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			{{Short description|Units defined only by physical constants}}
	{{original research|date=June 2019}}
			{{pp-sock|small=yes}}
	{{Use dmy dates|date=May 2019}}		{{Use dmy dates|date=May 2019}}
	In ] and ], '''Planck units''' are a set of ] defined exclusively in terms of five universal ], in such a manner that these five physical constants take on the numerical value of ] when expressed in terms of these units.
			In ] and ], '''Planck units''' are a ] defined exclusively in terms of four universal ]s: '']'', '']'', '']'', and ] (described further below). Expressing one of these physical constants in terms of Planck units yields a numerical value of ]. They are a system of ], defined using fundamental properties of ] (specifically, properties of ]) rather than properties of a chosen ]. Originally proposed in 1899 by German physicist ], they are relevant in research on unified theories such as ].
	Originally proposed in 1899 by German physicist ], these units are also known as ] because the origin of their definition comes only from properties of ] and not from any ] (e.g. ] (]), ] (]), and ] (])) nor any quality of ] or ] (e.g. ], ], and ]) nor any quality of a given ] (e.g. ] of ], ] of ], and ] of ]). Planck units are only one system of several systems of ], but Planck units are not based on properties of any ] or ] (e.g. ], ], and ]) (that would be arbitrarily chosen), but rather on only the properties of ] (e.g. ] is ], ] is ], ] is ], ] is ], all are properties of free space). Planck units have significance for theoretical physics since they simplify several recurring ]s of ] by ]. They are relevant in research on unified theories such as ].
	The term ] refers to the magnitudes of space, time, energy and other units, below which (or beyond which) the predictions of the ], ] and ] are no longer reconcilable, and ] are expected to dominate. This region may be characterized by ] around {{val|5.52|e=8|u=J}} (or {{val|3.44|e=27|u=eV}}) or {{val|1.96|e=9|u=J}} (or {{val|1.22|e=28|u=eV}}) (the ]), ] intervals around {{val|1.91|e=-43|u=s}} or {{val|5.39|e=−44|u=s}} (the ]) and ]s around {{val|5.73|e=-35|u=m}} or {{val|1.62|e=-35|u=m}} (the ]). At the Planck scale, current models are not expected to be a useful guide to the cosmos, and ]s have no ] to suggest how the physical universe behaves. The best known example is represented by the conditions in the first 10<sup>−43</sup> seconds ] after the ], approximately 13.8 billion years ago.		The term '''Planck scale''' refers to quantities of space, time, energy and other units that are similar in magnitude to corresponding Planck units. This region may be characterized by particle ] of around {{val|e=19|u=GeV}} or {{val|e=9|u=J}}, ] intervals of around {{val|5|e=−44|u=s}} and ]s of around {{val|e=-35|u=m}} (approximately the energy-equivalent of the Planck mass, the Planck time and the Planck length, respectively). At the Planck scale, the predictions of the ], ] and ] are not expected to apply, and ] are expected to dominate. One example is represented by the conditions in the ] of our universe after the ], approximately 13.8 billion years ago.
			The four ]s that, by definition, have a numeric value 1 when expressed in these units are:
	There are two versions of Planck units, ] (also called "rationalized") and ] (also called "non-rationalized").
			* ''c'', the ] in vacuum,
			* ''G'', the ],
			* ''ħ'', the ], and
			* ''k''<sub>B</sub>, the ].
			Variants of the basic idea of Planck units exist, such as alternate choices of normalization that give other numeric values to one or more of the four constants above.
	The five ]s that Planck units, by definition, ] to 1 are:
	* the ] in vacuum, ''c'', (also known as '''Planck speed''')
	* the ], ''G'',
	** ''G'' for the Gaussian version, 4{{pi}}''G'' for the Lorentz–Heaviside version
	* the ], ''ħ'', (also known as '''Planck action''')
	* the ], ''ε''<sub>0</sub> (also known as '''Planck permittivity''')
	** ''ε''<sub>0</sub> for the Lorentz–Heaviside version, 4{{pi}}''ε''<sub>0</sub> for the Gaussian version
	* the ], ''k''<sub>B</sub> (also known as '''Planck heat capacity''')
	Each of these constants can be associated with a fundamental physical theory or concept: ''c'' with ], ''G'' with ], ''ħ'' with ], ''ε''<sub>0</sub> with ], and ''k''<sub>B</sub> with the notion of ]/] (] and ]).
	== Introduction ==		== Introduction ==
	Any system of measurement may be assigned a mutually independent set of base quantities and associated ], from which all other quantities and units may be derived. In the ], for example, the ] include length with the associated unit of the ]. 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 ], the base unit of time is the ], 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 ],		Any system of measurement may be assigned a mutually independent set of base quantities and associated ], from which all other quantities and units may be derived. In the ], for example, the ] include length with the associated unit of the ]. In the system of Planck units, a similar set of base quantities and associated units may be selected, in terms of which other quantities and coherent units may be expressed.<ref name=":2" /><ref name="Gravitation">{{Cite book |last1=Misner |first1=Charles W. |title=Gravitation |title-link=Gravitation (book) |last2=Thorne |first2=Kip S. |last3=Wheeler |first3=John A. |date=1973 |isbn=0-7167-0334-3 |location=New York |oclc=585119 |author-link=Charles W. Misner |author-link2=Kip Thorne |author-link3=John Archibald Wheeler}}</ref>{{rp|page=1215}} The Planck unit of length has become known as the Planck length, and the Planck unit of time is known as the Planck time, but this nomenclature has not been established as extending to all quantities.
	: <math> \begin{align}
	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{align}</math>
			All Planck units are derived from the dimensional universal physical constants that define the system, and in a convention in which these units are omitted (i.e. treated as having the dimensionless value 1), these constants are then eliminated from equations of physics in which they appear. For example, ],
			<math display="block">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},</math>
	can be expressed as:		can be expressed as:
			<math display="block">\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}.</math>Both equations are ] and equally valid in ''any'' system of quantities, but the second equation, with {{mvar|G}} absent, is relating only ] since any ratio of two like-dimensioned quantities is a dimensionless quantity. If, by a shorthand convention, it is understood that each physical quantity is the corresponding ratio with a coherent Planck unit (or "expressed in Planck units"), the ratios above may be expressed simply with the symbols of physical quantity, without being scaled explicitly by their corresponding unit:
			<math display="block">F' = \frac{m_1' m_2'}{r'^2}.</math>This last equation (without {{mvar|G}}) is valid with {{math|''F''{{′}}}}, {{math|''m''<sub>1</sub>′}}, {{math|''m''<sub>2</sub>′}}, and {{math|''r''{{′}}}} being the dimensionless ratio quantities ''corresponding&nbsp;to'' the standard quantities, written e.g. {{nowrap|{{math|''F''{{′}} ≘ ''F''}}}} or {{nowrap|{{math|1=''F''{{′}} = ''F''/''F''{{sub|P}}}}}}, but not as a direct equality of quantities. This may seem to be "setting the constants {{mvar|c}}, {{mvar|G}}, etc., to 1" if the correspondence of the quantities is thought of as equality. For this reason, Planck or other natural units should be employed with care. Referring to "{{nowrap|{{math|1=''G'' = ''c'' {{=}} 1}}}}", ] wrote that, "Mathematically it is an acceptable trick which saves labour. Physically it represents a loss of information and can lead to confusion."<ref>{{cite journal |last1=Wesson |first1=P. S. |author-link=Paul S. Wesson |year=1980 |title=The application of dimensional analysis to cosmology |journal=] |volume=27 |issue=2 |page=117 |bibcode=1980SSRv...27..109W |doi=10.1007/bf00212237 |s2cid=120784299}}</ref>
			== History and definition ==
	: <math> \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}.</math>
			]
			The concept of ] was introduced in 1874, when ], noting that electric charge is quantized, derived units of length, time, and mass, now named ] in his honor. Stoney chose his units so that ''G'', ''c'', and the ] ''e'' would be numerically equal to 1.<ref>
			{{cite journal
			|last=Barrow |first=J. D. |author-link=John D. Barrow
			|date=1983-03-01
			|title=Natural Units Before Planck
			|url=https://ui.adsabs.harvard.edu/abs/1983QJRAS..24...24B
			|journal=Quarterly Journal of the Royal Astronomical Society
			|volume=24
			|page=24
			|bibcode=1983QJRAS..24...24B |issn=0035-8738
			|access-date=16 April 2022 |archive-date=20 January 2022 |archive-url=https://web.archive.org/web/20220120030835/https://ui.adsabs.harvard.edu/abs/1983QJRAS..24...24B
			|url-status=live
			}}</ref> In 1899, one year before the advent of quantum theory, ] introduced what became later known as the Planck constant.<ref name="planck-1899">{{cite journal |last=Planck |first=Max |author-link=Max Planck |year=1899 |title=Über irreversible Strahlungsvorgänge |journal=Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin |volume=5 |pages=440–480 |url=https://www.biodiversitylibrary.org/item/93034#page/498/mode/1up |language=de |access-date=23 May 2020 |archive-date=17 November 2020 |archive-url=https://web.archive.org/web/20201117200137/https://www.biodiversitylibrary.org/item/93034#page/498/mode/1up |url-status=live }} pp.&nbsp;478–80 contain the first appearance of the Planck base units, and of the ], which Planck denoted by ''b''. ''a'' and ''f'' in this paper correspond to the ] and ] in this article.</ref><ref name="TOM">
			{{cite conference
			| last = Tomilin
			| first = K. A.
			| date = 1999
			| title = Natural Systems of Units. To the Centenary Anniversary of the Planck System
			| url = http://old.ihst.ru/personal/tomilin/papers/tomil.pdf
			| conference = Proceedings Of The XXII Workshop On High Energy Physics And Field Theory
			| conference-url = https://inspirehep.net/literature/1766318
			| pages = 287–296
			| access-date = 31 December 2019
			| archive-date = 12 December 2020
			| archive-url = https://web.archive.org/web/20201212041222/http://old.ihst.ru/personal/tomilin/papers/tomil.pdf
			| url-status = dead
			}}</ref> At the end of the paper, he proposed the base units that were later named in his honor. The Planck units are based on the quantum of ], now usually known as the Planck constant, which appeared in the ] for ]. Planck underlined the universality of the new unit system, writing:<ref name="planck-1899" />
			{{blockquote|{{lang|de|... die Möglichkeit gegeben ist, Einheiten für Länge, Masse, Zeit und Temperatur aufzustellen, welche, unabhängig von speciellen Körpern oder Substanzen, ihre Bedeutung für alle Zeiten und für alle, auch ausserirdische und aussermenschliche Culturen nothwendig behalten und welche daher als »natürliche Maasseinheiten« bezeichnet werden können.}}}}
	Both equations are ] and equally valid in ''any'' system of units, but the second equation, with ''G'' missing, is relating only ] 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:
			{{blockquote|... it is possible to set up units for length, mass, time and temperature, which are independent of special bodies or substances, necessarily retaining their meaning for all times and for all civilizations, including extraterrestrial and non-human ones, which can be called "natural units of measure".}}
			Planck considered only the units based on the universal constants <math>G</math>, <math>h</math>, <math>c</math>, and <math>k_{\rm B}</math> to arrive at natural units for ], ], ], and ].<ref name="TOM" /> His definitions differ from the modern ones by a factor of <math>\sqrt{2 \pi}</math>, because the modern definitions use <math>\hbar</math> rather than <math>h</math>.<ref name="planck-1899" /><ref name="TOM" />
	: <math> F = \frac{m_1 m_2}{r^2} \ .</math>
	This last equation (without ''G'') is valid only if ''F'', ''m''<sub>1</sub>, ''m''<sub>2</sub>, 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 {{nowrap|''G'' {{=}} ''c'' {{=}} 1}}, ] wrote that, "Mathematically it is an acceptable trick which saves labour. Physically it represents a loss of information and can lead to confusion."<ref>{{cite journal | last1 = Wesson | first1 = P. S. | year = 1980 | title = The application of dimensional analysis to cosmology | journal = Space Science Reviews | volume = 27 | issue = 2| page = 117 | bibcode=1980SSRv...27..109W | doi=10.1007/bf00212237}}</ref>
	==Definition==
	{| class="wikitable" style="margin:1em auto 1em auto; background:#fff;"		{| class="wikitable" style="margin:1em auto 1em auto; background:#fff;"
	|+Table 1: Dimensional universal physical constants normalized with Planck units		|+Table 1: Modern values for Planck's original choice of quantities
	! Constant
	! Symbol
	! ]
	! Value (] units)<ref name="CODATA">{{cite web|url=http://physics.nist.gov/cuu/Constants/index.html|title=Fundamental Physical Constants from NIST|website=physics.nist.gov}}</ref>
	|-
	| ] in vacuum
	| ''c''
	| L T<sup>−1</sup>
	| {{physconst|c}} <br /> ''(exact by definition of ])''
	|-
	| ]
	| ''G''<br>(1 for the Gaussian version, {{sfrac|4{{pi}}}} for the Lorentz–Heaviside version)
	| L<sup>3</sup> M<sup>−1</sup> T<sup>−2</sup>
	| {{physconst|G}}
	|-
	| ]
	| ''ħ'' = {{sfrac|''h''|2{{pi}}}}<br> where ''h'' is the ]
	| L<sup>2</sup> M T<sup>−1</sup>
	| {{physconst|hbar}}<br /> ''(exact by definition of the ] since ])''
	|-
	| ]
	| ''ε''<sub>0</sub><br>(1 for the Lorentz–Heaviside version, {{sfrac|4{{pi}}}} for the Gaussian version)
	| {{nowrap|L<sup>−3</sup> M<sup>−1</sup> T<sup>2</sup> Q<sup>2</sup>}}
	| {{physconst|eps0}}<br /> ''(exact by definitions of ] and ] until ])''
	|-
	| ]
	| ''k''<sub>B</sub>
	| L<sup>2</sup> M T<sup>−2</sup> Θ<sup>−1</sup>
	| {{physconst|k}}<br /> ''(exact by definition of the ] since ])''
	|}
	'''Key''': L = ], M = ], T = ], Q = ], Θ = ].
	As can be seen above, the ] of two bodies of 1 ] each, set apart by 1 ] is 1 ] in Gaussian version, or {{sfrac|4{{pi}}}} ] in Lorentz–Heaviside version. Likewise, the distance traveled by ] during 1 ] is 1 ]. 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:
	: <math> l_\text{P} = c \ t_\text{P} </math>
	: <math> 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} </math> (Lorentz–Heaviside version)
	: <math> F_\text{P} = \frac{l_\text{P} m_\text{P}}{t_\text{P}^2} = G \ \frac{m_\text{P}^2}{l_\text{P}^2} </math> (Gaussian version)
	: <math> E_\text{P} = \frac{l_\text{P}^2 m_\text{P}}{t_\text{P}^2} = \hbar \ \frac{1}{t_\text{P}} </math>
	: <math> C_\text{P} = \frac{t_\text{P}^2 q_\text{P}^2}{l_\text{P}^2 m_\text{P}} = \epsilon_0 \ l_\text{P} </math> (Lorentz–Heaviside version)
	: <math> 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} </math> (Gaussian version)
	: <math> E_\text{P} = \frac{l_\text{P}^2 m_\text{P}}{t_\text{P}^2} = k_\text{B} \ T_\text{P}</math>
	Solving the five equations above for the five unknowns results in a unique set of values for the five base Planck units:
	{| class="wikitable" style="margin:1em auto 1em auto; background:#fff;"
	|+Table 2: Base Planck units
	|-
	! rowspan=2| Quantity
	! colspan=2| Expression
	! colspan=2| Approximate ] equivalent
	! rowspan=2| Name
	|-		|-
			! Name
	! ]
			! Dimension
	! ]
			! Expression
	! ]
			! Value (] units)
	! ]
	|- style="text-align:left;"		|- style="text-align:left;"
			| Planck length
	| ] (L)
			| ] (L)
	| <math>l_\text{P} = \sqrt{\frac{4\pi \hbar G}{c^3}}</math>
	| <math>l_\text{P} = \sqrt{\frac{\hbar G}{c^3}}</math>		| <math>l_\text{P} = \sqrt{\frac{\hbar G}{c^3}}</math>
			| {{physconst|lP}}
	| ] ]
	| ] ]
	| ]
	|-		|-
			| Planck mass
	| ] (M)
			| ] (M)
	| <math>m_\text{P} = \sqrt{\frac{\hbar c}{4\pi G}}</math>
	| <math>m_\text{P} = \sqrt{\frac{\hbar c}{G}}</math>		| <math>m_\text{P} = \sqrt{\frac{\hbar c}{G}}</math>
			| {{physconst|mP}}
	| ] ]
	| ] ]
	| ]
	|-		|-
			| Planck time
	| ] (T)
			| ] (T)
	| <math>t_\text{P} = \sqrt{\frac{4\pi \hbar G}{c^5}}</math>
	| <math>t_\text{P} = \sqrt{\frac{\hbar G}{c^5}}</math>		| <math>t_\text{P} = \sqrt{\frac{\hbar G}{c^5}}</math>
			| {{physconst|tP}}
	| ] ]
	| ] ]
	| ]
	|-		|-
			| Planck temperature
	| ] (Q)
			| ] (Θ)
	| <math>q_\text{P} = \sqrt{\hbar c \epsilon_0}</math>
	| <math>q_\text{P} = \sqrt{4\pi \hbar c \epsilon_0}</math>		| <math>T_\text{P} = \sqrt{\frac{\hbar c^5}{G k_\text{B}^2}}</math>
			| {{physconst|TP}}
	| ] ]
	| ] ]
	| ]
	|-
	| ] (Θ)
	| <math>T_\text{P} = \sqrt{\frac{\hbar c^5}{4\pi G {k_\text{B}}^2}}</math>
	| <math>T_\text{P} = \sqrt{\frac{\hbar c^5}{G {k_\text{B}}^2}}</math>
	| ] ]
	| ] ]
	| ]
	|}		|}
			Unlike the case with the ], there is no official entity that establishes a definition of a Planck unit system. Some authors define the base Planck units to be those of mass, length and time, regarding an additional unit for temperature to be redundant.{{NoteTag|For example, both ] and ] do so,<ref name=":2">
	Table 2 clearly defines Planck units in terms of the fundamental constants. Yet relative to other units of measurement such as ], 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.
			{{cite journal
			| last = Wilczek | first = Frank | author-link = Frank Wilczek
	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 ], is now ''defined'' as the length of the path travelled by light in vacuum during a time interval of {{sfrac|{{gaps|299|792|458}}}} of a second. Hence the value of ''c'' is now exact by definition, and contributes no uncertainty to the SI equivalents of the Planck units. The same is true of the value of the vacuum permittivity ''ε''<sub>0</sub>, due to the definition of ] which sets the ] ''μ''<sub>0</sub> to {{nowrap|4{{pi}} × 10<sup>−7</sup> H/m}} and the fact that ''μ''<sub>0</sub>''ε''<sub>0</sub> = {{sfrac|''c''<sup>2</sup>}}. 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 {{val|21300}} (or {{val|47000}} parts per billion).<ref name="CODATA" /> ''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 ±{{sfrac|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 {{val|43500}}, or {{val|23000}} parts per billion.)
			| date = 2005
			| title = On Absolute Units, I: Choices
			| journal = ]
			| volume = 58
			| issue = 10
			| pages = 12–13
			| doi = 10.1063/1.2138392
			| publisher = ]
			| bibcode = 2005PhT....58j..12W
			}}</ref><ref name=":0">
			{{cite book
			|last=Zwiebach |first=Barton |author-link=Barton Zwiebach
			|title=A First Course in String Theory
			|publisher=Cambridge University Press
			|year=2004
			|isbn=978-0-521-83143-7 |oclc=58568857
			}}</ref>{{rp|page=54}} as does the textbook '']''.<ref name="Gravitation"/>{{rp|1215}}}} Other tabulations add, in addition to a unit for temperature, a unit for electric charge, so that either the ] <math>k_\text{e}</math><ref>
			{{cite book
			| last1 = Deza | first1 = Michel Marie
			| last2 = Deza | first2 = Elena | author2-link = Elena Deza
			| date = 2016
			| title = Encyclopedia of Distances
			| url = https://books.google.com/books?id=q_7FBAAAQBAJ&pg=PA602
			| publisher = ]
			| page = 602
			| isbn = 978-3662528433
			| access-date = 9 September 2020
			| archive-date = 6 March 2021
			| archive-url = https://web.archive.org/web/20210306212728/https://books.google.com/books?id=q_7FBAAAQBAJ&pg=PA602
			| url-status = live
			}}</ref><ref name="physics_hypertextbook">
			{{cite web
			| url = https://physics.info/planck/#mechanics
			| title = Blackbody Radiation
			| last = Elert | first = Glenn
			| website = The Physics Hypertextbook
			| access-date = 2021-02-22
			| archive-date = 3 March 2021
			| archive-url = https://web.archive.org/web/20210303061957/https://physics.info/planck/#mechanics
			| url-status = live
			}}</ref><ref name="PAV">
			{{cite book
			|last=Pavšic |first=Matej
			|title=The Landscape of Theoretical Physics: A Global View
			|volume=119
			|year=2001
			|publisher=Kluwer Academic
			|location=Dordrecht
			|isbn=978-0-7923-7006-2
			|pages=347–352
			|url=https://www.springer.com/gp/book/9781402003516#otherversion=9780792370062
			|series=Fundamental Theories of Physics
			|doi=10.1007/0-306-47136-1
			|arxiv=gr-qc/0610061
			|access-date=31 December 2019
			|archive-date=5 September 2021
			|archive-url=https://web.archive.org/web/20210905013239/https://www.springer.com/gp/book/9781402003516#otherversion=9780792370062
			|url-status=live
			}}</ref> or the ] <math>\epsilon_0</math><ref name="Zeidler"/> is normalized to 1. Thus, depending on the author's choice, this charge unit is given by
			<math display="block">q_\text{P} = \sqrt{4\pi\epsilon_0 \hbar c} \approx 1.875546 \times 10^{-18} \text{ C} \approx 11.7 \ e</math>
			for <math> k_\text{e} = 1</math>, or
			<math display="block">q_\text{P} = \sqrt{\epsilon_0 \hbar c} \approx 5.290818 \times 10^{-19} \text{ C} \approx 3.3 \ e</math>
			for <math> \varepsilon_0 = 1</math>. Some of these tabulations also replace mass with energy when doing so.<ref name="Zeidler">
			{{cite book
			| last = Zeidler | first = Eberhard
			| date = 2006
			| title = Quantum Field Theory I: Basics in Mathematics and Physics
