text{P}}^{2}}{\hbar }}={\sqrt {\frac {4\pi \hbar G\epsilon _{0}^{2}}{c^{3}}}}}$ ${\displaystyle C_{\text{P}}={\frac {q_{\text{P}}}{U_{\text{P}}}}={\frac {t_{\text{P}}q_{\text{P}}^{2}}{\hbar }}={\sqrt {\frac {16\pi ^{2}\hbar G\epsilon _{0}^{2}}{c^{3}}}}}$ F F
Planck inductance inductance (L2MQ-2) ${\displaystyle L_{\text{P}}={\frac {E_{\text{P}}}{i_{\text{P}}}}={\frac {m_{\text{P}}l_{\text{P}}^{2}}{q_{\text{P}}^{2}}}={\sqrt {\frac {4\pi \hbar G}{c^{7}\epsilon _{0}^{2}}}}}$ ${\displaystyle L_{\text{P}}={\frac {E_{\text{P}}}{i_{\text{P}}}}={\frac {m_{\text{P}}l_{\text{P}}^{2}}{q_{\text{P}}^{2}}}={\sqrt {\frac {\hbar G}{16\pi ^{2}c^{7}\epsilon _{0}^{2}}}}}$ H H
Planck electrical resistivity electrical resistivity (L3MT-1Q-2) ${\displaystyle r_{\text{P}}=Z_{\text{P}}l_{\text{P}}={\sqrt {\frac {4\pi \hbar G}{c^{5}\epsilon _{0}^{2}}}}}$ ${\displaystyle r_{\text{P}}=Z_{\text{P}}l_{\text{P}}={\sqrt {\frac {\hbar G}{16\pi ^{2}c^{5}\epsilon _{0}^{2}}}}}$ m m
Planck electrical conductivity electrical conductivity (L-3M-1TQ2) ${\displaystyle \kappa _{\text{P}}={\frac {1}{r_{\text{P}}}}={\sqrt {\frac {c^{5}\epsilon _{0}^{2}}{4\pi \hbar G}}}}$ ${\displaystyle \kappa _{\text{P}}={\frac {1}{r_{\text{P}}}}={\sqrt {\frac {16\pi ^{2}c^{5}\epsilon _{0}^{2}}{\hbar G}}}}$ S/m S/m
Planck charge-to-mass ratio charge-to-mass ratio (M-1Q) ${\displaystyle \xi _{\text{P}}={\frac {q_{\text{P}}}{m_{\text{P}}}}={\sqrt {4\pi G\epsilon _{0}}}={\sqrt {\frac {G}{k_{e}}}}}$ C/kg
Planck mass-to-charge ratio mass-to-charge ratio (MQ-1) ${\displaystyle \iota _{\text{P}}={\frac {1}{\xi _{\text{P}}}}={\frac {m_{\text{P}}}{q_{\text{P}}}}={\sqrt {\frac {1}{4\pi G\epsilon _{0}}}}={\sqrt {\frac {k_{e}}{G}}}}$ kg/C
Planck charge density charge density (L-3Q) ${\displaystyle \rho _{\text{P}}={\frac {q_{\text{P}}}{V_{\text{P}}}}={\sqrt {\frac {c^{10}\epsilon _{0}}{64\pi ^{3}\hbar ^{2}G^{3}}}}}$ ${\displaystyle \rho _{\text{P}}={\frac {q_{\text{P}}}{V_{\text{P}}}}={\sqrt {\frac {4\pi c^{10}\epsilon _{0}}{\hbar ^{2}G^{3}}}}}$ C/m3 C/m3
Planck current density current density (L-2T-1Q) ${\displaystyle j_{\text{P}}={\frac {i_{\text{P}}}{A_{\text{P}}}}=\rho _{\text{P}}v_{\text{P}}={\sqrt {\frac {c^{12}\epsilon _{0}}{64\pi ^{3}\hbar ^{2}G^{3}}}}}$ ${\displaystyle j_{\text{P}}={\frac {i_{\text{P}}}{A_{\text{P}}}}=\rho _{\text{P}}v_{\text{P}}={\sqrt {\frac {4\pi c^{12}\epsilon _{0}}{\hbar ^{2}G^{3}}}}}$ A/m2 A/m2
Planck magnetic charge magnetic charge (LT-1Q) ${\displaystyle b_{\text{P}}=q_{\text{P}}v_{\text{P}}={\sqrt {\hbar c^{3}\epsilon _{0}}}}$ ${\displaystyle b_{\text{P}}=q_{\text{P}}v_{\text{P}}={\sqrt {4\pi \hbar c^{3}\epsilon _{0}}}}$ A?m A?m
Planck magnetic current magnetic current (L2MT-2Q-1) ${\displaystyle k_{\text{P}}=U_{\text{P}}={\sqrt {\frac {c^{4}}{4\pi G\epsilon _{0}}}}}$ V
Planck magnetic current density magnetic current density (MT-2Q-1) ${\displaystyle \delta _{\text{P}}={\frac {k_{\text{P}}}{A_{\text{P}}}}={\sqrt {\frac {c^{10}}{64\pi ^{3}\hbar ^{2}G^{3}\epsilon _{0}}}}}$ ${\displaystyle \delta _{\text{P}}={\frac {k_{\text{P}}}{A_{\text{P}}}}={\sqrt {\frac {c^{10}}{4\pi \hbar ^{2}G^{3}\epsilon _{0}}}}}$ V/m2 V/m2
Planck electric field intensity electric field intensity (LMT-2Q-1) ${\displaystyle e_{\text{P}}={\frac {F_{\text{P}}}{q_{\text{P}}}}={\sqrt {\frac {c^{7}}{16\pi ^{2}\hbar G^{2}\epsilon _{0}}}}}$ ${\displaystyle e_{\text{P}}={\frac {F_{\text{P}}}{q_{\text{P}}}}={\sqrt {\frac {c^{7}}{4\pi \hbar G^{2}\epsilon _{0}}}}}$ V/m V/m
Planck magnetic field intensity magnetic field intensity (L-1T-1Q) ${\displaystyle H_{\text{P}}={\frac {i_{\text{P}}}{l_{\text{P}}}}={\sqrt {\frac {c^{9}\epsilon _{0}}{16\pi ^{2}\hbar G^{2}}}}}$ ${\displaystyle H_{\text{P}}={\frac {i_{\text{P}}}{l_{\text{P}}}}={\sqrt {\frac {4\pi c^{9}\epsilon _{0}}{\hbar G^{2}}}}}$ A/m A/m
Planck electric induction electric induction (L-2Q) ${\displaystyle D_{\text{P}}={\frac {q_{\text{P}}}{l_{\text{P}}^{2}}}={\sqrt {\frac {c^{7}\epsilon _{0}}{16\pi ^{2}\hbar G^{2}}}}}$ ${\displaystyle D_{\text{P}}={\frac {q_{\text{P}}}{l_{\text{P}}^{2}}}={\sqrt {\frac {4\pi c^{7}\epsilon _{0}}{\hbar G^{2}}}}}$ C/m2 C/m2
