|
|
| Line 1: |
Line 1: |
| In [[physics]], '''Planck units''' are physical [[units of measurement]] defined exclusively in terms of five universal [[physical constants]] listed below, in such a manner that these five physical constants take on the numerical value of [[1 (number)|1]] when expressed in terms of these units. Planck units have profound significance for theoretical physics since they elegantly simplify several recurring [[algebraic expression]]s of [[physical law]] by [[nondimensionalization]]. They are particularly relevant in research on unified theories such as [[quantum gravity]]. | | Make sure the center channel is "anchored" for the screen. Function of the middle channel should be to communicate the dialog of the movie. In case the person's mouth is moving on the screen, but the sound is away from off on the side the dialog will seem not naturally made. A center channel speaker will be shielded certain that you are listed it near or at top of the tv.<br><br>For will be the home threare, light factor likewise important. You have to consider that how much amount of light you want inside the theatre. There is no need to make it entirely dark but you ought to have control on light inside to make the right situation. You should choose the curtains usually are opaque which enable eliminate the sunlight entirely from the outside through windows. Could possibly choose many colors of curtain based your Home theatre Decor like cherry, avocado, pink, camel, indigo, melon, mink, scarlet, sequin and even more. Once you will be succeeded in managing outside light by drawing appropriate curtains, you could very well control the inner illumination in the room.<br><br>10) Have they got a home theatre showroom that you can do come and play the crazy things that latest home theatre equipment yourself? Can you bring simple . action movies to hear your favorite sound effects on the various surround sound systems they've on company? Can you bring your favorite CD's and pay attention to the songs & do a comparison with several speaker packages so frequency higher the differences?<br><br>Create Customer Goodwill: Promotional products build the consumer goodwill towards the company with their positive attitude or feelings about transmit mail service, their product quality or even because of the promotional merchandise is given without spending a dime.<br><br>This KEF FiveTwo Series Model 7 is distinctive from the other home theatre speakers packages. This model carries only two annonceur. However, it gives you a surround sound which is audiophile-grade. Now, just imagine, with only two speakers and still produce awesome home Theater ([http://onlog.com.br/ajuda/index.php?title=How_you_Ought_To_Hire_A_Dj_For_Your_Wedding onlog.com.Br]) surround sound. It is a very uncommon type of virtual multichannel audio system. Priced around $1200, its overall rating is 'very good'.<br><br>Shipping - It significant to understand several reasons for having shipping. Exactly what are the shipping costs? Will the product get damaged in shipping and if so who is responsible. These directory sites feel that you've a much greater chance of receiving a damaged product when buying online and achieving it shipped, however frequently there is less handling of a tv from a large online retailer who ships the tv from their warehouse there is in a variety of the local stores where tvs are shipped from one warehouse distinct until it eventually enters the stores warehouse thereafter delivered to your house. Is white glove delivery available to bring the LCD directly to your home? Are you able to have someone setup the tv for as well as if so what exactly is the final price.<br><br>A swing set or play ground in your yard end up being the ideal if you are living in young friendly regional. Even a small box along with sand will do wonders to plant the idea of family in possible buyers' minds. Simple tricks something like this will help set apart your house from the opposite dozen houses they probably have looked during that day.<br><br>This "magic box" has aluminum cooling fans, with, for example, the Athlon 64 AMD having an easy design. Features the familiar built-in fans and a is simple, rather than being very complex. These fans keep the CPU cool by blowing out hot air, even though Athlon 64 will need some improvement the hho booster wants always be competitive with other, newer products. |
| | |
| Originally proposed in 1899 by German physicist [[Max Planck]], these units are also known as ''natural units'' because the origin of their definition comes only from properties of the fundamental physical theories and not from interchangeable experimental parameters. Planck units are only one system of [[natural units]] among other systems, but are considered unique in that these units are not based on properties of any [[Prototype#Metrology|prototype object]] or [[elementary particle|particle]] (that would be arbitrarily chosen), but rather on properties of [[free space]] alone.
| |
| | |
| The universal constants that Planck units, by definition, normalize to 1 are:
| |
| *the [[gravitational constant]], ''G'',
| |
| *the [[reduced Planck constant]], ''ħ'',
| |
| *the [[speed of light]] in a vacuum, ''c'',
| |
| *the [[Coulomb constant]], (4π''ε''<sub>0</sub>)<sup>−1</sup> (sometimes ''k''<sub>e</sub> or ''k''), and
| |
| *the [[Boltzmann constant]], ''k''<sub>B</sub> (sometimes ''k'').
| |
| | |
| Each of these constants can be associated with at least one fundamental physical theory: ''c'' with [[electromagnetism]] and [[special relativity]], ''G'' with [[general relativity]] and [[Newtonian gravity]], ''ħ'' with [[quantum mechanics]], ''ε''<sub>0</sub> with [[electrostatics]], and ''k''<sub>B</sub> with [[statistical mechanics]] and [[thermodynamics]].