			| url = https://cds.cern.ch/record/988238/files/978-3-540-34764-4_BookBackMatter.pdf
			| publisher = ]
			| page = 953
			| isbn = 978-3540347620
			| access-date = 31 May 2020
			| archive-date = 19 June 2020
			| archive-url = https://web.archive.org/web/20200619210529/https://cds.cern.ch/record/988238/files/978-3-540-34764-4_BookBackMatter.pdf
			| url-status = live
			}}</ref>
			In SI units, the values of ''c'', ''h'', ''e'' and ''k''<sub>B</sub> are exact and the values of ''ε''<sub>0</sub> and ''G'' in SI units respectively have relative uncertainties of {{physconst|eps0|runc=yes}} and {{physconst|G|runc=yes|after=.}} Hence, the uncertainties in the SI values of the Planck units derive almost entirely from uncertainty in the SI value of ''G''.
			Compared to ], Planck base units are all larger by a factor <math display="inline">\sqrt{{1}/{\alpha}} \approx 11.7</math>, where <math>\alpha</math> is the ].<ref>{{cite journal|last=Ray |first=T. P. |title=Stoney's Fundamental Units |journal=Irish Astronomical Journal |volume=15 |page=152 |year=1981 |bibcode=1981IrAJ...15..152R}}</ref>
	After ], ''h'' (and thus <math>\hbar=\frac{h}{2\pi}</math>) is exact, ''k''<sub>B</sub> is also exact, but since ''G'' is still not exact, the values of ''l''<sub>P</sub>, ''m''<sub>P</sub>, ''t''<sub>P</sub>, and ''T''<sub>P</sub> are also not exact. Besides, ''μ''<sub>0</sub> (and thus <math>\epsilon_0=\frac{1}{c^2 \mu_0}</math>) is no longer exact (only ''e'' is exact), thus ''q''<sub>P</sub> is also not exact.
	== Derived units ==		== Derived units ==
			In any system of measurement, units for many physical quantities can be derived from base units. Table 2 offers a sample of derived Planck units, some of which 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.
	{{unreferenced section|date=February 2020}}
	In any system of measurement, units for many physical quantities can be derived from base units. Table 3 offers a sample of ] 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.
	{| class="wikitable" style="margin:1em auto 1em auto; background:#fff;"		{| class="wikitable" style="margin:1em auto 1em auto; background:#fff;"
	|+ Table 3: Derived Planck units		|+ Table 2: Coherent derived units of Planck units
	|-		|-
			! Derived unit of
	! rowspan=2| Name
			! Expression
	! rowspan=2| ]
			! Approximate ] equivalent
	! colspan=2| Expression
			|- style="text-align:left;"
	! colspan=2| Approximate ] equivalent
			| ] (L<sup>2</sup>)
	|-
			| <math> l_\text{P}^2 = \frac{\hbar G}{c^3}</math>
	! ]
			| {{val|2.6121|e=-70|ul=m2}}
	! ]
			|- style="text-align:left;"
	! ]
			| ] (L<sup>3</sup>)
	! ]
			| <math> l_\text{P}^3 = \left( \frac{\hbar G}{c^3} \right)^{\frac{3}{2}} = \sqrt{\frac{(\hbar G)^3}{c^9}}</math>
	|-
			| {{val|4.2217|e=-105|ul=m3}}
	| ]
	| ] (L<sup>2</sup>)
	| <math>A_\text{P} = l_\text{P}^2 = \frac{4\pi \hbar G}{c^3}</math>
	| <math>A_\text{P} = l_\text{P}^2 = \frac{\hbar G}{c^3}</math>
	| ] ]
	| ] ]
	|-
	| ]
	| ] (L<sup>3</sup>)
	| <math>V_\text{P} = l_\text{P}^3 = \sqrt{\frac{64\pi^3 \hbar^3 G^3}{c^9}}</math>
	| <math>V_\text{P} = l_\text{P}^3 = \sqrt{\frac{\hbar^3 G^3}{c^9}}</math>
	| ] ]
	| ] ]
	|-
	| Planck wavenumber
	| ] (L<sup>−1</sup>)
	| <math>N_\text{P} = \frac{1}{l_\text{P}} = \sqrt{\frac{c^3}{4\pi \hbar G}}</math>
	| <math>N_\text{P} = \frac{1}{l_\text{P}} = \sqrt{\frac{c^3}{\hbar G}}</math>
	| ] ]
	| ] ]
	|-
	| ]
	| ] (L<sup>−3</sup>M)
	| <math>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}</math>
	| <math>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}</math>
	| ] ]
	| ] ]
	|-
	| Planck specific volume
	| ] (L<sup>3</sup>M<sup>−1</sup>)
	| <math>\beta_\text{P} = \frac{1}{d_\text{P}} = \frac{16\pi^2 \hbar G^2}{c^5}</math>
	| <math>\beta_\text{P} = \frac{1}{d_\text{P}} = \frac{\hbar G^2}{c^5}</math>
	| ] ]
	| ] ]
	|-
	| ]
	| ] (T<sup>−1</sup>)
	| <math>f_\text{P} = \frac{1}{t_\text{P}} = \sqrt{\frac{c^5}{4\pi \hbar G}}</math>
	| <math>f_\text{P} = \frac{1}{t_\text{P}} = \sqrt{\frac{c^5}{\hbar G}}</math>
	| ] ]
	| ] ]
	|-
	| ]
	| ] (LT<sup>−1</sup>)
	| colspan=2| <math>v_\text{P} = \frac{l_\text{P}}{t_\text{P}} = c</math>
	| colspan=2| ] ]
	|-
	| ]
	| ] (LT<sup>−2</sup>)
	| <math>a_\text{P} = \frac{v_\text{P}}{t_\text{P}} = \sqrt{\frac{c^7}{4\pi \hbar G}}</math>
	| <math>a_\text{P} = \frac{v_\text{P}}{t_\text{P}} = \sqrt{\frac{c^7}{\hbar G}}</math>
	| ] ]
	| ] ]
	|-
	| Planck jerk
	| ] (LT<sup>−3</sup>)
	| <math>\mathcal{J}_\text{P} = \frac{a_\text{P}}{t_\text{P}} = \frac{c^6}{4\pi \hbar G}</math>
	| <math>\mathcal{J}_\text{P} = \frac{a_\text{P}}{t_\text{P}} = \frac{c^6}{\hbar G}</math>
	| ] ]
	| ] ]
	|-
	| Planck snap
	| ] (LT<sup>−4</sup>)
	| <math>\mathcal{N}_\text{P} = \frac{\mathcal{J}_\text{P}}{t_\text{P}} = \sqrt{\frac{c^{17}}{64\pi^3 \hbar^3 G^3}}</math>
	| <math>\mathcal{N}_\text{P} = \frac{\mathcal{J}_\text{P}}{t_\text{P}} = \sqrt{\frac{c^{17}}{\hbar^3 G^3}}</math>
	| ] ]
	| ] ]
	|-
	| Planck crackle
	| ] (LT<sup>−5</sup>)
	| <math>\mathcal{K}_\text{P} = \frac{\mathcal{N}_\text{P}}{t_\text{P}} = \frac{c^{11}}{16\pi^2 \hbar^2 G^2}</math>
	| <math>\mathcal{K}_\text{P} = \frac{\mathcal{N}_\text{P}}{t_\text{P}} = \frac{c^{11}}{\hbar^2 G^2}</math>
	| ] ]
	| ] ]
	|-
	| Planck pop
	| ] (LT<sup>−6</sup>)
	| <math>\mathcal{P}_\text{P} = \frac{\mathcal{K}_\text{P}}{t_\text{P}} = \sqrt{\frac{c^{27}}{1024\pi^5 \hbar^5 G^5}}</math>
	| <math>\mathcal{P}_\text{P} = \frac{\mathcal{K}_\text{P}}{t_\text{P}} = \sqrt{\frac{c^{27}}{\hbar^5 G^5}}</math>
	| ] ]
	| ] ]
	|-
	| ]
	| ] (LMT<sup>−1</sup>)
	| <math>p_\text{P} = m_\text{P}v_\text{P} = \frac{\hbar}{l_\text{P}} = \sqrt{\frac{\hbar c^3}{4\pi G}}</math>
	| <math>p_\text{P} = m_\text{P}v_\text{P} = \frac{\hbar}{l_\text{P}} = \sqrt{\frac{\hbar c^3}{G}}</math>
	| ] ]
	| ] ]
	|-
	| ]
	| ] (LMT<sup>−2</sup>)
	| <math>F_\text{P} = m_\text{P}a_\text{P} = \frac{p_\text{P}}{t_\text{P}} = \frac{c^4}{4\pi G}</math>
	| <math>F_\text{P} = m_\text{P}a_\text{P} = \frac{p_\text{P}}{t_\text{P}} = \frac{c^4}{G}</math>
	| ] ]
	| ] ]
	|-
	| ]
	| ] (L<sup>2</sup>MT<sup>−2</sup>)
	| <math>E_\text{P} = m_\text{P}v_\text{P}^2 = \frac{\hbar}{t_\text{P}} = \sqrt{\frac{\hbar c^5}{4\pi G}}</math>
	| <math>E_\text{P} = m_\text{P}v_\text{P}^2 = \frac{\hbar}{t_\text{P}} = \sqrt{\frac{\hbar c^5}{G}}</math>
	| ] ]
	| ] ]
	|-
	| ]
	| ] (L<sup>2</sup>MT<sup>−3</sup>)
	| <math>P_\text{P} = \frac{E_\text{P}}{t_\text{P}} = \frac{\hbar}{t_\text{P}^2} = \frac{c^5}{4\pi G}</math>
	| <math>P_\text{P} = \frac{E_\text{P}}{t_\text{P}} = \frac{\hbar}{t_\text{P}^2} = \frac{c^5}{G}</math>
	| ] ]
	| ] ]
	|-
	| ]
	| ] (L<sup>2</sup>T<sup>−2</sup>)
	| colspan=2| <math>h_\text{P} = \frac{E_\text{P}}{m_\text{P}} = c^2</math>
	| colspan=2| ] ]
	|-
	| Planck energy density
	| ] (L<sup>−1</sup>MT<sup>−2</sup>)
	| <math>u_\text{P} = \frac{E_\text{P}}{V_\text{P}} = \frac{c^7}{16\pi^2 \hbar G^2}</math>
	| <math>u_\text{P} = \frac{E_\text{P}}{V_\text{P}} = \frac{c^7}{\hbar G^2}</math>
	| ] ]
	| ] ]
	|-
	| Planck intensity
	| ] (MT<sup>−3</sup>)
	| <math>\mathcal{I}_\text{P} = \frac{P_\text{P}}{A_\text{P}} = \frac{c^8}{16\pi^2 \hbar G^2}</math>
	| <math>\mathcal{I}_\text{P} = \frac{P_\text{P}}{A_\text{P}} = \frac{c^8}{\hbar G^2}</math>
	| ] ]
	| ] ]
	|-
	| ]
	| ] (L<sup>2</sup>MT<sup>−1</sup>)
	| colspan=2| <math>\mathcal{S}_\text{P} = l_\text{P}p_\text{P} = E_\text{P}t_\text{P} = \hbar</math>
	| colspan=2| ] ]
	|-
	| Planck gravitational induction
	| ] (LT<sup>−2</sup>)
	| <math>g_\text{P} = \frac{F_\text{P}}{m_\text{P}} = \sqrt{\frac{c^7}{4\pi \hbar G}}</math>
	| <math>g_\text{P} = \frac{F_\text{P}}{m_\text{P}} = \sqrt{\frac{c^7}{\hbar G}}</math>
	| ] ]
	| ] ]
	|-
	| Planck angle
	| ] (dimensionless)
	| colspan=2| <math>\theta_\text{P} = \frac{l_\text{P}}{l_\text{P}} = 1</math>
	| colspan=2| ] ]
	|-
	| ]
	| ] (T<sup>−1</sup>)
	| <math>\omega_\text{P} = \frac{\theta_\text{P}}{t_\text{P}} = \sqrt{\frac{c^5}{4\pi \hbar G}}</math>
	| <math>\omega_\text{P} = \frac{\theta_\text{P}}{t_\text{P}} = \sqrt{\frac{c^5}{\hbar G}}</math>
	| ] ]
	| ] ]
	|-
	| Planck angular acceleration
	| ] (T<sup>−2</sup>)
	| <math>\alpha_\text{P} = \frac{\omega_\text{P}}{t_\text{P}} = \frac{c^5}{4\pi \hbar G}</math>
	| <math>\alpha_\text{P} = \frac{\omega_\text{P}}{t_\text{P}} = \frac{c^5}{\hbar G}</math>
	| ] ]
	| ] ]
	|-
	| Planck angular jerk
	| ] (T<sup>−3</sup>)
	| <math>\zeta_\text{P} = \frac{\alpha_\text{P}}{t_\text{P}} = \sqrt{\frac{c^{15}}{64\pi^3 \hbar^3 G^3}}</math>
	| <math>\zeta_\text{P} = \frac{\alpha_\text{P}}{t_\text{P}} = \sqrt{\frac{c^{15}}{\hbar^3 G^3}}</math>
	| ] ]
	| ] ]
	|-
	| Planck rotational inertia
	| ] (L<sup>2</sup>M)
	| <math>I_\text{P} = m_\text{P}l_\text{P}^2 = \sqrt{\frac{4\pi \hbar^3 G}{c^5}}</math>
	| <math>I_\text{P} = m_\text{P}l_\text{P}^2 = \sqrt{\frac{\hbar^3 G}{c^5}}</math>
	| ] ]
	| ] ]
	|-
	| ]
	| ] (L<sup>2</sup>MT<sup>−1</sup>)
	| colspan=2| <math>\Lambda_\text{P} = I_\text{P}\omega_\text{P} = \hbar</math>
	| colspan=2| ] ]
	|-
	| Planck torque
	| ] (L<sup>2</sup>MT<sup>−2</sup>)
	| <math>\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}}</math>
	| <math>\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}}</math>
	| ] ]
	| ] ]
	|-
	| Planck specific angular momentum
	| ] (L<sup>2</sup>T<sup>−1</sup>)
	| <math>\pi_\text{P} = \frac{\Lambda_\text{P}}{m_\text{P}} = \sqrt{\frac{4\pi \hbar G}{c}}</math>
	| <math>\pi_\text{P} = \frac{\Lambda_\text{P}}{m_\text{P}} = \sqrt{\frac{\hbar G}{c}}</math>
	| ] ]
	| ] ]
	|-
	| Planck solid angle
	| ] (dimensionless)
	| colspan=2| <math>\Omega_\text{P} = \theta_\text{P}^2 = \frac{l_\text{P}^2}{l_\text{P}^2} = 1</math>
	| colspan=2| ] ]
	|-
	| Planck radiant intensity
	| ] (L<sup>2</sup>MT<sup>−3</sup>)
	| <math>\iota_\text{P} = \frac{P_\text{P}}{\Omega_\text{P}} = \frac{c^5}{4\pi G}</math>
	| <math>\iota_\text{P} = \frac{P_\text{P}}{\Omega_\text{P}} = \frac{c^5}{G}</math>
	| ] ]
	| ] ]
	|-
	| Planck radiance
	| ] (MT<sup>−3</sup>)
	| <math>\mathcal{L}_\text{P} = \frac{P_\text{P}}{A_\text{P}\Omega_\text{P}} = \frac{c^8}{16\pi^2 \hbar G^2}</math>
	| <math>\mathcal{L}_\text{P} = \frac{P_\text{P}}{A_\text{P}\Omega_\text{P}} = \frac{c^8}{\hbar G^2}</math>
	| ] ]
	| ] ]
	|-
	| Planck pressure
	| ] (L<sup>−1</sup>MT<sup>−2</sup>)
	| <math>\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}</math>
	| <math>\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}</math>
	| ] ]
	| ] ]
	|-
	| Planck surface tension
	| ] (MT<sup>−2</sup>)
	| <math>\sigma_\text{P} = \frac{F_\text{P}}{l_\text{P}} = \sqrt{\frac{c^{11}}{64\pi^3 \hbar G^3}}</math>
	| <math>\sigma_\text{P} = \frac{F_\text{P}}{l_\text{P}} = \sqrt{\frac{c^{11}}{\hbar G^3}}</math>
	| ] ]
	| ] ]
	|-
	| Planck volumetric flow rate
	| ] (L<sup>3</sup>T<sup>−1</sup>)
	| <math>Q_\text{P} = \frac{V_\text{P}}{t_\text{P}} = l_\text{P}^2v_\text{P} = \frac{4\pi \hbar G}{c^2}</math>
	| <math>Q_\text{P} = \frac{V_\text{P}}{t_\text{P}} = l_\text{P}^2v_\text{P} = \frac{\hbar G}{c^2}</math>
	| ] ]
	| ] ]
	|-
	| Planck mass flow rate
	| ] (MT<sup>−1</sup>)
	| <math>M_\text{P} = \frac{m_\text{P}}{t_\text{P}} = \frac{c^3}{4\pi G}</math>
	| <math>M_\text{P} = \frac{m_\text{P}}{t_\text{P}} = \frac{c^3}{G}</math>
	| ] ]
	| ] ]
	|-
	| Planck mass flux
	| ] (L<sup>−2</sup>MT<sup>−1</sup>)
	| <math>J_\text{P} = \frac{M_\text{P}}{A_\text{P}} = \frac{c^6}{16\pi^2 \hbar G^2}</math>
	| <math>J_\text{P} = \frac{M_\text{P}}{A_\text{P}} = \frac{c^6}{\hbar G^2}</math>
	| ] ]
	| ] ]
	|-
	| Planck stiffness
	| ] (MT<sup>−2</sup>)
	| <math>K_\text{P} = \frac{F_\text{P}}{l_\text{P}} = \sqrt{\frac{c^{11}}{64\pi^3 \hbar G^3}}</math>
	| <math>K_\text{P} = \frac{F_\text{P}}{l_\text{P}} = \sqrt{\frac{c^{11}}{\hbar G^3}}</math>
	| ] ]
	| ] ]
	|-
	| Planck flexibility
	| ] (M<sup>−1</sup>T<sup>2</sup>)
	| <math>x_\text{P} = \frac{1}{K_\text{P}} = \sqrt{\frac{64\pi^3 \hbar G^3}{c^{11}}}</math>
	| <math>x_\text{P} = \frac{1}{K_\text{P}} = \sqrt{\frac{\hbar G^3}{c^{11}}}</math>
	| ] ]
	| ] ]
	|-
	| Planck rotational stiffness
	| ] (L<sup>2</sup>MT<sup>−2</sup>)
	| <math>O_\text{P} = \frac{\tau_\text{P}}{\theta_\text{P}} = \sqrt{\frac{\hbar c^5}{4\pi G}}</math>
	| <math>O_\text{P} = \frac{\tau_\text{P}}{\theta_\text{P}} = \sqrt{\frac{\hbar c^5}{G}}</math>
	| ] ]
	| ] ]
	|-
	| Planck rotational flexibility
	| ] (L<sup>−2</sup>M<sup>−1</sup>T<sup>2</sup>)
	| <math>y_\text{P} = \frac{1}{O_\text{P}} = \sqrt{\frac{4\pi G}{\hbar c^5}}</math>
	| <math>y_\text{P} = \frac{1}{O_\text{P}} = \sqrt{\frac{G}{\hbar c^5}}</math>
	| ] ]
	| ] ]
	|-
	| Planck ultimate tensile strength
	| ] (L<sup>−1</sup>MT<sup>−2</sup>)
	| <math>\Sigma_\text{P} = \frac{F_\text{P}}{A_\text{P}} = \frac{c^7}{16\pi^2 \hbar G^2}</math>
	| <math>\Sigma_\text{P} = \frac{F_\text{P}}{A_\text{P}} = \frac{c^7}{\hbar G^2}</math>
	| ] ]
	| ] ]
	|-
	| Planck indentation hardness
	| ] (L<sup>−1</sup>MT<sup>−2</sup>)
	| <math>\mathcal{H}_\text{P} = \frac{F_\text{P}}{A_\text{P}} = \frac{c^7}{16\pi^2 \hbar G^2}</math>
	| <math>\mathcal{H}_\text{P} = \frac{F_\text{P}}{A_\text{P}} = \frac{c^7}{\hbar G^2}</math>
	| ] ]
	| ] ]
	|-
	| Planck absolute hardness
	| ] (M)
	| <math>\mathcal{G}_\text{P} = \frac{F_\text{P}}{g_\text{P}} = \sqrt{\frac{\hbar c}{4\pi G}}</math>
	| <math>\mathcal{G}_\text{P} = \frac{F_\text{P}}{g_\text{P}} = \sqrt{\frac{\hbar c}{G}}</math>
	| ] ]
	| ] ]
	|-
	| Planck viscosity
	| ] (L<sup>−1</sup>MT<sup>−1</sup>)
	| <math>\eta_\text{P} = P_\text{P}t_\text{P} = \sqrt{\frac{c^9}{64\pi^3 \hbar G^3}}</math>
	| <math>\eta_\text{P} = P_\text{P}t_\text{P} = \sqrt{\frac{c^9}{\hbar G^3}}</math>
	| ] ]
	| ] ]
	|-
	| Planck kinematic viscosity
	| ] (L<sup>2</sup>T<sup>−1</sup>)
	| <math>\nu_\text{P} = \frac{A_\text{P}}{t_\text{P}} = \frac{\eta_\text{P}}{d_\text{P}} = \sqrt{\frac{4\pi \hbar G}{c}}</math>
	| <math>\nu_\text{P} = \frac{A_\text{P}}{t_\text{P}} = \frac{\eta_\text{P}}{d_\text{P}} = \sqrt{\frac{\hbar G}{c}}</math>
	| ] ]
	| ] ]
	|-
	| Planck toughness
	| ] (L<sup>−1</sup>MT<sup>−2</sup>)
	| <math>\mathcal{T}_\text{P} = \frac{E_\text{P}}{V_\text{P}} = \frac{c^7}{16\pi^2 \hbar G^2}</math>
	| <math>\mathcal{T}_\text{P} = \frac{E_\text{P}}{V_\text{P}} = \frac{c^7}{\hbar G^2}</math>
	| ] ]
	| ] ]
	|-
	| Planck current
	| ] (T<sup>−1</sup>Q)
	| <math>i_\text{P} = \frac{q_\text{P}}{t_\text{P}} = \sqrt{\frac{c^6 \epsilon_0}{4\pi G}}</math>
	| <math>i_\text{P} = \frac{q_\text{P}}{t_\text{P}} = \sqrt{\frac{4\pi c^6 \epsilon_0}{G}}</math>
	| ] ]
	| ] ]
	|-
	| Planck voltage
	| ] (L<sup>2</sup>MT<sup>−2</sup>Q<sup>−1</sup>)
	| colspan=2| <math>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}}</math>
	| colspan=2| ] ]
	|-
	| ]
	| ] (L<sup>2</sup>MT<sup>−1</sup>Q<sup>−2</sup>)
	| <math>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}</math>
	| <math>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}</math>
	| ] ]
	| ] ]
	|-
	| ]
	| ] (L<sup>−2</sup>M<sup>−1</sup>TQ<sup>2</sup>)
	| <math>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}</math>
	| <math>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}</math>
	| ] ]
	| ] ]
	|-
	| Planck capacitance
	| ] (L<sup>−2</sup>M<sup>−1</sup>T<sup>2</sup>Q<sup>2</sup>)
	| <math>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}}</math>
	| <math>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}}</math>
	| ] ]
	| ] ]
	|-
	| Planck inductance
	| ] (L<sup>2</sup>MQ<sup>−2</sup>)
	| <math>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}}</math>
	| <math>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}}</math>
	| ] ]
	| ] ]
	|-
	| Planck electrical resistivity
	| ] (L<sup>3</sup>MT<sup>−1</sup>Q<sup>−2</sup>)
	| <math>r_\text{P} = Z_\text{P}l_\text{P} = \sqrt{\frac{4\pi \hbar G}{c^5 \epsilon_0^2}}</math>
	| <math>r_\text{P} = Z_\text{P}l_\text{P} = \sqrt{\frac{\hbar G}{16\pi^2 c^5 \epsilon_0^2}}</math>
	| ] ]
	| ] ]
	|-
	| Planck electrical conductivity
	| ] (L<sup>−3</sup>M<sup>−1</sup>TQ<sup>2</sup>)
	| <math>\kappa_\text{P} = \frac{1}{r_\text{P}} = \sqrt{\frac{c^5 \epsilon_0^2}{4\pi \hbar G}}</math>
	| <math>\kappa_\text{P} = \frac{1}{r_\text{P}} = \sqrt{\frac{16\pi^2 c^5 \epsilon_0^2}{\hbar G}}</math>
	| ] ]
	| ] ]
	|-
	| Planck charge-to-mass ratio
	| ] (M<sup>−1</sup>Q)
	| colspan=2| <math>\xi_\text{P} = \frac{q_\text{P}}{m_\text{P}} = \sqrt{4\pi G \epsilon_0} = \sqrt{\frac{G}{k_e}}</math>
	| colspan=2| ] ]
	|-
	| Planck mass-to-charge ratio
	| ] (MQ<sup>−1</sup>)
	| colspan=2| <math>\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}}</math>
	| colspan=2| ] ]
	|-
	| Planck charge density
	| ] (L<sup>−3</sup>Q)
	| <math>\rho_\text{P} = \frac{q_\text{P}}{V_\text{P}} = \sqrt{\frac{c^{10} \epsilon_0}{64\pi^3 \hbar^2 G^3}}</math>
	| <math>\rho_\text{P} = \frac{q_\text{P}}{V_\text{P}} = \sqrt{\frac{4\pi c^{10} \epsilon_0}{\hbar^2 G^3}}</math>
	| ] ]
	| ] ]
	|-
	| Planck current density
	| ] (L<sup>−2</sup>T<sup>−1</sup>Q)
	| <math>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}}</math>
	| <math>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}}</math>
	| ] ]
	| ] ]
	|-
	| Planck magnetic charge
	| ] (LT<sup>−1</sup>Q)
	| <math>b_\text{P} = q_\text{P}v_\text{P} = \sqrt{\hbar c^3 \epsilon_0}</math>
	| <math>b_\text{P} = q_\text{P}v_\text{P} = \sqrt{4\pi \hbar c^3 \epsilon_0}</math>
	| ] ]
	| ] ]
	|-
	| Planck magnetic current
	| ] (L<sup>2</sup>MT<sup>−2</sup>Q<sup>−1</sup>)
	| colspan=2| <math>k_\text{P} = U_\text{P} = \sqrt{\frac{c^4}{4\pi G \epsilon_0}}</math>
	| colspan=2| ] ]
	|-
	| Planck magnetic current density
	| ] (MT<sup>−2</sup>Q<sup>−1</sup>)
	| <math>\delta_\text{P} = \frac{k_\text{P}}{A_\text{P}} = \sqrt{\frac{c^{10}}{64\pi^3 \hbar^2 G^3 \epsilon_0}}</math>
	| <math>\delta_\text{P} = \frac{k_\text{P}}{A_\text{P}} = \sqrt{\frac{c^{10}}{4\pi \hbar^2 G^3 \epsilon_0}}</math>
	| ] ]
	| ] ]
	|-
	| Planck electric field intensity
	| ] (LMT<sup>−2</sup>Q<sup>−1</sup>)
	| <math>e_\text{P} = \frac{F_\text{P}}{q_\text{P}} = \sqrt{\frac{c^7}{16\pi^2 \hbar G^2 \epsilon_0}}</math>
	| <math>e_\text{P} = \frac{F_\text{P}}{q_\text{P}} = \sqrt{\frac{c^7}{4\pi \hbar G^2 \epsilon_0}}</math>
	| ] ]
	| ] ]
	|-
	| Planck magnetic field intensity
	| ] (L<sup>−1</sup>T<sup>−1</sup>Q)
	| <math>H_\text{P} = \frac{i_\text{P}}{l_\text{P}} = \sqrt{\frac{c^9 \epsilon_0}{16\pi^2 \hbar G^2}}</math>
	| <math>H_\text{P} = \frac{i_\text{P}}{l_\text{P}} = \sqrt{\frac{4\pi c^9 \epsilon_0}{\hbar G^2}}</math>
	| ] ]
	| ] ]
	|-
	| Planck electric induction
	| ] (L<sup>−2</sup>Q)
	| <math>D_\text{P} = \frac{q_\text{P}}{l_\text{P}^2} = \sqrt{\frac{c^7 \epsilon_0}{16\pi^2 \hbar G^2}}</math>
	| <math>D_\text{P} = \frac{q_\text{P}}{l_\text{P}^2} = \sqrt{\frac{4\pi c^7 \epsilon_0}{\hbar G^2}}</math>
	| ] ]
	| ] ]
	|-
	| Planck magnetic induction
	| ] (MT<sup>−1</sup>Q<sup>−1</sup>)
	| <math>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}}</math>
	| <math>B_\text{P} = \frac{F_\text{P}}{l_\text{P}i_\text{P}} = \sqrt{\frac{c^5}{4\pi \hbar G^2 \epsilon_0}}</math>
	| ] ]
	| ] ]
	|-
	| Planck electric potential
	| ] (L<sup>2</sup>MT<sup>−2</sup>Q<sup>−1</sup>)
	| colspan=2| <math>\phi_\text{P} = \frac{E_\text{P}}{q_\text{P}} = U_\text{P} = \sqrt{\frac{c^4}{4\pi G \epsilon_0}}</math>
	| colspan=2| ] ]
	|-
	| Planck magnetic potential
	| ] (LMT<sup>−1</sup>Q<sup>−1</sup>)
	| colspan=2|<math>\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}}</math>
	| colspan=2|] ]
	|-
	| Planck electromotive force
	| ] (L<sup>2</sup>MT<sup>−2</sup>Q<sup>−1</sup>)
	| colspan=2| <math>\mathcal{E}_\text{P} = \frac{E_\text{P}}{q_\text{P}} = \sqrt{\frac{c^4}{4\pi G \epsilon_0}}</math>
	| colspan=2| ] ]
	|-
	| Planck magnetomotive force
	| ] (T<sup>−1</sup>Q)
	| <math>\mathcal{F}_\text{P} = i_\text{P} = \sqrt{\frac{c^6 \epsilon_0}{4\pi G}}</math>
	| <math>\mathcal{F}_\text{P} = i_\text{P} = \sqrt{\frac{4\pi c^6 \epsilon_0}{G}}</math>
	| ] ]
	| ] ]
	|-
	| ]
	| ] (L<sup>−3</sup>M<sup>−1</sup>T<sup>2</sup>Q<sup>2</sup>)
	| <math>\epsilon_\text{P} = \frac{C_\text{P}}{l_\text{P}} = \epsilon_0</math>
	| <math>\epsilon_\text{P} = \frac{C_\text{P}}{l_\text{P}} = 4\pi \epsilon_0</math>
	| ] ]
	| ] ]
	|-
	| ]
	| ] (LMQ<sup>−2</sup>)
	| <math>\mu_\text{P} = \frac{L_\text{P}}{l_\text{P}} = \frac{1}{c^2 \epsilon_0} = \mu_0</math>
	| <math>\mu_\text{P} = \frac{L_\text{P}}{l_\text{P}} = \frac{1}{4\pi c^2 \epsilon_0} = \frac{\mu_0}{4\pi}</math>
	| ] ]
	| ] ]
	|-
	| Planck electric dipole moment
	| ] (LQ)
	| colspan=2| <math>\mathcal{Q}_\text{P} = q_\text{P}l_\text{P} = \sqrt{\frac{4\pi \hbar^2 G \epsilon_0}{c^2}}</math>
	| colspan=2| ] ]
	|-
	| Planck magnetic dipole moment
	| ] (L<sup>2</sup>T<sup>−1</sup>Q)
	| colspan=2| <math>\mathcal{M}_\text{P} = \frac{E_\text{P}}{b_\text{P}} = \sqrt{4\pi \hbar^2 G \epsilon_0}</math>
	| colspan=2| ] ]
	|-
	| Planck electric flux
	| ] (L<sup>3</sup>MT<sup>−2</sup>Q<sup>−1</sup>)
	| <math>\Phi_{\text{P}} = e_\text{P}A_\text{P} = \sqrt{\frac{\hbar c}{\epsilon_0}}</math>
	| <math>\Phi_{\text{P}} = e_\text{P}A_\text{P} = \sqrt{\frac{\hbar c}{4\pi \epsilon_0}}</math>
	| ] ]
	| ] ]
	|-
	| Planck magnetic flux
	| ] (L<sup>2</sup>MT<sup>−1</sup>Q<sup>−1</sup>)
	| <math>\Psi_{\text{P}} = B_\text{P}A_\text{P} = \sqrt{\frac{\hbar}{c \epsilon_0}}</math>
	| <math>\Psi_{\text{P}} = B_\text{P}A_\text{P} = \sqrt{\frac{\hbar}{4\pi c \epsilon_0}}</math>
	| ] ]
	| ] ]
	|-
	| Planck electric polarizability
	| ] (M<sup>−1</sup>T<sup>2</sup>Q<sup>2</sup>)
	| <math>\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}}</math>
	| <math>\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}}</math>