Planck magnetic induction magnetic induction (MT-1Q-1) ${\displaystyle B_{\text{P}}={\frac {F_{\text{P}}}{l_{\text{P}}i_{\text{P}}}}={\sqrt {\frac {c^{5}}{16\pi ^{2}\hbar G^{2}\epsilon _{0}}}}}$ ${\displaystyle B_{\text{P}}={\frac {F_{\text{P}}}{l_{\text{P}}i_{\text{P}}}}={\sqrt {\frac {c^{5}}{4\pi \hbar G^{2}\epsilon _{0}}}}}$ T T
Planck electric potential electric potential (L2MT-2Q-1) ${\displaystyle \phi _{\text{P}}={\frac {E_{\text{P}}}{q_{\text{P}}}}=U_{\text{P}}={\sqrt {\frac {c^{4}}{4\pi G\epsilon _{0}}}}}$ V
Planck magnetic potential magnetic potential (LMT-1Q-1) ${\displaystyle \psi _{\text{P}}={\frac {F_{\text{P}}}{i_{\text{P}}}}=B_{\text{P}}l_{\text{P}}={\frac {U_{\text{P}}}{v_{\text{P}}}}={\sqrt {\frac {c^{2}}{4\pi G\epsilon _{0}}}}}$ T?m
Planck electromotive force electromotive force (L2MT-2Q-1) ${\displaystyle {\mathcal {E}}_{\text{P}}={\frac {E_{\text{P}}}{q_{\text{P}}}}={\sqrt {\frac {c^{4}}{4\pi G\epsilon _{0}}}}}$ V
Planck magnetomotive force magnetomotive force (T-1Q) ${\displaystyle {\mathcal {F}}_{\text{P}}=i_{\text{P}}={\sqrt {\frac {c^{6}\epsilon _{0}}{4\pi G}}}}$ ${\displaystyle {\mathcal {F}}_{\text{P}}=i_{\text{P}}={\sqrt {\frac {4\pi c^{6}\epsilon _{0}}{G}}}}$ A A
Planck permittivity permittivity (L-3M-1T2Q2) ${\displaystyle \epsilon _{\text{P}}={\frac {C_{\text{P}}}{l_{\text{P}}}}=\epsilon _{0}}$ ${\displaystyle \epsilon _{\text{P}}={\frac {C_{\text{P}}}{l_{\text{P}}}}=4\pi \epsilon _{0}}$ F/m F/m
Planck permeability permeability (LMQ-2) ${\displaystyle \mu _{\text{P}}={\frac {L_{\text{P}}}{l_{\text{P}}}}={\frac {1}{c^{2}\epsilon _{0}}}=\mu _{0}}$ ${\displaystyle \mu _{\text{P}}={\frac {L_{\text{P}}}{l_{\text{P}}}}={\frac {1}{4\pi c^{2}\epsilon _{0}}}={\frac {\mu _{0}}{4\pi }}}$ H/m H/m
Planck electric dipole moment electric dipole moment (LQ) ${\displaystyle {\mathcal {Q}}_{\text{P}}=q_{\text{P}}l_{\text{P}}={\sqrt {\frac {4\pi \hbar ^{2}G\epsilon _{0}}{c^{2}}}}}$ C?m
Planck magnetic dipole moment magnetic dipole moment (L2T-1Q) ${\displaystyle {\mathcal {M}}_{\text{P}}={\frac {E_{\text{P}}}{b_{\text{P}}}}={\sqrt {4\pi \hbar ^{2}G\epsilon _{0}}}}$ J/T
Planck electric flux electric flux (L3MT-2Q-1) ${\displaystyle \Phi _{\text{P}}=e_{\text{P}}A_{\text{P}}={\sqrt {\frac {\hbar c}{\epsilon _{0}}}}}$ ${\displaystyle \Phi _{\text{P}}=e_{\text{P}}A_{\text{P}}={\sqrt {\frac {\hbar c}{4\pi \epsilon _{0}}}}}$ V?m V?m
Planck magnetic flux magnetic flux (L2MT-1Q-1) ${\displaystyle \Psi _{\text{P}}=B_{\text{P}}A_{\text{P}}={\sqrt {\frac {\hbar }{c\epsilon _{0}}}}}$ ${\displaystyle \Psi _{\text{P}}=B_{\text{P}}A_{\text{P}}={\sqrt {\frac {\hbar }{4\pi c\epsilon _{0}}}}}$ Wb Wb
Planck gyromagnetic ratio gyromagnetic ratio (M-1Q) ${\displaystyle \Theta _{\text{P}}={\frac {\theta _{\text{P}}}{t_{\text{P}}B_{\text{P}}}}={\sqrt {4\pi G\epsilon _{0}}}={\sqrt {\frac {G}{k_{e}}}}}$ rad/s/T
Planck magnetogyric ratio magnetogyric ratio (MQ-1) ${\displaystyle \Xi _{\text{P}}={\frac {1}{\Theta _{\text{P}}}}={\frac {t_{\text{P}}B_{\text{P}}}{\theta _{\text{P}}}}={\sqrt {\frac {1}{4\pi G\epsilon _{0}}}}={\sqrt {\frac {k_{e}}{G}}}}$ s?T/rad
Planck magnetic reluctance magnetic reluctance (L-2M-1Q2) ${\displaystyle {\mathcal {R}}_{\text{P}}={\frac {{\mathcal {F}}_{\text{P}}}{\Psi _{\text{P}}}}={\sqrt {\frac {c^{7}\epsilon _{0}^{2}}{4\pi \hbar G}}}}$ ${\displaystyle {\mathcal {R}}_{\text{P}}={\frac {{\mathcal {F}}_{\text{P}}}{\Psi _{\text{P}}}}={\sqrt {\frac {16\pi ^{2}c^{7}\epsilon _{0}^{2}}{\hbar G}}}}$ H-1 H-1
Planck thermal expansion coefficient thermal expansion coefficient (?-1) ${\displaystyle \gamma _{\text{P}}={\frac {1}{T_{\text{P}}}}={\sqrt {\frac {4\pi G{k_{\text{B}}}^{2}}{\hbar c^{5}}}}}$ ${\displaystyle \gamma _{\text{P}}={\frac {1}{T_{\text{P}}}}={\sqrt {\frac {G{k_{\text{B}}}^{2}}{\hbar c^{5}}}}}$ K-1 K-1
Planck heat capacity heat capacity (L2MT-2?-1) ${\displaystyle \Gamma _{\text{P}}={\frac {E_{\text{P}}}{T_{\text{P}}}}=k_{\text{B}}}$ J/K
Planck specific heat capacity specific heat capacity (L2T-2?-1) ${\displaystyle c_{\text{P}}={\frac {E_{\text{P}}}{m_{\text{P}}T_{\text{P}}}}={\frac {\Gamma _{\text{P}}}{m_{\text{P}}}}={\sqrt {\frac {4\pi Gk_{\text{B}}^{2}}{\hbar c}}}}$ ${\displaystyle c_{\text{P}}={\frac {E_{\text{P}}}{m_{\text{P}}T_{\text{P}}}}={\frac {\Gamma _{\text{P}}}{m_{\text{P}}}}={\sqrt {\frac {Gk_{\text{B}}^{2}}{\hbar c}}}}$ J/kg?K J/kg?K