| |
| | |
| Planck units are sometimes called "God's units",<ref>{{Cite book
| |
| | last = Collins
| |
| | first = Joseph
| |
| | title = Intelligent Computer Mathematics : 16th Symposium, Calculemus 2009, 8th International Conference, MKM 2009, Grand Bend, Canada, July 6-12, 2009, Proceedings
| |
| | chapter = OpenMath Content Dictionaries for SI Quantities and Units
| |
| | chapter-url = http://books.google.com/books?id=IKNmax4dZ70C&pg=PA257&lpg=PA257&dq=%22God%27s+units%22&source=bl&ots=hH1f8x9x-Q&sig=sXwblHjyINz9NpZRv2o2zAyQxrc&hl=en&sa=X&ei=VrA6T_D7GeORiQK99O2TDA&ved=0CFMQ6AEwBQ#v=onepage&q=%22God%27s%20units%22&f=false
| |
| | publisher = Springer Verlag
| |
| | page = 257
| |
| | editor-last = Dixon
| |
| | editor-first = Lucas
| |
| | editor2-last = Carette
| |
| | editor2-first = Jacques
| |
| | others = Sacerdoti Coen, Watt
| |
| | volume = 5625
| |
| | series = Lecture Notes in Computer Science
| |
| | edition = 2nd
| |
| | year = 2009
| |
| | isbn = 978-3-642-02613-3
| |
| | url = http://www.springerlink.com/content/n3416j826585/#section=181260&page=1
| |
| | postscript = <!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}
| |
| }}</ref><ref>Clifford A. Pickover [http://books.google.com/books?id=SQXcpvjcJBUC&pg=PA419&lpg=PA419&dq=Planck+%22God%27s+units%22&source=bl&ots=QfZQL4Sqpz&sig=97V-jDZM3I1U8y6JTN1V1WEOCnk&hl=en&sa=X&ei=llZQT-TvHYbW0QHUpMi9DQ&ved=0CDUQ6AEwAg#v=onepage&q=Planck%20%22God%27s%20units%22&f=false Archimedes to Hawking: laws of science and the great minds behind them]</ref>
| |
| since Planck units are free of [[anthropocentric]] arbitrariness. Some physicists argue that communication with [[extraterrestrial life|extraterrestrial intelligence]] would have to employ such a system of units in order to be understood.<ref>Michael W. Busch, Rachel M. Reddick (2010) "[http://www.lpi.usra.edu/meetings/abscicon2010/pdf/5070.pdf Testing SETI Message Designs,]" [http://www.lpi.usra.edu/meetings/abscicon2010/ ''Astrobiology Science Conference 2010''], April 26–29, 2010, League City, Texas.</ref> Unlike the [[metre]] and [[second]], which exist as fundamental units in the [[SI]] system for historical reasons, the [[Planck length]] and [[Planck time]] are conceptually linked at a fundamental physical level.
| |
| | |
| Natural units help physicists to reframe questions. [[Frank Wilczek]] puts it succinctly:
| |
| {{bq|We see that the question [posed] is not, "Why is gravity so feeble?" but rather, "Why is the proton's mass so small?" For in natural (Planck) units, the strength of gravity simply is what it is, a primary quantity, while the proton's mass is the tiny number [1/(13 quintillion)].<ref>[http://phys.columbia.edu/~millis/1601/supplementaryreading/WilczekScales.pdf June 2001 Physics Today]</ref>}}
| |
| | |
| 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. From the point of view of Planck units, however, this is not a statement about the relative strengths of the two forces; rather, it is a manifestation of the fact that the [[elementary charge|charge on the protons]] is approximately the [[Planck charge]] but the [[Proton mass|mass of the protons]] is far less than the [[Planck mass]].
| |
| | |
| == Base units ==
| |
| | |
| All systems of measurement feature [[Fundamental unit|base unit]]s: in the [[International System of Units]] (SI), for example, the base unit of length is the [[metre]]. In the system of Planck units, the Planck base unit of length is known simply as the [[Planck length]], the base unit of time is the [[Planck time]], and so on. These units are derived from the five dimensional universal physical constants of Table 1, in such a manner that these constants are eliminated from fundamental equations of physical law when physical quantities are expressed in terms of Planck units. For example, Newton's [[law of universal gravitation]],
| |
| | |
| : <math> F = G \frac{m_1 m_2}{r^2},</math>
| |
| | |
| can be expressed as
| |
| | |
| : <math> \frac{F}{F_\text{P}} = \frac{\left(m_1/m_\text{P}\right) \left(m_2/m_\text{P}\right)}{\left(r/ l_\text{P}\right)^2}.</math>
| |
| | |
| Both equations are [[dimensional analysis|dimensionally consistent]] and equally valid in ''any'' system of units, but the second equation, with ''G'' missing, is relating only [[dimensionless quantities]] since any ratio of two like-dimensioned quantities is a dimensionless quantity. If, by a shorthand convention, it is axiomatically understood that all physical quantities are expressed in terms of Planck units, the ratios above may be expressed simply with the symbols of physical quantity, without being scaled by their corresponding unit:
| |
| | |
| : <math> F = \frac{m_1 m_2}{r^2} \ .</math>
| |
| | |