	| ] ]
	| ] ]
	|-
	| ]
	| ] (L<sup>−3</sup>M<sup>−1</sup>T<sup>2</sup>Q<sup>2</sup>)
	| <math>\mathcal{O}_\text{P} = \frac{\mathcal{Y}_\text{P}}{V_\text{P}} = \frac{1}{\epsilon_0}</math>
	| <math>\mathcal{O}_\text{P} = \frac{\mathcal{Y}_\text{P}}{V_\text{P}} = \frac{1}{4\pi \epsilon_0}</math>
	| ] ]
	| ] ]
	|-
	| Planck electric field gradient
	| ] (MT<sup>−2</sup>Q<sup>−1</sup>)
	| <math>\mathcal{Z}_\text{P} = \frac{k_\text{P}}{A_\text{P}} = \sqrt{\frac{c^{10}}{64\pi^3 \hbar^2 G^3 \epsilon_0}}</math>
	| <math>\mathcal{Z}_\text{P} = \frac{k_\text{P}}{A_\text{P}} = \sqrt{\frac{c^{10}}{4\pi \hbar^2 G^3 \epsilon_0}}</math>
	| ] ]
	| ] ]
	|-
	| Planck gyromagnetic ratio
	| ] (M<sup>−1</sup>Q)
	| colspan=2| <math>\Theta_\text{P} = \frac{\theta_\text{P}}{t_\text{P}B_\text{P}} = \sqrt{4\pi G \epsilon_0} = \sqrt{\frac{G}{k_e}}</math>
	| colspan=2| ] ]
	|-
	| Planck magnetogyric ratio
	| ] (MQ<sup>−1</sup>)
	| colspan=2| <math>\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}}</math>
	| colspan=2| ] ]
	|-
	| Planck magnetic reluctance
	| ] (L<sup>−2</sup>M<sup>−1</sup>Q<sup>2</sup>)
	| <math>\mathcal{R}_\text{P} = \frac{\mathcal{F}_\text{P}}{\Psi_\text{P}} = \sqrt{\frac{c^7 \epsilon_0^2}{4\pi \hbar G}}</math>
	| <math>\mathcal{R}_\text{P} = \frac{\mathcal{F}_\text{P}}{\Psi_\text{P}} = \sqrt{\frac{16\pi^2 c^7 \epsilon_0^2}{\hbar G}}</math>
	| ] ]
	| ] ]
	|-
	| Planck specific activity
	| ] (T<sup>−1</sup>)
	| <math>\mathcal{A}_\text{P} = \frac{1}{t_\text{P}} = \sqrt{\frac{c^5}{4\pi \hbar G}}</math>
	| <math>\mathcal{A}_\text{P} = \frac{1}{t_\text{P}} = \sqrt{\frac{c^5}{\hbar G}}</math>
	| ] ]
	| ] ]
	|-
	| Planck radiation exposure
	| ] (M<sup>−1</sup>Q)
	| colspan=2| <math>X_\text{P} = \frac{q_\text{P}}{m_\text{P}} = \sqrt{4\pi G \epsilon_0} = \sqrt{\frac{G}{k_e}}</math>
	| colspan=2| ] ]
	|-
	| ]
	| ] (L<sup>2</sup>T<sup>−2</sup>)
	| colspan=2| <math>\mathcal{D}_\text{P} = \frac{E_\text{P}}{m_\text{P}} = c^2</math>
	| colspan=2| ] ]
	|-
	| Planck absorbed dose rate
	| ] (L<sup>2</sup>T<sup>−3</sup>)
	| <math>W_\text{P} = \frac{\mathcal{D}_\text{P}}{t_\text{P}} = \sqrt{\frac{c^9}{4\pi \hbar G}}</math>
	| <math>W_\text{P} = \frac{\mathcal{D}_\text{P}}{t_\text{P}} = \sqrt{\frac{c^9}{\hbar G}}</math>
	| ] ]
	| ] ]
	|-
	| Planck thermal expansion coefficient
	| ] (Θ<sup>−1</sup>)
	| <math>\gamma_\text{P} = \frac{1}{T_\text{P}} = \sqrt{\frac{4\pi G {k_\text{B}}^2}{\hbar c^5}}</math>
	| <math>\gamma_\text{P} = \frac{1}{T_\text{P}} = \sqrt{\frac{G {k_\text{B}}^2}{\hbar c^5}}</math>
	| ] ]
	| ] ]
	|-
	| ]
	| ] (L<sup>2</sup>MT<sup>−2</sup>Θ<sup>−1</sup>)
	| colspan=2| <math>\Gamma_\text{P} = \frac{E_\text{P}}{T_\text{P}} = k_\text{B}</math>
	| colspan=2| ] ]
	|-
	| Planck specific heat capacity
	| ] (L<sup>2</sup>T<sup>−2</sup>Θ<sup>−1</sup>)
	| <math>c_\text{P} = \frac{E_\text{P}}{m_\text{P} T_\text{P}} = \frac{\Gamma_\text{P}}{m_\text{P}} = \sqrt{\frac{4\pi G k_\text{B}^2}{\hbar c}}</math>
	| <math>c_\text{P} = \frac{E_\text{P}}{m_\text{P} T_\text{P}} = \frac{\Gamma_\text{P}}{m_\text{P}} = \sqrt{\frac{G k_\text{B}^2}{\hbar c}}</math>
	| ] ]
	| ] ]
	|-
	| Planck volumetric heat capacity
	| ] (L<sup>−1</sup>MT<sup>−2</sup>Θ<sup>−1</sup>)
	| <math>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}}</math>
	| <math>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}}</math>
	| ] ]
	| ] ]
	|-
	| Planck thermal resistance
	| ] (L<sup>−2</sup>M<sup>−1</sup>T<sup>3</sup>Θ)
	| <math>R_\text{P} = \frac{T_\text{P}}{P_\text{P}} = \sqrt{\frac{4\pi \hbar G}{c^5 k_\text{B}^2}}</math>
	| <math>R_\text{P} = \frac{T_\text{P}}{P_\text{P}} = \sqrt{\frac{\hbar G}{c^5 k_\text{B}^2}}</math>
	| ] ]
	| ] ]
	|-
	| Planck thermal conductance
	| ] (L<sup>2</sup>MT<sup>−3</sup>Θ<sup>−1</sup>)
	| <math>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}}</math>
	| <math>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}}</math>
	| ] ]
	| ] ]
	|-
	| Planck thermal resistivity
	| ] (L<sup>−1</sup>M<sup>−1</sup>T<sup>3</sup>Θ)
	| <math>\chi_\text{P} = R_\text{P}l_\text{P} = \sqrt{\frac{16\pi^2 \hbar^2 G^2}{c^8k_\text{B}^2}}</math>
	| <math>\chi_\text{P} = R_\text{P}l_\text{P} = \sqrt{\frac{\hbar^2 G^2}{c^8k_\text{B}^2}}</math>
	| ] ]
	| ] ]
	|-
	| Planck thermal conductivity
	| ] (LMT<sup>−3</sup>Θ<sup>−1</sup>)
	| <math>\lambda_\text{P} = \frac{P_\text{P}}{l_\text{P}T_\text{P}} = \frac{1}{\chi_\text{P}} = \sqrt{\frac{c^8k_\text{B}^2}{16\pi^2 \hbar^2 G^2}}\neq B_\text{P} \frac{2 \pi}{\sqrt{\alpha}}</math>
	| <math>\lambda_\text{P} = \frac{P_\text{P}}{l_\text{P}T_\text{P}} = \frac{1}{\chi_\text{P}} = \sqrt{\frac{c^8k_\text{B}^2}{\hbar^2 G^2}}\neq B_\text{P} \frac{2 \pi}{\sqrt{\alpha}}</math>
	| ] ]
	| ] ]
	|-
	| Planck thermal insulance
	| ] (M<sup>−1</sup>T<sup>3</sup>Θ)
	| <math>o_\text{P} = R_\text{P}A_\text{P} = \sqrt{\frac{64\pi^3 \hbar^3 G^3}{c^{11} k_\text{B}^2}}</math>
	| <math>o_\text{P} = R_\text{P}A_\text{P} = \sqrt{\frac{\hbar^3 G^3}{c^{11} k_\text{B}^2}}</math>
	| ] ]
	| ] ]
	|-
	| Planck thermal transmittance
	| ] (MT<sup>−3</sup>Θ<sup>−1</sup>)
	| <math>w_\text{P} = \frac{1}{o_\text{P}} = \sqrt{\frac{c^{11} k_\text{B}^2}{64\pi^3 \hbar^3 G^3}}</math>
	| <math>w_\text{P} = \frac{1}{o_\text{P}} = \sqrt{\frac{c^{11} k_\text{B}^2}{\hbar^3 G^3}}</math>
	| ] ]
	| ] ]
	|-
	| ]
	| ] (L<sup>2</sup>MT<sup>−2</sup>Θ<sup>−1</sup>)
	| colspan=2| <math>S_\text{P} = \frac{E_\text{P}}{T_\text{P}} = k_\text{B}</math>
	| colspan=2| ] ]
	|-
	| Planck amount of substance
	| ] (N)
	| colspan=2| <math>n_\text{P} = \frac{1}{N_\text{A}}</math>
	| colspan=2| ] ]
	|-
	| Planck molar mass
	| ] (MN<sup>−1</sup>)
	| <math>\mathcal{M}_\text{P} = \frac{m_\text{P}}{n_\text{P}} = \sqrt{\frac{\hbar c N_\text{A}^2}{4\pi G}}</math>
	| <math>\mathcal{M}_\text{P} = \frac{m_\text{P}}{n_\text{P}} = \sqrt{\frac{\hbar c N_\text{A}^2}{G}}</math>
	| ] ]
	| ] ]
	|-
	| Planck molar volume
	| ] (L<sup>3</sup>N<sup>−1</sup>)
	| <math>\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}}</math>
	| <math>\mathcal{V}_\text{P} = \frac{V_\text{P}}{n_\text{P}} = \sqrt{\frac{\hbar^3 G^3 N_\text{A}^2}{c^9}}</math>
	| ] ]
	| ] ]
	|-
	| ]
	| ] (L<sup>2</sup>MT<sup>−2</sup>Θ<sup>−1</sup>N<sup>−1</sup>)
	| colspan=2| <math>\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}</math>
	| colspan=2| ] ]
	|-
	| Planck mass fraction
	| ] (dimensionless)
	| colspan=2| <math>\mathcal{W}_\text{P} = \frac{m_\text{P}}{m_\text{P}} = 1</math>
	| colspan=2| ] ]
	|-		|-
			| ] (LMT<sup>−1</sup>)
	| Planck volume fraction
			| <math>m_\text{P} c = \frac{\hbar}{l_\text{P}} = \sqrt{\frac{\hbar c^3}{G}} </math>
	| ] (dimensionless)
			| {{val|6.5249|u=]}}
	| colspan=2| <math>\varphi_\text{P} = \frac{V_\text{P}}{V_\text{P}} = 1</math>
	| colspan=2| ] ]
	|-		|-
			| ] (L<sup>2</sup>MT<sup>−2</sup>)
	| Planck molality
			| <math>E_\text{P} = m_\text{P} c^2 = \frac{\hbar}{t_\text{P}} = \sqrt{\frac{\hbar c^5}{G}} </math>
	| ] (M<sup>−1</sup>N)
			| {{val|1.9561|e=9|ul=J}}
	| <math>\mathcal{B}_\text{P} = \frac{n_\text{P}}{m_\text{P}} = \sqrt{\frac{4\pi G}{\hbar c N_\text{A}^2}}</math>
	| <math>\mathcal{B}_\text{P} = \frac{n_\text{P}}{m_\text{P}} = \sqrt{\frac{G}{\hbar c N_\text{A}^2}}</math>
	| ] ]
	| ] ]
	|-		|-
			| ] (LMT<sup>−2</sup>)
	| Planck molarity
			| <math>F_\text{P} = \frac{E_\text{P}}{l_\text{P}} = \frac{\hbar}{l_\text{P} t_\text{P}} = \frac{c^4}{G} </math>
	| ] (L<sup>−3</sup>N)
			| {{val|1.2103|e=44|ul=N}}
	| <math>\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}}</math>
	| <math>\mathcal{C}_\text{P} = \frac{n_\text{P}}{V_\text{P}} = \sqrt{\frac{c^9}{\hbar^3 G^3 N_\text{A}^2}}</math>
	| ] ]
	| ] ]
	|-		|-
			| ] (L<sup>−3</sup>M)
	| Planck mole fraction
			| <math>\rho_\text{P} = \frac{m_\text{P}}{l_\text{P}^3} = \frac{\hbar t_\text{P}}{l_\text{P}^5} = \frac{c^5}{\hbar G^2} </math>
	| ] (dimensionless)
			| {{val|5.1550|e=96|ul=kg/m3}}
	| colspan=2| <math>\mathcal{X}_\text{P} = \frac{n_\text{P}}{n_\text{P}} = 1</math>
	| colspan=2| ]
	|-		|-
			| ] (LT<sup>−2</sup>)
	| Planck catalytic activity
			| <math> a_\text{P} = \frac{c}{t_\text{P}} = \sqrt{\frac{c^7}{\hbar G}}</math>
	| ] (T<sup>−1</sup>N)
			| {{val|5.5608|e=51|ul=m/s2}}
	| <math>z_\text{P} = \frac{n_\text{P}}{t_\text{P}} = \sqrt{\frac{c^5}{4\pi \hbar G N_\text{A}^2}}</math>
	| <math>z_\text{P} = \frac{n_\text{P}}{t_\text{P}} = \sqrt{\frac{c^5}{\hbar G N_\text{A}^2}}</math>
	| ] ]
	| ] ]
	|}		|}
			Some Planck units, such as of time and length, are many ] too large or too small to be of practical use, so that Planck units as a system are typically only relevant to theoretical physics. In some cases, a Planck unit may suggest a limit to a range of a physical quantity where present-day theories of physics apply.<ref name=":1">{{cite book|last=Zee|first=Anthony|title=Quantum Field Theory in a Nutshell|title-link=Quantum Field Theory in a Nutshell|publisher=]|year=2010|isbn=978-0-691-14034-6|edition=second|pages=|oclc=659549695|quote=Just as in our discussion of the Fermi theory, the nonrenormalizability of quantum gravity tells us that at the Planck energy scale ... new physics must appear. Fermi's theory cried out, and the new physics turned out to be the electroweak theory. Einstein's theory is now crying out.|author-link=Anthony Zee}}</ref> For example, our understanding of the ] does not extend to the ], i.e., when the universe was less than one Planck time old. Describing the universe during the Planck epoch requires a theory of ] that would incorporate quantum effects into ]. Such a theory does not yet exist.
	(Note: <math>k_e</math> is the ], <math>\mu_0</math> is the ], <math>Z_0</math> is the ], <math>Y_0</math> is the ])
			Several quantities are not "extreme" in magnitude, such as the Planck mass, which is about ]: very large in comparison with subatomic particles, and within the mass range of living organisms.<ref name="Penrose2005">{{cite book |title=The Road to Reality |last=Penrose |first=Roger |author-link=Roger Penrose |year=2005 |publisher=Alfred A. Knopf |location=New York |isbn=978-0-679-45443-4 |title-link=The Road to Reality }}</ref>{{rp|872}} Similarly, the related units of energy and of momentum are in the range of some everyday phenomena.
	(Note: <math>N_\text{A}</math> is the ], which is also normalized to 1 in (both two versions of) Planck units)
			== Significance ==
	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.<ref></ref><ref>{{cite web|url=http://www.atlantecircuitale.com/energie2006/pdf/labor-ingl.pdf|title=Electromagnetic Unification Electronic Conception of the Space, the Energy and the Matter}}</ref> The ] <math>e</math>, measured in terms of the Planck charge, is
			Planck units have little ] arbitrariness, but do still involve some arbitrary choices in terms of the defining constants. Unlike the ] and ], which exist as ] in the ] system for historical reasons, the ] and Planck time are conceptually linked at a fundamental physical level. Consequently, natural units help physicists to reframe questions. ] puts it succinctly:
			{{blockquote|We see that the question is not, "Why is gravity so feeble?" but rather, "Why is the proton's mass so small?" For in natural (Planck) units, the strength of gravity simply is what it is, a primary quantity, while the proton's mass is the tiny number 1/13 ].<ref>{{cite journal|title=Scaling Mount Planck I: A View from the Bottom|journal=]|volume=54|issue=6|pages=12–13|year=2001|last=Wilczek|first=Frank|author-link=Frank Wilczek |doi=10.1063/1.1387576|bibcode=2001PhT....54f..12W|doi-access=free}}</ref>}}
			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 ], because ] and ] are ] quantities. Rather, the disparity of magnitude of force is a manifestation of that the ] is approximately the unit charge but the ] is far less than the unit mass in a system that treats both forces as having the same form.
	: <math>e = \sqrt{4\pi \alpha} \cdot q_{\text{P}} \approx 0.302822121 \cdot q_{\text{P}} \, </math> (Lorentz–Heaviside version)
			When Planck proposed his units, the goal was only that of establishing a universal ("natural") way of measuring objects, without giving any special meaning to quantities that measured one single unit. In 1918, ] suggested that the Planck length could have a special significance for understanding gravitation, but this suggestion was not influential.<ref name="Gorelik"/><ref>{{cite book|last=Stachel |first=John |author-link=John Stachel |year=1986 |chapter=Eddington and Einstein |editor-last=Ullmann-Margalit |editor-first=Edna |title=The Prism of Science |series=Boston Studies in the Philosophy of Science |volume=95 |publisher=Springer |doi=10.1007/978-94-009-4566-1_18 |isbn=978-90-277-2161-7 |pages=225–250}}</ref> During the 1950s, multiple authors including ] and ] argued that quantities on the order of the Planck scale indicated the limits of the validity of quantum field theory. ] proposed in 1955 that quantum fluctuations of spacetime become significant at the Planck scale, though at the time he was unaware of Planck's unit system.<ref name="Gorelik"/><ref name="Wheeler"/> In 1959, C. A. Mead showed that distances that measured of the order of one Planck length, or, similarly, times that measured of the order of Planck time, did carry special implications related to ]'s ]:<ref>
	: <math>e = \sqrt{\alpha} \cdot q_{\text{P}} \approx 0.085424543 \cdot q_{\text{P}} \, </math> (Gaussian version)
			{{cite journal
			| last = Mead
			| first = Chester Alden
			| date = 1964-08-10
			| title = Possible Connection Between Gravitation and Fundamental Length
			| journal = ]
			| volume = 135
			| issue = 3B
			| pages = B849–B862
			| doi = 10.1103/PhysRev.135.B849
			| publisher = American Physical Society
			| bibcode = 1964PhRv..135..849M
			}}</ref>
			{{blockquote|An analysis of the effect of gravitation on hypothetical experiments indicates that it is impossible to measure the position of a particle with error less than {{math|1=𝛥⁢𝑥 ≳ √𝐺 = 1.6 × 10<sup>−33</sup> cm}}, where 𝐺 is the gravitational constant in natural units. A similar limitation applies to the precise synchronization of clocks.}}
	where <math> {\alpha} </math> is the ]
			== Planck scale ==
	: <math> \alpha = \frac{k_e e^2}{\hbar c} \approx \frac{1}{137.03599911} </math>
			In ] and ], the Planck scale is an ] around {{val|1.22|e=28|ul=eV}} (the Planck energy, corresponding to the ] of the Planck mass, {{val|2.17645|e=−8|u=kg}}) at which ] of ] become significant. At this scale, present descriptions and theories of sub-atomic particle interactions in terms of ] break down and become inadequate, due to the impact of the apparent ] of gravity within current theories.<ref name=":1" />
			=== Relationship to gravity ===
	: <math> \alpha = \frac{1}{4 \pi} \left ( \frac{e}{q_{\text{P}}} \right )^2</math> (Lorentz–Heaviside version)
			At the Planck length scale, the strength of gravity is expected to become comparable with the other forces, and it has been theorized that all the fundamental forces are unified at that scale, but the exact mechanism of this unification remains unknown.<ref>{{cite arXiv|eprint=hep-ph/0207124 |first=Ed |last=Witten |author-link=Ed Witten |title=Quest For Unification |year=2002}}</ref> The Planck scale is therefore the point at which the effects of quantum gravity can no longer be ignored in other ], where current calculations and approaches begin to break down, and a means to take account of its impact is necessary.<ref>{{cite web |url=https://cerncourier.com/a/can-experiment-access-planck-scale-physics/ |title=Can experiment access Planck-scale physics? |website=] |date=2006-10-04 |access-date=2021-11-04 |first=Robert |last=Bingham |archive-date=30 November 2020 |archive-url=https://web.archive.org/web/20201130035047/https://cerncourier.com/a/can-experiment-access-planck-scale-physics/ |url-status=live }}</ref> On these grounds, it has been speculated that it may be an ] at which a black hole could be formed by collapse.<ref name="particlecreate">{{cite journal |last=Hawking |first=Stephen W. |author-link=Stephen Hawking |date=1975 |title=Particle Creation by Black Holes |url=http://projecteuclid.org/euclid.cmp/1103899181 |url-status=live |journal=] |volume=43 |issue=3 |pages=199–220 |bibcode=1975CMaPh..43..199H |doi=10.1007/BF02345020 |s2cid=55539246 |archive-url=https://web.archive.org/web/20140705012739/http://projecteuclid.org/euclid.cmp/1103899181 |archive-date=5 July 2014 |access-date=20 March 2022}}</ref>
			While physicists have a fairly good understanding of the other fundamental interactions of forces on the quantum level, ] is problematic, and cannot be integrated with ] at very high energies using the usual framework of quantum field theory. At lesser energy levels it is usually ignored, while for energies approaching or exceeding the Planck scale, a new theory of ] is necessary. Approaches to this problem include ] and ], ], ], and ].<ref name="scholarpedia">{{cite journal |last1=Rovelli |first1=Carlo |author-link=Carlo Rovelli |year=2008 |title=Quantum gravity |journal=] |volume=3 |issue=5 |page=7117 |bibcode=2008SchpJ...3.7117R |doi=10.4249/scholarpedia.7117 |doi-access=free}}</ref>
	: <math> \alpha = \left ( \frac{e}{q_{\text{P}}} \right )^2</math> (Gaussian version)
			=== <span class="anchor" id="Cosmology"></span><span class="anchor" id="Planck epoch"></span> In cosmology ===
	The ] <math>\alpha</math> is also called the '''electromagnetic coupling constant''', thus comparing with the ] <math>\alpha_G</math>. The ] <math>m_e</math> measured in terms of the Planck mass, is
			{{Main|Chronology of the universe}}
			{{Further|Time-variation of fundamental constants}}
			In ], the '''Planck epoch''' or '''Planck era''' is the earliest stage of the ], before the ] was equal to the Planck time, ''t''<sub>P</sub>, or approximately 10<sup>−43</sup> seconds.<ref name="Planck-UOregon">{{cite web |first=James |last=Schombert |title=Birth of the Universe |url=http://abyss.uoregon.edu/~js/cosmo/lectures/lec20.html |website=HC 441: Cosmology |publisher=] |access-date=March 20, 2022 |archive-date=28 November 2018 |archive-url=https://web.archive.org/web/20181128045313/http://abyss.uoregon.edu/~js/cosmo/lectures/lec20.html |url-status=live }} Discusses "Planck time" and "]" at the very beginning of the ]</ref> There is no currently available physical theory to describe such short times, and it is not clear in what sense the concept of ] is meaningful for values smaller than the Planck time. It is generally assumed that ] dominate physical interactions at this time scale. At this scale, the ] of the ] is assumed to be ]. Immeasurably hot and dense, the state of the Planck epoch was succeeded by the ], where gravitation is separated from the unified force of the Standard Model, in turn followed by the ], which ended after about 10<sup>−32</sup> seconds (or about 10<sup>11</sup>&nbsp;''t''<sub>P</sub>).<ref name="KolbTurner1994">{{cite book|first1=Edward W.|last1=Kolb|first2=Michael S.|last2=Turner|title=The Early Universe|url=https://books.google.com/books?id=l6Z8W33JWGQC&pg=PA447|access-date=10 April 2010|year=1994|publisher=Basic Books|isbn=978-0-201-62674-2|page=447|archive-date=6 March 2021|archive-url=https://web.archive.org/web/20210306212708/https://books.google.com/books?id=l6Z8W33JWGQC&pg=PA447|url-status=live}}</ref>
			Table 3 lists properties of the observable universe today expressed in Planck units.<ref name="John D 2002">{{cite book|first=John D. |last=Barrow |author-link=John D. Barrow |year=2002 |title=The Constants of Nature: From Alpha to Omega – The Numbers that Encode the Deepest Secrets of the Universe |publisher=Pantheon Books |isbn=0-375-42221-8}}</ref><ref>{{BarrowTipler1986}}</ref>
	: <math>m_e = \sqrt{4\pi \alpha_G} \cdot m_{\text{P}} \approx 1.48368 \times 10^{-22} \cdot m_{\text{P}} \, </math> (Lorentz–Heaviside version)
			{| class="wikitable" style="margin:1em auto 1em auto;"
	: <math>m_e = \sqrt{\alpha_G} \cdot m_{\text{P}} \approx 4.18539 \times 10^{-23} \cdot m_{\text{P}} \, </math> (Gaussian version)
			|+Table 3: Today's universe in Planck units
			! Property of{{br}} present-day ]
	where <math> {\alpha_G} </math> is the ]
			! Approximate number{{br}} of Planck units
			! Equivalents
	: <math> \alpha_G = \frac{G m_e^2}{\hbar c} \approx 1.7518 \times 10^{-45} </math>
			|- style="text-align:left;"
			| ]
	: <math> \alpha_G = \frac{1}{4 \pi} \left ( \frac{m_e}{m_{\text{P}}} \right )^2</math> (Lorentz–Heaviside version)
			| 8.08 × 10<sup>60</sup> ''t''<sub>P</sub>
			| 4.35 × 10<sup>17</sup> s or 1.38 × 10<sup>10</sup> years
	: <math> \alpha_G = \left ( \frac{m_e}{m_{\text{P}}} \right )^2</math> (Gaussian version)
	Some Planck units are suitable for measuring quantities that are familiar from daily experience. For example:
	* 1 ] is about ] (Lorentz–Heaviside version) or ] (Gaussian version);
	* 1 ] is about 1.84 ] (Lorentz–Heaviside version) or 6.52 ] (Gaussian version);
	* 1 ] is about ] (Lorentz–Heaviside version) or ] (Gaussian version);
	* 1 ] is 1 ] (both versions);
	* 1 ] is 1 ] (both versions);
	* 1 ] is about 3.3 ]s (Lorentz–Heaviside version) or 11.7 ]s (Gaussian version);
	* 1 ] is about 377 ]s (Lorentz–Heaviside version) or 30 ]s (Gaussian version);
	* 1 ] is about 2.65 ] (Lorentz–Heaviside version) or 33.4 ] (Gaussian version);
	* 1 ] is about 1.26 ] (Lorentz–Heaviside version) or 0.1 ] (Gaussian version);
