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{{Short description|Units defined only by physical constants}}
{{pp-sock|small=yes}}
{{Use dmy dates|date=May 2019}}

In [[particle physics]] and [[physical cosmology]], '''Planck units''' are a [[system of units of measurement]] defined exclusively in terms of four universal [[physical constant]]s: ''[[Speed of light|c]]'', ''[[Gravitational constant|G]]'', ''[[Reduced Planck constant|ħ]]'', and [[Boltzmann constant|''k''<sub>B</sub>]] (described further below). Expressing one of these physical constants in terms of Planck units yields a numerical value of [[1 (number)|1]]. They are a system of [[natural units]], defined using fundamental properties of [[Nature#Matter and energy|nature]] (specifically, properties of [[vacuum|free space]]) rather than properties of a chosen [[Prototype (metrology)|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 [[energy|energies]] of around {{val|e=19|u=GeV}} or {{val|e=9|u=J}}, [[time]] intervals of around {{val|5|e=−44|u=s}} and [[length]]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 [[Standard Model]], [[quantum field theory]] and [[general relativity]] are not expected to apply, and [[Quantum Gravity|quantum effects of gravity]] are expected to dominate. One example is represented by the conditions in the [[Chronology of the universe|first 10<sup>−43</sup> seconds]] of our universe after the [[Big Bang]], approximately 13.8 billion years ago.

The four [[universal constant]]s 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''<sub>B</sub>, 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 unit (measurement)|base units]], from which all other quantities and units may be derived. In the [[International System of Units]], for example, the [[SI base quantity|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.<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.

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]],
<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:
<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 [[dimensional analysis|dimensionally consistent]] and equally valid in ''any'' system of quantities, but the second equation, with {{mvar|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:
<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}}}}", [[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."<ref>{{cite journal |last1=Wesson |first1=P. S. |author-link=Paul S. Wesson |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 |s2cid=120784299}}</ref>

== 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 [[elementary charge|electron charge]] ''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, [[Max Planck]] 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 [[Planck constant]], which Planck denoted by ''b''. ''a'' and ''f'' in this paper correspond to the [[Boltzmann constant|''k'']] and [[gravitation constant|''G'']] 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 [[Action (physics)|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:<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.}}}}
{{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 [[length]], [[time]], [[mass]], and [[temperature]].<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" />

{| class="wikitable" style="margin:1em auto 1em auto; background:#fff;"
|+Table 1: Modern values for Planck's original choice of quantities
|-
 ! Name
 ! Dimension
 ! Expression
 ! Value ([[International System of Units|SI]] units)
|- style="text-align:left;"
 | Planck length
 | [[length]] (L)
 | <math>l_\text{P} = \sqrt{\frac{\hbar G}{c^3}}</math>
 | {{physconst|lP}}
|-
 | Planck mass
 | [[mass]] (M)
 | <math>m_\text{P} = \sqrt{\frac{\hbar c}{G}}</math>
 | {{physconst|mP}}
|-
 | Planck time
 | [[time]] (T)
 | <math>t_\text{P} = \sqrt{\frac{\hbar G}{c^5}}</math>
 | {{physconst|tP}}
|-
 | Planck temperature
 | [[temperature]] (Θ)
 | <math>T_\text{P} = \sqrt{\frac{\hbar c^5}{G k_\text{B}^2}}</math>
 | {{physconst|TP}}
|}

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.{{NoteTag|For example, both [[Frank Wilczek]] and [[Barton Zwiebach]] do so,<ref name=":2">
{{cite journal
 | last = Wilczek | first = Frank | author-link = Frank Wilczek
 | date = 2005
 | title = On Absolute Units, I: Choices
 | journal = [[Physics Today]]
 | volume = 58
 | issue = 10
 | pages = 12–13
 | doi = 10.1063/1.2138392
 | publisher = [[American Institute of Physics]]
 | 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 ''[[Gravitation (book)|Gravitation]]''.<ref name="Gravitation"/>{{rp|1215}}}} Other tabulations add, in addition to a unit for temperature, a unit for electric charge, so that the [[Coulomb constant]] <math>k_\text{e}</math> is added to the list of constants used by Planck.<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 = [[Springer Science+Business Media|Springer]]
 | 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><ref>
{{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 = [[Springer Science+Business Media|Springer]]
 | 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>
Thus, this charge unit is typically 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>

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 [[Stoney units]], 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 [[fine-structure constant]].<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>

== 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.