| In order for this last equation to be valid (without ''G'' present), ''F'', ''m''<sub>1</sub>, ''m''<sub>2</sub>, and ''r'' are understood to be the dimensionless numerical values of these quantities measured in terms of Planck units. This is why Planck units or any other use of natural units should be employed with care; referring to {{nowrap|''G'' {{=}} ''c'' {{=}} 1}}, [[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>Wesson P. S. (1980) "[http://articles.adsabs.harvard.edu//full/1980SSRv...27..109W/0000117.000.html The application of dimensional analysis to cosmology,]" ''Space Science Reviews'' '''27''': 117.</ref>
| |
| | |
| {| class="wikitable"
| |
| |+Table 1: Fundamental physical constants
| |
| ! Constant
| |
| ! Symbol
| |
| ! Dimension
| |
| ! Value in [[International System of Units|SI]] units with [[Uncertainty#Measurements|uncertainties]]<ref name="CODATA">[http://physics.nist.gov/cuu/Constants/index.html Fundamental Physical Constants from NIST]</ref>
| |
| |-
| |
| | {{nowrap|[[Speed of light]] in vacuum}}
| |
| | ''c''
| |
| | L T <sup>−1</sup>
| |
| | {{val|2.99792458|e=8|u=m s<sup>−1</sup>}}<br/> ''(exact by definition of [[metre]])''
| |
| |-
| |
| | [[Gravitational constant]]
| |
| | ''G''
| |
| | L<sup>3</sup> M<sup>−1</sup> T <sup>−2</sup>
| |
| | {{physconst|G}}
| |
| |-
| |
| | {{nowrap|Reduced [[Planck constant]]}}
| |
| | ''ħ'' = ''h''/2π<br/> where ''h'' is Planck constant
| |
| | L<sup>2</sup> M T <sup>−1</sup>
| |
| | {{physconst|hbar}}
| |
| |-
| |
| | [[Coulomb constant]]
| |
| | (4π''ε''<sub>0</sub>)<sup>−1</sup><br/> where ''ε''<sub>0</sub> is the [[permittivity of free space]]
| |
| | {{nowrap|L<sup>3</sup> M T <sup>−2</sup> Q<sup>−2</sup>}}
| |
| | {{gaps|8.987|551|787|368|1764|e=9|u=kg m<sup>3</sup> s<sup>−2</sup> C<sup>−2</sup>}}<br/> ''(exact by definitions of [[ampere]] and metre)''
| |
| |-
| |
| | [[Boltzmann constant]]
| |
| | ''k''<sub>B</sub>
| |
| | L<sup>2</sup> M T <sup>−2</sup> Θ<sup>−1</sup>
| |
| | {{physconst|k}}
| |
| |}
| |
| '''Key''': L = [[length]], M = [[mass]], T = [[time]], Q = [[electric charge]], Θ = [[temperature]].
| |
| | |
| As can be seen above, the gravitational attractive force of two bodies of 1 [[Planck mass]] each, set apart by 1 Planck length is 1 [[Planck force]]. Likewise, the distance traveled by light during 1 Planck time is 1 Planck length. To determine, in terms of SI or another existing system of units, the quantitative values of the five base Planck units, those two equations and three others must be satisfied to determine the five unknown quantities that define the base Planck units:
| |
| | |
| : <math> l_\text{P} = c t_\text{P} </math>
| |
| | |
| : <math> F_\text{P} = \frac{m_\text{P} l_\text{P}}{t_\text{P}^2} = G \frac{m_\text{P}^2}{l_\text{P}^2} </math>
| |
| | |
| : <math> E_\text{P} = \frac{m_\text{P} l_\text{P}^2}{t_\text{P}^2} = \hbar \frac{1}{t_\text{P}} </math>
| |
| | |
| : <math> F_\text{P} = \frac{m_\text{P} l_\text{P}}{t_\text{P}^2} = \frac{1}{4 \pi \varepsilon_0} \frac{q_\text{P}^2}{l_\text{P}^2} </math>
| |
| | |
| : <math> E_\text{P} = \frac{m_\text{P} l_\text{P}^2}{t_\text{P}^2} = k_\text{B} T_\text{P}.</math>
| |
| | |
| Solving the five equations above for the five unknowns results in a unique set of values for the five base Planck units:
| |
| | |
| <center>'''Table 2: Base Planck units'''</center>
| |
| {{Base Planck units}}
| |
| | |
| == Derived units ==
| |
| In any system of measurement, units for many physical quantities can be derived from base units. Table 3 offers a sample of derived Planck units, some of which in fact are seldom used. As with the base units, their use is mostly confined to theoretical physics because most of them are too large or too small for empirical or practical use and there are large uncertainties in their values (see ''[[#Discussion|Discussion]]'' and ''[[#Uncertainties in values|Uncertainties in values]]'' below).
| |
| <!-- TO DO: the P in subscripts should be in upright type (\mathrm{P} or \text{P}); the ² characters should be changed to exponents (<sup>2</sup>), links redirecting back here should be removed (or articles should be written for them), etc. -->
| |
| | |
| <center>'''Table 3: Derived Planck units'''</center>
| |
| {{Derived Planck units}}
| |
| | |
| ==Simplification of physical equations==
| |
| Physical quantities that have different dimensions (such as time and length) cannot be equated even if they are numerically equal (1 second is not the same as 1 metre). In theoretical physics, however, this scruple can be set aside, by a process called [[nondimensionalization]]. Table 4 shows how Planck units, by setting the numerical values of five fundamental constants to the number one, nondimensionalizes and simplifies many fundamental equations of physics.
| |
| | |
| {| class="wikitable"
| |
| |+Table 4: How Planck units simplify the key equations of physics
| |
| !