	* 1 ] is about 59.8 ] (Lorentz–Heaviside version) or 16.9 ] (Gaussian version).
	However, most Planck units are many ] 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 ] in a vacuum, the maximum possible physical speed in ];<ref>{{cite book |last1=Feynman |first1=R. P. |authorlink1=Richard Feynman |last2=Leighton |first2=R. B. |authorlink2=Robert B. Leighton |last3=Sands |first3=M. |title=The Feynman Lectures on Physics |volume=1 "Mainly mechanics, radiation, and heat" |publisher=Addison-Wesley |pages=15–9 |chapter=The Special Theory of Relativity |year=1963 |isbn=978-0-7382-0008-8 |lccn=63020717|title-link=The Feynman Lectures on Physics }}</ref> 1 ]Planck speed is about 1.079 ].
	* Our understanding of the ] begins with the ], when the universe was 1 Planck time old and 1 Planck length in diameter, and had a Planck temperature of 1. At that moment, ] as presently understood becomes applicable. Understanding the universe when it was less than 1 Planck time old requires a theory of ] that would incorporate quantum effects into ]. Such a theory does not yet exist.
	In Planck units, we have:
	:<math>\alpha=\frac{e^2}{4\pi}</math> (Lorentz–Heaviside version)
	:<math>\alpha=e^2</math> (Gaussian version)
	:<math>\alpha_G=\frac{m_e^2}{4\pi}</math> (Lorentz–Heaviside version)
	:<math>\alpha_G=m_e^2</math> (Gaussian version)
	where
	:<math>\alpha</math> is the ]
	:<math>e</math> is the ]
	:<math>\alpha_G</math> is the ]
	:<math>m_e</math> is the ]
	Hence the ] of ] (<math>\frac{e}{m_e}</math>) is <math>\sqrt{\frac{\alpha}{\alpha_G}}</math> ], in both two versions of Planck units.
	== Significance ==
	Planck units are free of ] arbitrariness. Some physicists argue that communication with ] would have to employ such a system of units in order to be understood.<ref>Michael W. Busch, Rachel M. Reddick (2010) "" , 26–29 April 2010, League City, Texas.</ref> Unlike the ] and ], which exist as ] in the ] system for historical reasons, the ] and ] are conceptually linked at a fundamental physical level.
	Natural units help physicists to reframe questions. ] puts it succinctly:
	{{bq|We see that the question is not, "Why is gravity so feeble?" but rather, "Why is the proton's mass so small?" For in natural (Planck) units, the strength of gravity simply is what it is, a primary quantity, while the proton's mass is the tiny number ])].<ref>{{cite journal|title=Scaling Mount Planck I: A View from the Bottom|journal=Physics Today|volume=54|issue=6|pages=12–13|year=2001|last=Wilczek|first=Frank|doi=10.1063/1.1387576|bibcode=2001PhT....54f..12W}}</ref>}}
	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 ], because ] and ] are ] quantities. Rather, the disparity of magnitude of force is a manifestation of the fact that the ] is approximately the ] but the ] is far less than the ].
	=== Cosmology ===
	{{Main|Chronology of the Universe}}
	{{anchor|Planck epoch}}
	In ], the '''Planck epoch''' or '''Planck era''' is the earliest stage of the ], before the ] was equal to the ], ''t''<sub>P</sub>, or approximately 10<sup>−43</sup> seconds.<ref name="Planck-UOregon">{{cite web |author=Staff |title=Birth of the Universe |url=http://abyss.uoregon.edu/~js/cosmo/lectures/lec20.html |date= |work=] |accessdate=September 24, 2016 }} - discusses "]" and "]" at the very beginning of the ]</ref> There is no currently available physical theory to describe such short times, and it is not clear in what sense the concept of ] is meaningful for values smaller than the Planck time. It is generally assumed that ] dominate physical interactions at this time scale. At this scale, the ] of the ] is assumed to be ]. Immeasurably ] and ], the state of the Planck epoch was succeeded by the ], where gravitation is separated from the unified force of the Standard Model, in turn followed by the ], which ended after about 10<sup>−32</sup> seconds (or about 10<sup>10</sup>&nbsp;''t''<sub>P</sub>).<ref name="KolbTurner1994">{{cite book|author1=Edward W. Kolb|author2=Michael S. Turner|title=The Early Universe|url=https://books.google.com/?id=l6Z8W33JWGQC&pg=PA447|accessdate=10 April 2010|year=1994|publisher=Basic Books|isbn=978-0-201-62674-2|page=447}}</ref>
	Relative to the Planck epoch, the observable universe today looks extreme when expressed in Planck units, as in this set of approximations:<ref name="John D 2002">], 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}}.</ref><ref>{{BarrowTipler1986}}</ref>
	{{further|Time-variation of physical constants|Dirac large numbers hypothesis}}
	The recurrence of large numbers close or related to 10<sup>60</sup> in the above table is a coincidence that intrigues some theorists. It is an example of the kind of ] that led theorists such as ] and ] to develop alternative physical theories (e.g. a ] or ]).<ref>{{cite journal|author=P.A.M. Dirac|year=1938 |title=A New Basis for Cosmology |journal=] |volume=165 |issue=921 |pages=199–208 |doi=10.1098/rspa.1938.0053 |bibcode = 1938RSPSA.165..199D }}</ref>
	After the measurement of the ] in 1998, estimated at 10<sup>−122</sup> in Planck units, it was noted that this is suggestively close to the reciprocal of the ] squared.<ref>J.D. Barrow and F.J. Tipler, ''The Anthropic Cosmological'' Principle, Oxford UP, Oxford (1986), chapter 6.9.</ref> Barrow and Shaw (2011) proposed a modified theory in which ] is a field evolving in such a way that its value remains Λ ~ ''T''<sup>−2</sup> throughout the history of the universe.<ref>{{cite journal|doi= 10.1007/s10714-011-1199-1|arxiv=1105.3105|title=The value of the cosmological constant|journal=General Relativity and Gravitation|volume=43|issue=10|pages=2555–2560|year=2011|last1=Barrow|first1=John D.|last2=Shaw|first2=Douglas J.|bibcode = 2011GReGr..43.2555B }}</ref>
	{|class="wikitable" style="margin:1em auto 1em auto; background:#fff;"
	|+Table 4: Some common physical quantities
	! Quantities
	! In Lorentz–Heaviside version Planck units
	! In Gaussian version Planck units
	|-
	! ] (<math>g</math>)
	| {{val|6.25154|e=-51}} <math>g_\text{P}</math>
	| {{val|1.76353|e=-51}} <math>g_\text{P}</math>
	|-
	! ] (<math>atm</math>)
	| {{val|3.45343|e=-108}} <math>\Pi_\text{P}</math>
	| {{val|2.18691|e=-109}} <math>\Pi_\text{P}</math>
	|-
	! ]
	| {{val|4.52091|e=47}} <math>t_\text{P}</math>
	| {{val|1.60262|e=48}} <math>t_\text{P}</math>
	|-
	! ]
	| {{val|1.11323|e=41}} <math>l_\text{P}</math>
	| {{val|3.94629|e=41}} <math>l_\text{P}</math>
	|-
	! ]
	| {{val|6.99465|e=41}} <math>l_\text{P}</math>
	| {{val|2.47954|e=42}} <math>l_\text{P}</math>
	|-
	! ]
	| {{val|1.53594|e=61}} <math>l_\text{P}</math>
	| {{val|5.44477|e=61}} <math>l_\text{P}</math>
	|-
	! ]
	| {{val|1.89062|e=55}} <math>V_\text{P}</math>
	| {{val|6.70208|e=55}} <math>V_\text{P}</math>
	|-
	! ]
	| {{val|6.98156|e=114}} <math>V_\text{P}</math>
	| {{val|2.47490|e=115}} <math>V_\text{P}</math>
	|-
	! ]
	| {{val|9.72717|e=32}} <math>m_\text{P}</math>
	| {{val|2.74398|e=32}} <math>m_\text{P}</math>
	|-
	! ]
	| {{val|2.37796|e=61}} <math>m_\text{P}</math>
	| {{val|6.70811|e=60}} <math>m_\text{P}</math>
	|-
	! ]
	| {{val|1.68905|e=-91}} <math>d_\text{P}</math>
	| {{val|1.06960|e=-93}} <math>d_\text{P}</math>
	|-
	! ]
	| {{val|3.03257|e=-121}} <math>d_\text{P}</math>
	| {{val|1.92040|e=-123}} <math>d_\text{P}</math>
	|-
	! ]
	| {{val|7.49657|e=59}} <math>t_\text{P}</math>
	| {{val|2.65747|e=60}} <math>t_\text{P}</math>
	|-
	! ]
	| {{val|2.27853|e=60}} <math>t_\text{P}</math>
	| {{val|8.07719|e=60}} <math>t_\text{P}</math>
	|-
	! ]
	| {{val|7.18485|e=-30}} <math>T_\text{P}</math>
	| {{val|2.02681|e=-30}} <math>T_\text{P}</math>
	|-
	! ]
	| {{val|6.81806|e=-32}} <math>T_\text{P}</math>
	| {{val|1.92334|e=-32}} <math>T_\text{P}</math>
	|-
	! ] (<math>H_0</math>)
	| {{val|4.20446|e=-61}} <math>t_\text{P}^{-1}</math>
	| {{val|1.18605|e=-61}} <math>t_\text{P}^{-1}</math>
	|-
	! ] (<math>\Lambda</math>)
	| {{val|3.62922|e=-121}} <math>l_\text{P}^{-2}</math>
	| {{val|2.88805|e=-122}} <math>l_\text{P}^{-2}</math>
	|-
	! ] (<math>\rho_\text{vacuum}</math>)
	| {{val|1.82567|e=-121}} <math>\rho_\text{P}</math>
	| {{val|1.15612|e=-123}} <math>\rho_\text{P}</math>
	|-
	! ] of ]
	| {{val|6.83432|e=-30}} <math>T_\text{P}</math>
	| {{val|1.92793|e=-30}} <math>T_\text{P}</math>
	|-
	! ] of ]
	| {{val|9.33636|e=-30}} <math>T_\text{P}</math>
	| {{val|2.63374|e=-30}} <math>T_\text{P}</math>
	|-
	! ] of ] of ]
	| {{val|2.08469|e=-109}} <math>\Pi_\text{P}</math>
	| {{val|1.32015|e=-111}} <math>\Pi_\text{P}</math>
	|-
	! ] of ] of ]
	| {{val|6.83457|e=-30}} <math>T_\text{P}</math>
	| {{val|1.92800|e=-30}} <math>T_\text{P}</math>
	|-
	! ] of ]
	| {{val|3.06320|e=-92}} <math>d_\text{P}</math>
	| {{val|1.93980|e=-94}} <math>d_\text{P}</math>
	|-
	! ] of ]
	| {{val|1.86061|e=18}} <math>c_\text{P}</math>
	| {{val|6.59570|e=18}} <math>c_\text{P}</math>
	|-
	! ] (<math>V_m</math>)
	| {{val|2.00522|e=77}} <math>\mathcal{V}_\text{P}</math>
	| {{val|8.93256|e=78}} <math>\mathcal{V}_\text{P}</math>
	|-
	! ] (<math>e</math>)
	| {{val|3.02822|e=-1}} <math>q_\text{P}</math>
	| {{val|8.54245|e=-2}} <math>q_\text{P}</math>
	|-
	! ] (<math>m_e</math>)
	| {{val|1.48368|e=-22}} <math>m_\text{P}</math>
	| {{val|4.18539|e=-23}} <math>m_\text{P}</math>
	|-
	! ] (<math>m_p</math>)
	| {{val|2.72427|e=-19}} <math>m_\text{P}</math>
	| {{val|7.68502|e=-20}} <math>m_\text{P}</math>
	|-
	! ] (<math>m_n</math>)
	| {{val|2.72802|e=-19}} <math>m_\text{P}</math>
	| {{val|7.69562|e=-20}} <math>m_\text{P}</math>
	|-
	! ] (<math>u</math>)
	| {{val|2.70459|e=-19}} <math>m_\text{P}</math>
	| {{val|7.62951|e=-20}} <math>m_\text{P}</math>
	|-
	! ] of ] (<math>\xi_e</math>)
	| colspan=2| {{val|&minus;2.04102|e=21}} <math>\xi_\text{P}</math>
	|-
	! ] of ] (<math>\xi_p</math>)
	| colspan=2| {{val|1.11157|e=18}} <math>\xi_\text{P}</math>
	|-
	! ] of ] (<math>\gamma_p</math>)
	| colspan=2| {{val|3.10445|e=18}} <math>\Theta_\text{P}</math>
	|-
	! ]
	| colspan=2| {{val|&minus;1.02169|e=21}} <math>\mathcal{M}_\text{P}</math>
	|-
	! ]
	| colspan=2| {{val|1.55223|e=18}} <math>\mathcal{M}_\text{P}</math>
	|-
	! ] (<math>F</math>)
	| {{val|3.02822|e=-1}} <math>q_\text{P}/n_\text{P}</math>
	| {{val|8.54245|e=-2}} <math>q_\text{P}/n_\text{P}</math>
	|-
	! ] (<math>a_0</math>)
	| {{val|9.23620|e=23}} <math>l_\text{P}</math>
	| {{val|3.27415|e=24}} <math>l_\text{P}</math>
	|-
	! ] (<math>\mu_B</math>)
	| colspan=2| {{val|1.02051|e=21}} <math>E_\text{P}/B_\text{P}</math>
	|-
	! ] (<math>\varphi_0</math>)
	| {{val|10.3744}} <math>\Psi_\text{P}</math>
	| {{val|36.7762}} <math>\Psi_\text{P}</math>
	|-		|-
			| ]
	! ] (<math>r_e</math>)
			| 5.4 × 10<sup>61</sup> ''l''<sub>P</sub>
	| {{val|4.91840|e=19}} <math>l_\text{P}</math>
			| 8.7 × 10<sup>26</sup> m or 9.2 × 10<sup>10</sup> ]
	| {{val|1.74353|e=20}} <math>l_\text{P}</math>
	|-		|-
			| ]
	! ] (<math>\lambda_c</math>)
			| approx. 10<sup>60</sup> ''m''<sub>P</sub>
	| {{val|2.71873|e=28}} <math>l_\text{P}</math>
			| 3 × 10<sup>52</sup> kg or 1.5 × 10<sup>22</sup> ]es (only counting stars){{br}} ] (sometimes known as the ])
	| {{val|9.63763|e=28}} <math>l_\text{P}</math>
	|-		|-
			| ]
	! ] (<math>R_\infty</math>)
			| 1.8 × 10<sup>−123</sup> ''m''<sub>P</sub>⋅''l''<sub>P</sub><sup>−3</sup>
	| {{val|6.28727|e=-28}} <math>l_\text{P}^{-1}</math>
			| 9.9 × 10<sup>−27</sup> kg⋅m<sup>−3</sup>
	| {{val|1.77361|e=-28}} <math>l_\text{P}^{-1}</math>
	|-		|-
			| ]
	! ] (<math>K_J</math>)
			| 1.9 × 10<sup>−32</sup> ''T''<sub>P</sub>
	| {{val|9.63913|e=-2}} <math>f_\text{P}/U_\text{P}</math>
			| 2.725 K{{br}} temperature of the ]
	| {{val|2.71915|e=-2}} <math>f_\text{P}/U_\text{P}</math>
	|-		|-
	! ] (<math>R_K</math>)		| ]
			| ≈ 10<sup>−122</sup> ''l''{{su|b=P|p=&nbsp;−2|lh=1em}}
	| 68.5180 <math>Z_\text{P}</math>
			| ≈ 10<sup>−52</sup> m<sup>−2</sup>
	| 861.023 <math>Z_\text{P}</math>
	|-		|-
	! ] (<math>\sigma</math>)		| ]
			| ≈ 10<sup>−61</sup> ''t''{{su|b=P|p=&nbsp;−1|lh=1em}}
	| colspan=2| {{val|1.64493|e=-1}} <math>P_\text{P}/l_\text{P}^2/T_\text{P}^4</math>
			| ≈ 10<sup>−18</sup> s<sup>−1</sup> ≈ 10<sup>2</sup> (km/s)/]
	|}		|}
			After the measurement of the cosmological constant (Λ) in 1998, estimated at 10<sup>−122</sup> in Planck units, it was noted that this is suggestively close to the reciprocal of the ] (''T'') squared. Barrow and Shaw proposed a modified theory in which ] is a field evolving in such a way that its value remains {{nowarp|Λ ~ ''T''<sup>−2</sup>}} throughout the history of the universe.<ref>{{cite journal|doi= 10.1007/s10714-011-1199-1|arxiv=1105.3105|title=The value of the cosmological constant|journal=]|volume=43|issue=10|pages=2555–2560|year=2011|last1=Barrow|first1=John D.|last2=Shaw|first2=Douglas J.|bibcode = 2011GReGr..43.2555B |s2cid=55125081}}</ref>
	== History ==
	] began in 1881, when ], noting that ] is quantized, derived units of ], ], and ], now named ] in his honor, by normalizing ''G'', ''c'', {{sfrac|4{{pi}}''ε''<sub>0</sub>}}, ''k''<sub>B</sub>, and the ], ''e'', to 1.
			=== Analysis of the units ===
	Already in 1899 (i.e. one year before the advent of quantum theory) ] introduced what became later known as Planck's constant.<ref>Planck (1899), p. 479.</ref><ref name="TOM">*Tomilin, K. A., 1999, "", 287–296.</ref> 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 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:
	{{bq|''...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''<sub>B</sub> to arrive at natural units for ], ], ], and ].<ref name="TOM" /> Planck did not adopt any electromagnetic units. However, since the ] gravitational constant, ''G'', is set to 1, a natural extension of Planck units to a unit of ] is to also set the non-rationalized Coulomb constant, ''k''<sub>e</sub>, to 1 as well (as well as the Stoney units).<ref name="PAV">{{cite book|last=Pavšic|first=Matej|title=The Landscape of Theoretical Physics: A Global View|volume=119|year=2001|publisher=Kluwer Academic|location=Dordrecht|isbn=978-0-7923-7006-2|pages=347–352|url=https://www.springer.com/gp/book/9781402003516#otherversion=9780792370062|series=Fundamental Theories of Physics | doi = 10.1007/0-306-47136-1 |arxiv=gr-qc/0610061}}</ref> 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{{pi}}''G'' and ''ε''<sub>0</sub> (instead of ''G'' and ''k''<sub>e</sub>) to 1, which may be less convenient but is rationalized. Another convention is to use the ] as the basic unit of ] in the Planck system.<ref name="Tolin1999">{{cite journal | title=Fine-structure constant and dimension analysis | author=Tomilin, K. | journal=Eur. J. Phys. | year=1999 | volume=20 | issue=5 | pages=L39–L40 | doi=10.1088/0143-0807/20/5/404|bibcode = 1999EJPh...20L..39T | url=https://semanticscholar.org/paper/1a874406c6f191a4b18d3e7f30657ff6faa25ca4 }}</ref> Such a system is convenient for ] physics. The two conventions for unit charge differ by a factor of the square root of the ]. Planck's paper also gave numerical values for the base units that were close to modern values.
			==== Planck length ====
	== List of physical equations ==
			The Planck length, denoted {{math|<var>ℓ</var><sub>P</sub>}}, is a unit of ] defined as:<math display="block">\ell_\mathrm{P} = \sqrt\frac{\hbar G}{c^3}</math>It is equal to {{physconst|lP}} (the two digits enclosed by parentheses are the estimated ] associated with the reported numerical value) or about {{val|e=-20}} times the diameter of a ].<ref name="baez1999">{{cite book |last=Baez |first=John |author-link=John Baez |chapter-url=https://math.ucr.edu/home/baez/planck/node2.html |access-date=2022-03-20 |chapter=Higher-Dimensional Algebra and Planck-Scale Physics |title=Physics Meets Philosophy at the Planck Scale |editor-first1=Craig |editor-last1=Callender |editor-link1=Craig Callender |editor-first2=Nick |editor-last2=Huggett |publisher=Cambridge University Press |year=2001 |pages=172–195 |isbn=978-0-521-66280-2 |oclc=924701824 |archive-date=8 July 2021 |archive-url=https://web.archive.org/web/20210708045541/https://math.ucr.edu/home/baez/planck/node2.html |url-status=live }}</ref> It can be motivated in various ways, such as considering a particle whose ] is comparable to its ],<ref name="baez1999" /><ref name="Adler2010">{{cite journal |arxiv=1001.1205|doi=10.1119/1.3439650|title=Six easy roads to the Planck scale|year=2010|last1=Adler|first1=Ronald J.|journal=] |volume=78|issue=9|pages=925–932|bibcode=2010AmJPh..78..925A|s2cid=55181581}}</ref><ref>{{cite web |last=Siegel |first=Ethan |author-link=Ethan Siegel |date=2019-06-26 |title=What Is The Smallest Possible Distance In The Universe? |url=https://www.forbes.com/sites/startswithabang/2019/06/26/what-is-the-smallest-possible-distance-in-the-universe/?sh=2b99cbc848a1 |access-date=2019-06-26 |work=Starts with a Bang |publisher=] |archive-date=18 September 2021 |archive-url=https://web.archive.org/web/20210918054216/https://www.forbes.com/sites/startswithabang/2019/06/26/what-is-the-smallest-possible-distance-in-the-universe/?sh=2b99cbc848a1 |url-status=live }}</ref> though whether those concepts are in fact simultaneously applicable is open to debate.<ref>{{Cite journal |last=Faraoni |first=Valerio |date=November 2017 |title=Three new roads to the Planck scale |url=http://aapt.scitation.org/doi/10.1119/1.4994804 |journal=] |language=en |volume=85 |issue=11 |pages=865–869 |arxiv=1705.09749 |bibcode=2017AmJPh..85..865F |doi=10.1119/1.4994804 |s2cid=119022491 |issn=0002-9505 |quote=Like all orders of magnitude estimates, this procedure is not rigorous since it extrapolates the concepts of black hole and of Compton wavelength to a new regime in which both concepts would probably lose their accepted meanings and would, strictly speaking, cease being valid. However, this is how one gains intuition into a new physical regime. |access-date=9 April 2022 |archive-date=30 December 2017 |archive-url=https://web.archive.org/web/20171230103939/http://aapt.scitation.org/doi/10.1119/1.4994804 |url-status=live }}</ref> (The same heuristic argument simultaneously motivates the Planck mass.<ref name="Adler2010" />)
	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 ]. 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.
			The Planck length is a distance scale of interest in speculations about quantum gravity. The ] is one-fourth the area of its ] in units of Planck length squared.<ref name=":0" />{{rp|page=370}} Since the 1950s, it has been conjectured that quantum fluctuations of the spacetime metric might make the familiar notion of distance inapplicable below the Planck length.<ref name="Wheeler">{{cite journal |last=Wheeler |first=J. A. |author-link=John Archibald Wheeler |date=January 1955 |title=Geons |journal=] |volume=97 |issue=2 |pages=511–536 |bibcode=1955PhRv...97..511W |doi=10.1103/PhysRev.97.511}}</ref><ref>{{Cite journal |last=Regge |first=T. |author-link=Tullio Regge |date=1958-01-01 |title=Gravitational fields and quantum mechanics |url=https://doi.org/10.1007/BF02744199 |journal=Il Nuovo Cimento |language=en |volume=7 |issue=2 |pages=215–221 |doi=10.1007/BF02744199 |bibcode=1958NCim....7..215R |s2cid=123012079 |issn=1827-6121 |access-date=22 March 2022 |archive-date=24 March 2022 |archive-url=https://web.archive.org/web/20220324150912/https://link.springer.com/article/10.1007/BF02744199 |url-status=live }}</ref><ref name="Gorelik">{{Cite book |last=Gorelik |first=Gennady |url=http://people.bu.edu/gorelik/cGh_FirstSteps92_MPB_36/cGh_FirstSteps92_text.htm |title=Studies in the History of General Relativity: Based on the proceedings of the 2nd International Conference on the History of General Relativity, Luminy, France, 1988 |date=1992 |publisher=Birkhäuser |isbn=0-8176-3479-7 |editor-last=Eisenstaedt |editor-first=Jean |location=Boston |pages=364–379 |chapter=First Steps of Quantum Gravity and the Planck Values |oclc=24011430 |author-link=Gennady Gorelik |editor-last2=Kox |editor-first2=Anne J. |archive-url=https://web.archive.org/web/20190425200045/http://people.bu.edu/gorelik/cGh_FirstSteps92_MPB_36/cGh_FirstSteps92_text.htm |archive-date=2019-04-25}}</ref> This is sometimes expressed by saying that "spacetime becomes a ]".<ref>{{Cite journal |last=Mermin |first=N. David |author-link=N. David Mermin |date=May 2009 |title=What's bad about this habit |url=http://physicstoday.scitation.org/doi/10.1063/1.3141952 |journal=] |language=en |volume=62 |issue=5 |pages=8–9 |bibcode=2009PhT....62e...8M |doi=10.1063/1.3141952 |issn=0031-9228 |access-date=22 March 2022 |archive-date=22 March 2022 |archive-url=https://web.archive.org/web/20220322074715/https://physicstoday.scitation.org/doi/10.1063/1.3141952 |url-status=live }}</ref> It is possible that the Planck length is the shortest physically measurable distance, since any attempt to investigate the possible existence of shorter distances, by performing higher-energy collisions, would result in black hole production. Higher-energy collisions, rather than splitting matter into finer pieces, would simply produce bigger black holes.<ref>{{cite journal |last1=Carr |first1=Bernard J.|author2-link=Steven Giddings |last2=Giddings |first2=Steven B. |date=May 2005 |title=Quantum Black Holes |url=https://pdfs.semanticscholar.org/c2fe/378fdddf98b3726b1ac12788e9cae03b884a.pdf |journal=] |volume=292 |pages=48–55 |number=5|doi=10.1038/scientificamerican0505-48 |pmid=15882021 |bibcode=2005SciAm.292e..48C |s2cid=10872062 |archive-url=https://web.archive.org/web/20190214174427/https://pdfs.semanticscholar.org/c2fe/378fdddf98b3726b1ac12788e9cae03b884a.pdf |archive-date=14 February 2019 }}</ref>