{| class="wikitable" style="margin:1em auto 1em auto; background:#fff;"
|+ Table 2: Coherent derived units of Planck units
|-
 ! Derived unit of
 ! Expression
 ! Approximate [[SI]] equivalent
|- style="text-align:left;"
 | [[area]] (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;"
 | [[volume]] (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}}
|-
 | [[momentum]] (LMT<sup>−1</sup>)
 | <math>m_\text{P} c = \frac{\hbar}{l_\text{P}} = \sqrt{\frac{\hbar c^3}{G}} </math>
 | {{val|6.5249|u=[[Newton-second|kg⋅m/s]]}}
|-
 | [[energy]] (L<sup>2</sup>MT<sup>−2</sup>)
 | <math>E_\text{P} = m_\text{P} c^2 = \frac{\hbar}{t_\text{P}} = \sqrt{\frac{\hbar c^5}{G}} </math>
 | {{val|1.9561|e=9|ul=J}}
|-
 | [[force]] (LMT<sup>−2</sup>)
 | <math>F_\text{P} = \frac{E_\text{P}}{l_\text{P}} = \frac{\hbar}{l_\text{P} t_\text{P}} = \frac{c^4}{G} </math>
 | {{val|1.2103|e=44|ul=N}}
|-
 | [[density]] (L<sup>−3</sup>M)
 | <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>
 | {{val|5.1550|e=96|ul=kg/m3}}
|-
 | [[acceleration]] (LT<sup>−2</sup>)
 | <math> a_\text{P} = \frac{c}{t_\text{P}} = \sqrt{\frac{c^7}{\hbar G}}</math>
 | {{val|5.5608|e=51|ul=m/s2}}
|}

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.<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=[[Princeton University Press]]|year=2010|isbn=978-0-691-14034-6|edition=second|pages=[https://archive.org/details/isbn_9780691140346/page/172 172,434&ndash;435]|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 [[Big Bang]] does not extend to the [[Chronology of the universe#Planck epoch|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 [[1 E-8 kg|22 micrograms]]: 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.

== 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 [[SI base unit|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:
{{blockquote|We see that the question [posed] is not, "Why is gravity so feeble?" but rather, "Why is the proton's mass so small?" For in natural (Planck) units, the strength of gravity simply is what it is, a primary quantity, while the proton's mass is the tiny number 1/13  [[Names of large numbers|quintillion]].<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|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 [[apples and oranges|comparing apples with oranges]], because [[mass]] and [[electric charge]] are [[Dimensional analysis|incommensurable]] quantities. Rather, the disparity of magnitude of force is a manifestation of that the [[elementary charge|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. However, 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 [[Werner Heisenberg|Heisenberg]]'s [[uncertainty principle]]:<ref>
{{cite journal
 | last = Mead
 | first = Chester Alden
 | date = 1964-08-10
 | title = Possible Connection Between Gravitation and Fundamental Length
 | journal = [[Physical Review]]
 | 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.}}

== Planck scale ==
In [[particle physics]] and [[physical cosmology]], the Planck scale is an [[energy scale]] around {{val|1.22|e=28|ul=eV}} (the Planck energy, corresponding to the [[mass–energy equivalence|energy equivalent]] of the Planck mass, {{val|2.17645|e=−8|u=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 [[renormalization|non-renormalizability]] of gravity within current theories.<ref name=":1" />

=== 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.<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 [[fundamental interactions]], 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=[[CERN Courier]] |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 [[Micro black hole#Minimum mass of a black hole|approximate lower limit]] at which a black hole could be formed by collapse.<ref name="particlecreate">{{cite journal |author-link=Stephen Hawking |first=S. W. |last=Hawking |title=Particle Creation by Black Holes |journal=[[Communications in Mathematical Physics]] |volume=43 |issue=3 |date=1975 |pages=199–220 |doi=10.1007/BF02345020 |bibcode=1975CMaPh..43..199H |s2cid=55539246 |url=http://projecteuclid.org/euclid.cmp/1103899181 |access-date=20 March 2022 |archive-date=5 July 2014 |archive-url=https://web.archive.org/web/20140705012739/http://projecteuclid.org/euclid.cmp/1103899181 |url-status=live }}</ref>

