Revision 800040979 of "Planck length" on enwiki

{{Infobox unit
| symbol   = {{math|<var>ℓ</var><sub>P</sub>}}
| standard = [[Planck units]]
| quantity = [[length]]
| units1   = [[SI units]]
| inunits1 = {{val|1.616229|(38)|e=-35|ul=m}}
| units2   = [[natural units]]
| inunits2 = 11.706&nbsp;[[Stoney units|{{math|<var>ℓ</var><sub>S</sub>}}]]<br /><!--
      -->&emsp;{{val|3.0542|e=-25}}&nbsp;[[Bohr radius|''a''<sub>0</sub>]]
| units3   = [[imperial units|imperial]]/[[US customary units|US]]&nbsp;units
| inunits3 = {{convert|1.616229e-35|m|in|disp=out|lk=on|sigfig=5|comma=gaps}}
}}
In [[physics]], the '''Planck length''', denoted {{math|<var>ℓ</var><sub>P</sub>}}, is a unit of [[length]], equal to {{val|1.616229|(38)|e=-35}} [[metre]]s. It is a [[Base unit (measurement)|base unit]] in the system of [[Planck units]], developed by physicist [[Max Planck]]. The Planck length can be defined from three [[physical constant|fundamental physical constants]]: the [[speed of light]] in a vacuum, the [[Planck constant]], and the [[gravitational constant]].

==Value==
The Planck length {{math|<var>ℓ</var><sub>P</sub>}} is defined as

:<math>\ell_\mathrm{P} =\sqrt\frac{\hbar G}{c^3} \approx 1.616\;229 (38) \times 10^{-35}\ \mathrm{m}</math>

where <math>c</math> is the [[speed of light]] in a vacuum, {{mvar|G}} is the [[gravitational constant]], and {{mvar|ħ}} is the [[reduced Planck constant]]. The two digits enclosed by [[Bracket|parentheses]] are the estimated [[standard error (statistics)|standard error]] associated with the reported numerical value.<ref>[[John Baez]], [http://math.ucr.edu/home/baez/planck/node2.html The Planck Length]</ref><ref>[[NIST]], "[http://physics.nist.gov/cgi-bin/cuu/Value?plkl Planck length]", [http://physics.nist.gov/cuu/Constants/index.html NIST's published] [[CODATA]] constants</ref>

The Planck length is about 10<sup>−20</sup> times the diameter of a [[proton]].
It can be defined using the radius of [[Planck particle]]

==Theoretical significance==
Much like the rest of the Planck units, there is currently no proven physical significance of the Planck length. However, it is theoretically considered to be the quantization of space which makes up the fabric of the universe by quantum gravity theorists, also referred to as [[quantum foam]].

The Planck length is believed to be the shortest meaningful length, the limiting distance below which the very notions of space and length  cease to exist. Any attempt to investigate the possible existence of shorter distances, by performing higher-energy collisions, would inevitably result in black hole production. Higher-energy collisions, rather than splitting matter into finer pieces, would simply produce bigger black holes.<ref>[http://www.nature.com/scientificamerican/journal/v292/n5/full/scientificamerican0505-48.html?foxtrotcallback=true Bernard J. Carr & Steven B. Giddings "Quantum Black Holes" Scientific American 292, 48 - 55 (2005)]</ref>

In some forms of [[quantum gravity]], the Planck length is the length scale at which the structure of spacetime becomes dominated by quantum effects, and it is impossible to determine the difference between two locations less than one Planck length apart.