	{| class="wikitable"
	|+Table 5: How Planck units simplify the key equations of physics
			The strings of ] are modeled to be on the order of the Planck length.<ref>{{Cite book |last=Manoukian |first=Edouard B. |url=http://link.springer.com/10.1007/978-3-319-33852-1 |title=Quantum Field Theory II: Introductions to Quantum Gravity, Supersymmetry and String Theory |date=2016 |publisher=Springer International Publishing |isbn=978-3-319-33851-4 |series=Graduate Texts in Physics |location=Cham |pages=187 |language=en |doi=10.1007/978-3-319-33852-1 |access-date=22 March 2022 |archive-date=24 March 2022 |archive-url=https://web.archive.org/web/20220324150830/https://link.springer.com/book/10.1007/978-3-319-33852-1 |url-status=live }}</ref><ref>{{cite book |last=Schwarz |first=John H. |title=Geoffrey Chew: Architect of the Bootstrap |date=December 2021 |publisher=World Scientific |isbn=978-981-12-1982-5 |pages=72–83 |language=en |chapter=From the S Matrix to String Theory |doi=10.1142/9789811219832_0013 |s2cid=245575026 |author-link=John Henry Schwarz}}</ref> In theories with ]s, the Planck length calculated from the observed value of <math>G</math> can be smaller than the true, fundamental Planck length.<ref name=":0" />{{rp|61}}<ref>{{Cite journal |last=Hossenfelder |first=Sabine |author-link=Sabine Hossenfelder |date=December 2013 |title=Minimal Length Scale Scenarios for Quantum Gravity |journal=] |language=en |volume=16 |issue=1 |pages=2 |arxiv=1203.6191 |bibcode=2013LRR....16....2H |doi=10.12942/lrr-2013-2 |doi-access=free |issn=2367-3613 |pmc=5255898 |pmid=28179841}}</ref>
	!
	! SI form
	! Lorentz–Heaviside version Planck form
	! Gaussian version Planck form
	|-
	| ] in ]
	| <math>{ E = m c^2} \ </math>
	| colspan=2| <math>{ E = m } \ </math>
	|-
	| ]
	| <math> E^2 = m^2 c^4 + p^2 c^2 \;</math>
	| colspan=2| <math> E^2 = m^2 + p^2 \;</math>
	|-
	| ]
	| <math> F = -G \frac{m_1 m_2}{r^2} </math>
	| <math> F = -\frac{m_1 m_2}{4\pi r^2} </math>
	| <math> F = -\frac{m_1 m_2}{r^2} </math>
	|-
	| ] in ]
	| <math>{ G_{\mu \nu} = {8 \pi G \over c^4} T_{\mu \nu} } \ </math>
	| <math>{ G_{\mu \nu} = 2 T_{\mu \nu} } \ </math>
	| <math>{ G_{\mu \nu} = 8 \pi T_{\mu \nu} } \ </math>
	|-
	| The formula of ]
	| <math> r_s = \frac{2Gm}{c^2} </math>
	| <math> r_s = \frac{m}{2\pi} </math>
	| <math> r_s = 2m </math>
	|-
	| ]
	| <math>\mathbf{g}\cdot d\mathbf{A} = -4 \pi GM</math><br><math>\nabla\cdot \mathbf{g} = -4\pi G\rho</math>
	| <math>\mathbf{g}\cdot d\mathbf{A} = -M</math><br><math>\nabla\cdot \mathbf{g} = -\rho</math>
	| <math>\mathbf{g}\cdot d\mathbf{A} = -4 \pi M</math><br><math>\nabla\cdot \mathbf{g} = -4\pi \rho</math>
	|-
	| ]
	| <math>{\nabla}^2 \phi = 4\pi G \rho</math><br> <math>\phi(r) = \dfrac {-G m}{r}</math>
	| <math>{\nabla}^2 \phi = \rho</math><br> <math>\phi(r) = \dfrac {-m}{4\pi r}</math>
	| <math>{\nabla}^2 \phi = 4\pi \rho</math><br> <math>\phi(r) = \dfrac {-m}{r}</math>
	|-
	| The ]
	| <math>Z_0=\frac{4\pi G}{c}</math>
	| <math>Z_0=1</math>
	| <math>Z_0=4\pi</math>
	|-
	| The ]
	| <math>Y_0=\frac{c}{4\pi G}</math>
	| <math>Y_0=1</math>
	| <math>Y_0=\frac{1}{4\pi}</math>
	|-
	| ]
	| <math>\nabla \cdot \mathbf{E_g} = -4\pi G \rho_g</math><br />
	<math>\nabla \cdot \mathbf{D_g} = \rho_gf</math><br />
	<math>\nabla \cdot \mathbf{B_g} = 0 \ </math><br />
	<math>\nabla \times \mathbf{E_g} = -\frac{\partial \mathbf{B_g}} {\partial t}</math><br />
	<math>\nabla \times \mathbf{B_g} = \frac{1}{c^2} \left(-4\pi G \mathbf{J_g} + \frac{\partial \mathbf{E_g}} {\partial t} \right)</math><br />
	<math>\nabla \times \mathbf{H_g} = \mathbf{J_g}f + \frac{\partial \mathbf{D_g}} {\partial t}</math>
	| <math>\nabla \cdot \mathbf{E_g} = \rho_g</math><br />
	<math>\nabla \cdot \mathbf{D_g} = \rho_gf</math><br />
	<math>\nabla \cdot \mathbf{B_g} = 0 \ </math><br />
	<math>\nabla \times \mathbf{E_g} = -\frac{\partial \mathbf{B_g}} {\partial t}</math><br />
	<math>\nabla \times \mathbf{B_g} = \mathbf{J_g} + \frac{\partial \mathbf{E_g}} {\partial t}</math><br />
	<math>\nabla \times \mathbf{H_g} = \mathbf{J_g}f + \frac{\partial \mathbf{D_g}} {\partial t}</math>
	| <math>\nabla \cdot \mathbf{E_g} = 4 \pi \rho_g \ </math><br/>
	<math>\nabla \cdot \mathbf{D_g} = \rho_gf</math><br />
	<math>\nabla \cdot \mathbf{B_g} = 0 \ </math><br />
	<math>\nabla \times \mathbf{E_g} = -\frac{\partial \mathbf{B_g}} {\partial t}</math><br />
	<math>\nabla \times \mathbf{B_g} = 4 \pi \mathbf{J_g} + \frac{\partial \mathbf{E_g}} {\partial t}</math><br />
	<math>\nabla \times \mathbf{H_g} = \mathbf{J_g}f + \frac{\partial \mathbf{D_g}} {\partial t}</math>
	|-
	| ]
	| <math>{ E = h \nu } \ </math><br><math>{ E = \hbar \omega } \ </math>
	| colspan=2| <math>{ E = 2\pi \nu } \ </math><br><math>{ E = \omega } \ </math>
	|-
	| ]
	| <math>\Delta x \cdot \Delta p \ge \frac{\hbar}{2}</math>
	| colspan=2| <math>\Delta x \cdot \Delta p \ge \frac{1}{2}</math>
	|-
	| ] of ]
	| <math>E=\hbar\omega=h\nu=\frac{hc}{\lambda}</math> {{math| {{=}} {{sfrac|''ħc''|''ƛ''}}}}
	| colspan=2| <math>E=\omega=2\pi \nu=\frac{2\pi}{\lambda}</math> {{math| {{=}} {{sfrac|''ƛ''}}}}
	|-
	| ] of ]
	| <math>p=\hbar k=\frac{h\nu}{c}=\frac{h}{\lambda}</math> {{math| {{=}} {{sfrac|''ħ''|''ƛ''}}}}
	| colspan=2| <math>p=k=2\pi \nu=\frac{2\pi}{\lambda}</math> {{math| {{=}} {{sfrac|''ƛ''}}}}
	|-
	| ] and ] of ]
	| <math>\lambda = \frac{h}{mv} = \frac{2\pi \hbar}{mv}</math><br> {{math|''ƛ'' {{=}} {{sfrac|''ħ''|''mv''}}}}
	| colspan=2| <math>\lambda = \frac{2\pi}{mv}</math><br> {{math|''ƛ'' {{=}} {{sfrac|''mv''}}}}
	|-
	| The formula of ] and ]
	| <math>\lambda = \frac{h}{mc} = \frac{2\pi \hbar}{mc}</math><br> {{math|''ƛ'' {{=}} {{sfrac|''ħ''|''mc''}}}}
	| colspan=2| <math>\lambda = \frac{2\pi}{m}</math><br> {{math|''ƛ'' {{=}} {{sfrac|''m''}}}}
	|-
	| ]
	| <math>- \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}</math>
	| colspan=2| <math>- \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}</math>
	|-
	| ]
	| <math>i \hbar \frac{\partial}{\partial t} \psi = H \cdot \psi</math>
	| colspan=2| <math>i \frac{\partial}{\partial t} \psi = H \cdot \psi</math>
	|-
	| ] form of ]
	| <math> H \left| \psi_t \right\rangle = i \hbar \frac{\partial}{\partial t} \left| \psi_t \right\rangle</math>
	| colspan=2| <math> H \left| \psi_t \right\rangle = i \frac{\partial}{\partial t} \left| \psi_t \right\rangle</math>
	|-
	| Covariant form of the ]
	| <math>\ ( i\hbar \gamma^\mu \partial_\mu - mc) \psi = 0</math>
	| colspan=2| <math>\ ( i\gamma^\mu \partial_\mu - m) \psi = 0</math>
	|-
	| The main role in quantum gravity
	| <math>\Delta r_s\Delta r\ge \frac{\hbar G}{c^3}</math>
	| <math>\Delta r_s\Delta r\ge \frac{1}{4\pi}</math>
	| <math>\Delta r_s\Delta r\ge 1</math>
	|-
	| The ]
	| <math>\mu_0=\frac{1}{\epsilon_0 c^2}</math>
	| <math>\mu_0=1</math>
	| <math>\mu_0=4\pi</math>
	|-
	| The ]
	| <math>Z_0=\frac{\mathbf{E}}{\mathbf{H}}=\sqrt{\frac{\mu_0}{\epsilon_0}}=\frac{1}{\epsilon_0 c}=\mu_0 c</math>
	| <math>Z_0=1</math>
	| <math>Z_0=4\pi</math>
	|-
	| The ]
	| <math>Y_0=\frac{\mathbf{H}}{\mathbf{E}}=\sqrt{\frac{\epsilon_0}{\mu_0}}=\epsilon_0 c=\frac{1}{\mu_0 c}</math>
	| <math>Y_0=1</math>
	| <math>Y_0=\frac{1}{4\pi}</math>
	|-
	| The ]
	| <math>k_e=\frac{1}{4\pi\epsilon_0}</math>
	| <math>k_e=\frac{1}{4\pi}</math>
	| <math>k_e=1</math>
	|-
	| ]
	| <math> F = k_e \frac{q_1 q_2}{r^2} = \frac{1}{4 \pi \epsilon_0} \frac{q_1 q_2}{r^2} </math>
	| <math> F = \frac{q_1 q_2}{4 \pi r^2} </math>
	| <math> F = \frac{q_1 q_2}{r^2} </math>
	|-
	| ] for two stationary ]
	| <math> F = k_m \frac{b_1 b_2}{r^2} = \frac{\mu_0}{4 \pi} \frac{b_1 b_2}{r^2} </math>
	| <math> F = \frac{b_1 b_2}{4 \pi r^2} </math>
	| <math> F = \frac{b_1 b_2}{r^2} </math>
	|-
	| ]
	| <math>\Delta B = \frac{\mu_0 I}{4\pi} \frac{\Delta L}{r^2} \sin \theta</math>
	| <math>\Delta B = \frac{I}{4\pi} \frac{\Delta L}{r^2} \sin \theta</math>
	| <math>\Delta B = I \frac{\Delta L}{r^2} \sin \theta</math>
	|-
	| ]
	| <math>\mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi} \int_C \frac{I \, d\boldsymbol \ell\times\mathbf{r'}}{|\mathbf{r'}|^3}</math>
	| <math>\mathbf{B}(\mathbf{r}) = \frac{1}{4\pi} \int_C \frac{I \, d\boldsymbol \ell\times\mathbf{r'}}{|\mathbf{r'}|^3}</math>
	| <math>\mathbf{B}(\mathbf{r}) = \int_C \frac{I \, d\boldsymbol \ell\times\mathbf{r'}}{|\mathbf{r'}|^3}</math>
	|-
	| Equation of ] and ]
	| <math>\mathbf{D}=\epsilon_0 \mathbf{E}</math>
	| <math>\mathbf{D}=\mathbf{E}</math>
	| <math>\mathbf{D}=\frac{\mathbf{E}}{4\pi}</math>
	|-
	| Equation of ] and ]
	| <math>\mathbf{B}=\mu_0 \mathbf{H}</math>
	| <math>\mathbf{B}=\mathbf{H}</math>
	| <math>\mathbf{B}=4\pi \mathbf{H}</math>
	|-
	| ]
	| <math>\nabla \cdot \mathbf{E} = \frac{1}{\epsilon_0} \rho</math><br />
	<math>\nabla \cdot \mathbf{D} = \rho_f</math><br />
	<math>\nabla \cdot \mathbf{B} = 0 \ </math><br />
	<math>\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}</math><br />
	<math>\nabla \times \mathbf{B} = \frac{1}{c^2} \left(\frac{1}{\epsilon_0} \mathbf{J} + \frac{\partial \mathbf{E}} {\partial t} \right)</math><br />
	<math>\nabla \times \mathbf{H} = \mathbf{J}_f + \frac{\partial \mathbf{D}} {\partial t}</math>
	| <math>\nabla \cdot \mathbf{E} = \rho</math><br />
	<math>\nabla \cdot \mathbf{D} = \rho_f</math><br />
	<math>\nabla \cdot \mathbf{B} = 0 \ </math><br />
	<math>\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}</math><br />
	<math>\nabla \times \mathbf{B} = \mathbf{J} + \frac{\partial \mathbf{E}} {\partial t}</math><br />
	<math>\nabla \times \mathbf{H} = \mathbf{J}_f + \frac{\partial \mathbf{D}} {\partial t}</math>
	| <math>\nabla \cdot \mathbf{E} = 4 \pi \rho \ </math><br />
	<math>\nabla \cdot \mathbf{D} = \rho_f</math><br />
	<math>\nabla \cdot \mathbf{B} = 0 \ </math><br />
	<math>\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}</math><br />
	<math>\nabla \times \mathbf{B} = 4 \pi \mathbf{J} + \frac{\partial \mathbf{E}} {\partial t}</math><br />
	<math>\nabla \times \mathbf{H} = \mathbf{J}_f + \frac{\partial \mathbf{D}} {\partial t}</math>
	|-
	| ] ''K''<sub>''J''</sub> defined
	| <math>K_J=\frac{e}{\pi \hbar}</math>
	| <math>K_J=\sqrt{\frac{4\alpha}{\pi}}</math>
	| <math>K_J=\frac{\sqrt{\alpha}}{\pi}</math>
	|-
	| ] ''R''<sub>''K''</sub> defined
	| <math>R_K=\frac{2\pi \hbar}{e^2}</math>
	| <math>R_K=\frac{1}{2\alpha}</math>
	| <math>R_K=\frac{2\pi}{\alpha}</math>
	|-
	| The ] of ]
	| <math>\xi_e=\frac{e}{m_e}=\sqrt{\frac{G \alpha}{k_e \alpha_G}}=\sqrt{\frac{4\pi G \epsilon_0 \alpha}{\alpha_G}}</math>
	| colspan=2| <math>\xi_e=\sqrt{\frac{\alpha}{\alpha_G}}</math>
	|-
	| The ]
	| <math>a_0=\frac{4\pi \epsilon_0 \hbar^2}{m_{\text{e}} e^2} = \frac{\hbar}{m_{\text{e}} c \alpha}</math>
	| <math>a_0=\frac{1}{\alpha\sqrt{4\pi\alpha_G}}</math>
	| <math>a_0=\frac{1}{\alpha\sqrt{\alpha_G}}</math>
	|-
	| The ]
	| <math>\mu_B=\frac{e \hbar}{2 m_e}</math>
	| colspan=2| <math>\mu_B=\sqrt{\frac{\alpha}{4\alpha_G}}</math>
	|-
	| ] ''R''<sub>∞</sub> defined
	| <math>R_\infty={\frac {m_{\text{e}}e^{4}}{8\epsilon_0^2h^3c}}=\frac{\alpha^2 m_\text{e} c}{4 \pi \hbar}</math>
	| <math>R_\infty=\sqrt{\frac{\alpha^4\alpha_G}{4 \pi}}</math>
	| <math>R_\infty=\frac{\sqrt{\alpha^4\alpha_G}}{4 \pi}</math>
	|-
	| ]
	| <math>PV = nRT = Nk_\text{B}T</math>
	| colspan=2| <math>PV = NT</math>
	|-
	| Equation of the ]
	| <math>v_{rms} = \sqrt{\frac{3RT}{M}} = \sqrt{\frac{3k_\text{B}T}{m}}</math>
	| colspan=2| <math>v_{rms} = \sqrt{\frac{3T}{m}}</math>
	|-
	| ]
	| <math>\Sigma \frac{1}{2} mv^2 = \frac{3}{2}Nk_\text{B}T</math>
	| colspan=2| <math>\Sigma \frac{1}{2} mv^2 = \frac{3}{2}NT</math>
	|-
	| ]
	| <math>T=\frac{\hbar a}{2\pi c k_B}</math>
	| colspan=2| <math>T=\frac{a}{2\pi}</math>
	|-
	| ] per particle per ]
	| <math>{ E = \tfrac12 k_\text{B} T} \ </math>
	| colspan=2| <math>{ E = \tfrac12 T} \ </math>
	|-
	| Boltzmann's ] formula
	| <math>{ S = k_\text{B} \ln \Omega } \ </math>
	| colspan=2| <math>{ S = \ln \Omega } \ </math>
	|-
	| ] ''σ'' defined
	| <math> \sigma = \frac{\pi^2 k_\text{B}^4}{60 \hbar^3 c^2} </math>
	| colspan=2| <math> \sigma = \frac{\pi^2}{60} </math>
	|-
	| ] (surface ] per unit ] per unit ]) for ] at ] ''T''.
	| <math> I(\omega,T) = \frac{\hbar \omega^3 }{4 \pi^3 c^2}~\frac{1}{e^{\frac{\hbar \omega}{k_\text{B} T}}-1} </math>
	| colspan=2| <math> I(\omega,T) = \frac{\omega^3 }{4 \pi^3}~\frac{1}{e^{\omega/T}-1} </math>
	|-
	| The formula of ]
	| <math>T=\frac{\hbar a}{2\pi c k_B}</math>
	| colspan=2| <math>T=\frac{a}{2\pi}</math>
	|-
	| ] of a black hole
	| <math>T_H=\frac{\hbar c^3}{8\pi G M k_B}</math>
	| <math>T_H=\frac{1}{2 M}</math>
	| <math>T_H=\frac{1}{8\pi M}</math>
	|-
	| ]–] ]<ref>Also see ] (1989) '']''. Oxford Univ. Press: 714-17. Knopf.</ref>
	| <math>S_\text{BH} = \frac{A_\text{BH} k_\text{B} c^3}{4 G \hbar} = \frac{4\pi G k_\text{B} m^2_\text{BH}}{\hbar c}</math>
	| <math>S_\text{BH} = \pi A_\text{BH} = m^2_\text{BH}</math>
	| <math>S_\text{BH} = \frac{A_\text{BH}}{4} = 4\pi m^2_\text{BH}</math>
	|}
			==== Planck time ====
	Note:
			The Planck time, denoted {{math|<var>t</var><sub>P</sub>}}, is defined as: <math display="block">t_\mathrm{P} = \frac{\ell_\mathrm{P}}{c} = \sqrt\frac{\hbar G}{c^5}</math>This is the ] required for ] to travel a distance of 1&nbsp;Planck length in ], which is a time interval of approximately {{val|5.39|e=−44|ul=s}}. No current physical theory can describe timescales shorter than the Planck time, such as the earliest events after the Big Bang.<ref name="Planck-UOregon" /> Some conjectures state that the structure of time need not remain smooth on intervals comparable to the Planck time.<ref name="1e-33">
			{{cite journal
			| last1 = Wendel | first1 = Garrett
			| last2 = Martínez | first2 = Luis
			| last3 = Bojowald | first3 = Martin
			| date = 19 June 2020
			| title = Physical Implications of a Fundamental Period of Time
			| journal = ]
			| volume = 124
			| issue = 24
			| pages = 241301
			| doi = 10.1103/PhysRevLett.124.241301
			| pmid = 32639827
			| arxiv = 2005.11572
			| bibcode = 2020PhRvL.124x1301W
			| s2cid = 218870394
			}}</ref><!--
			==== Planck mass ====
			-- editor note: A section on the Planck mass would fit here, but let's not put in the heading without content
			-->
			==== Planck energy ====
	* For the ] <math>e</math>:
			The Planck energy ''E''<sub>P</sub> is approximately equal to the energy released in the combustion of the fuel in an automobile fuel tank (57.2&nbsp;L at 34.2&nbsp;MJ/L of chemical energy). The ] ] had a measured energy of about 50&nbsp;J, equivalent to about {{val|2.5|e=−8|u=''E''<sub>P</sub>}}.<ref>{{Cite web|url=http://www.cosmic-ray.org/reading/flyseye.html#SEC10|title=HiRes – The High Resolution Fly's Eye Ultra High Energy Cosmic Ray Observatory|website=www.cosmic-ray.org|access-date=2016-12-21|archive-date=15 August 2009|archive-url=https://web.archive.org/web/20090815102123/http://www.cosmic-ray.org/reading/flyseye.html#SEC10|url-status=live}}</ref><ref>{{cite journal |last1=Bird |first1=D. J. |last2=Corbato |first2=S. C. |last3=Dai |first3=H. Y. |last4=Elbert |first4=J. W. |last5=Green |first5=K. D. |last6=Huang |first6=M. A. |last7=Kieda |first7=D. B. |last8=Ko |first8=S. |last9=Larsen |first9=C. G. |last10=Loh |first10=E. C. |last11=Luo |first11=M. Z. |last12=Salamon |first12=M. H. |last13=Smith |first13=J. D. |last14=Sokolsky |first14=P. |last15=Sommers |first15=P. |date=March 1995 |title=Detection of a cosmic ray with measured energy well beyond the expected spectral cutoff due to cosmic microwave radiation |journal=] |volume=441 |pages=144 |arxiv=astro-ph/9410067 |bibcode=1995ApJ...441..144B |doi=10.1086/175344 |s2cid=119092012 |last17=Thomas |last16=Tang |first16=J. K. K. |first17=S. B.}}</ref>
			Proposals for theories of ] posit that, in addition to the speed of light, an energy scale is also invariant for all inertial observers. Typically, this energy scale is chosen to be the Planck energy.<ref>{{Cite journal |last1=Judes |first1=Simon |last2=Visser |first2=Matt |author-link2=Matt Visser |date=2003-08-04 |title=Conservation laws in "doubly special relativity" |url=https://link.aps.org/doi/10.1103/PhysRevD.68.045001 |journal=Physical Review D |language=en |volume=68 |issue=4 |pages=045001 |arxiv=gr-qc/0205067 |bibcode=2003PhRvD..68d5001J |doi=10.1103/PhysRevD.68.045001 |s2cid=119094398 |issn=0556-2821}}</ref><ref>{{Cite journal |last=Hossenfelder |first=Sabine |author-link=Sabine Hossenfelder |date=2014-07-09 |title=The Soccer-Ball Problem |url=http://www.emis.de/journals/SIGMA/2014/074/ |journal=Symmetry, Integrability and Geometry: Methods and Applications |volume=10 |pages=74 |arxiv=1403.2080 |bibcode=2014SIGMA..10..074H |doi=10.3842/SIGMA.2014.074 |s2cid=14373748 |access-date=16 April 2022 |archive-date=19 March 2022 |archive-url=https://web.archive.org/web/20220319004139/https://www.emis.de/journals/SIGMA/2014/074/ |url-status=live }}</ref>
	: <math>e = \sqrt{4\pi \alpha}</math> (Lorentz–Heaviside version)
	: <math>e = \sqrt{\alpha}</math> (Gaussian version)
			==== Planck unit of force ====
	where <math>\alpha</math> is the ].
			The Planck unit of force may be thought of as the derived unit of ] in the Planck system if the Planck units of time, length, and mass are considered to be base units.<math display="block">F_\text{P} = \frac{m_\text{P} c}{t_\text{P}} = \frac{c^4}{G} \approx \mathrm{1.2103 \times 10^{44} ~N}</math>It is the gravitational attractive force of two bodies of 1 Planck mass each that are held 1 Planck length apart. One convention for the Planck charge is to choose it so that the electrostatic repulsion of two objects with Planck charge and mass that are held 1 Planck length apart balances the Newtonian attraction between them.<ref>{{cite book|last=Kiefer |first=Claus |title=Quantum Gravity |publisher=Oxford University Press |year=2012 |series=International series of monographs on physics |volume=155 |isbn=978-0-191-62885-6 |oclc=785233016 |page=5}}</ref>
			Some authors have argued that the Planck force is on the order of the maximum force that can occur between two bodies.<ref>{{cite journal |first1=Venzo |last1=de Sabbata |first2=C. |last2=Sivaram |title=On limiting field strengths in gravitation |journal=] |date=1993 |volume=6 |issue=6 |pages=561–570 |doi=10.1007/BF00662806|bibcode=1993FoPhL...6..561D |s2cid=120924238 }}</ref><ref>{{cite journal |first1=G. W. |last1=Gibbons |title=The Maximum Tension Principle in General Relativity |journal=] |date=2002 |volume=32 |issue=12 |pages=1891–1901 |doi=10.1023/A:1022370717626|arxiv=hep-th/0210109|bibcode=2002FoPh...32.1891G |s2cid=118154613 }}</ref> However, the validity of these conjectures has been disputed.<ref>{{cite journal|arxiv=2102.01831 |title=Counterexamples to the maximum force conjecture |first1=Aden |last1=Jowsey |first2=Matt |last2=Visser |journal=] |author-link2=Matt Visser |date=2021-02-03 |volume=7 |issue=11 |page=403 |doi=10.3390/universe7110403 |bibcode=2021Univ....7..403J |doi-access=free }}</ref><ref>{{cite journal|first1=Niayesh|last1=Afshordi|bibcode=2012BASI...40....1A|title=Where will Einstein fail? Leasing for Gravity and cosmology|journal=Bulletin of the Astronomical Society of India|volume=40|issue=1|page=5|date=March 1, 2012|publisher=], NASA ]|arxiv=1203.3827|oclc=810438317|quote=However, for most experimental physicists, approaching energies comparable to Planck energy is little more than a distant fantasy. The most powerful accelerators on Earth miss Planck energy of 15 orders of magnitude, while ultra high energy cosmic rays are still 9 orders of magnitude short of ''M''<sub>p</sub>.}}</ref>
	* For the ] <math>m_e</math>:
			==== Planck temperature ====
	: <math>m_e = \sqrt{4\pi \alpha_G}</math> (Lorentz–Heaviside version)
			The Planck temperature ''T''<sub>P</sub> is {{physconst|TP|after=.}} At this temperature, the wavelength of light emitted by ] reaches the Planck length. There are no known physical models able to describe temperatures greater than ''T''<sub>P</sub>; a quantum theory of gravity would be required to model the extreme energies attained.<ref>
	: <math>m_e = \sqrt{\alpha_G}</math> (Gaussian version)
			{{cite book
			|first=Hubert |last=Reeves
			|title=The Hour of Our Delight
			|quote=The point at which our physical theories run into most serious difficulties is that where matter reaches a temperature of approximately 10<sup>32</sup> degrees, also known as Planck's temperature. The extreme density of radiation emitted at this temperature creates a disproportionately intense field of gravity. To go even farther back, a ] would be necessary, but such a theory has yet to be written.
			|page=117
			|publisher=W. H. Freeman Company
			|year=1991
			|isbn=978-0-7167-2220-5
			}}</ref> Hypothetically, a system in ] at the Planck temperature might contain Planck-scale black holes, constantly being formed from thermal radiation and decaying via ]. Adding energy to such a system might ''decrease'' its temperature by creating larger black holes, whose Hawking temperature is lower.<ref>{{cite arXiv |eprint=1807.04363 |first=Peter W. |last=Shor |author-link=Peter Shor |title=Scrambling Time and Causal Structure of the Photon Sphere of a Schwarzschild Black Hole |date=17 July 2018|class=gr-qc }}</ref>
			== Nondimensionalized equations ==
	where <math>\alpha_G</math> is the ].
