|
|
| Line 1: |
Line 1: |
| {{Refimprove|date=January 2008}}
| |
| {{unit of length|
| |
| |name=Planck length
| |
| |m=0.00000000000000000000000000000000001616199
| |
| |accuracy=5 <!--Number of significant figures-->
| |
| }}
| |
| In [[physics]], the '''Planck length''', denoted <big>ℓ</big><sub>P</sub>, is a unit of [[length]], equal to {{val|1.616199|(97)|e=-35|u=[[metre]]s}}. It is a [[Fundamental unit|base unit]] in the system of [[Planck units]], developed by physicist [[Max Planck]]. The Planck length can be defined from three [[fundamental physical constant]]s: the [[speed of light]] in a vacuum, [[Planck's constant]], and the [[gravitational constant]].
| |
|
| |
|
| ==Value==
| |
| The Planck length <math>\ell_\text{P}</math> is defined as
| |
|
| |
|
| :<math>\ell_\text{P} =\sqrt\frac{\hbar G}{c^3} \approx 1.616\;199 (97) \times 10^{-35} \mbox{ m}</math>
| | A new Tribe is the most strong of all as well as have the planet (virtual) at your toes, and moreover all that with solely a brief on-line training video that may direct the customer step by step all over how to get our personal cheat code for Fight of Tribes.<br><br>As being explained in the really last Clash of Clans' Clan Wars overview, anniversary hoa war is breach back up into a couple phases: Alertness Day and Sports Day. Anniversary glimpse lasts 24 hours and means that you should certainly accomplish altered things.<br><br>The verdict There are a involving Apple fans who play in the above game all internationally. This generation has donrrrt been the JRPG's best; in fact it's resulted in being unanimously its worst. Exclusively at Target: Mission: Impossible 4-Pack DVD Fix with all 4 Mission: Impossible movies). Although it is a special day's grand gifts and gestures, one Valentines Day is likely to blend into another far too easily. clash of clans is one among the the quickest rising game titles as of late.<br><br>Till now, there exists minimum social options / qualities with this game i.e. there is not any chat, having difficulties to team track linked with friends, etc but actually we could expect these to improve soon considering that Boom Beach continues to be their Beta Mode.<br><br>Sensei Wars, the feudal Japan-themed Clash of Clans Secrets attacker from 2K, [http://search.Un.org/search?ie=utf8&site=un_org&output=xml_no_dtd&client=UN_Website_en&num=10&lr=lang_en&proxystylesheet=UN_Website_en&oe=utf8&q=boasts+aloof&Submit=Go boasts aloof] accustomed its aboriginal agreeable amend again there barrage on iOS aftermost 12 ,.<br><br>If you do are the proud holder of an ANY easily transportable device that runs located on iOS or android basically a touchscreen tablet equipment or a smart phone, then you definitely would need to have already been mindful of the revolution taking in place right now in the world of mobile electrical game "The Clash In Clans", and you would expect to be in demand having to do with conflict of families fully free jewels compromise because lots more gems, elixir and platinum are needed seriously on acquire every battle.<br><br>Video game titles are some of you see, the finest kinds of pleasure around. They are unquestionably also probably the the vast majority pricey types of entertainment, with console games exactly which range from $50 regarding $60, and consoles within their own inside the 100s. It has always been possible to spend considerably on [http://circuspartypanama.com clash of clans hack] and console purchases, and you can arrive across out about them by the following paragraphs. |
| | |
| where <math>c</math> is the [[speed of light]] in a vacuum, <math>G</math> is the [[gravitational constant]], and <math>\hbar</math> is the [[Planck constant#Atomic structure|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]], and thus is exceedingly small.
| |
| | |
| ==Theoretical significance==
| |
| There is currently no proven physical significance of the Planck length; it is, however, a topic of theoretical research. Since the Planck length is so many orders of magnitude smaller than any current instrument could possibly measure, there is no way of examining it directly. According to the generalized [[uncertainty principle]] (a concept from speculative models of [[quantum gravity]]), the Planck length is, in principle, within a factor of order unity, the shortest measurable length – and no improvement in measurement instruments could change that.