Planck volumetric heat capacity volumetric heat capacity (L-1MT-2?-1) ${\displaystyle s_{\text{P}}={\frac {E_{\text{P}}}{V_{\text{P}}T_{\text{P}}}}={\frac {\Gamma _{\text{P}}}{V_{\text{P}}}}=c_{\text{P}}d_{\text{P}}={\sqrt {\frac {c^{9}k_{\text{B}}^{2}}{64\pi ^{3}\hbar ^{3}G^{3}}}}}$ ${\displaystyle s_{\text{P}}={\frac {E_{\text{P}}}{V_{\text{P}}T_{\text{P}}}}={\frac {\Gamma _{\text{P}}}{V_{\text{P}}}}=c_{\text{P}}d_{\text{P}}={\sqrt {\frac {c^{9}k_{\text{B}}^{2}}{\hbar ^{3}G^{3}}}}}$ J/m3?K J/m3?K
Planck thermal resistance thermal resistance (L-2M-1T3?) ${\displaystyle R_{\text{P}}={\frac {T_{\text{P}}}{P_{\text{P}}}}={\sqrt {\frac {4\pi \hbar G}{c^{5}k_{\text{B}}^{2}}}}}$ ${\displaystyle R_{\text{P}}={\frac {T_{\text{P}}}{P_{\text{P}}}}={\sqrt {\frac {\hbar G}{c^{5}k_{\text{B}}^{2}}}}}$ K/W K/W
Planck thermal conductance thermal conductance (L2MT-3?-1) ${\displaystyle G_{\text{P}}={\frac {1}{R_{\text{P}}}}={\sqrt {\frac {c^{5}k_{\text{B}}^{2}}{4\pi \hbar G}}}\neq \psi _{\text{P}}{\frac {2\pi }{\sqrt {\alpha }}}}$ ${\displaystyle G_{\text{P}}={\frac {1}{R_{\text{P}}}}={\sqrt {\frac {c^{5}k_{\text{B}}^{2}}{\hbar G}}}\neq \psi _{\text{P}}{\frac {2\pi }{\sqrt {\alpha }}}}$ W/K W/K
Planck thermal resistivity thermal resistivity (L-1M-1T3?) ${\displaystyle \chi _{\text{P}}=R_{\text{P}}l_{\text{P}}={\sqrt {\frac {16\pi ^{2}\hbar ^{2}G^{2}}{c^{8}k_{\text{B}}^{2}}}}}$ ${\displaystyle \chi _{\text{P}}=R_{\text{P}}l_{\text{P}}={\sqrt {\frac {\hbar ^{2}G^{2}}{c^{8}k_{\text{B}}^{2}}}}}$ m?K/W m?K/W
Planck thermal conductivity thermal conductivity (LMT-3?-1) ${\displaystyle \lambda _{\text{P}}={\frac {P_{\text{P}}}{l_{\text{P}}T_{\text{P}}}}={\frac {1}{\chi _{\text{P}}}}={\sqrt {\frac {c^{8}k_{\text{B}}^{2}}{16\pi ^{2}\hbar ^{2}G^{2}}}}\neq B_{\text{P}}{\frac {2\pi }{\sqrt {\alpha }}}}$ ${\displaystyle \lambda _{\text{P}}={\frac {P_{\text{P}}}{l_{\text{P}}T_{\text{P}}}}={\frac {1}{\chi _{\text{P}}}}={\sqrt {\frac {c^{8}k_{\text{B}}^{2}}{\hbar ^{2}G^{2}}}}\neq B_{\text{P}}{\frac {2\pi }{\sqrt {\alpha }}}}$ W/m?K W/m?K
Planck thermal insulance thermal insulance (M-1T3?) ${\displaystyle o_{\text{P}}=R_{\text{P}}A_{\text{P}}={\sqrt {\frac {64\pi ^{3}\hbar ^{3}G^{3}}{c^{11}k_{\text{B}}^{2}}}}}$ ${\displaystyle o_{\text{P}}=R_{\text{P}}A_{\text{P}}={\sqrt {\frac {\hbar ^{3}G^{3}}{c^{11}k_{\text{B}}^{2}}}}}$ m2?K/W m2?K/W
Planck thermal transmittance thermal transmittance (MT-3?-1) ${\displaystyle w_{\text{P}}={\frac {1}{o_{\text{P}}}}={\sqrt {\frac {c^{11}k_{\text{B}}^{2}}{64\pi ^{3}\hbar ^{3}G^{3}}}}}$ ${\displaystyle w_{\text{P}}={\frac {1}{o_{\text{P}}}}={\sqrt {\frac {c^{11}k_{\text{B}}^{2}}{\hbar ^{3}G^{3}}}}}$ W/m2?K W/m2?K
Planck entropy entropy (L2MT-2?-1) ${\displaystyle S_{\text{P}}={\frac {E_{\text{P}}}{T_{\text{P}}}}=k_{\text{B}}}$ J/K
Planck amount of substance amount of substance (N) ${\displaystyle n_{\text{P}}={\frac {1}{N_{\text{A}}}}}$ mol
Planck molar mass molar mass (MN-1) ${\displaystyle {\mathcal {M}}_{\text{P}}={\frac {m_{\text{P}}}{n_{\text{P}}}}={\sqrt {\frac {\hbar cN_{\text{A}}^{2}}{4\pi G}}}}$ ${\displaystyle {\mathcal {M}}_{\text{P}}={\frac {m_{\text{P}}}{n_{\text{P}}}}={\sqrt {\frac {\hbar cN_{\text{A}}^{2}}{G}}}}$ kg/mol kg/mol
Planck molar volume molar volume (V3N-1) ${\displaystyle {\mathcal {V}}_{\text{P}}={\frac {V_{\text{P}}}{n_{\text{P}}}}={\sqrt {\frac {64\pi ^{3}\hbar ^{3}G^{3}N_{\text{A}}^{2}}{c^{9}}}}}$ ${\displaystyle {\mathcal {V}}_{\text{P}}={\frac {V_{\text{P}}}{n_{\text{P}}}}={\sqrt {\frac {\hbar ^{3}G^{3}N_{\text{A}}^{2}}{c^{9}}}}}$ m3/mol m3/mol
Planck mass fraction mass fraction (dimensionless)
Planck volume fraction volume fraction (dimensionless)
Planck molality molality (M-1N)
Planck molarity molarity (V-3N)
Planck mole fraction mole fraction (dimensionless)
Planck catalytic activity catalytic activity (T-1N) ${\displaystyle z_{\text{P}}={\frac {n_{\text{P}}}{t_{\text{P}}}}={\sqrt {\frac {c^{5}}{4\pi \hbar GN_{\text{A}}^{2}}}}}$ ${\displaystyle z_{\text{P}}={\frac {n_{\text{P}}}{t_{\text{P}}}}={\sqrt {\frac {c^{5}}{\hbar GN_{\text{A}}^{2}}}}}$ kat kat