{{Hider|
  title = Examples |
  content = 
<!------------------------------------------------------------------------------------->
[[Einstein's equation]] is <center><math>G_{\mu\nu}+\Lambda g_{\mu\nu}={8\pi G\over c^4}T_{\mu\nu}</math></center>
where <math>G_{\mu\nu}= R_{\mu\nu}-(1/2)g_{\mu\nu}\,R</math> is the Einstein tensor, which combines the [[Ricci tensor]], the [[scalar curvature]] and the [[metric tensor]], <math>\Lambda</math> is the [[cosmological constant]],  <math>T_{\mu\nu}</math> is energy-momentum tensor of matter, <math>\pi</math> is the number, <math>c</math> is the [[speed of light]], <math>G</math> is Newton's [[gravitational constant]].
This equation can be written as 
:<center><math>\frac{1}{4\pi}(G_{\mu\nu}+\Lambda g_{\mu\nu})=2\left ({G\over c^3}\right)\left({1\over c}\,T_{\mu\nu}\right)</math></center>
where <math>\left({1\over c}\,T_{\mu\nu}\right)</math> is the density of the energy-momentum of matter.  
In the derivation of his equations, [[Einstein]] suggested that physical spacetime is [[Riemannian]], ie curved. A small domain of it is approximately flat spacetime. 

For any tensor field <math>N_{\mu\nu...}</math> value <math>N_{\mu\nu...}\sqrt{-g}</math> we may call a tensor density, where <math>g</math> is the determinant of the metric tensor <math>g_{\mu\nu}</math>. The integral <math>\int N_{\mu\nu...}\sqrt{-g}\,d^4x</math> is a tensor if the domain of integration is small. It is not a tensor if the domain of integration is not small, because it then consists of a sum of tensors located at different points and it does not transform in any simple way under a transformation of coordinates.<ref>P.A.M.Dirac(1975), General Theory of Relativity, A Wilay Interscience Publication, p.37</ref> 

Here we consider only small domains. This is also true for the integration over the three-dimensional [[hypersurface]] <math>S^{\nu}</math>.
Thus, Einstein's equations for small spacetime domain can be integrated by the three-dimensional hypersurface <math>S^{\nu}</math>. Have <ref>[https://www.amazon.com/Postigaya-mirozdanie-Fiziko-filosofskie-ocherki-Russian/dp/3659163457  A.P.Klimets(2012) "Postigaya mirozdanie", LAP LAMBERT Academic Publishing, Deutschland], [http://apklimets.narod.ru/Mirozdanie.pdf PDF, p.82 — 87]</ref>  
:<center><math>\frac{1}{4\pi}\int\left(G_{\mu\nu}+\Lambda g_{\mu\nu}\right)\sqrt{-g}\,dS^{\nu}={2G\over c^3}\int\left(\frac{1}{c}T_{\mu\nu}\right)\sqrt{-g}\,dS^{\nu}</math></center> 
Since integrable spacetime domain is small, we obtain the tensor equation
<center>
{{Equation box 1
|indent =::
|equation = <math>R_{\mu}=\frac{2G}{c^3}P_{\mu}</math>
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F5FFFA}}
</center>
where <math>P_{\mu}=\frac{1}{c}\int T_{\mu\nu}\sqrt{-g}\,dS^{\nu}</math> is the <math>{\mu}</math>-component of the 4-momentum of matter, <math>R_{\mu}=\frac{1}{4\pi}\int\left(G_{\mu\nu}+\Lambda g_{\mu\nu}\right)\sqrt{-g}\,dS^{\nu}</math> is the <math>{\mu}</math>-component of the radius of curvature small domain. Since <math>P_{\mu}=Mc\,U_{\mu}</math> then 
:<center><math>R_{\mu}=\frac{2G}{c^3}Mc\,U_{\mu}=r_s\,U_{\mu}</math></center> 
where <math>r_s</math> is the [[Schwarzschild radius]], <math>U_{\mu}</math> is the [[four-velocity]], <math>M</math> is the  gravitational mass. This entry reveals the physical meaning of the <math>R_{\mu}</math> values as a component of the gravitational radius <math>r_s</math>. Here <math>R_{\mu}R^{\mu}=r_s^2</math>  (compare <math>dx_{\mu}dx^{\mu}=dS^2</math>).