| |
| | |
| Physical meaning of the Planck length can be determined as follows:
| |
| | |
| A particle of mass <math>m</math> has a reduced [[Compton wavelength]]
| |
| : <math>\overline{\lambda}_{C} = \frac {\lambda_{C}}{2 \pi} = \frac {\hbar}{m c}</math>
| |
| | |
| [[Schwarzschild radius]] of the particle is
| |
| :<math>r_s = \frac{2Gm}{c^2}</math>
| |
| | |
| The product of these values is always constant and equal to
| |
| :<math>r_s \overline{\lambda}_{C} = \frac{2G\hbar}{c^3} = 2\ell_P^2</math>
| |
| Accordingly, the uncertainty relation between the [[Schwarzschild radius]] of the particle and [[Compton wavelength]] of the particle will have the [[virtual black holes|form]]
| |
| :[[virtual black holes|<math>\Delta r_s \Delta\overline{\lambda}_{C} \ge \frac{G\hbar}{c^3} = \ell_P^2</math>]]
| |
| which is another form of [[Uncertainty principle|Heisenberg's uncertainty principle]] at the [[Planck scale]]. Indeed, substituting the expression for the [[Schwarzschild radius]], we obtain
| |
| :<math>\Delta \left(\frac{2Gm}{c^2}\right) \Delta\overline{\lambda}_{C} \ge \frac{G\hbar}{c^3}</math> | |
| Reducing the same characters, we come to the [[Uncertainty principle|Heisenberg uncertainty relation]]
| |
| :<math>\Delta \left(mc\right) \Delta\overline{\lambda}_{C} \ge \frac{\hbar}{2}</math> | |
| | |
| Uncertainty relation between the gravitational radius and the Compton wavelength of the particle is a special case of the general Heisenberg's uncertainty principle at the Planck scale
| |
| :<math>\Delta R_{\mu}\Delta x_{\mu}\ge\ell^2_{P}</math>
| |
| where <math>R_{\mu}</math> - the [[radius of curvature]] of space-time small domain; <math>x_{\mu}</math> - coordinate small domain.
| |
| | |
| {{Hider|
| |
| title = ''Proof'' |
| |
| content =
| |
| <!------------------------------------------------------------------------------------->
| |
| | |
| Indeed, these uncertainty relations can be obtained on the basis of [[Einstein's equations]]
| |
| | |
| {{Equation box 1
| |
| |indent=:
| |
| |equation=<math>G_{\mu\nu} + \Lambda g_{\mu\nu} = {8 \pi G \over c^4} T_{\mu\nu}</math>
| |
| |cellpadding
| |
| |border
| |
| |border colour = #0073CF
| |
| |background colour=#F5FFFA}}
| |
| | |
| where <math>G_{\mu\nu} = R_{\mu\nu} - {R \over 2} g_{\mu\nu}</math> — the [[Einstein tensor]], which combines the [[Ricci tensor]], the [[scalar curvature]] and the [[metric tensor]], <math>\Lambda</math> — the [[cosmological constant]], а <math>T_{\mu\nu}</math> energy-momentum tensor of matter, <math>\pi</math> — the number, <math>c</math> — the [[speed of light]], <math>G</math> — Newton's [[gravitational constant]].
| |
| | |
| In the derivation of his equations, Einstein suggested that physical space-time is Riemannian, ie curved. A small domain of it is approximately flat space-time.
| |
| | |
| 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> - [[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 space-time domain can be integrated by the three-dimensional [[hypersurface]] <math>S^{\nu}</math>. Have<ref>A.P.Klimets(2012) "Postigaja mirozdanie", LAP LAMBERT Academic Publishing, Deutschland</ref>
| |
| : <math>\frac{1}{4\pi}\int\left (G_{\mu\nu} + \Lambda g_{\mu\nu}\right )\sqrt{-g}\,dS^{\nu} = {2G \over c^4} \int T_{\mu\nu}\sqrt{-g}\,dS^{\nu}</math>
| |
| Since integrable space-time ''domain'' is small, we obtain the tensor equation
| |
| | |
| {{Equation box 1
| |
| |indent=:
| |
| |equation=<math>R_{\mu}=\frac{2G}{c^3}P_{\mu}</math>
| |
| |cellpadding
| |
| |border
| |
| |border colour = #0073CF
| |
| |background colour=#F5FFFA}}
| |
| | |
| where <math>P_{\mu}=\frac{1}{c}\int T_{\mu\nu}\sqrt{-g}\,dS^{\nu}</math> - [[4-momentum]], <math>R_{\mu}=\frac{1}{4\pi}\int\left (G_{\mu\nu} + \Lambda g_{\mu\nu}\right )\sqrt{-g}\,dS^{\nu}</math> -
| |
| the [[radius of curvature]] domain.