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

(Note: ${\displaystyle N_{\text{A}}}$ 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.[8][9] The elementary charge ${\displaystyle e}$, measured in terms of the Planck charge, is

${\displaystyle e={\sqrt {4\pi \alpha }}\cdot q_{\text{P}}\approx 0.302822121\cdot q_{\text{P}}\,}$ (Lorentz-Heaviside version)
${\displaystyle e={\sqrt {\alpha }}\cdot q_{\text{P}}\approx 0.085424543\cdot q_{\text{P}}\,}$ (Gaussian version)

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

${\displaystyle \alpha ={\frac {k_{e}e^{2}}{\hbar c}}\approx {\frac {1}{137.03599911}}}$
${\displaystyle \alpha ={\frac {1}{4\pi }}\left({\frac {e}{q_{\text{P}}}}\right)^{2}}$ (Lorentz-Heaviside version)
${\displaystyle \alpha =\left({\frac {e}{q_{\text{P}}}}\right)^{2}}$ (Gaussian version)

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

${\displaystyle m_{e}={\sqrt {4\pi \alpha _{G}}}\cdot m_{\text{P}}\approx 1.48368\times 10^{-22}\cdot m_{\text{P}}\,}$ (Lorentz-Heaviside version)
${\displaystyle m_{e}={\sqrt {\alpha _{G}}}\cdot m_{\text{P}}\approx 4.18539\times 10^{-23}\cdot m_{\text{P}}\,}$ (Gaussian version)

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

${\displaystyle \alpha _{G}={\frac {Gm_{e}^{2}}{\hbar c}}\approx 1.5718\times 10^{-45}}$
${\displaystyle \alpha _{G}={\frac {1}{4\pi }}\left({\frac {m_{e}}{m_{\text{P}}}}\right)^{2}}$ (Lorentz-Heaviside version)
${\displaystyle \alpha _{G}=\left({\frac {m_{e}}{m_{\text{P}}}}\right)^{2}}$ (Gaussian version)

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

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;[10] 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:

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

where

${\displaystyle \alpha }$ is the fine-structure constant
${\displaystyle e}$ is the elementary charge
${\displaystyle \alpha _{G}}$ is the gravitational coupling constant
${\displaystyle m_{e}}$ is the electron rest mass

Hence the specific charge of electron (${\displaystyle {\frac {e}{m_{e}}}}$) is ${\displaystyle {\sqrt {\frac {\alpha }{\alpha _{G}}}}}$ 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.[11] 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)].[12]

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.

Table 4: Some common physical quantities
Quantities In Lorentz-Heaviside version Planck units In Gaussian version Planck units
Standard gravity (${\displaystyle g}$) ${\displaystyle g_{\text{P}}}$ ${\displaystyle g_{\text{P}}}$
Standard atmosphere (${\displaystyle atm}$) ${\displaystyle \Pi _{\text{P}}}$ ${\displaystyle \Pi _{\text{P}}}$
Mean solar time ${\displaystyle t_{\text{P}}}$ ${\displaystyle t_{\text{P}}}$
Diameter of the Earth ${\displaystyle l_{\text{P}}}$ ${\displaystyle l_{\text{P}}}$
Diameter of the observable universe ${\displaystyle l_{\text{P}}}$ ${\displaystyle l_{\text{P}}}$
Volume of the Earth ${\displaystyle V_{\text{P}}}$ ${\displaystyle V_{\text{P}}}$
Volume of the observable universe ${\displaystyle V_{\text{P}}}$ ${\displaystyle V_{\text{P}}}$
Mass of the Earth ${\displaystyle m_{\text{P}}}$ ${\displaystyle m_{\text{P}}}$
Mass of the observable universe ${\displaystyle m_{\text{P}}}$ ${\displaystyle m_{\text{P}}}$
Age of the Earth ${\displaystyle t_{\text{P}}}$ ${\displaystyle t_{\text{P}}}$
Age of the universe ${\displaystyle t_{\text{P}}}$ ${\displaystyle t_{\text{P}}}$
Hubble constant (${\displaystyle H_{0}}$) ${\displaystyle t_{\text{P}}^{-1}}$ ${\displaystyle t_{\text{P}}^{-1}}$
Cosmological constant (${\displaystyle \Lambda }$) ${\displaystyle l_{\text{P}}^{-2}}$ ${\displaystyle l_{\text{P}}^{-2}}$
vacuum energy density (${\displaystyle \rho _{\text{vacuum}}}$) ${\displaystyle \rho _{\text{P}}}$ ${\displaystyle \rho _{\text{P}}}$
Melting point of water ${\displaystyle T_{\text{P}}}$ ${\displaystyle T_{\text{P}}}$
Boiling point of water ${\displaystyle T_{\text{P}}}$ ${\displaystyle T_{\text{P}}}$
Pressure of the triple point of water
Temperature of the triple point of water
Density of water ${\displaystyle d_{\text{P}}}$ ${\displaystyle d_{\text{P}}}$
Specific heat capacity of water ${\displaystyle c_{\text{P}}}$ ${\displaystyle c_{\text{P}}}$
Elementary charge (${\displaystyle e}$) ${\displaystyle q_{\text{P}}}$ ${\displaystyle q_{\text{P}}}$
Electron rest mass (${\displaystyle m_{e}}$) ${\displaystyle m_{\text{P}}}$ ${\displaystyle m_{\text{P}}}$
Proton rest mass (${\displaystyle m_{p}}$) ${\displaystyle m_{\text{P}}}$ ${\displaystyle m_{\text{P}}}$
Neutron rest mass (${\displaystyle m_{n}}$) ${\displaystyle m_{\text{P}}}$ ${\displaystyle m_{\text{P}}}$
Charge-to-mass ratio of electron (${\displaystyle \xi _{e}}$) ${\displaystyle \xi _{\text{P}}}$
Charge-to-mass ratio of proton (${\displaystyle \xi _{p}}$) ${\displaystyle \xi _{\text{P}}}$
Bohr radius (${\displaystyle a_{0}}$) ${\displaystyle l_{\text{P}}}$ ${\displaystyle l_{\text{P}}}$
Bohr magneton (${\displaystyle \mu _{B}}$)
Rydberg constant (${\displaystyle R_{\infty }}$) ${\displaystyle l_{\text{P}}^{-1}}$ ${\displaystyle l_{\text{P}}^{-1}}$
Josephson constant (${\displaystyle K_{J}}$) ${\displaystyle f_{\text{P}}/U_{\text{P}}}$ ${\displaystyle f_{\text{P}}/U_{\text{P}}}$
von Klitzing constant (${\displaystyle R_{K}}$) 68.5180 ${\displaystyle Z_{\text{P}}}$ 861.023 ${\displaystyle Z_{\text{P}}}$
Stefan-Boltzmann constant (${\displaystyle \sigma }$) ${\displaystyle P_{\text{P}}/l_{\text{P}}^{2}/T_{\text{P}}^{4}}$

### Cosmology

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, tP, or approximately 10-43 seconds.[13] 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 tP).[14]

Relative to the Planck epoch, the observable universe today looks extreme when expressed in Planck units, as in this set of approximations:[15][16]

Table 5: Today's universe in Planck units (Gaussian version).
Property of
present-day Observable Universe
Approximate number
of Planck units
Equivalents
Age 8.08 × 1060tP 4.35 × 1017 s, or 13.8 × 109 years
Diameter 5.4 × 1061lP 8.7 × 1026 m or 9.2 × 1010ly
Volume 8.4559 × 10184VP 3.57 × 1080 m3 or 4.22 × 1032 ly3
Mass approx. 1060mP 3 × 1052 kg or 1.5 × 1022solar masses (only counting stars)
1080 protons (sometimes known as the Eddington number)
Density 1.8 × 10-123dP 9.9 × 10-27 kg m-3
Temperature 1.9 × 10-32TP 2.725 K
temperature of the cosmic microwave background radiation
Cosmological constant 5.6 × 10-122t-2
P
1.9 × 10-35 s-2
Hubble constant 1.18 × 10-61t-1
P
2.2 × 10-18 s-1 or 67.8 (km/s)/Mpc