| |
| ! SI form
| |
| ! Nondimensionalized form
| |
| |-
| |
| | [[Newton's law of universal gravitation]]
| |
| | <math> F = - G \frac{m_1 m_2}{r^2} </math>
| |
| | <math> F = - \frac{m_1 m_2}{r^2} </math>
| |
| |-
| |
| | [[Einstein field equations]] in [[general relativity]]
| |
| | <math>{ G_{\mu \nu} = 8 \pi {G \over c^4} T_{\mu \nu} } \ </math>
| |
| | <math>{ G_{\mu \nu} = 8 \pi T_{\mu \nu} } \ </math>
| |
| |-
| |
| | [[Mass–energy equivalence]] in [[special relativity]]
| |
| | <math>{ E = m c^2} \ </math>
| |
| | <math>{ E = m } \ </math>
| |
| |-
| |
| | [[Energy–momentum relation]]
| |
| |<math> E^2 = m^2 c^4 + p^2 c^2 \;</math>
| |
| |<math> E^2 = m^2 + p^2 \;</math>
| |
| |-
| |
| | [[Thermal energy]] per particle per [[Degrees of freedom (physics and chemistry)|degree of freedom]]
| |
| | <math>{ E = \tfrac12 k_\text{B} T} \ </math>
| |
| | <math>{ E = \tfrac12 T} \ </math>
| |
| |-
| |
| | Boltzmann's [[entropy (statistical thermodynamics)|entropy]] formula
| |
| | <math>{ S = k_\text{B} \ln \Omega } \ </math>
| |
| | <math>{ S = \ln \Omega } \ </math>
| |
| |-
| |
| | [[Planck's relation]] for energy and [[angular frequency]]
| |
| | <math>{ E = \hbar \omega } \ </math>
| |
| | <math>{ E = \omega } \ </math>
| |
| |-
| |
| | [[Planck's law]] (surface [[intensity (physics)|intensity]] per unit [[solid angle]] per unit [[angular frequency]]) for [[black body]] at [[temperature]] ''T''.
| |
| | <math> I(\omega,T) = \frac{\hbar \omega^3 }{4 \pi^3 c^2}~\frac{1}{e^{\frac{\hbar \omega}{k_\text{B} T}}-1} </math>
| |
| | <math> I(\omega,T) = \frac{\omega^3 }{4 \pi^3}~\frac{1}{e^{\omega/T}-1} </math>
| |
| |-
| |
| |[[Stefan–Boltzmann constant]] σ defined
| |
| |<math> \sigma = \frac{\pi^2 k_\text{B}^4}{60 \hbar^3 c^2} </math>
| |
| |<math>\ \sigma = \pi^2/60 </math>
| |
| |-
| |
| |[[Jacob Bekenstein|Bekenstein]]–[[Stephen Hawking|Hawking]] [[Black hole thermodynamics|black hole entropy]]<ref>Also see [[Roger Penrose]] (1989) ''[[The Road to Reality]]''. Oxford Univ. Press: 714-17. Knopf.</ref>
| |
| |<math>S_\text{BH} = \frac{A_\text{BH} k_\text{B} c^3}{4 G \hbar} = \frac{4\pi G k_\text{B} m^2_\text{BH}}{\hbar c}</math>
| |
| |<math>S_\text{BH} = A_\text{BH}/4 = 4\pi m^2_\text{BH}</math>
| |
| |-
| |
| | [[Schrödinger's equation]]
| |
| | <math>
| |
| - \frac{\hbar^2}{2m} \nabla^2 \psi(\mathbf{r}, t) + V(\mathbf{r}) \psi(\mathbf{r}, t) = i \hbar \dot{\psi}(\mathbf{r}, t)</math>
| |
| | <math>
| |
| - \frac{1}{2m} \nabla^2 \psi(\mathbf{r}, t) + V(\mathbf{r}) \psi(\mathbf{r}, t) = i \dot{\psi}(\mathbf{r}, t)</math>
| |
| |-
| |
| | [[Hamiltonian (quantum mechanics)|Hamiltonian]] form of [[Schrödinger's equation]]
| |
| | <math> H \left| \psi_t \right\rangle = i \hbar \partial \left| \psi_t \right\rangle/\partial t</math>
| |
| | <math> H \left| \psi_t \right\rangle = i \partial \left| \psi_t \right\rangle/\partial t</math>
| |
| |-
| |
| | Covariant form of the [[Dirac equation]]
| |
| |<math>\ ( i\hbar \gamma^\mu \partial_\mu - mc) \psi = 0</math>
| |
| |<math>\ ( i\gamma^\mu \partial_\mu - m) \psi = 0</math>
| |
| |-
| |
| | [[Coulomb's law]]
| |
| | <math> F = \frac{1}{4 \pi \epsilon_0} \frac{q_1 q_2}{r^2} </math>
| |
| | <math> F = \frac{q_1 q_2}{r^2} </math>
| |
| |-
| |
| | [[Maxwell's equations]]
| |
| | <math>\nabla \cdot \mathbf{E} = \frac{1}{\epsilon_0} \rho</math>
| |
| <math>\nabla \cdot \mathbf{B} = 0 \ </math><br />
| |
| <math>\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}</math><br />
| |
| <math>\nabla \times \mathbf{B} = \frac{1}{c^2} \left(\frac{1}{\epsilon_0} \mathbf{J} + \frac{\partial \mathbf{E}} {\partial t} \right)</math>
| |
| | <math>\nabla \cdot \mathbf{E} = 4 \pi \rho \ </math>
| |
| <math>\nabla \cdot \mathbf{B} = 0 \ </math><br />
| |
| <math>\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}</math><br />
| |
| <math>\nabla \times \mathbf{B} = 4 \pi \mathbf{J} + \frac{\partial \mathbf{E}} {\partial t}</math>
| |
| |}
| |
| | |
| ==Other possible normalizations==
| |
| 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.