There is a similarity between the obtained tensor equation and the expression for the [[gravitational radius]] of the body. Indeed, for static spherically symmetric field and static distribution of matter have <math>U_{0}=1, U_i=0 \,(i=1,2,3)</math>. We obtain
:<center><math>R_0=\frac{2G}{c^3}Mc\,U_0=\frac{2G}{c^3}M\,c=r_s</math> </center>
In a small area of spacetime is almost flat and this equation can be written in the operator form 
:<center><math>\hat R_{\mu}=\frac{2G}{c^3}\hat P_{\mu}=\frac{2G}{c^3}(-i\hbar)\frac{\partial}{\partial\,x^{\mu}}=-2i\,\ell^2_{P}\frac{\partial}{\partial\,x^{\mu}}</math></center> 
where <math>\hbar</math> is the [[Dirac constant]]. Then commutator  of operators <math>\hat R_{\mu}</math> and <math>\hat x_{\mu}</math> is 
:<center><math>[\hat R_{\mu},\hat x_{\mu}]=-2i\ell^2_{P}</math></center> 
From here follow the specified uncertainty relations
<center> 
{{Equation box 1
|indent =::
|equation = <math>\Delta{R_{\mu}}\Delta{x_{\mu}}\ge\ell^2_{P}</math>
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F5FFFA}}
</center>  
Substituting the values of <math>R_{\mu}=\frac{2G}{c^3}M\,c\,U_{\mu}</math>  and  <math>\ell^2_{P}=\frac{\hbar\,G}{c^3}</math> and cutting right and left of the same symbols, we obtain the [[Heisenberg uncertainty principle]] 
:<center><math>\Delta{P_{\mu}}\Delta{x_{\mu}}=\Delta{(Mc\,U_{\mu})}\Delta{x_{\mu}}\ge\frac{\hbar}{2}</math></center> 
Since <math>R_{\mu}=({2G}/{c^3})\,P_{\mu}</math> and <math>P_{\mu}=\hbar\, k_{\mu}</math>, then <math>R_{\mu}=\ell^2_P\, k_{\mu}</math> , where <math>k_{\mu}</math> is the wave 4-vector. That is, <math>R_{\mu}</math> ([[Schwarzschild radius]]) is quantized, but the quantization step is extremely small.

In the particular case of a static spherically symmetric field  and static distribution of matter  <math>U_{0}=1,U_i=0\,(i=1,2,3)</math> and have remained
:<center><math>\Delta{R_{0}}\Delta{x_{0}}=\Delta{r_s}\Delta{r}\ge\ell^2_{P}</math></center>  
where <math>r_s</math> is the [[Schwarzschild radius]], <math>r</math> is [[radial coordinate]]. Here <math>R_0=r_s</math> and <math>x_0=c\,t=r</math>, since the matter moves with velocity of light in the [[Planck scale]].
Last uncertainty relation allows make us some estimates of the equations of [[general relativity]] at the Planck scale.

For example, the equation for the invariant interval <math>dS</math> in the Schwarzschild solution has the form 
:<center><math>dS^2=\left(1-\frac{r_s}{r}\right)c^2dt^2-\frac{dr^2}{1-{r_s}/{r}}-r^2(d\Omega^2+\sin^2\Omega d\varphi^2)</math></center> 
Substitute according to the uncertainty relations <math>r_s\approx\ell^2_P/r</math>. We obtain 
:<center><math>dS^2\approx\left(1-\frac{\ell^2_{P}}{r^2}\right)c^2dt^2-\frac{dr^2}{1-{\ell^2_{P}}/{r^2}}-r^2(d\Omega^2+\sin^2\Omega d\varphi^2)</math></center>
It is seen that at the [[Planck scale]] <math>r=\ell_P</math> spacetime metric is bounded below by the Planck length, and on this scale, there are real and [[virtual black holes]].