			Physical quantities that have different dimensions (such as time and length) cannot be equated even if they are numerically equal (e.g., 1 second is not the same as 1 metre). In theoretical physics, however, this scruple may be set aside, by a process called ]. The effective result is that many fundamental equations of physics, which often include some of the constants used to define Planck units, become equations where these constants are replaced by a 1.
			Examples include the ] {{nowrap|1=<math>E^2 = (mc^2)^2 + (pc)^2</math>}} (which becomes {{nowrap|1=<math>E^2=m^2+p^2</math>)}} and the ] {{math|1=<math>(i\hbar \gamma ^{\mu}\partial_{\mu} - mc)\psi = 0</math>}} (which becomes {{math|1=<math>(i\gamma ^{\mu}\partial_{\mu} - m)\psi = 0</math>}}).
	== Alternative choices of normalization ==		== Alternative choices of normalization ==
			<!-- ] links to this section -->
	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.		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{{pi}} is ubiquitous in ] because the surface area of a ] of radius ''r'' is 4{{pi}}''r''<sup>2</sup>. This, along with the concept of ], are the basis for the ], ], and the ] operator applied to ]. For example, ] and ]s produced by point charges have spherical symmetry (Barrow 2002: 214–15). The 4{{pi}}''r''<sup>2</sup> appearing in the denominator of Coulomb's law in ], 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{{pi}} would have to be changed according to the geometry of the ].)		The factor 4{{pi}} is ubiquitous in ] because in three-dimensional space, the surface area of a ] of radius ''r'' is 4{{pi}}''r''{{i sup|2}}. This, along with the concept of ], are the basis for the ], ], and the ] operator applied to ]. For example, ] and ]s produced by point objects have spherical symmetry, and so the electric flux through a sphere of radius ''r'' around a point charge will be distributed uniformly over that sphere. From this, it follows that a factor of 4{{pi}}''r''{{i sup|2}} will appear in the denominator of Coulomb's law in ].<ref name="John D 2002" />{{rp|pages=214–15}} (Both the numerical factor and the power of the dependence on ''r'' would change if space were higher-dimensional; the correct expressions can be deduced from the geometry of ].<ref name=":0" />{{rp|page=51}}) Likewise for Newton's law of universal gravitation: a factor of 4{{pi}} naturally appears in ] when relating the gravitational potential to the distribution of matter.<ref name=":0" />{{rp|page=56}}
	Hence a substantial body of physical theory developed since Planck (1899) suggests normalizing not ''G'' but either 4{{pi}}''G'' (or 8{{pi}}''G'' or 16{{pi}}''G'') to 1. Doing so would introduce a factor of {{sfrac|4{{pi}}}} (or {{sfrac|8{{pi}}}} or {{sfrac|16{{pi}}}}) 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 {{sfrac|4{{pi}}}} in the nondimensionalized form of Coulomb's law as well, so that the nondimensionalized Maxwell's equations for electromagnetism and ] both take the same form as those for electromagnetism in SI, which do not have any factors of 4{{pi}}. When this is applied to electromagnetic constants, ε<sub>0</sub>, this unit system is called ''"rationalized"'' ]. When applied additionally to gravitation and Planck units, these are called '''rationalized Planck units'''<ref>{{cite journal|title=Kaluza-Klein Monopole|journal=Physical Review Letters|volume=51|issue=2|pages=87–90|year=1983|last=Sorkin|first=Rafael|doi=10.1103/PhysRevLett.51.87|bibcode=1983PhRvL..51...87S}}</ref> and are seen in .		Hence a substantial body of physical theory developed since Planck's 1899 paper suggests normalizing not ''G'' but 4{{pi}}''G'' (or 8{{pi}}''G'') to 1. Doing so would introduce a factor of {{sfrac|4{{pi}}}} (or {{sfrac|8{{pi}}}}) 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 {{sfrac|4{{pi}}}} in the nondimensionalized form of Coulomb's law as well, so that the nondimensionalized Maxwell's equations for electromagnetism and ] both take the same form as those for electromagnetism in SI, which do not have any factors of 4{{pi}}. When this is applied to electromagnetic constants, ''ε''<sub>0</sub>, this unit system is called "''rationalized''{{-"}}. When applied additionally to gravitation and Planck units, these are called '''rationalized Planck units'''<ref>{{cite journal|title=Kaluza-Klein Monopole|journal=]|volume=51|issue=2|pages=87–90|year=1983|last=Sorkin|first=Rafael|author-link=Rafael Sorkin |doi=10.1103/PhysRevLett.51.87|bibcode=1983PhRvL..51...87S}}</ref> and are seen in high-energy physics.<ref>{{cite book |last=Rañada |first=Antonio F. |chapter=A Model of Topological Quantization of the Electromagnetic Field |title=Fundamental Problems in Quantum Physics |date=31 October 1995 |editor=M. Ferrero |editor2=Alwyn van der Merwe |chapter-url=https://books.google.com/books?id=1wUFoP7HC38C&pg=PA271 |page=271 |publisher=Springer |isbn=9780792336709 |access-date=16 January 2018 |archive-date=1 September 2020 |archive-url=https://web.archive.org/web/20200901153435/https://books.google.com/books?id=1wUFoP7HC38C&pg=PA271 |url-status=live }}</ref>
	The rationalized Planck units are defined so that <math>c=4\pi G=\hbar=\epsilon_0=k_\text{B}=1</math>. These are the Planck units based on ] (instead of on the more conventional ]) as depicted above.		The rationalized Planck units are defined so that {{nowrap|1=''c'' = 4''πG'' = ''ħ'' = ''ε''<sub>0</sub> = ''k''<sub>B</sub> = 1}}.
	There are several possible alternative normalizations.		There are several possible alternative normalizations.
	=== Gravity ===		=== Gravitational constant ===
	<!-- ], ] and ] link to this section -->
	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 ] in 1915). Hence Planck normalized to 1 the ] ''G'' in Newton's law. In theories emerging after 1899, ''G'' nearly always appears in formulae multiplied by 4{{pi}} 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{{pi}} appearing in the equations of physics are to be eliminated via the normalization.		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 ] in 1915). Hence Planck normalized to 1 the ] ''G'' in Newton's law. In theories emerging after 1899, ''G'' nearly always appears in formulae multiplied by 4{{pi}} 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{{pi}} appearing in the equations of physics are to be eliminated via the normalization.
	* Normalizing 4{{pi}}''G'' to 1: (like the Lorentz–Heaviside version Planck units)		* Normalizing 4{{pi}}''G'' to 1 (and therefore setting {{nowrap|1=''G'' = {{sfrac|1|4{{pi}}}}}}):
			** ] becomes {{nowrap|1=''Φ''<sub>'''g'''</sub> = −''M''}} (rather than {{nowrap|1=''Φ''<sub>'''g'''</sub> = −4{{pi}}''M''}} in Planck units).
	** ] has 4{{pi}}''r''<sup>2</sup> remaining in the denominator (which is the surface area of the enclosing sphere at radius ''r'').
	** ] becomes {{nowrap|1=''Φ''<sub>'''g'''</sub> = −''M''}} (rather than {{nowrap|1=''Φ''<sub>'''g'''</sub> = −4{{pi}}''M''}} in Gaussian version Planck units).
	** Eliminates 4{{pi}}''G'' from the ].		** Eliminates 4{{pi}}''G'' from the ].
	** Eliminates 4{{pi}}''G'' in the ] (GEM) equations, which hold in weak ]s or ]. These equations have the same form as Maxwell's equations (and the ] equation) of ], with ] replacing ], and with {{sfrac|4{{pi}}''G''}} replacing ε<sub>0</sub>.		** Eliminates 4{{pi}}''G'' in the ] (GEM) equations, which hold in weak ]s or ]. These equations have the same form as Maxwell's equations (and the ] equation) of ], with ] replacing ], and with {{sfrac|4{{pi}}''G''}} replacing ''ε''<sub>0</sub>.
	** Normalizes the ] ''Z''<sub>0</sub> of ] in free space to 1. (Normally expressed as {{sfrac|4{{pi}}''G''|''c''}}){{NoteTag|] predicts that ] propagates at the same speed as ].}}		** Normalizes the ] ''Z''<sub>'''g'''</sub> of ] in free space to 1 (normally expressed as {{sfrac|4{{pi}}''G''|''c''}}).{{NoteTag|] predicts that ] propagates at the same speed as ].<ref>{{Cite book |last=Choquet-Bruhat |first=Yvonne |url=https://www.worldcat.org/oclc/317496332 |title=General Relativity and the Einstein Equations |date=2009 |publisher=Oxford University Press |isbn=978-0-19-155226-7 |location=Oxford |oclc=317496332 |author-link=Yvonne Choquet-Bruhat}}</ref>{{rp|page=60}}<ref>{{Cite book |last=Stavrov |first=Iva |title=Curvature of Space and Time, with an Introduction to Geometric Analysis |title-link=Curvature of Space and Time, with an Introduction to Geometric Analysis |date=2020 |publisher=American Mathematical Society |isbn=978-1-4704-6313-7 |location=Providence, Rhode Island |oclc=1202475208}}</ref>{{rp|page=158}}}}
	** Eliminates 4{{pi}}''G'' from the ] (for the ] in terms of its mass ''m''<sub>BH</sub> and the area of its ] ''A''<sub>BH</sub>) which is simplified to {{nowrap|1=''S''<sub>BH</sub> = {{pi}}''A''<sub>BH</sub> = (''m''<sub>BH</sub>)<sup>2</sup>}}.		** Eliminates 4{{pi}}''G'' from the Bekenstein–Hawking formula (for the ] in terms of its mass ''m''<sub>BH</sub> and the area of its ] ''A''<sub>BH</sub>) which is simplified to {{nowrap|1=''S''<sub>BH</sub> = {{pi}}''A''<sub>BH</sub> = (''m''<sub>BH</sub>)<sup>2</sup>}}.
			* Setting {{nowrap|1=8{{pi}}''G'' = 1}} (and therefore setting ''G'' = {{sfrac|1|8{{pi}}}}). This would eliminate 8{{pi}}''G'' from the ], ], and the ], for gravitation. Planck units modified so that {{nowrap|1=8{{pi}}''G'' = 1}} are known as ''reduced Planck units'', because the Planck mass is divided by {{sqrt|8{{pi}}}}. Also, the ] for the entropy of a black hole simplifies to {{nowrap|1=''S''<sub>BH</sub> = (''m''<sub>BH</sub>)<sup>2</sup>/2 = 2{{pi}}''A''<sub>BH</sub>}}.
	** In this case the ], measured in terms of this ] Planck mass, is
	::: <math> m_e = \sqrt{4 \pi \alpha_G} \cdot m_{\text{P}} \approx 1.48368 \times 10^{-22} \cdot m_{\text{P}} \, </math>
	:: where <math> {\alpha_G} \ </math> is the ]. This convention is seen in high-energy physics.
	* Setting {{nowrap|1=8{{pi}}''G'' = 1}}. This would eliminate 8{{pi}}''G'' from the ], ], and the ], for gravitation. Planck units modified so that {{nowrap|1=8{{pi}}''G'' = 1}} are known as ''reduced Planck units'', because the ] is divided by {{sqrt|8{{pi}}}}. Also, the Bekenstein–Hawking formula for the entropy of a black hole simplifies to {{nowrap|1=''S''<sub>BH</sub> = (''m''<sub>BH</sub>)<sup>2</sup>/2 = 2{{pi}}''A''<sub>BH</sub>}}.
	* Setting {{nowrap|1=16{{pi}}''G'' = 1}}. This would eliminate the constant {{sfrac|''c''<sup>4</sup>|16{{pi}}''G''}} from the Einstein–Hilbert action. The form of the Einstein field equations with ] ''Λ'' becomes {{nowrap|1=''R<sub>μν</sub>'' + ''Λg<sub>μν</sub>'' = {{sfrac|2}}(''Rg<sub>μν</sub>'' + ''T<sub>μν</sub>'')}}.
	=== Electromagnetism ===
	In order to build natural units in electromagnetism one can use:
	*''']''' (classified as a '''rationalized''' system of electromagnetism units).
	*''']''' (classified as a '''non-rationalized''' system of electromagnetism units).
	Of these, Lorentz–Heaviside is somewhat more common,<ref name="GreinerNeise1995">{{cite book|author1=Walter Greiner|author2=Ludwig Neise|author3=Horst Stöcker|title=Thermodynamics and Statistical Mechanics|url=https://books.google.com/books?id=12DKsFtFTgYC&pg=PA385|year=1995|publisher=Springer-Verlag|isbn=978-0-387-94299-5|page=385}}</ref> mainly because ] are simpler in Lorentz–Heaviside units than they are in Gaussian units.
	In the two unit systems, the Planck unit charge {{math|''q''<sub>P</sub>}} is:
	*{{math|''q''<sub>P</sub> {{=}} {{sqrt|4π''αħc''}}}} (Lorentz–Heaviside),
	*{{math|''q''<sub>P</sub> {{=}} {{sqrt|''αħc''}}}} (Gaussian)
	where {{math|''ħ''}} is the ], {{math|''c''}} is the ], and {{math|''α'' ≈ {{sfrac|137.036}}}} is the ].
	In a natural unit system where {{math|1=] = 1}}, Lorentz–Heaviside units can be derived from units by setting {{math|1=] = ] = 1}}. Gaussian units can be derived from units by a more complicated set of transformations, such as multiplying all ]s by {{math|(4π''ε''<sub>0</sub>)<sup>−{{frac|2}}</sup>}}, multiplying all ] by {{math|4π}}, and so on.<ref>See ] and references therein.</ref>
	Planck units normalize to 1 the ] ''k''<sub>e</sub> = {{sfrac|4{{pi}}''ε''<sub>0</sub>}} (as does the ] system of units and the ]). This sets the ], ''Z''<sub>P</sub> equal to {{sfrac|''Z''<sub>0</sub>|4{{pi}}}}, where ''Z''<sub>0</sub> is the ].
	* Normalizing the ] ''ε''<sub>0</sub> to 1: (as does the ]) (like the Lorentz–Heaviside version Planck units)
	** Sets the ] ''μ''<sub>0</sub> = 1 (because ''c'' = 1).
	** Sets the ] to the ], ''Z''<sub>P</sub> = ''Z''<sub>0</sub> (or sets the characteristic impedance of free space ''Z''<sub>0</sub> to 1).
	** Eliminates 4{{pi}} from the nondimensionalized form of ].
	** ] has 4{{pi}}''r''<sup>2</sup> remaining in the denominator (which is the surface area of the enclosing sphere at radius ''r'').
	** Equates the notions of ] and ] in free space (] '''E''' and ] '''D''', ] '''H''' and ] '''B''')
	** In this case the ], measured in terms of this ] Planck charge, is
	::: <math> e = \sqrt{4 \pi \alpha} \cdot q_{\text{P}} \approx 0.302822121 \cdot q_{\text{P}} \, </math>
	:: where <math> {\alpha} \ </math> is the ]. This convention is seen in high-energy physics.
	=== Temperature ===
	Planck normalized to 1 the ] ''k''<sub>B</sub>.
	* Normalizing {{sfrac|2}}''k''<sub>B</sub> to 1:
	** Removes the factor of {{sfrac|2}} in the nondimensionalized equation for the ] per particle per ].
	** 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 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 ] and ]) have proposed ] that conjecture that physical "constants" might actually change over time (e.g. a ] or ]). 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 ] that is not ], such as the ], ''did'' in fact change, would we be able to notice it or measure it unambiguously? – a question examined by ] in his paper "Comment on time-variation of fundamental constants".<ref name="hep-th0208093">{{cite journal|author=Michael Duff |title=How fundamental are fundamental constants?|arxiv=1412.2040|doi=10.1080/00107514.2014.980093|author-link=Michael Duff (physicist)|doi-broken-date=2020-03-21|url=https://www.tandfonline.com/doi/abs/10.1080/00107514.2014.980093|journal=Contemporary Physics|volume=56|issue=1|pages=35–47|year=2015}}</ref>
	] argued in his book '']'' 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:
	{{quotation| 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 you cannot tell because all the pure numbers defined by the ratios of any pair of masses are unchanged.|Barrow 2002<ref name="John D 2002" />}}
	Referring to Duff's "Comment on time-variation of fundamental constants"<ref name="hep-th0208093" /> and Duff, Okun, and ]'s paper "Trialogue on the number of fundamental constants",<ref name="DOV">{{cite journal | last1 = Duff | first1 = Michael | authorlink = Michael Duff (physicist) | authorlink3 = Gabriele Veneziano | last2 = Okun | first2 = Lev | last3 = Veneziano | first3 = Gabriele | year = 2002 | title = Trialogue on the number of fundamental constants | arxiv=physics/0110060 | journal = ] | volume = 2002 | issue = 3| page = 023 |bibcode = 2002JHEP...03..023D |doi = 10.1088/1126-6708/2002/03/023 }}</ref> 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 ], ''α'', changes or the ], {{sfrac|''m''<sub>p</sub>|''m''<sub>e</sub>}}, 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 ], ''c'', has changed. And, indeed, the Tompkins concept becomes meaningless in our perception of reality if a dimensional quantity such as ''c'' ], even drastically.
	If the speed of light ''c'', were somehow suddenly cut in half and changed to {{sfrac|2}}''c'' (but with the axiom that ''all'' dimensionless physical quantities remain the same), then the Planck length would ''increase'' by a factor of 2{{sqrt|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 ]) are related to the Planck length by an unchanging dimensionless constant of proportionality:
	: <math>a_0 = \frac{4 \pi \epsilon_0 \hbar^2}{m_e e^2} = \frac{m_\text{P}}{m_e \alpha} l_\text{P}. </math>
	Then atoms would be bigger (in one dimension) by 2{{sqrt|2}}, each of us would be taller by 2{{sqrt|2}}, and so would our metre sticks be taller (and wider and thicker) by a factor of 2{{sqrt|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{{sqrt|2}} (from the point of view of this unaffected observer on the outside) because the Planck time has increased by 4{{sqrt|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 {{val|299792458}} of our ''new'' metres in the time elapsed by one of our ''new'' seconds ({{sfrac|2}}''c'' × 4{{sqrt|2}} ÷ 2{{sqrt|2}} continues to equal {{val|299792458|u=m/s}}). We would not notice any difference.
	This contradicts what ] writes in his book '']''; 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)&nbsp;all other ''dimensionless'' constants were kept the same, or whether (2)&nbsp;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 ], is unjustified. The very same operational difference in measurement or perceived reality could just as well be caused by a change in ] or ] 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.<ref name="hep-th0208093" /><ref>] </ref>
	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 ]s. One such dimensionless physical constant is the ]. There are some experimental physicists who assert they have in fact measured a change in the fine structure constant<ref>{{cite journal | last1 = Webb | first1 = J. K. | display-authors = etal | year = 2001 | title = Further evidence for cosmological evolution of the fine structure constant | arxiv=astro-ph/0012539v3 | journal = ] | volume = 87 | issue = 9| page = 884 |bibcode = 2001PhRvL..87i1301W |doi = 10.1103/PhysRevLett.87.091301 | pmid=11531558}}</ref> and this has intensified the debate about the measurement of physical constants. According to some theorists<ref>{{cite journal | last1 = Davies | first1 = Paul C. | authorlink = Paul C. Davies | last2 = Davis | first2 = T. M. | last3 = Lineweaver | first3 = C. H. | year = 2002 | title = Cosmology: Black Holes Constrain Varying Constants | url = | journal = ] | volume = 418 | issue = 6898| pages = 602–3 | doi=10.1038/418602a | pmid=12167848|bibcode = 2002Natur.418..602D }}</ref> 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.<ref name="hep-th0208093" /> 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.<ref name="DOV" />
	== See also ==		== See also ==
	{{div col|colwidth=30em}}
	* ]		* ]
	* ]		* ]
	* ]		* ]
	* ]		* ]
	* ]
	* ]
	* ]		* ]
	{{div col end}}
	== Notes ==		== Explanatory notes ==
	{{NoteFoot}}		{{NoteFoot}}
	== References ==		== References ==
			{{reflist}}
	=== Citations ===
	{{Reflist}}
	=== Sources ===
	{{refbegin}}
	* {{cite book |title=The Constants of Nature; From Alpha to Omega – The Numbers that Encode the Deepest Secrets of the Universe |url=https://archive.org/details/constantsofnatur0000barr |url-access=registration |last=Barrow |first=John D. |author-link=John D. Barrow |year=2002 |publisher=Pantheon Books |location=New York |isbn=978-0-375-42221-8 }} Easier.
	* {{cite book |title=The Anthropic Cosmological Principle |last1=Barrow |first1=John D. |author1-link=John D. Barrow |last2=Tipler |first2 = Frank J. |year=1986 |publisher=Claredon Press |location=Oxford |isbn = 978-0-19-851949-2 |title-link=Anthropic Principle#The Anthropic Cosmological Principle |author2-link=Frank J. Tipler }} Harder.
	* {{cite book |title=The Road to Reality |last=Penrose |first=Roger |author-link=Roger Penrose |year=2005 |publisher=Alfred A. Knopf |location=New York |isbn=978-0-679-45443-4 |section = Section 31.1 |nopp=true |title-link=The Road to Reality }}
	* {{cite journal |last=Planck |first=Max |author-link=Max Planck |year=1899 |title=Über irreversible Strahlungsvorgänge |journal=Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin |volume=5 |pages=440–480 |url = http://bibliothek.bbaw.de/bibliothek-digital/digitalequellen/schriften/anzeige/index_html?band=10-sitz/1899-1&seite:int=454 |language = de }} pp.&nbsp;478–80 contain the first appearance of the Planck base units other than the ], and of ], which Planck denoted by ''b''. ''a'' and ''f'' in this paper correspond to '']'' and '']'' in this entry.
	* {{cite book |last=Tomilin |first=K. A. |title=Natural Systems of Units: To the Centenary Anniversary of the Planck System |pages=287–296 |year=1999 |url = http://dbserv.ihep.su/~pubs/tconf99/ps/tomil.pdf |series=Proceedings Of The XXII Workshop On High Energy Physics And Field Theory |url-status=dead |archive-url = https://web.archive.org/web/20060617063055/http://dbserv.ihep.su/~pubs/tconf99/ps/tomil.pdf |archive-date=17 June 2006 }}
	{{refend}}
	== External links ==		== External links ==
	* , including the Planck base units, as reported by the ] (NIST).		* , including the Planck units, as reported by the ] (NIST).
			* from 'Einstein Light' at UNSW
	* Sections C-E of bear on Planck units. As of 2011, those pages had been removed from the planck.org web site. Use the to access pre-2011 versions of the website. Good discussion of why 8{{pi}}''G'' should be normalized to 1 when doing ] and ]. Many links.
	* (up to 10<sup>−43</sup> seconds after ] of ]) (]).
	* offers a different set of Planck units and defines 31 physical constants in terms of them.
	*
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Latest revision as of 22:16, 5 December 2024