| |
| | |
| There are several possible alternative normalizations.
| |
| | |
| ===Gravity===
| |
| In 1899, Newton's law of universal gravitation was still seen as exact,{{Citation needed|date=May 2012}} rather than as a convenient approximation holding for "small" velocities and distances (the non-fundamental nature of Newton's law was shown to be true 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 multiplied by 4π or a small integer multiple thereof. Hence a fundamental choice that has to be made when designing a system of natural units is which, if any, instances of 4π appearing in the equations of physics are to be eliminated via the normalization.
| |
| | |
| *Normalizing 4π''G'' to 1:
| |
| :* [[Gauss's law for gravity]] becomes Φ<sub>'''g'''</sub> = −''M'' (rather than Φ<sub>'''g'''</sub> = −4π''M'' in Planck units).
| |
| :* 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> simplifies to ''S''<sub>BH</sub> = π''A''<sub>BH</sub> = (''m''<sub>BH</sub>)<sup>2</sup>, where ''A''<sub>BH</sub> and ''m''<sub>BH</sub> are both measured in a slight modification of reduced Planck units, described below.
| |
| :* The [[characteristic impedance]] ''Z''<sub>0</sub> of [[gravitational radiation]] in free space becomes equal to 1. (It is equal to 4π''G''/''c'' in any system of units.)<ref>Chiao, Raymond Y. (2007) "[http://arxiv.org/abs/0710.1378v4 Generation and detection of gravitational waves at microwave frequencies by means of a superconducting two-body system.]"</ref><ref>General relativity predicts that gravitational radiation propagates at the same speed as electromagnetic radiation.</ref>
| |
| :* No factors of 4π appear in the [[Gravitoelectromagnetism|gravitoelectromagnetic]] (GEM) equations, which hold in weak [[gravitational field]]s or [[Minkowski space#Locally flat spacetime|locally flat space-time]]. These equations have the same form as Maxwell's equations (and the [[Lorentz force]] equation) of [[electromagnetism]], with [[mass density]] replacing [[charge density]], and with 1/(4π''G'') replacing ε<sub>0</sub>.
| |
| *Setting 8π''G'' = 1. This would eliminate 8π''G'' from the [[Einstein field equations]], [[Einstein–Hilbert action]], [[Friedmann equations]], and the [[Poisson equation]] for gravitation. Planck units modified so that 8π''G'' = 1 are known as ''reduced Planck units'', because the Planck mass is divided by {{sqrt|8π}}. Also, the Bekenstein–Hawking formula for the entropy of a black hole simplifies to ''S''<sub>BH</sub> = 2(''m''<sub>BH</sub>)<sup>2</sup> = 2π''A''<sub>BH</sub>.
| |
| *Setting 16π''G'' = 1. This would eliminate the constant ''c''<sup>4</sup>/(16π''G'') from the Einstein–Hilbert action. The form of the Einstein field equations with [[cosmological constant]] Λ becomes ''R<sub>μν</sub>'' − Λ''g<sub>μν</sub>'' = (''Rg<sub>μν</sub>'' − ''T<sub>μν</sub>'')/2.
| |
| | |
| ===Electromagnetism===
| |
| Planck normalized to 1 the [[Coulomb force constant]] 1/(4π''ε''<sub>0</sub>) (as does the [[cgs]] system of units). This sets the [[Planck impedance]], ''Z''<sub>P</sub> equal to ''Z''<sub>0</sub>/4π, where ''Z''<sub>0</sub> is the [[characteristic impedance of free space]].
| |
| | |
| *Normalizing the [[permittivity of free space]] ''ε''<sub>0</sub> to 1:
| |
| :*Sets the [[permeability of free space]] ''µ''<sub>0</sub> = 1, (because ''c'' = 1).
| |
| :*Sets the unit impedance, ''Z''<sub>P</sub> = [[characteristic impedance of free space|''Z''<sub>0</sub>]].
| |
| :*Eliminates 4π from the nondimensionalized form of [[Maxwell's equations]].
| |
| :*Eliminates ''ε''<sub>0</sub> from the nondimensionalized form of [[Coulomb's law]], leaving 1/4π
| |
| | |
| ===Temperature===
| |
| Planck normalized to 1 the [[Boltzmann constant]] ''k''<sub>B</sub>.
| |
| | |
| *Normalizing 1/2 ''k''<sub>B</sub> to 1:
| |
| **Removes the factor of 1/2 in the nondimensionalized equation for the [[thermal energy]] per particle per [[Degrees of freedom (physics and chemistry)|degree of freedom]].
| |
| **Introduces a factor of 2 into the nondimensionalized form of Boltzmann's entropy formula.
| |
| **Does not affect the value of any base or derived Planck unit other than the [[Planck temperature]], which it doubles.