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).[17] 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.[18] Barrow and Shaw (2011) proposed a modified theory in which ? is a field evolving in such a way that its value remains ? ~ T-2 throughout the history of the universe.[19]

## 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, , kB, 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.[20][21] 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 ?) 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, ?, c, and kB to arrive at natural units for length, time, mass, and temperature.[21] 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, ke, to 1 as well (as well as the Stoney units).[22] 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 4?G and ?0 (instead of G and ke) 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.[23] 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 6: How Planck units simplify the key equations of physics
SI form Lorentz-Heaviside version Planck form Gaussian version Planck form
Mass-energy equivalence in special relativity ${\displaystyle {E=mc^{2}}\ }$ ${\displaystyle {E=m}\ }$
Energy-momentum relation ${\displaystyle E^{2}=m^{2}c^{4}+p^{2}c^{2}\;}$ ${\displaystyle E^{2}=m^{2}+p^{2}\;}$
Newton's law of universal gravitation ${\displaystyle F=-G{\frac {m_{1}m_{2}}{r^{2}}}}$ ${\displaystyle F=-{\frac {m_{1}m_{2}}{4\pi r^{2}}}}$ ${\displaystyle F=-{\frac {m_{1}m_{2}}{r^{2}}}}$
Einstein field equations in general relativity ${\displaystyle {G_{\mu \nu }={8\pi G \over c^{4}}T_{\mu \nu }}\ }$ ${\displaystyle {G_{\mu \nu }=2T_{\mu \nu }}\ }$ ${\displaystyle {G_{\mu \nu }=8\pi T_{\mu \nu }}\ }$
The formula of Schwarzschild radius ${\displaystyle r_{s}={\frac {2Gm}{c^{2}}}}$ ${\displaystyle r_{s}={\frac {m}{2\pi }}}$ ${\displaystyle r_{s}=2m}$
Gauss's law for gravity ${\displaystyle \mathbf {g} \cdot d\mathbf {A} =-4\pi GM}$
${\displaystyle \nabla \cdot \mathbf {g} =-4\pi G\rho }$
${\displaystyle \mathbf {g} \cdot d\mathbf {A} =-M}$
${\displaystyle \nabla \cdot \mathbf {g} =-\rho }$
${\displaystyle \mathbf {g} \cdot d\mathbf {A} =-4\pi M}$
${\displaystyle \nabla \cdot \mathbf {g} =-4\pi \rho }$
Poisson's equation ${\displaystyle {\nabla }^{2}\phi =4\pi G\rho }$
${\displaystyle \phi (r)={\dfrac {-Gm}{r}}}$
${\displaystyle {\nabla }^{2}\phi =\rho }$
${\displaystyle \phi (r)={\dfrac {-m}{4\pi r}}}$
${\displaystyle {\nabla }^{2}\phi =4\pi \rho }$
${\displaystyle \phi (r)={\dfrac {-m}{r}}}$
The characteristic impedance ${\displaystyle Z_{0}={\frac {4\pi G}{c}}}$ ${\displaystyle Z_{0}=1}$ ${\displaystyle Z_{0}=4\pi }$
The characteristic admittance ${\displaystyle Y_{0}={\frac {c}{4\pi G}}}$ ${\displaystyle Y_{0}=1}$ ${\displaystyle Y_{0}={\frac {1}{4\pi }}}$
GEM equations ${\displaystyle \nabla \cdot \mathbf {E_{g}} =-4\pi G\rho _{g}}$

${\displaystyle \nabla \cdot \mathbf {D_{g}} =\rho _{g}f}$
${\displaystyle \nabla \cdot \mathbf {B_{g}} =0\ }$
${\displaystyle \nabla \times \mathbf {E_{g}} =-{\frac {\partial \mathbf {B_{g}} }{\partial t}}}$
${\displaystyle \nabla \times \mathbf {B_{g}} ={\frac {1}{c^{2}}}\left(-4\pi G\mathbf {J_{g}} +{\frac {\partial \mathbf {E_{g}} }{\partial t}}\right)}$
${\displaystyle \nabla \times \mathbf {H_{g}} =\mathbf {J_{g}} f+{\frac {\partial \mathbf {D_{g}} }{\partial t}}}$

${\displaystyle \nabla \cdot \mathbf {E_{g}} =\rho _{g}}$

${\displaystyle \nabla \cdot \mathbf {D_{g}} =\rho _{g}f}$
${\displaystyle \nabla \cdot \mathbf {B_{g}} =0\ }$
${\displaystyle \nabla \times \mathbf {E_{g}} =-{\frac {\partial \mathbf {B_{g}} }{\partial t}}}$
${\displaystyle \nabla \times \mathbf {B_{g}} =\mathbf {J_{g}} +{\frac {\partial \mathbf {E_{g}} }{\partial t}}}$
${\displaystyle \nabla \times \mathbf {H_{g}} =\mathbf {J_{g}} f+{\frac {\partial \mathbf {D_{g}} }{\partial t}}}$

${\displaystyle \nabla \cdot \mathbf {E_{g}} =4\pi \rho _{g}\ }$

${\displaystyle \nabla \cdot \mathbf {D_{g}} =\rho _{g}f}$
${\displaystyle \nabla \cdot \mathbf {B_{g}} =0\ }$
${\displaystyle \nabla \times \mathbf {E_{g}} =-{\frac {\partial \mathbf {B_{g}} }{\partial t}}}$
${\displaystyle \nabla \times \mathbf {B_{g}} =4\pi \mathbf {J_{g}} +{\frac {\partial \mathbf {E_{g}} }{\partial t}}}$
${\displaystyle \nabla \times \mathbf {H_{g}} =\mathbf {J_{g}} f+{\frac {\partial \mathbf {D_{g}} }{\partial t}}}$