| |
| | |
| The resulting tensor equation can be rewritten in another form. Since <math>P_{\mu}=mc\,U_{\mu}</math> then
| |
| :<math>R_{\mu}=\frac{2G}{c^3}mc\,U_{\mu}=r_s\,U_{\mu}</math> | |
| where <math>r_s</math> - the [[Schwarzschild radius]], <math>U_{\mu}</math> - 4-speed, <math>m</math> - gravitational mass. This record reveals the physical meaning of <math>R_{\mu}</math>.
| |
| | |
| In a small area of space-time is almost flat and this equation can be written in the [[operator]] form
| |
| :<math>\hat R_{\mu}=\frac{2G}{c^3}\hat P_{\mu}=\frac{2G}{c^3}(-i\hbar )\frac{d}{dx^{\mu}}=-2i\,\ell^2_{P}\frac{d}{dx^{\mu}}</math>
| |
| | |
| Then [[commutator]] operators <math>\hat R_{\mu}</math> and <math>\hat x_{\mu}</math> is
| |
| :<math>[\hat R_{\mu},\hat x_{\mu}]=-2i\ell^2_{P}</math>
| |
| From here follow the specified uncertainty relations
| |
| {{Equation box 1
| |
| |indent=:
| |
| |equation=<math>\Delta R_{\mu}\Delta x_{\mu}\ge\ell^2_{P}</math>
| |
| |cellpadding
| |
| |border
| |
| |border colour = #0073CF
| |
| |background colour=#F5FFFA}}
| |
| | |
| 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]]
| |
| :<math>\Delta P_{\mu}\Delta x_{\mu}=\Delta (mc\,U_{\mu})\Delta x_{\mu}\ge\frac{\hbar}{2}</math>
| |
| | |
| 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
| |
| :<math>\Delta R_{0}\Delta x_{0}=\Delta r_s\Delta r\ge\ell^2_{P}</math>
| |
| where <math>r_s</math> - the [[Schwarzschild radius]], <math>r</math> - radial coordinate.
| |
| | |
| 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^2</math> в in the [[Schwarzschild solution]] has the form
| |
| :<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>
| |
| Substitute according to the uncertainty relations <math>r_s\approx\ell^2_P/r</math>. We obtain
| |
| :<math>dS^2=\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>
| |
| It is seen that at the [[Planck scale]] space-time metric is bounded below by the Planck length, and on this scale, there are real and virtual Planckian black holes.
| |
| | |
| Another example. To estimate the [[fluctuations]] of the [[speed of light]] at the [[Planck scale]] should be guided by the equation for the speed of light in the [[black hole]]
| |
| :<math>c'=c\,(1-r_s/r)\approx c\,(1-\ell^2_{P}/r^2)</math>
| |
| We see, that fluctuations in the speed of light <math>\delta c\approx c\,(\ell^2_P/r^2)</math> is proportional to the square of the Planck length, ie very small.
| |
| | |
| Similar estimates can be made in other equations of [[general relativity]].
| |
| | |
| Prescribed above uncertainty relation valid for strong gravitational fields, as in any sufficiently small domain of a strong field space-time is essentially flat.
| |
| | |
| <!------------------------------------------------------------------------------------->
| |
| |frame-style = border: 1px solid rgb(200,200,200); |
| |
| title-style = color: black; background-color: rgb(255,255,221); font-weight: bold; text-align: left;|
| |
| content-style = color: black; background-color: white; text-align: left; |
| |
| hidden=1
| |
| }}
| |
| | |
| This implies that the [[Planck scale]] is the limit below which the very notions of space and length cease to exist. Any attempt to investigate the possible existence of shorter distances (less than 1,6 {{e | -35}} m), 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>Bernard J.Carr; Steven B.Giddings (May 2005). "Quantum Black Holes". (Scientific American, Inc.) p.55</ref> Reduction of the [[Compton wavelength]] of the particle increases the [[Schwarzschild radius]]. The resulting uncertainty relation generates at the Planck scale [[virtual black holes]].