| |
| | |
| The factor 4π is ubiquitous in [[theoretical physics]] because the surface area of a [[sphere]] is 4π''r''<sup>2</sup>. This, along with the concept of [[flux]] is the basis for the [[inverse-square law]]. For example, [[Gravitational field|gravitational]] and [[electrostatic field]]s produced by point charges have spherical symmetry (Barrow 2002: 214–15). The 4π''r''<sup>2</sup> appearing in the denominator of Coulomb's law, for example, follows from the flux of an electrostatic field being distributed uniformly on the surface of a sphere. If space had more than three spacial dimensions, the factor 4π would have to be changed according to the geometry of the [[Sphere#Generalization to other dimensions|sphere in higher dimensions]]. Likewise for Newton's law of universal gravitation.
| |
| | |
| Hence a substantial body of physical theory discovered since Planck (1899) suggests normalizing to 1 not ''G'' but 4''n''π''G'', for one of ''n'' = 1, 2, or 4. Doing so would introduce a factor of 1/(4''n''π) into the nondimensionalized form of the law of universal gravitation, consistent with the modern formulation of Coulomb's law in terms of the vacuum permittivity. In fact, alternative normalizations frequently preserve the factor of 1/(4π) in the nondimensionalized form of Coulomb's law as well, so that the nondimensionalized Maxwell's equations for electromagnetism and gravitomagnetism both take the same form as those for electromagnetism in SI, which does not have any factors of 4π.
| |
| | |
| ==Uncertainties in measured values==
| |
| Table 2 clearly defines Planck units in terms of the fundamental constants. Yet relative to other units of measurement such as [[International System of Units|SI]], the values of the Planck units are only known ''approximately.'' This is mostly due to uncertainty in the value of the gravitational constant ''G''.
| |
| | |
| Today the value of the speed of light ''c'' in SI units is not subject to measurement error, because the SI base unit of length, the [[metre]], is now ''defined'' as the length of the path travelled by light in vacuum during a time interval of {{frac|{{gaps|299|792|458}}}} of a second. Hence the value of ''c'' is now exact by definition, and contributes no uncertainty to the SI equivalents of the Planck units. The same is true of the value of the vacuum permittivity ''ε''<sub>0</sub>, due to the definition of [[ampere]] which sets the [[vacuum permeability]] ''μ''<sub>0</sub> to {{nowrap|4π × 10<sup>−7</sup> H/m}} and the fact that ''μ''<sub>0</sub>''ε''<sub>0</sub> = 1/''c''<sup>2</sup>. The numerical value of the reduced Planck constant ℏ has been determined experimentally to 44 parts per billion, while that of ''G'' has been determined experimentally to no better than 1 part in 8300 (or 120000 parts per billion).<ref name="CODATA" /> ''G'' appears in the definition of almost every Planck unit in Tables 2 and 3. Hence the uncertainty in the values of the Table 2 and 3 SI equivalents of the Planck units derives almost entirely from uncertainty in the value of ''G''. (The propagation of the error in ''G'' is a function of the exponent of ''G'' in the algebraic expression for a unit. Since that exponent is ±{{Frac|1|2}} for every base unit other than Planck charge, the relative uncertainty of each base unit is about one half that of ''G''. This is indeed the case; according to CODATA, the experimental values of the SI equivalents of the base Planck units are known to about 1 part in 16600, or 60000 parts per billion.)
| |
| | |
| == Discussion ==
| |
| Some Planck units are suitable for measuring quantities that are familiar from daily experience. For example:
| |
| *1 [[Planck mass]] is about [[1 E-8 kg|22 micrograms]];
| |
| *1 [[Planck momentum]] is about 6.5 kg m/s;
| |
| *1 [[Planck energy]] is about [[1 E9 J|500 kWh]];
| |
| *1 [[Planck charge]] is slightly more than 11.7 [[elementary charge]]s;
| |
| *1 [[Planck impedance]] is very nearly 30 [[ohm]]s.
| |
| | |
| However, most Planck units are many [[orders of magnitude]] too large or too small to be of any practical use, so that Planck units as a system are really only relevant to theoretical physics. In fact, 1 Planck unit is often the largest or smallest value of a physical quantity that makes sense according to our current understanding. For example:
| |
| *A speed of 1 Planck length per Planck time is the [[speed of light]] in a vacuum, the maximum possible speed in [[special relativity]];<ref>{{cite book |last1=Feynman |first1=R. P. |authorlink1=Richard Feynman |last2=Leighton |first2=R. B. |authorlink2=Robert B. Leighton |last3=Sands |first3=M. |title=[[The Feynman Lectures on Physics]] |volume=1 "Mainly mechanics, radiation, and heat" |publisher=Addison-Wesley |pages=15–9 |chapter=The Special Theory of Relativity |year=1963 |isbn=0-7382-0008-5 |lccn=6320717}}</ref>
| |
| *Our understanding of the [[Big Bang]] begins with the [[Planck epoch]], when the universe was 1 Planck time old and 1 Planck length in diameter, and had a Planck temperature of 1. At that moment, [[Quantum mechanics|quantum theory]] as presently understood becomes applicable. Understanding the universe when it was less than 1 Planck time old requires a theory of [[quantum gravity]] that would incorporate quantum effects into [[general relativity]]. Such a theory does not yet exist;
| |
| *At a Planck temperature of 1, all [[symmetry breaking|symmetries broke]] since the early Big Bang would be restored, and the four [[fundamental interaction|fundamental forces]] of contemporary physical theory would become one force.{{fact|date=September 2013}}
| |
| | |
| Relative to the [[Planck Epoch]], the universe today looks extreme when expressed in Planck units, as in this set of approximations:<ref name="John D 2002">[[John D. Barrow]], 2002. ''The Constants of Nature; From Alpha to Omega - The Numbers that Encode the Deepest Secrets of the Universe''. Pantheon Books. ISBN 0-375-42221-8.</ref><ref>{{BarrowTipler1986}}</ref>
| |
| | |
| {| class="wikitable" style="margin: 1em auto 1em auto; background-color: #ffffff"
| |
| |+Table 5: Today's universe in Planck units.