Planck-Einstein relation ${\displaystyle {E=h\nu }\ }$
${\displaystyle {E=\hbar \omega }\ }$
${\displaystyle {E=2\pi \nu }\ }$
${\displaystyle {E=\omega }\ }$
Heisenberg's uncertainty principle ${\displaystyle \Delta x\cdot \Delta p\geq {\frac {\hbar }{2}}}$ ${\displaystyle \Delta x\cdot \Delta p\geq {\frac {1}{2}}}$
Energy of photon ${\displaystyle E=\hbar \omega =h\nu ={\frac {hc}{\lambda }}}$ = ${\displaystyle E=\omega =2\pi \nu ={\frac {2\pi }{\lambda }}}$ =
Momentum of photon ${\displaystyle p=\hbar k={\frac {h\nu }{c}}={\frac {h}{\lambda }}}$ = ${\displaystyle p=k=2\pi \nu ={\frac {2\pi }{\lambda }}}$ =
Wavelength and reduced wavelength of matter wave ${\displaystyle \lambda ={\frac {h}{mv}}={\frac {2\pi \hbar }{mv}}}$
? =
${\displaystyle \lambda ={\frac {2\pi }{mv}}}$
? =
The formula of Compton wavelength and reduced Compton wavelength ${\displaystyle \lambda ={\frac {h}{mc}}={\frac {2\pi \hbar }{mc}}}$
? =
${\displaystyle \lambda ={\frac {2\pi }{m}}}$
? =
Schrödinger's equation ${\displaystyle -{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi (\mathbf {r} ,t)+V(\mathbf {r} ,t)\psi (\mathbf {r} ,t)=i\hbar {\frac {\partial \psi (\mathbf {r} ,t)}{\partial t}}}$ ${\displaystyle -{\frac {1}{2m}}\nabla ^{2}\psi (\mathbf {r} ,t)+V(\mathbf {r} ,t)\psi (\mathbf {r} ,t)=i{\frac {\partial \psi (\mathbf {r} ,t)}{\partial t}}}$
Schrödinger's equation ${\displaystyle i\hbar {\frac {\partial }{\partial t}}\psi =H\cdot \psi }$ ${\displaystyle i{\frac {\partial }{\partial t}}\psi =H\cdot \psi }$
Hamiltonian form of Schrödinger's equation ${\displaystyle H\left|\psi _{t}\right\rangle =i\hbar {\frac {\partial }{\partial t}}\left|\psi _{t}\right\rangle }$ ${\displaystyle H\left|\psi _{t}\right\rangle =i{\frac {\partial }{\partial t}}\left|\psi _{t}\right\rangle }$
Covariant form of the Dirac equation ${\displaystyle \ (i\hbar \gamma ^{\mu }\partial _{\mu }-mc)\psi =0}$ ${\displaystyle \ (i\gamma ^{\mu }\partial _{\mu }-m)\psi =0}$
The main role in quantum gravity ${\displaystyle \Delta r_{s}\Delta r\geq {\frac {\hbar G}{c^{3}}}}$ ${\displaystyle \Delta r_{s}\Delta r\geq {\frac {1}{4\pi }}}$ ${\displaystyle \Delta r_{s}\Delta r\geq 1}$
The vacuum permeability ${\displaystyle \mu _{0}={\frac {1}{\epsilon _{0}c^{2}}}}$ ${\displaystyle \mu _{0}=1}$ ${\displaystyle \mu _{0}=4\pi }$
The impedance of free space ${\displaystyle Z_{0}={\frac {\mathbf {E} }{\mathbf {H} }}={\sqrt {\frac {\mu _{0}}{\epsilon _{0}}}}={\frac {1}{\epsilon _{0}c}}=\mu _{0}c}$ ${\displaystyle Z_{0}=1}$ ${\displaystyle Z_{0}=4\pi }$
The admittance of free space ${\displaystyle Y_{0}={\frac {\mathbf {H} }{\mathbf {E} }}={\sqrt {\frac {\epsilon _{0}}{\mu _{0}}}}=\epsilon _{0}c={\frac {1}{\mu _{0}c}}}$ ${\displaystyle Y_{0}=1}$ ${\displaystyle Y_{0}={\frac {1}{4\pi }}}$
The Coulomb constant ${\displaystyle k_{e}={\frac {1}{4\pi \epsilon _{0}}}}$ ${\displaystyle k_{e}={\frac {1}{4\pi }}}$ ${\displaystyle k_{e}=1}$
Coulomb's law ${\displaystyle F=k_{e}{\frac {q_{1}q_{2}}{r^{2}}}={\frac {1}{4\pi \epsilon _{0}}}{\frac {q_{1}q_{2}}{r^{2}}}}$ ${\displaystyle F={\frac {q_{1}q_{2}}{4\pi r^{2}}}}$ ${\displaystyle F={\frac {q_{1}q_{2}}{r^{2}}}}$
Coulomb's law for two stationary magnetic charge ${\displaystyle F=k_{m}{\frac {b_{1}b_{2}}{r^{2}}}={\frac {\mu _{0}}{4\pi }}{\frac {b_{1}b_{2}}{r^{2}}}}$ ${\displaystyle F={\frac {b_{1}b_{2}}{4\pi r^{2}}}}$ ${\displaystyle F={\frac {b_{1}b_{2}}{r^{2}}}}$
Biot-Savart law ${\displaystyle \Delta B={\frac {\mu _{0}I}{4\pi }}{\frac {\Delta L}{r^{2}}}\sin \theta }$ ${\displaystyle \Delta B={\frac {I}{4\pi }}{\frac {\Delta L}{r^{2}}}\sin \theta }$ ${\displaystyle \Delta B=I{\frac {\Delta L}{r^{2}}}\sin \theta }$
Biot-Savart law ${\displaystyle \mathbf {B} (\mathbf {r} )={\frac {\mu _{0}}{4\pi }}\int _{C}{\frac {I\,d{\boldsymbol {\ell }}\times \mathbf {r'} }{|\mathbf {r'} |^{3}}}}$ ${\displaystyle \mathbf {B} (\mathbf {r} )={\frac {1}{4\pi }}\int _{C}{\frac {I\,d{\boldsymbol {\ell }}\times \mathbf {r'} }{|\mathbf {r'} |^{3}}}}$ ${\displaystyle \mathbf {B} (\mathbf {r} )=\int _{C}{\frac {I\,d{\boldsymbol {\ell }}\times \mathbf {r'} }{|\mathbf {r'} |^{3}}}}$
Equation of electric field intensity and electric induction ${\displaystyle \mathbf {D} =\epsilon _{0}\mathbf {E} }$ ${\displaystyle \mathbf {D} =\mathbf {E} }$ ${\displaystyle \mathbf {D} ={\frac {\mathbf {E} }{4\pi }}}$
Equation of magnetic field intensity and magnetic induction ${\displaystyle \mathbf {B} =\mu _{0}\mathbf {H} }$ ${\displaystyle \mathbf {B} =\mathbf {H} }$ ${\displaystyle \mathbf {B} =4\pi \mathbf {H} }$
Maxwell's equations ${\displaystyle \nabla \cdot \mathbf {E} ={\frac {1}{\epsilon _{0}}}\rho }$

${\displaystyle \nabla \cdot \mathbf {D} =\rho _{f}}$
${\displaystyle \nabla \cdot \mathbf {B} =0\ }$
${\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}$
${\displaystyle \nabla \times \mathbf {B} ={\frac {1}{c^{2}}}\left({\frac {1}{\epsilon _{0}}}\mathbf {J} +{\frac {\partial \mathbf {E} }{\partial t}}\right)}$
${\displaystyle \nabla \times \mathbf {H} =\mathbf {J} _{f}+{\frac {\partial \mathbf {D} }{\partial t}}}$

${\displaystyle \nabla \cdot \mathbf {E} =\rho }$

${\displaystyle \nabla \cdot \mathbf {D} =\rho _{f}}$
${\displaystyle \nabla \cdot \mathbf {B} =0\ }$
${\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}$
${\displaystyle \nabla \times \mathbf {B} =\mathbf {J} +{\frac {\partial \mathbf {E} }{\partial t}}}$
${\displaystyle \nabla \times \mathbf {H} =\mathbf {J} _{f}+{\frac {\partial \mathbf {D} }{\partial t}}}$