The space-time metric <math>g_{00}= 1-\Delta g\approx 1-\ell^2_P/(\Delta r)^2</math> fluctuates and generates a [[quantum foam]]. These fluctuations <math>\Delta g\sim\ell^2_P/(\Delta r)^2</math> in the macroworld and in the world of atoms are very small in comparison with <math>1</math> and become noticeable only on the Planck scale. The formula for the fluctuations of the gravitational potential <math>\Delta g\sim\ell^2_P/(\Delta r)^2</math> agrees with the Bohr-Rosenfeld uncertainty relation <math>\Delta g\,(\Delta r)^2\ge 2\ell^2_P</math>.<ref>[https://books.google.by/books?id=fPXwCAAAQBAJ&pg=PA33&lpg=PA33&dq=Bohr-Rosenfeld+uncertainty+relations&source=bl&ots=3gxdUleQBF&sig=OmkcThvEYl-y9nbVREag8imZFrw&hl=ru&sa=X&ved=0ahUKEwiL2KGj-f3VAhVBOBQKHRm8B5MQ6AEIRjAF#v=onepage&q=Bohr-Rosenfeld%20uncertainty%20relations&f=false H.H.von Borzeszkowski, H.J.Treder, The Meaning of Quantum Gravity, D.Reidel Publishing Company, 1987, p.36]</ref> and with the detailed analysis of gravity field measurements by T. Regge.<ref>T. Regge, Nuovo Cim. 7, 215 (1958). Gravitational fields and quantum mechanics</ref>

These small-scale fluctuations tell one that something like gravitational collapse is taking place everywhere in space and all the time; that gravitational collapse is in effect perpetually being done and undone; that in addition to the gravitational collapse of the universe, and of a star, one has also to deal with a third and, because it is constantly being undone, most significant level of gravitational collapse at the Planck scale of distances.<ref>[https://www.pdf-archive.com/2016/03/21/gravitation-misner-thorne-wheeler/gravitation-misner-thorne-wheeler.pdf Misner, Charles W.; Kip S. Thorne; John Archibald Wheeler (1973). Gravitation. San Francisco: W. H. Freeman., p.1194]</ref>  

The second example.The [[speed of light]] has the form in a gravitational field: <math>c\,'=c\,(1+2\,\varphi/c^2)=c\,(1-r_s/r)</math>. Therefore, the fluctuations in the speed of light on the Planck scale are <math>c'\approx c\,(1-\ell_P^2/\Delta\lambda^2)</math>. Here <math>\varphi</math> is the gravitational potential, <math>\lambda</math> is the wavelength of light. The photon velocity fluctuations are determined by the value of <math>\ell_P^2/\Delta\lambda^2</math> (but not by <math>\ell_P/\Delta\lambda</math>), so that these fluctuations are immeasurably small and the images of distant stars will be sharp even at metagalactic distances.

An analysis of the [[Hamilton-Jacobi equation]] in spaces of various dimensions on the Planck scale showed that the appearance of [[virtual black holes]] ([[quantum foam]], the basis of the "tissue" of the [[Universe]]) is energetically most profitable in three-dimensional space.<ref>[http://fizika.hfd.hr/fizika_b/bv00/b9p023.htm A.P.Klimets FIZIKA B (Zagreb) 9 (2000) 1, 23 — 42, § 4]</ref> This may have predetermined the three-dimensionality of the observed space. 
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The Planck area, equal to the square of the Planck length, plays a role in [[black hole entropy]]. The value of this entropy, in units of the [[Boltzmann constant]], is known to be given by <math>A/(4\ell_\mathrm{P}^2)</math>, where {{mvar|A}} is the area of the [[event horizon]]. The Planck area is the area by which the surface of a spherical [[black hole]] increases when the black hole swallows one bit of [[Physical information|information]], as was proven by [[Jacob Bekenstein]].<ref>{{cite web|url=http://prd.aps.org/abstract/PRD/v7/i8/p2333_1 |title=Phys. Rev. D 7, 2333 (1973): Black Holes and Entropy |publisher=Prd.aps.org |accessdate=2013-10-21}}</ref>

If [[large extra dimension]]s exist, the measured strength of gravity may be much smaller than its true (small-scale) value. In this case the Planck length would have no fundamental physical significance, and quantum gravitational effects would appear at other scales.