| |
| ! Property of<br/> present-day [[Universe]]
| |
| ! Approximate number<br/> of Planck units
| |
| ! Equivalents
| |
| |- 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 13.8 × 10<sup>9</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]]
| |
| |-
| |
| | [[Observable universe#Mass|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]])
| |
| |-
| |
| | [[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]]
| |
| | 5.6 × 10<sup>−122</sup> ''t''<sub>P</sub><sup>−2</sup>
| |
| | 1.9 × 10<sup>−35</sup> s<sup>−2</sup>
| |
| |-
| |
| | [[Hubble constant]]
| |
| | 1.24 × 10<sup>−61</sup> ''t''<sub>P</sub><sup>−1</sup>
| |
| | 67.8 (km/s)/[[parsec|Mpc]]
| |
| |}
| |
| | |
| The recurrence of large numbers close or related to 10<sup>60</sup> in the above table is a coincidence that intrigues some theorists. It is an example of the kind of [[Dirac large numbers hypothesis|large numbers coincidence]] that led theorists such as [[Arthur Stanley Eddington|Eddington]] and [[Paul Dirac|Dirac]] to develop alternative physical theories. Theories derived from such coincidences have sometimes been dismissed by mainstream physicists as "[[numerology]]".{{Citation needed|date=January 2013}}
| |
| | |
| ===History===
| |
| [[Natural units]] began in 1881, when [[George Johnstone Stoney]], noting that electric charge is quantized, derived units of length, time, and mass, now named [[Stoney units]] in his honor, by normalizing ''G'', ''c'', and the [[elementary charge|electron charge]], ''e'', to 1. In 1898, [[Max Planck]] discovered that [[Action (physics)|action]] is quantized, and published the result in a paper presented to the Prussian Academy of Sciences in May 1899.<ref>Planck (1899), p. 479.</ref><ref name="TOM">*Tomilin, K. A., 1999, "[http://dbserv.ihep.su/~pubs/tconf99/ps/tomil.pdf Natural Systems of Units: To the Centenary Anniversary of the Planck System]", 287-296.</ref> At the end of the paper, Planck introduced, as a consequence of his discovery, the base units later named in his honor. The Planck units are based on the quantum of action, now usually known as [[Planck's constant]]. Planck called the constant ''b'' in his paper, though ''h'' is now common. Planck underlined the universality of the new unit system, writing:
| |
| {{bq|''...ihre Bedeutung für alle Zeiten und für alle, auch außerirdische und außermenschliche Kulturen notwendig behalten und welche daher als »natürliche Maßeinheiten« bezeichnet werden können...''
| |
| ...These necessarily retain their meaning for all times and for all civilizations, even extraterrestrial and non-human ones, and can therefore be designated as "natural units"...}}
| |
| Planck considered only the units based on the universal constants ''G'', ''ħ'', ''c'', and ''k''<sub>B</sub> to arrive at natural units for [[length]], [[time]], [[mass]], and [[temperature]].<ref name="TOM"/> Planck did not adopt any electromagnetic units. However, since the [[Lorentz–Heaviside_units#Rationalization|non-rationalized]] gravitational constant, ''G'', is set to 1, a natural generalization of Planck units to a unit [[electric charge]] is to also set the non-rationalized Coulomb constant, ''k''<sub>e</sub>, to 1 as well.
| |
| <ref name="PAV">{{cite book|last=Pavšic|first=Matej|title=The Landscape of Theoretical Physics: A Global View|year=2001|publisher=Kluwer Academic|location=Dordrecht|isbn=0-7923-7006-6|pages=347–352|url=http://www.springer.com/physics/theoretical%2C+mathematical+%26+computational+physics/book/978-0-7923-7006-2?otherVersion=978-1-4020-0351-6}}</ref> Planck's paper also gave numerical values for the base units that were close to modern values.