Units defined only by physical constants

In particle physics and physical cosmology, Planck units are a system of units of measurement defined exclusively in terms of four universal physical constants: c, G, ħ, and k_B (described further below). Expressing one of these physical constants in terms of Planck units yields a numerical value of 1. They are a system of natural units, defined using fundamental properties of nature (specifically, properties of free space) rather than properties of a chosen prototype object. Originally proposed in 1899 by German physicist Max Planck, they are relevant in research on unified theories such as quantum gravity.

The term Planck scale refers to quantities of space, time, energy and other units that are similar in magnitude to corresponding Planck units. This region may be characterized by particle energies of around 10 GeV or 10 J, time intervals of around 5×10 s and lengths of around 10 m (approximately the energy-equivalent of the Planck mass, the Planck time and the Planck length, respectively). At the Planck scale, the predictions of the Standard Model, quantum field theory and general relativity are not expected to apply, and quantum effects of gravity are expected to dominate. One example is represented by the conditions in the first 10 seconds of our universe after the Big Bang, approximately 13.8 billion years ago.

The four universal constants that, by definition, have a numeric value 1 when expressed in these units are:

• c, the speed of light in vacuum,
• G, the gravitational constant,
• ħ, the reduced Planck constant, and
• k_B, the Boltzmann constant.

Variants of the basic idea of Planck units exist, such as alternate choices of normalization that give other numeric values to one or more of the four constants above.

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 and associated units may be selected, in terms of which other quantities and coherent units may be expressed. The Planck unit of length has become known as the Planck length, and the Planck unit of time is known as the Planck time, but this nomenclature has not been established as extending to all quantities.

All Planck units are derived from the dimensional universal physical constants that define the system, and in a convention in which these units are omitted (i.e. treated as having the dimensionless value 1), these constants are then eliminated from equations of physics in which they appear. For example, Newton's law of universal gravitation, $${\displaystyle F=G{\frac {m_{1}m_{2}}{r^{2}}}=\left({\frac {F_{\text{P}}l_{\text{P}}^{2}}{m_{\text{P}}^{2}}}\right){\frac {m_{1}m_{2}}{r^{2}}},}$$ 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 quantities, but the second equation, with G absent, 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 each physical quantity is the corresponding ratio with a coherent Planck unit (or "expressed in 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 with F′, m₁′, m₂′, and r′ being the dimensionless ratio quantities corresponding to the standard quantities, written e.g. F′ ≘ F or F′ = F/F_P, but not as a direct equality of quantities. This may seem to be "setting the constants c, G, etc., to 1" if the correspondence of the quantities is thought of as equality. For this reason, Planck or other 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."

History and definition

The concept of natural units was introduced in 1874, when George Johnstone Stoney, noting that electric charge is quantized, derived units of length, time, and mass, now named Stoney units in his honor. Stoney chose his units so that G, c, and the electron charge e would be numerically equal to 1. In 1899, one year before the advent of quantum theory, Max Planck introduced what became later known as the Planck constant. At the end of the paper, he proposed the base units that were later named in his honor. The Planck units are based on the quantum of action, now usually known as the Planck constant, which appeared in the Wien approximation for black-body radiation. Planck underlined the universality of the new unit system, writing:

... die Möglichkeit gegeben ist, Einheiten für Länge, Masse, Zeit und Temperatur aufzustellen, welche, unabhängig von speciellen Körpern oder Substanzen, ihre Bedeutung für alle Zeiten und für alle, auch ausserirdische und aussermenschliche Culturen nothwendig behalten und welche daher als »natürliche Maasseinheiten« bezeichnet werden können.

... it is possible to set up units for length, mass, time and temperature, which are independent of special bodies or substances, necessarily retaining their meaning for all times and for all civilizations, including extraterrestrial and non-human ones, which can be called "natural units of measure".

Planck considered only the units based on the universal constants ${\displaystyle G}$, ${\displaystyle h}$, ${\displaystyle c}$, and ${\displaystyle k_{\rm {B}}}$ to arrive at natural units for length, time, mass, and temperature. His definitions differ from the modern ones by a factor of ${\displaystyle {\sqrt {2\pi }}}$, because the modern definitions use ${\displaystyle \hbar }$ rather than ${\displaystyle h}$.

Table 1: Modern values for Planck's original choice of quantities

Name	Dimension	Expression	Value (SI units)
Planck length	length (L)	${\displaystyle l_{\text{P}}={\sqrt {\frac {\hbar G}{c^{3}}}}}$	1.616255(18)×10 m‍
Planck mass	mass (M)	${\displaystyle m_{\text{P}}={\sqrt {\frac {\hbar c}{G}}}}$	2.176434(24)×10 kg‍
Planck time	time (T)	${\displaystyle t_{\text{P}}={\sqrt {\frac {\hbar G}{c^{5}}}}}$	5.391247(60)×10 s‍
Planck temperature	temperature (Θ)	${\displaystyle T_{\text{P}}={\sqrt {\frac {\hbar c^{5}}{Gk_{\text{B}}^{2}}}}}$	1.416784(16)×10 K‍

Unlike the case with the International System of Units, there is no official entity that establishes a definition of a Planck unit system. Some authors define the base Planck units to be those of mass, length and time, regarding an additional unit for temperature to be redundant. Other tabulations add, in addition to a unit for temperature, a unit for electric charge, so that either the Coulomb constant ${\displaystyle k_{\text{e}}}$ or the vacuum permittivity ${\displaystyle \epsilon _{0}}$ is normalized to 1. Thus, depending on the author's choice, this charge unit is given by $${\displaystyle q_{\text{P}}={\sqrt {4\pi \epsilon _{0}\hbar c}}\approx 1.875546\times 10^{-18}{\text{ C}}\approx 11.7\ e}$$ for ${\displaystyle k_{\text{e}}=1}$, or $${\displaystyle q_{\text{P}}={\sqrt {\epsilon _{0}\hbar c}}\approx 5.290818\times 10^{-19}{\text{ C}}\approx 3.3\ e}$$ for ${\displaystyle \varepsilon _{0}=1}$. Some of these tabulations also replace mass with energy when doing so. In SI units, the values of c, h, e and k_B are exact and the values of ε₀ and G in SI units respectively have relative uncertainties of 1.6×10‍ and 2.2×10. Hence, the uncertainties in the SI values of the Planck units derive almost entirely from uncertainty in the SI value of G.

Compared to Stoney units, Planck base units are all larger by a factor ${\textstyle {\sqrt {{1}/{\alpha }}}\approx 11.7}$, where ${\displaystyle \alpha }$ is the fine-structure constant.

Derived units

In any system of measurement, units for many physical quantities can be derived from base units. Table 2 offers a sample of derived Planck units, some of which 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 2: Coherent derived units of Planck units

Derived unit of	Expression	Approximate SI equivalent
area (L)	${\displaystyle l_{\text{P}}^{2}={\frac {\hbar G}{c^{3}}}}$	2.6121×10 m
volume (L)	${\displaystyle l_{\text{P}}^{3}=\left({\frac {\hbar G}{c^{3}}}\right)^{\frac {3}{2}}={\sqrt {\frac {(\hbar G)^{3}}{c^{9}}}}}$	4.2217×10 m
momentum (LMT)	${\displaystyle m_{\text{P}}c={\frac {\hbar }{l_{\text{P}}}}={\sqrt {\frac {\hbar c^{3}}{G}}}}$	6.5249 kg⋅m/s
energy (LMT)	${\displaystyle E_{\text{P}}=m_{\text{P}}c^{2}={\frac {\hbar }{t_{\text{P}}}}={\sqrt {\frac {\hbar c^{5}}{G}}}}$	1.9561×10 J
force (LMT)	${\displaystyle F_{\text{P}}={\frac {E_{\text{P}}}{l_{\text{P}}}}={\frac {\hbar }{l_{\text{P}}t_{\text{P}}}}={\frac {c^{4}}{G}}}$	1.2103×10 N
density (LM)	${\displaystyle \rho _{\text{P}}={\frac {m_{\text{P}}}{l_{\text{P}}^{3}}}={\frac {\hbar t_{\text{P}}}{l_{\text{P}}^{5}}}={\frac {c^{5}}{\hbar G^{2}}}}$	5.1550×10 kg/m
acceleration (LT)	${\displaystyle a_{\text{P}}={\frac {c}{t_{\text{P}}}}={\sqrt {\frac {c^{7}}{\hbar G}}}}$	5.5608×10 m/s

Some Planck units, such as of time and length, are many orders of magnitude too large or too small to be of practical use, so that Planck units as a system are typically only relevant to theoretical physics. In some cases, a Planck unit may suggest a limit to a range of a physical quantity where present-day theories of physics apply. For example, our understanding of the Big Bang does not extend to the Planck epoch, i.e., when the universe was less than one Planck time old. Describing the universe during the Planck epoch requires a theory of quantum gravity that would incorporate quantum effects into general relativity. Such a theory does not yet exist.

Several quantities are not "extreme" in magnitude, such as the Planck mass, which is about 22 micrograms: very large in comparison with subatomic particles, and within the mass range of living organisms. Similarly, the related units of energy and of momentum are in the range of some everyday phenomena.

Significance

Planck units have little anthropocentric arbitrariness, but do still involve some arbitrary choices in terms of the defining constants. 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. Consequently, natural units help physicists to reframe questions. Frank Wilczek puts it succinctly:

We see that the question is not, "Why is gravity so feeble?" but rather, "Why is the proton's mass so small?" For in natural (Planck) units, the strength of gravity simply is what it is, a primary quantity, while the proton's mass is the tiny number 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 with oranges, because mass and electric charge are incommensurable quantities. Rather, the disparity of magnitude of force is a manifestation of that the proton charge is approximately the unit charge but the proton mass is far less than the unit mass in a system that treats both forces as having the same form.

When Planck proposed his units, the goal was only that of establishing a universal ("natural") way of measuring objects, without giving any special meaning to quantities that measured one single unit. In 1918, Arthur Eddington suggested that the Planck length could have a special significance for understanding gravitation, but this suggestion was not influential. During the 1950s, multiple authors including Lev Landau and Oskar Klein argued that quantities on the order of the Planck scale indicated the limits of the validity of quantum field theory. John Archibald Wheeler proposed in 1955 that quantum fluctuations of spacetime become significant at the Planck scale, though at the time he was unaware of Planck's unit system. In 1959, C. A. Mead showed that distances that measured of the order of one Planck length, or, similarly, times that measured of the order of Planck time, did carry special implications related to Heisenberg's uncertainty principle:

An analysis of the effect of gravitation on hypothetical experiments indicates that it is impossible to measure the position of a particle with error less than 𝛥⁢𝑥 ≳ √𝐺 = 1.6 × 10 cm, where 𝐺 is the gravitational constant in natural units. A similar limitation applies to the precise synchronization of clocks.