${\displaystyle \nabla \cdot \mathbf {E} =4\pi \rho \ }$

${\displaystyle \nabla \cdot \mathbf {D} =\rho _{f}}$
${\displaystyle \nabla \cdot \mathbf {B} =0\ }$
${\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}$
${\displaystyle \nabla \times \mathbf {B} =4\pi \mathbf {J} +{\frac {\partial \mathbf {E} }{\partial t}}}$
${\displaystyle \nabla \times \mathbf {H} =\mathbf {J} _{f}+{\frac {\partial \mathbf {D} }{\partial t}}}$

Josephson constant KJ defined ${\displaystyle K_{J}={\frac {e}{\pi \hbar }}}$ ${\displaystyle K_{J}={\sqrt {\frac {4\alpha }{\pi }}}}$ ${\displaystyle K_{J}={\frac {\sqrt {\alpha }}{\pi }}}$
von Klitzing constant RK defined ${\displaystyle R_{K}={\frac {2\pi \hbar }{e^{2}}}}$ ${\displaystyle R_{K}={\frac {1}{2\alpha }}}$ ${\displaystyle R_{K}={\frac {2\pi }{\alpha }}}$
The charge-to-mass ratio of electron ${\displaystyle \xi _{e}={\frac {e}{m_{e}}}={\sqrt {\frac {G\alpha }{k_{e}\alpha _{G}}}}={\sqrt {\frac {4\pi G\epsilon _{0}\alpha }{\alpha _{G}}}}}$ ${\displaystyle \xi _{e}={\sqrt {\frac {\alpha }{\alpha _{G}}}}}$
The Bohr radius ${\displaystyle a_{0}={\frac {4\pi \epsilon _{0}\hbar ^{2}}{m_{\text{e}}e^{2}}}={\frac {\hbar }{m_{\text{e}}c\alpha }}}$ ${\displaystyle a_{0}={\frac {1}{\alpha {\sqrt {4\pi \alpha _{G}}}}}}$ ${\displaystyle a_{0}={\frac {1}{\alpha {\sqrt {\alpha _{G}}}}}}$
The Bohr magneton ${\displaystyle \mu _{B}={\frac {e\hbar }{2m_{e}}}}$ ${\displaystyle \mu _{B}={\sqrt {\frac {\alpha }{4\alpha _{G}}}}}$
Rydberg constant R? defined ${\displaystyle R_{\infty }={\frac {m_{\text{e}}e^{4}}{8\epsilon _{0}^{2}h^{3}c}}={\frac {\alpha ^{2}m_{\text{e}}c}{4\pi \hbar }}}$ ${\displaystyle R_{\infty }={\sqrt {\frac {\alpha ^{4}\alpha _{G}}{4\pi }}}}$ ${\displaystyle R_{\infty }={\frac {\sqrt {\alpha ^{4}\alpha _{G}}}{4\pi }}}$
Ideal gas law ${\displaystyle PV=nRT=Nk_{\text{B}}T}$ ${\displaystyle PV=NT}$
Equation of the root-mean-square speed ${\displaystyle v_{rms}={\sqrt {\frac {3RT}{M}}}={\sqrt {\frac {3k_{\text{B}}T}{m}}}}$ ${\displaystyle v_{rms}={\sqrt {\frac {3T}{m}}}}$
Kinetic theory of gases ${\displaystyle \Sigma {\frac {1}{2}}mv^{2}={\frac {3}{2}}Nk_{\text{B}}T}$ ${\displaystyle \Sigma {\frac {1}{2}}mv^{2}={\frac {3}{2}}NT}$
Unruh temperature ${\displaystyle T={\frac {\hbar a}{2\pi ck_{B}}}}$ ${\displaystyle T={\frac {a}{2\pi }}}$
Thermal energy per particle per degree of freedom ${\displaystyle {E={\tfrac {1}{2}}k_{\text{B}}T}\ }$ ${\displaystyle {E={\tfrac {1}{2}}T}\ }$
Boltzmann's entropy formula ${\displaystyle {S=k_{\text{B}}\ln \Omega }\ }$ ${\displaystyle {S=\ln \Omega }\ }$
Stefan-Boltzmann constant ? defined ${\displaystyle \sigma ={\frac {\pi ^{2}k_{\text{B}}^{4}}{60\hbar ^{3}c^{2}}}}$ ${\displaystyle \sigma ={\frac {\pi ^{2}}{60}}}$
Planck's law (surface intensity per unit solid angle per unit angular frequency) for black body at temperature T. ${\displaystyle I(\omega ,T)={\frac {\hbar \omega ^{3}}{4\pi ^{3}c^{2}}}~{\frac {1}{e^{\frac {\hbar \omega }{k_{\text{B}}T}}-1}}}$ ${\displaystyle I(\omega ,T)={\frac {\omega ^{3}}{4\pi ^{3}}}~{\frac {1}{e^{\omega /T}-1}}}$
The formula of Unruh temperature ${\displaystyle T={\frac {\hbar a}{2\pi ck_{B}}}}$ ${\displaystyle T={\frac {a}{2\pi }}}$
Hawking temperature of a black hole ${\displaystyle T_{H}={\frac {\hbar c^{3}}{8\pi GMk_{B}}}}$ ${\displaystyle T_{H}={\frac {1}{2M}}}$ ${\displaystyle T_{H}={\frac {1}{8\pi M}}}$
Bekenstein-Hawking black hole entropy[24] ${\displaystyle S_{\text{BH}}={\frac {A_{\text{BH}}k_{\text{B}}c^{3}}{4G\hbar }}={\frac {4\pi Gk_{\text{B}}m_{\text{BH}}^{2}}{\hbar c}}}$ ${\displaystyle S_{\text{BH}}=\pi A_{\text{BH}}=m_{\text{BH}}^{2}}$ ${\displaystyle S_{\text{BH}}={\frac {A_{\text{BH}}}{4}}=4\pi m_{\text{BH}}^{2}}$

Note:

• For the elementary charge ${\displaystyle e}$:
${\displaystyle e={\sqrt {4\pi \alpha }}}$ (Lorentz-Heaviside version)
${\displaystyle e={\sqrt {\alpha }}}$ (Gaussian version)

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

• For the electron rest mass ${\displaystyle m_{e}}$:
${\displaystyle m_{e}={\sqrt {4\pi \alpha _{G}}}}$ (Lorentz-Heaviside version)
${\displaystyle m_{e}={\sqrt {\alpha _{G}}}}$ (Gaussian version)

where ${\displaystyle \alpha _{G}}$ 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 4? is ubiquitous in theoretical physics because the surface area of a sphere of radius r is 4?r2. 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 4?r2 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 4? 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 4?G (or 8?G or 16?G) to 1. Doing so would introduce a factor of (or or ) 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 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 4?. When this is applied to electromagnetic constants, ?0, this unit system is called "rationalized" Lorentz-Heaviside units. When applied additionally to gravitation and Planck units, these are called rationalized Planck units[25] and are seen in high-energy physics.

The rationalized Planck units are defined so that ${\displaystyle c=4\pi G=\hbar =\epsilon _{0}=k_{\text{B}}=1}$. 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 4? 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 4? appearing in the equations of physics are to be eliminated via the normalization.