In [[string theory]], the Planck length is the order of magnitude of the oscillating strings that form elementary particles, and shorter lengths do not make physical sense.<ref name="Burgess_and_Quevedo">{{cite news | author=[[Cliff Burgess]] |author2=[[Fernando Quevedo]]  | title=The Great Cosmic Roller-Coaster Ride | url= | type=print | work=[[Scientific American]] | publisher=Scientific American, Inc. | page=55 | date=November 2007 }}</ref> The string scale {{mvar|l<sub>s</sub>}} is related to the Planck scale by {{math|<var>ℓ</var><sub>P</sub> {{=}} <var>g<sub>s</sub></var><sup>{{sfrac|4}}</sup><var>l<sub>s</sub></var>}}, where {{mvar|g<sub>s</sub>}} is the string coupling constant. Contrary to what the name suggests, the string coupling constant is not constant, but depends on the value of a scalar field known as the [[dilaton]].

In [[loop quantum gravity]], area is quantized, and the Planck area is, within a factor of 10, the smallest possible area value.

In [[doubly special relativity]], the Planck length is observer-invariant.

The search for the laws of physics valid at the Planck length is a part of the search for the [[theory of everything]].{{clarify|date=July 2015}}

== Visualization ==

The size of the Planck length can be visualized as follows: if a particle or dot about 0.005&nbsp;mm in size (which is the same size as a [[Silt#Grain_size_criteria|small grain of silt]]) were magnified in size to be as large as the [[observable universe]], then inside that universe-sized "dot", the Planck length would be roughly the size of an actual 0.005&nbsp;mm dot. In other words, a 0.005&nbsp;mm dot is halfway between the Planck length and the size of the observable universe on a [[logarithmic scale]].<ref>[[Wolfram Alpha]][http://m.wolframalpha.com/input/?i=size+of+observable+universe+in+mm+%2F+1.616x10%5E32&x=0&y=0]</ref> All said, the attempt to visualize to an arbitrary scale of a 0.005 mm dot is only for a hinge point.  With no fixed frame of reference for time or space, where the spatial units shrink toward infinitesimally small spatial sections and time stretches toward infinity, scale breaks down.  Inverted, where space is stretched and time is shrunk, the scale adjusts the other way according to the ratio V-squared/C-squared ([[Lorentz transformation]]).{{what|date=July 2017}}

==See also==
{{col-list|colwidth=20em|
* [[Fock–Lorentz symmetry]]
* [[Orders of magnitude (length)]]
* [[Planck energy]]
* [[Planck mass]]
* [[Planck epoch]]
* [[Planck temperature]]
* [[Planck time]]
}}

== Notes and references==
{{reflist}}

==Bibliography==
* {{cite journal
|last=Garay |first=Luis J.
|date=January 1995
|title=Quantum gravity and minimum length
|journal=[[International Journal of Modern Physics A]]
|volume=10 |issue= 2|pages=145 ff.
|arxiv=gr-qc/9403008v2
|bibcode= 1995IJMPA..10..145G
|doi=10.1142/S0217751X95000085
}}

==External links==
* {{cite web|last=Bowley|first=Roger|title=Planck Length|url=http://www.sixtysymbols.com/videos/plancklength.htm|work=Sixty Symbols|publisher=[[Brady Haran]] for the [[University of Nottingham]]|author2=Eaves, Laurence |authorlink2=Laurence Eaves |year=2010}}
{{Planck's natural units}}
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