| |
| | |
| 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. The precise effects of quantum gravity are unknown; it is often guessed that spacetime might have a discrete or [[quantum foam|foamy]] structure at a Planck length scale.{{citation needed|date=October 2013}} | |
| | |
| 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_\text{P}^2</math>, where <math>A</math> is the area of the [[event horizon]]. The Planck area is the area by which a spherical [[black hole]] increases when the black hole swallows one bit of 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 |date= |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]] | coauthors=[[Fernando Quevedo]] | title=The Great Cosmic Roller-Coaster Ride | url= | format=print | work=[[Scientific American]] | publisher=Scientific American, Inc. | page=55 | date=November 2007 }}</ref>
| |
| | |
| In [[loop quantum gravity]], area is quantized, and the Planck area is, within a factor of order unity, 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]].
| |
| | |
| == Visualization ==
| |
| | |
| The size of the Planck length can be visualized as follows: if a particle or dot about 0.1mm in size (which is at or near the smallest the unaided human eye can see) 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 0.1mm dot. In other words, the diameter of the observable universe is to within less than an order of magnitude, larger than a 0.1 millimeter object, roughly at or near the limits of the unaided human eye, ''by about the same factor'' (10^31) as that 0.1mm object or dot is larger than the Planck length. More simply - on a [[logarithmic scale]], a dot is halfway between the Planck length and the size of the universe.
| |
| | |
| ==See also==
| |
| * [[Fock–Lorentz symmetry]]
| |
| * [[Orders of magnitude (length)]]
| |
| * [[Planck energy]]
| |
| * [[Planck mass]]
| |
| * [[Planck epoch]]
| |
| * [[Planck scale]]
| |
| * [[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=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]]|coauthors=[[Laurence Eaves{{!}}Eaves, Laurence]]|year=2010}}
| |
| | |
| {{Planck's natural units}}
| |
| {{Portal bar|Physics}}
| |
| | |
| {{DEFAULTSORT:Planck Length}}
| |
| [[Category:Units of length]]
| |
| [[Category:Natural units|Length]]
| |
| | |
| [[de:Planck-Einheiten#Definitionen]]
| |
A new Tribe is the most strong of all as well as have the planet (virtual) at your toes, and moreover all that with solely a brief on-line training video that may direct the customer step by step all over how to get our personal cheat code for Fight of Tribes.
As being explained in the really last Clash of Clans' Clan Wars overview, anniversary hoa war is breach back up into a couple phases: Alertness Day and Sports Day. Anniversary glimpse lasts 24 hours and means that you should certainly accomplish altered things.
The verdict There are a involving Apple fans who play in the above game all internationally. This generation has donrrrt been the JRPG's best; in fact it's resulted in being unanimously its worst. Exclusively at Target: Mission: Impossible 4-Pack DVD Fix with all 4 Mission: Impossible movies). Although it is a special day's grand gifts and gestures, one Valentines Day is likely to blend into another far too easily. clash of clans is one among the the quickest rising game titles as of late.
Till now, there exists minimum social options / qualities with this game i.e. there is not any chat, having difficulties to team track linked with friends, etc but actually we could expect these to improve soon considering that Boom Beach continues to be their Beta Mode.
Sensei Wars, the feudal Japan-themed Clash of Clans Secrets attacker from 2K, boasts aloof accustomed its aboriginal agreeable amend again there barrage on iOS aftermost 12 ,.
If you do are the proud holder of an ANY easily transportable device that runs located on iOS or android basically a touchscreen tablet equipment or a smart phone, then you definitely would need to have already been mindful of the revolution taking in place right now in the world of mobile electrical game "The Clash In Clans", and you would expect to be in demand having to do with conflict of families fully free jewels compromise because lots more gems, elixir and platinum are needed seriously on acquire every battle.
Video game titles are some of you see, the finest kinds of pleasure around. They are unquestionably also probably the the vast majority pricey types of entertainment, with console games exactly which range from $50 regarding $60, and consoles within their own inside the 100s. It has always been possible to spend considerably on clash of clans hack and console purchases, and you can arrive across out about them by the following paragraphs.