${\displaystyle m_{e}={\sqrt {4\pi \alpha _{G}}}\cdot m_{\text{P}}\approx 1.48368\times 10^{-22}\cdot m_{\text{P}}\,}$
where ${\displaystyle {\alpha _{G}}\ }$ is the gravitational coupling constant. This convention is seen in high-energy physics.

### Electromagnetism

Planck units normalize to 1 the Coulomb force constant ke = (as does the cgs system of units). This sets the Planck impedance, ZP equal to , where Z0 is the characteristic impedance of free space.

${\displaystyle e={\sqrt {4\pi \alpha }}\cdot q_{\text{P}}\approx 0.302822121\cdot q_{\text{P}}\,}$
where ${\displaystyle {\alpha }\ }$ is the fine-structure constant. This convention is seen in high-energy physics.

### Temperature

Planck normalized to 1 the Boltzmann constant kB.

• Normalizing kB to 1:
• Removes the factor of 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.
• Does not affect the value of any of the base or derived Planck units listed in Tables 2 and 3 other than the Planck temperature, Planck entropy, Planck specific heat capacity, and Planck thermal conductivity, which Planck temperature doubles, and the other three become their half.

## 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".[26]

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 ? 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 ? 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 ? remained the same, this new world would be observationally indistinguishable from our world. The only thing that counts in the definition of worlds are the values of the dimensionless constants of Nature. If all masses were doubled in value [including the Planck mass mP ] you cannot tell because all the pure numbers defined by the ratios of any pair of masses are unchanged.

-- Barrow 2002[15]

Referring to Duff's "Comment on time-variation of fundamental constants"[26] and Duff, Okun, and Veneziano's paper "Trialogue on the number of fundamental constants",[27] 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, ?, changes or the proton-to-electron mass ratio, , 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 c (but with the axiom that all dimensionless physical quantities remain the same), then the Planck length would increase by a factor of 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:

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

Then atoms would be bigger (in one dimension) by 2, each of us would be taller by 2, and so would our metre sticks be taller (and wider and thicker) by a factor of 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 (from the point of view of this unaffected observer on the outside) because the Planck time has increased by 4 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 of our new metres in the time elapsed by one of our new seconds (c × 4 ÷ 2 continues to equal ). 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. ?, 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 ? is changed and no other dimensionless constants are changed. It is only the dimensionless physical constants that ultimately matter in the definition of worlds.[26][28]

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[29] and this has intensified the debate about the measurement of physical constants. According to some theorists[30] 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.[26] 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.[27]

## Notes

1. ^ General relativity predicts that gravitational radiation propagates at the same speed as electromagnetic radiation.

## References

### Citations

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3. ^ "2018 CODATA Value: speed of light in vacuum". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019.
4. ^ "2018 CODATA Value: Newtonian constant of gravitation". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019.
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6. ^ "2018 CODATA Value: vacuum electric permittivity". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019.
7. ^ "2018 CODATA Value: Boltzmann constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019.
8. ^ [Theory of Quantized Space - Date of registration 21/9/1994 N. 344146 protocol 4646 Presidency of the Council of Ministers - Italy - Dep. Information and Publishing, literary, artistic and scientific property]
9. ^
10. ^ Feynman, R. P.; Leighton, R. B.; Sands, M. (1963). "The Special Theory of Relativity". The Feynman Lectures on Physics. 1 "Mainly mechanics, radiation, and heat". Addison-Wesley. pp. 15-9. ISBN 978-0-7382-0008-8. LCCN 63020717.
11. ^ Michael W. Busch, Rachel M. Reddick (2010) "Testing SETI Message Designs," Astrobiology Science Conference 2010, 26-29 April 2010, League City, Texas.
12. ^ Wilczek, Frank (2001). "Scaling Mount Planck I: A View from the Bottom". Physics Today. 54 (6): 12-13. Bibcode:2001PhT....54f..12W. doi:10.1063/1.1387576.
13. ^ Staff. "Birth of the Universe". University of Oregon. Retrieved 2016. - discusses "Planck time" and "Planck era" at the very beginning of the Universe
14. ^ Edward W. Kolb; Michael S. Turner (1994). The Early Universe. Basic Books. p. 447. ISBN 978-0-201-62674-2. Retrieved 2010.
15. ^ a b John D. Barrow, 2002. The Constants of Nature; From Alpha to Omega - The Numbers that Encode the Deepest Secrets of the Universe. Pantheon Books. ISBN 0-375-42221-8.
16. ^
17. ^ P.A.M. Dirac (1938). "A New Basis for Cosmology". Proceedings of the Royal Society A. 165 (921): 199-208. Bibcode:1938RSPSA.165..199D. doi:10.1098/rspa.1938.0053.
18. ^ J.D. Barrow and F.J. Tipler, The Anthropic Cosmological Principle, Oxford UP, Oxford (1986), chapter 6.9.
19. ^ Barrow, John D.; Shaw, Douglas J. (2011). "The value of the cosmological constant". General Relativity and Gravitation. 43 (10): 2555-2560. arXiv:1105.3105. Bibcode:2011GReGr..43.2555B. doi:10.1007/s10714-011-1199-1.
20. ^ Planck (1899), p. 479.
21. ^ a b *Tomilin, K. A., 1999, "Natural Systems of Units: To the Centenary Anniversary of the Planck System", 287-296.
22. ^ Pav?ic, Matej (2001). The Landscape of Theoretical Physics: A Global View. Fundamental Theories of Physics. 119. Dordrecht: Kluwer Academic. pp. 347-352. arXiv:gr-qc/0610061. doi:10.1007/0-306-47136-1. ISBN 978-0-7923-7006-2.
23. ^ Tomilin, K. (1999). "Fine-structure constant and dimension analysis". Eur. J. Phys. 20 (5): L39-L40. Bibcode:1999EJPh...20L..39T. doi:10.1088/0143-0807/20/5/404.
24. ^ Also see Roger Penrose (1989) The Road to Reality. Oxford Univ. Press: 714-17. Knopf.
25. ^ Sorkin, Rafael (1983). "Kaluza-Klein Monopole". Physical Review Letters. 51 (2): 87-90. Bibcode:1983PhRvL..51...87S. doi:10.1103/PhysRevLett.51.87.
26. ^ a b c d Michael Duff (2015). "How fundamental are fundamental constants?". Contemporary Physics. 56 (1): 35-47. arXiv:1412.2040. doi:10.1080/00107514.2014.980093 (inactive 21 March 2020).
27. ^ a b Duff, Michael; Okun, Lev; Veneziano, Gabriele (2002). "Trialogue on the number of fundamental constants". Journal of High Energy Physics. 2002 (3): 023. arXiv:physics/0110060. Bibcode:2002JHEP...03..023D. doi:10.1088/1126-6708/2002/03/023.
28. ^
29. ^ Webb, J. K.; et al. (2001). "Further evidence for cosmological evolution of the fine structure constant". Phys. Rev. Lett. 87 (9): 884. arXiv:astro-ph/0012539v3. Bibcode:2001PhRvL..87i1301W. doi:10.1103/PhysRevLett.87.091301. PMID 11531558.
30. ^ Davies, Paul C.; Davis, T. M.; Lineweaver, C. H. (2002). "Cosmology: Black Holes Constrain Varying Constants". Nature. 418 (6898): 602-3. Bibcode:2002Natur.418..602D. doi:10.1038/418602a. PMID 12167848.