Planck scale

In particle physics and physical cosmology, the Planck scale is an energy scale around 1.22×10 eV (the Planck energy, corresponding to the energy equivalent of the Planck mass, 2.17645×10 kg) at which quantum effects of gravity become significant. At this scale, present descriptions and theories of sub-atomic particle interactions in terms of quantum field theory break down and become inadequate, due to the impact of the apparent non-renormalizability of gravity within current theories.

Relationship to gravity

At the Planck length scale, the strength of gravity is expected to become comparable with the other forces, and it has been theorized that all the fundamental forces are unified at that scale, but the exact mechanism of this unification remains unknown. The Planck scale is therefore the point at which the effects of quantum gravity can no longer be ignored in other fundamental interactions, where current calculations and approaches begin to break down, and a means to take account of its impact is necessary. On these grounds, it has been speculated that it may be an approximate lower limit at which a black hole could be formed by collapse.

While physicists have a fairly good understanding of the other fundamental interactions of forces on the quantum level, gravity is problematic, and cannot be integrated with quantum mechanics at very high energies using the usual framework of quantum field theory. At lesser energy levels it is usually ignored, while for energies approaching or exceeding the Planck scale, a new theory of quantum gravity is necessary. Approaches to this problem include string theory and M-theory, loop quantum gravity, noncommutative geometry, and causal set theory.

In cosmology

Main article: Chronology of the universe Further information: Time-variation of fundamental constants

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 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 seconds (or about 10 t_P).

Table 3 lists properties of the observable universe today expressed in Planck units.

Table 3: Today's universe in Planck units

Property of present-day observable universe	Approximate number of Planck units	Equivalents
Age	8.08 × 10 tP	4.35 × 10 s or 1.38 × 10 years
Diameter	5.4 × 10 lP	8.7 × 10 m or 9.2 × 10 light-years
Mass	approx. 10 mP	3 × 10 kg or 1.5 × 10 solar masses (only counting stars) 10 protons (sometimes known as the Eddington number)
Density	1.8 × 10 mP⋅lP	9.9 × 10 kg⋅m
Temperature	1.9 × 10 TP	2.725 K temperature of the cosmic microwave background radiation
Cosmological constant	≈ 10 l P	≈ 10 m
Hubble constant	≈ 10 t P	≈ 10 s ≈ 10 (km/s)/Mpc

After the measurement of the cosmological constant (Λ) in 1998, estimated at 10 in Planck units, it was noted that this is suggestively close to the reciprocal of the age of the universe (T) squared. Barrow and Shaw proposed a modified theory in which Λ is a field evolving in such a way that its value remains Λ ~ T throughout the history of the universe.

Analysis of the units

Planck length

The Planck length, denoted ℓ_P, is a unit of length defined as:$${\displaystyle \ell _{\mathrm {P} }={\sqrt {\frac {\hbar G}{c^{3}}}}}$$It is equal to 1.616255(18)×10 m‍ (the two digits enclosed by parentheses are the estimated standard error associated with the reported numerical value) or about 10 times the diameter of a proton. It can be motivated in various ways, such as considering a particle whose reduced Compton wavelength is comparable to its Schwarzschild radius, though whether those concepts are in fact simultaneously applicable is open to debate. (The same heuristic argument simultaneously motivates the Planck mass.)

The Planck length is a distance scale of interest in speculations about quantum gravity. The Bekenstein–Hawking entropy of a black hole is one-fourth the area of its event horizon in units of Planck length squared. Since the 1950s, it has been conjectured that quantum fluctuations of the spacetime metric might make the familiar notion of distance inapplicable below the Planck length. This is sometimes expressed by saying that "spacetime becomes a foam at the Planck scale". It is possible that the Planck length is the shortest physically measurable distance, since any attempt to investigate the possible existence of shorter distances, by performing higher-energy collisions, would result in black hole production. Higher-energy collisions, rather than splitting matter into finer pieces, would simply produce bigger black holes.

The strings of string theory are modeled to be on the order of the Planck length. In theories with large extra dimensions, the Planck length calculated from the observed value of ${\displaystyle G}$ can be smaller than the true, fundamental Planck length.

Planck time

The Planck time, denoted t_P, is defined as: $${\displaystyle t_{\mathrm {P} }={\frac {\ell _{\mathrm {P} }}{c}}={\sqrt {\frac {\hbar G}{c^{5}}}}}$$This is the time required for light to travel a distance of 1 Planck length in vacuum, which is a time interval of approximately 5.39×10 s. No current physical theory can describe timescales shorter than the Planck time, such as the earliest events after the Big Bang. Some conjectures state that the structure of time need not remain smooth on intervals comparable to the Planck time.

Planck energy

The Planck energy E_P is approximately equal to the energy released in the combustion of the fuel in an automobile fuel tank (57.2 L at 34.2 MJ/L of chemical energy). The ultra-high-energy cosmic ray observed in 1991 had a measured energy of about 50 J, equivalent to about 2.5×10 E_P.

Proposals for theories of doubly special relativity posit that, in addition to the speed of light, an energy scale is also invariant for all inertial observers. Typically, this energy scale is chosen to be the Planck energy.

Planck unit of force

The Planck unit of force may be thought of as the derived unit of force in the Planck system if the Planck units of time, length, and mass are considered to be base units.$${\displaystyle F_{\text{P}}={\frac {m_{\text{P}}c}{t_{\text{P}}}}={\frac {c^{4}}{G}}\approx \mathrm {1.2103\times 10^{44}~N} }$$It is the gravitational attractive force of two bodies of 1 Planck mass each that are held 1 Planck length apart. One convention for the Planck charge is to choose it so that the electrostatic repulsion of two objects with Planck charge and mass that are held 1 Planck length apart balances the Newtonian attraction between them.

Some authors have argued that the Planck force is on the order of the maximum force that can occur between two bodies. However, the validity of these conjectures has been disputed.

Planck temperature

The Planck temperature T_P is 1.416784(16)×10 K. At this temperature, the wavelength of light emitted by thermal radiation reaches the Planck length. There are no known physical models able to describe temperatures greater than T_P; a quantum theory of gravity would be required to model the extreme energies attained. Hypothetically, a system in thermal equilibrium at the Planck temperature might contain Planck-scale black holes, constantly being formed from thermal radiation and decaying via Hawking evaporation. Adding energy to such a system might decrease its temperature by creating larger black holes, whose Hawking temperature is lower.

Nondimensionalized equations

Physical quantities that have different dimensions (such as time and length) cannot be equated even if they are numerically equal (e.g., 1 second is not the same as 1 metre). In theoretical physics, however, this scruple may be set aside, by a process called nondimensionalization. The effective result is that many fundamental equations of physics, which often include some of the constants used to define Planck units, become equations where these constants are replaced by a 1.

Examples include the energy–momentum relation ${\displaystyle E^{2}=(mc^{2})^{2}+(pc)^{2}}$ (which becomes ${\displaystyle E^{2}=m^{2}+p^{2}}$) and the Dirac equation ${\displaystyle (i\hbar \gamma ^{\mu }\partial _{\mu }-mc)\psi =0}$ (which becomes ${\displaystyle (i\gamma ^{\mu }\partial _{\mu }-m)\psi =0}$).

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 in three-dimensional space, the surface area of a sphere of radius r is 4πr. 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 objects have spherical symmetry, and so the electric flux through a sphere of radius r around a point charge will be distributed uniformly over that sphere. From this, it follows that a factor of 4πr will appear in the denominator of Coulomb's law in rationalized form. (Both the numerical factor and the power of the dependence on r would change if space were higher-dimensional; the correct expressions can be deduced from the geometry of higher-dimensional spheres.) Likewise for Newton's law of universal gravitation: a factor of 4π naturally appears in Poisson's equation when relating the gravitational potential to the distribution of matter.

Hence a substantial body of physical theory developed since Planck's 1899 paper suggests normalizing not G but 4πG (or 8πG) to 1. Doing so would introduce a factor of ⁠1/4π⁠ (or ⁠1/8π⁠) 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". 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 = 4πG = ħ = ε₀ = k_B = 1.

There are several possible alternative normalizations.

Gravitational constant

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 (and therefore setting G = ⁠1/4π⁠):
  • Gauss's law for gravity becomes Φ_g = −M (rather than Φ_g = −4πM in Planck units).
  • Eliminates 4πG from the Poisson equation.
  • Eliminates 4πG in the gravitoelectromagnetic (GEM) equations, which hold in weak gravitational fields or locally flat spacetime. 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 Z_g 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).
• Setting 8πG = 1 (and therefore setting G = ⁠1/8π⁠). 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 = 2πA_BH.

See also

• cGh physics
• Dimensional analysis
• Doubly special relativity
• Trans-Planckian problem
• Zero-point energy

Explanatory notes

1. For example, both Frank Wilczek and Barton Zwiebach do so, as does the textbook Gravitation.
2. General relativity predicts that gravitational radiation propagates at the same speed as electromagnetic radiation.

References

1. ^ Wilczek, Frank (2005). "On Absolute Units, I: Choices". Physics Today. 58 (10). American Institute of Physics: 12–13. Bibcode:2005PhT....58j..12W. doi:10.1063/1.2138392.
2. ^ Misner, Charles W.; Thorne, Kip S.; Wheeler, John A. (1973). Gravitation. New York. ISBN 0-7167-0334-3. OCLC 585119.{{cite book}}: CS1 maint: location missing publisher (link)
3. Wesson, P. S. (1980). "The application of dimensional analysis to cosmology". Space Science Reviews. 27 (2): 117. Bibcode:1980SSRv...27..109W. doi:10.1007/bf00212237. S2CID 120784299.
4. Barrow, J. D. (1 March 1983). "Natural Units Before Planck". Quarterly Journal of the Royal Astronomical Society. 24: 24. Bibcode:1983QJRAS..24...24B. ISSN 0035-8738. Archived from the original on 20 January 2022. Retrieved 16 April 2022.
5. ^ Planck, Max (1899). "Über irreversible Strahlungsvorgänge". Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin (in German). 5: 440–480. Archived from the original on 17 November 2020. Retrieved 23 May 2020. pp. 478–80 contain the first appearance of the Planck base units, and of the Planck constant, which Planck denoted by b. a and f in this paper correspond to the k and G in this article.
6. ^ 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 12 December 2020. Retrieved 31 December 2019.
7. ^ "2022 CODATA Value: Planck length". The NIST Reference on Constants, Units, and Uncertainty. NIST. May 2024. Retrieved 18 May 2024.
8. "2022 CODATA Value: Planck mass". The NIST Reference on Constants, Units, and Uncertainty. NIST. May 2024. Retrieved 18 May 2024.
9. "2022 CODATA Value: Planck time". The NIST Reference on Constants, Units, and Uncertainty. NIST. May 2024. Retrieved 18 May 2024.
10. ^ "2022 CODATA Value: Planck temperature". The NIST Reference on Constants, Units, and Uncertainty. NIST. May 2024. Retrieved 18 May 2024.
11. ^ Zwiebach, Barton (2004). A First Course in String Theory. Cambridge University Press. ISBN 978-0-521-83143-7. OCLC 58568857.
12. Deza, Michel Marie; Deza, Elena (2016). Encyclopedia of Distances. Springer. p. 602. ISBN 978-3662528433. Archived from the original on 6 March 2021. Retrieved 9 September 2020.
13. Elert, Glenn. "Blackbody Radiation". The Physics Hypertextbook. Archived from the original on 3 March 2021. Retrieved 22 February 2021.
14. Pavšic, Matej (2001). The Landscape of Theoretical Physics: A Global View. Fundamental Theories of Physics. Vol. 119. Dordrecht: Kluwer Academic. pp. 347–352. arXiv:gr-qc/0610061. doi:10.1007/0-306-47136-1. ISBN 978-0-7923-7006-2. Archived from the original on 5 September 2021. Retrieved 31 December 2019.
15. ^ Zeidler, Eberhard (2006). Quantum Field Theory I: Basics in Mathematics and Physics (PDF). Springer. p. 953. ISBN 978-3540347620. Archived (PDF) from the original on 19 June 2020. Retrieved 31 May 2020.
16. "2022 CODATA Value: vacuum electric permittivity". The NIST Reference on Constants, Units, and Uncertainty. NIST. May 2024. Retrieved 18 May 2024.
17. "2022 CODATA Value: Newtonian constant of gravitation". The NIST Reference on Constants, Units, and Uncertainty. NIST. May 2024. Retrieved 18 May 2024.
18. Ray, T. P. (1981). "Stoney's Fundamental Units". Irish Astronomical Journal. 15: 152. Bibcode:1981IrAJ...15..152R.
19. ^ Zee, Anthony (2010). Quantum Field Theory in a Nutshell (second ed.). Princeton University Press. pp. 172, 434–435. ISBN 978-0-691-14034-6. OCLC 659549695. Just as in our discussion of the Fermi theory, the nonrenormalizability of quantum gravity tells us that at the Planck energy scale ... new physics must appear. Fermi's theory cried out, and the new physics turned out to be the electroweak theory. Einstein's theory is now crying out.
20. Penrose, Roger (2005). The Road to Reality. New York: Alfred A. Knopf. ISBN 978-0-679-45443-4.
21. 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.
22. ^ Gorelik, Gennady (1992). "First Steps of Quantum Gravity and the Planck Values". In Eisenstaedt, Jean; Kox, Anne J. (eds.). Studies in the History of General Relativity: Based on the proceedings of the 2nd International Conference on the History of General Relativity, Luminy, France, 1988. Boston: Birkhäuser. pp. 364–379. ISBN 0-8176-3479-7. OCLC 24011430. Archived from the original on 25 April 2019.
23. Stachel, John (1986). "Eddington and Einstein". In Ullmann-Margalit, Edna (ed.). The Prism of Science. Boston Studies in the Philosophy of Science. Vol. 95. Springer. pp. 225–250. doi:10.1007/978-94-009-4566-1_18. ISBN 978-90-277-2161-7.
24. ^ Wheeler, J. A. (January 1955). "Geons". Physical Review. 97 (2): 511–536. Bibcode:1955PhRv...97..511W. doi:10.1103/PhysRev.97.511.
25. Mead, Chester Alden (10 August 1964). "Possible Connection Between Gravitation and Fundamental Length". Physical Review. 135 (3B). American Physical Society: B849 – B862. Bibcode:1964PhRv..135..849M. doi:10.1103/PhysRev.135.B849.
26. Witten, Ed (2002). "Quest For Unification". arXiv:hep-ph/0207124.
27. Bingham, Robert (4 October 2006). "Can experiment access Planck-scale physics?". CERN Courier. Archived from the original on 30 November 2020. Retrieved 4 November 2021.
28. Hawking, Stephen W. (1975). "Particle Creation by Black Holes". Communications in Mathematical Physics. 43 (3): 199–220. Bibcode:1975CMaPh..43..199H. doi:10.1007/BF02345020. S2CID 55539246. Archived from the original on 5 July 2014. Retrieved 20 March 2022.
29. Rovelli, Carlo (2008). "Quantum gravity". Scholarpedia. 3 (5): 7117. Bibcode:2008SchpJ...3.7117R. doi:10.4249/scholarpedia.7117.
30. ^ Schombert, James. "Birth of the Universe". HC 441: Cosmology. University of Oregon. Archived from the original on 28 November 2018. Retrieved 20 March 2022. Discusses "Planck time" and "Planck era" at the very beginning of the Universe
31. Kolb, Edward W.; Turner, Michael S. (1994). The Early Universe. Basic Books. p. 447. ISBN 978-0-201-62674-2. Archived from the original on 6 March 2021. Retrieved 10 April 2010.
32. ^ Barrow, John D. (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.
33. Barrow, John D.; Tipler, Frank J. (1986). The Anthropic Cosmological Principle (1st ed.). Oxford University Press. ISBN 978-0-19-282147-8. LCCN 87028148.
34. 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. S2CID 55125081.
35. ^ Baez, John (2001). "Higher-Dimensional Algebra and Planck-Scale Physics". In Callender, Craig; Huggett, Nick (eds.). Physics Meets Philosophy at the Planck Scale. Cambridge University Press. pp. 172–195. ISBN 978-0-521-66280-2. OCLC 924701824. Archived from the original on 8 July 2021. Retrieved 20 March 2022.
36. ^ Adler, Ronald J. (2010). "Six easy roads to the Planck scale". American Journal of Physics. 78 (9): 925–932. arXiv:1001.1205. Bibcode:2010AmJPh..78..925A. doi:10.1119/1.3439650. S2CID 55181581.
37. Siegel, Ethan (26 June 2019). "What Is The Smallest Possible Distance In The Universe?". Starts with a Bang. Forbes. Archived from the original on 18 September 2021. Retrieved 26 June 2019.
38. Faraoni, Valerio (November 2017). "Three new roads to the Planck scale". American Journal of Physics. 85 (11): 865–869. arXiv:1705.09749. Bibcode:2017AmJPh..85..865F. doi:10.1119/1.4994804. ISSN 0002-9505. S2CID 119022491. Archived from the original on 30 December 2017. Retrieved 9 April 2022. Like all orders of magnitude estimates, this procedure is not rigorous since it extrapolates the concepts of black hole and of Compton wavelength to a new regime in which both concepts would probably lose their accepted meanings and would, strictly speaking, cease being valid. However, this is how one gains intuition into a new physical regime.
39. Regge, T. (1 January 1958). "Gravitational fields and quantum mechanics". Il Nuovo Cimento. 7 (2): 215–221. Bibcode:1958NCim....7..215R. doi:10.1007/BF02744199. ISSN 1827-6121. S2CID 123012079. Archived from the original on 24 March 2022. Retrieved 22 March 2022.
40. Mermin, N. David (May 2009). "What's bad about this habit". Physics Today. 62 (5): 8–9. Bibcode:2009PhT....62e...8M. doi:10.1063/1.3141952. ISSN 0031-9228. Archived from the original on 22 March 2022. Retrieved 22 March 2022.
41. Carr, Bernard J.; Giddings, Steven B. (May 2005). "Quantum Black Holes" (PDF). Scientific American. 292 (5): 48–55. Bibcode:2005SciAm.292e..48C. doi:10.1038/scientificamerican0505-48. PMID 15882021. S2CID 10872062. Archived from the original (PDF) on 14 February 2019.
42. Manoukian, Edouard B. (2016). Quantum Field Theory II: Introductions to Quantum Gravity, Supersymmetry and String Theory. Graduate Texts in Physics. Cham: Springer International Publishing. p. 187. doi:10.1007/978-3-319-33852-1. ISBN 978-3-319-33851-4. Archived from the original on 24 March 2022. Retrieved 22 March 2022.
43. Schwarz, John H. (December 2021). "From the S Matrix to String Theory". Geoffrey Chew: Architect of the Bootstrap. World Scientific. pp. 72–83. doi:10.1142/9789811219832_0013. ISBN 978-981-12-1982-5. S2CID 245575026.
44. Hossenfelder, Sabine (December 2013). "Minimal Length Scale Scenarios for Quantum Gravity". Living Reviews in Relativity. 16 (1): 2. arXiv:1203.6191. Bibcode:2013LRR....16....2H. doi:10.12942/lrr-2013-2. ISSN 2367-3613. PMC 5255898. PMID 28179841.
45. Wendel, Garrett; Martínez, Luis; Bojowald, Martin (19 June 2020). "Physical Implications of a Fundamental Period of Time". Physical Review Letters. 124 (24): 241301. arXiv:2005.11572. Bibcode:2020PhRvL.124x1301W. doi:10.1103/PhysRevLett.124.241301. PMID 32639827. S2CID 218870394.
46. "HiRes – The High Resolution Fly's Eye Ultra High Energy Cosmic Ray Observatory". www.cosmic-ray.org. Archived from the original on 15 August 2009. Retrieved 21 December 2016.
47. Bird, D. J.; Corbato, S. C.; Dai, H. Y.; Elbert, J. W.; Green, K. D.; Huang, M. A.; Kieda, D. B.; Ko, S.; Larsen, C. G.; Loh, E. C.; Luo, M. Z.; Salamon, M. H.; Smith, J. D.; Sokolsky, P.; Sommers, P.; Tang, J. K. K.; Thomas, S. B. (March 1995). "Detection of a cosmic ray with measured energy well beyond the expected spectral cutoff due to cosmic microwave radiation". The Astrophysical Journal. 441: 144. arXiv:astro-ph/9410067. Bibcode:1995ApJ...441..144B. doi:10.1086/175344. S2CID 119092012.
48. Judes, Simon; Visser, Matt (4 August 2003). "Conservation laws in "doubly special relativity"". Physical Review D. 68 (4): 045001. arXiv:gr-qc/0205067. Bibcode:2003PhRvD..68d5001J. doi:10.1103/PhysRevD.68.045001. ISSN 0556-2821. S2CID 119094398.
49. Hossenfelder, Sabine (9 July 2014). "The Soccer-Ball Problem". Symmetry, Integrability and Geometry: Methods and Applications. 10: 74. arXiv:1403.2080. Bibcode:2014SIGMA..10..074H. doi:10.3842/SIGMA.2014.074. S2CID 14373748. Archived from the original on 19 March 2022. Retrieved 16 April 2022.
50. Kiefer, Claus (2012). Quantum Gravity. International series of monographs on physics. Vol. 155. Oxford University Press. p. 5. ISBN 978-0-191-62885-6. OCLC 785233016.
51. de Sabbata, Venzo; Sivaram, C. (1993). "On limiting field strengths in gravitation". Foundations of Physics Letters. 6 (6): 561–570. Bibcode:1993FoPhL...6..561D. doi:10.1007/BF00662806. S2CID 120924238.
52. Gibbons, G. W. (2002). "The Maximum Tension Principle in General Relativity". Foundations of Physics. 32 (12): 1891–1901. arXiv:hep-th/0210109. Bibcode:2002FoPh...32.1891G. doi:10.1023/A:1022370717626. S2CID 118154613.
53. Jowsey, Aden; Visser, Matt (3 February 2021). "Counterexamples to the maximum force conjecture". Universe. 7 (11): 403. arXiv:2102.01831. Bibcode:2021Univ....7..403J. doi:10.3390/universe7110403.
54. Afshordi, Niayesh (1 March 2012). "Where will Einstein fail? Leasing for Gravity and cosmology". Bulletin of the Astronomical Society of India. 40 (1). Astronomical Society of India, NASA Astrophysics Data System: 5. arXiv:1203.3827. Bibcode:2012BASI...40....1A. OCLC 810438317. However, for most experimental physicists, approaching energies comparable to Planck energy is little more than a distant fantasy. The most powerful accelerators on Earth miss Planck energy of 15 orders of magnitude, while ultra high energy cosmic rays are still 9 orders of magnitude short of M_p.
55. Reeves, Hubert (1991). The Hour of Our Delight. W. H. Freeman Company. p. 117. ISBN 978-0-7167-2220-5. The point at which our physical theories run into most serious difficulties is that where matter reaches a temperature of approximately 10 degrees, also known as Planck's temperature. The extreme density of radiation emitted at this temperature creates a disproportionately intense field of gravity. To go even farther back, a quantum theory of gravity would be necessary, but such a theory has yet to be written.
56. Shor, Peter W. (17 July 2018). "Scrambling Time and Causal Structure of the Photon Sphere of a Schwarzschild Black Hole". arXiv:1807.04363 .
57. Sorkin, Rafael (1983). "Kaluza-Klein Monopole". Phys. Rev. Lett. 51 (2): 87–90. Bibcode:1983PhRvL..51...87S. doi:10.1103/PhysRevLett.51.87.
58. Rañada, Antonio F. (31 October 1995). "A Model of Topological Quantization of the Electromagnetic Field". In M. Ferrero; Alwyn van der Merwe (eds.). Fundamental Problems in Quantum Physics. Springer. p. 271. ISBN 9780792336709. Archived from the original on 1 September 2020. Retrieved 16 January 2018.
59. Choquet-Bruhat, Yvonne (2009). General Relativity and the Einstein Equations. Oxford: Oxford University Press. ISBN 978-0-19-155226-7. OCLC 317496332.
60. Stavrov, Iva (2020). Curvature of Space and Time, with an Introduction to Geometric Analysis. Providence, Rhode Island: American Mathematical Society. ISBN 978-1-4704-6313-7. OCLC 1202475208.

External links

• Value of the fundamental constants, including the Planck units, as reported by the National Institute of Standards and Technology (NIST).
• The Planck scale: relativity meets quantum mechanics meets gravity from 'Einstein Light' at UNSW

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