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 sets|causal set theory]].<ref name="scholarpedia">{{cite journal |last1=Rovelli |first1=Carlo |author-link=Carlo Rovelli |year=2008 |title=Quantum gravity |journal=[[Scholarpedia]] |volume=3 |issue=5 |page=7117 |bibcode=2008SchpJ...3.7117R |doi=10.4249/scholarpedia.7117 |doi-access=free}}</ref>

=== <span class="anchor" id="Cosmology"></span><span class="anchor" id="Planck epoch"></span> In cosmology ===
{{Main|Chronology of the universe}}
{{Further|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 [[cosmic time|time passed]] 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=[[University of Oregon]] |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 "[[Planck era]]" at the very beginning of the [[Universe]]</ref> 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 gravity|quantum effects of gravity]] dominate physical interactions at this time scale. At this scale, the [[grand unification|unified force]] of the [[Standard Model]] is assumed to be [[Theory of everything|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<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>

{| class="wikitable" style="margin:1em auto 1em auto;"
|+Table 3: Today's universe in Planck units
 ! Property of{{br}} present-day [[observable universe]]
 ! Approximate number{{br}} of Planck units
 ! Equivalents
|- style="text-align:left;"
 | [[Age of the universe|Age]]
 | 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
|-
 | [[Observable universe#Size|Diameter]]
 | 5.4 × 10<sup>61</sup> ''l''<sub>P</sub>
 | 8.7 × 10<sup>26</sup> m or 9.2 × 10<sup>10</sup> [[light-years]]
|-
 | [[Mass of the observable universe|Mass]]
 | approx. 10<sup>60</sup> ''m''<sub>P</sub>
 | 3 × 10<sup>52</sup> kg or 1.5 × 10<sup>22</sup> [[solar mass]]es (only counting stars){{br}} [[Observable universe#Matter content|10<sup>80</sup> protons]] (sometimes known as the [[Eddington number]])
|-
 | [[Observable Universe#Estimates based on critical density|Density]]
 | 1.8 × 10<sup>−123</sup> ''m''<sub>P</sub>⋅''l''<sub>P</sub><sup>−3</sup>
 | 9.9 × 10<sup>−27</sup> kg⋅m<sup>−3</sup>
|-
 | [[Cosmic microwave background radiation|Temperature]]
 | 1.9 × 10<sup>−32</sup> ''T''<sub>P</sub>
 | 2.725 K{{br}} temperature of the [[cosmic microwave background radiation]]
|-
 | [[Cosmological constant]]
 | ≈ 10<sup>−122</sup> ''l''{{su|b=P|p=&nbsp;−2|lh=1em}}
 | ≈ 10<sup>−52</sup> m<sup>−2</sup>
|-
 | [[Hubble constant]]
 | ≈ 10<sup>−61</sup> ''t''{{su|b=P|p=&nbsp;−1|lh=1em}}
 | ≈ 10<sup>−18</sup> s<sup>−1</sup> ≈ 10<sup>2</sup> (km/s)/[[parsec|Mpc]]
|}

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 [[age of the universe]] (''T'') squared. Barrow and Shaw proposed a modified theory in which [[Cosmological constant|Λ]] 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=[[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 |s2cid=55125081}}</ref>

=== Analysis of the units ===

==== Planck length ====
The Planck length, denoted {{math|<var>ℓ</var><sub>P</sub>}}, is a unit of [[length]] 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 [[Standard error (statistics)|standard error]] associated with the reported numerical value) or about {{val|e=-20}} times the diameter of a [[proton]].<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 [[reduced Compton wavelength]] is comparable to its [[Schwarzschild radius]],<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=[[American Journal of Physics]] |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=[[Forbes]] |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=[[American Journal of Physics]] |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"/>)

The Planck length is a distance scale of interest in speculations about quantum gravity. The [[Black hole thermodynamics|Bekenstein–Hawking entropy of a black hole]] is one-fourth the area of its [[event horizon]] 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>{{cite journal |last=Wheeler |first=J. A. |author-link=John Archibald Wheeler |date=January 1955 |title=Geons |journal=[[Physical Review]] |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>{{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 [[Quantum foam|foam at the Planck scale]]".<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=[[Physics Today]] |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=[[Scientific American]] |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>

The strings of [[string theory]] 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 [[large extra dimension]]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=[[Living Reviews in Relativity]] |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>

==== Planck time ====
The Planck time {{math|<var>t</var><sub>P</sub>}} is the [[time]] required for [[light]] to travel a distance of 1&nbsp;Planck length in [[vacuum]], 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 = [[Physical Review Letters|Phys. Rev. Lett.]]
 | 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 ====
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 [[ultra-high-energy cosmic ray]] [[Oh-My-God particle|observed in 1991]] 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=[[The Astrophysical 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 [[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.<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>

==== 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.
<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=[[Foundations of Physics Letters]] |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=[[Foundations of Physics]] |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=[[Universe (journal)|Universe]] |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=[[Astronomical Society of India]], NASA [[Astrophysics Data System]]|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>

==== Planck temperature ====
The Planck temperature ''T''<sub>P</sub> is {{physconst|TP|after=.}} 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''<sub>P</sub>; a quantum theory of gravity would be required to model the extreme energies attained.<ref>
{{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 [[Quantum gravity|quantum theory of gravity]] 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 [[thermal equilibrium]] at the Planck temperature might contain Planck-scale black holes, constantly being formed from thermal radiation and decaying via [[Hawking radiation|Hawking evaporation]]. 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 ==
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]] {{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 [[Dirac equation]] {{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 ==
<!-- [[Reduced Planck units]] 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.

The factor 4{{pi}} is ubiquitous in [[theoretical physics]] because in three-dimensional space, the surface area of a [[sphere]] of radius ''r'' is 4{{pi}}''r''{{i sup|2}}. 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 field|gravitational]] and [[electrostatic field]]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 [[Heaviside–Lorentz units#Maxwell's equations|rationalized form]].<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 [[N-sphere|higher-dimensional spheres]].<ref name=":0" />{{rp|page=51}}) Likewise for Newton's law of universal gravitation: a factor of 4{{pi}} naturally appears in [[Poisson's equation]] 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'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 [[gravitoelectromagnetism]] 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|Phys. Rev. Lett.]]|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 {{nowrap|1=''c'' = 4''πG'' = ''ħ'' = ''ε''<sub>0</sub> = ''k''<sub>B</sub> = 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 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 (and therefore setting {{nowrap|1=''G'' = {{sfrac|1|4{{pi}}}}}}):
** [[Gauss's law for gravity]] becomes {{nowrap|1=''Φ''<sub>'''g'''</sub> = −''M''}} (rather than {{nowrap|1=''Φ''<sub>'''g'''</sub> = −4{{pi}}''M''}} in Planck units).
** Eliminates 4{{pi}}''G'' from the [[Poisson equation]].
** Eliminates 4{{pi}}''G'' in the [[Gravitoelectromagnetism|gravitoelectromagnetic]] (GEM) equations, which hold in weak [[gravitational field]]s 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 {{sfrac|4{{pi}}''G''}} replacing ''ε''<sub>0</sub>.
** Normalizes the [[characteristic impedance]] ''Z''<sub>'''g'''</sub> of [[gravitational radiation]] in free space to 1 (normally expressed as {{sfrac|4{{pi}}''G''|''c''}}).{{NoteTag|[[General relativity]] predicts that [[gravitational radiation]] propagates at the same speed as [[electromagnetic radiation]].<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 Bekenstein–Hawking formula (for the [[black hole thermodynamics|entropy of a black hole]] in terms of its mass ''m''<sub>BH</sub> and the area of its [[event horizon]] ''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 [[Einstein field equations]], [[Einstein–Hilbert action]], and the [[Friedmann equations]], 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 [[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>}}.

== See also ==
* [[cGh physics]]
* [[Dimensional analysis]]
* [[Doubly special relativity]]
* [[Trans-Planckian problem]]
* [[Zero-point energy]]

== Explanatory notes ==
{{NoteFoot}}

== References ==
{{reflist}}

== External links ==
* [http://physics.nist.gov/cuu/Constants/index.html Value of the fundamental constants], including the Planck units, as reported by the [[National Institute of Standards and Technology]] (NIST).
* [http://www.phys.unsw.edu.au/einsteinlight/jw/module6_Planck.htm The Planck scale: relativity meets quantum mechanics meets gravity] from 'Einstein Light' at UNSW

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[[Category:Max Planck]]
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