| |
| | |
| === Planck units and the invariant scaling of nature ===
| |
| Some theorists (such as [[Paul Dirac|Dirac]] and [[Edward Arthur Milne|Milne]]) have proposed [[Cosmology|cosmologies]] that conjecture that physical "constants" might actually change over time (e.g. [[Variable speed of light]] or [[Dirac large numbers hypothesis|Dirac varying-G theory]]). Such cosmologies have not gained mainstream acceptance and yet there is still considerable scientific interest in the possibility that physical "constants" might change, although such propositions introduce difficult questions. Perhaps the first question to address is: How would such a change make a noticeable operational difference in physical measurement or, more fundamentally, our perception of reality? If some particular physical constant had changed, how would we notice it, how would physical reality be different? Which changed constants result in a meaningful and measurable difference in physical reality? If a [[physical constant]] that is not [[dimensionless]], such as the [[speed of light]], ''did'' in fact change, would we be able to notice it or measure it unambiguously? – a question examined by [[Michael Duff (physicist)|Michael Duff]] in his paper "Comment on time-variation of fundamental constants".<ref name="hep-th0208093">[[Michael Duff (physicist)|Michael Duff]] (2002). [http://arxiv.org/abs/hep-th/0208093 "Comment on time-variation of fundamental constants"].</ref>
| |
| | |
| [[George Gamow]] argued in his book ''[[Mr Tompkins in Wonderland]]'' that a sufficient change in a dimensionful physical constant, such as the speed of light in a vacuum, would result in obvious perceptible changes. But this idea is challenged:
| |
| | |
| {{quotation|[An] important lesson we learn from the way that pure numbers like ''α'' define the world is what it really means for worlds to be different. The pure number we call the fine structure constant and denote by ''α'' is a combination of the electron charge, ''e'', the speed of light, ''c'', and Planck's constant, ''h''. At first we might be tempted to think that a world in which the speed of light was slower would be a different world. But this would be a mistake. If ''c'', ''h'', and ''e'' were all changed so that the values they have in metric (or any other) units were different when we looked them up in our tables of physical constants, but the value of ''α'' remained the same, this new world would be ''observationally indistinguishable'' from our world. The only thing that counts in the definition of worlds are the values of the dimensionless constants of Nature. If all masses were doubled in value [including the Planck mass ''m''<sub>P</sub> ] you cannot tell because all the pure numbers defined by the ratios of any pair of masses are unchanged.|Barrow 2002<ref name="John D 2002"/>}}
| |
| | |
| Referring to Duff's "Comment on time-variation of fundamental constants"<ref name="hep-th0208093"/> and Duff, Okun, and [[Gabriele Veneziano|Veneziano]]'s paper "Trialogue on the number of fundamental constants",<ref name="DOV">[[Michael Duff (physicist)|Michael Duff]], O. Okun and [[Gabriele Veneziano]] (2002). [http://arxiv.org/abs/physics/0110060 "Trialogue on the number of fundamental constants"]. ''[[Journal of High Energy Physics]]'' '''3''': 023.</ref> particularly the section entitled "The operationally indistinguishable world of Mr. Tompkins", if all physical quantities (masses and other properties of particles) were expressed in terms of Planck units, those quantities would be dimensionless numbers (mass divided by the Planck mass, length divided by the Planck length, etc.) and the only quantities that we ultimately measure in physical experiments or in our perception of reality are dimensionless numbers. When one commonly measures a length with a ruler or tape-measure, that person is actually counting tick marks on a given standard or is measuring the length relative to that given standard, which is a dimensionless value. It is no different for physical experiments, as all physical quantities are measured relative to some other like-dimensioned quantity.
| |
| | |
| We can notice a difference if some dimensionless physical quantity such as [[fine-structure constant]], ''α'', changes or the [[proton-to-electron mass ratio]], ''m''<sub>p</sub>/''m''<sub>e</sub>, changes (atomic structures would change) but if all dimensionless physical quantities remained unchanged (this includes all possible ratios of identically dimensioned physical quantity), we can not tell if a dimensionful quantity, such as the [[speed of light]], ''c'', has changed. And, indeed, the Tompkins concept becomes meaningless in our perception of reality if a dimensional quantity such as ''c'' [[Variable speed of light|has changed]], even drastically.
| |
| | |
| If the speed of light ''c'', were somehow suddenly cut in half and changed to ''c''/2, (but with the axiom that ''all'' dimensionless physical quantities continuing to remain the same), then the Planck Length would ''increase'' by a factor of <math>\scriptstyle 2 \sqrt{2}</math> from the point-of-view of some unaffected "god-like" observer on the outside.{{Citation needed|reason=statement is not trivial|date=July 2013}} Measured by "mortal" observers in terms of Planck units, the new speed of light would be remain as 1 new Planck length per 1 new Planck time – which is no different from the old measurement. But, since by axiom, the size of atoms (approximately the [[Bohr radius]]) are related to the Planck length by an unchanging dimensionless constant of proportionality:
| |
| | |
| : <math>a_0 = \frac{4 \pi \epsilon_0 \hbar^2}{m_e e^2} = \frac{m_\text{P}}{m_e \alpha} l_\text{P}. </math>
| |
| | |
| Then atoms would be bigger (in one dimension) by <math>\scriptstyle 2 \sqrt{2}</math>, each of us would be taller by <math>\scriptstyle 2 \sqrt{2}</math>, and so would our metre sticks be taller (and wider and thicker) by a factor of <math>\scriptstyle 2 \sqrt{2}</math>. Our perception of distance and lengths relative to the Planck length is, by axiom, an unchanging dimensionless constant.
| |
| | |
| Our clocks would tick slower by a factor of <math>\scriptstyle 4 \sqrt{2}</math> (from the point-of-view of this unaffected "god-like" observer) because the Planck time has increased by <math>\scriptstyle 4 \sqrt{2}</math> but we would not know the difference (our perception of durations of time relative to the Planck time is, by axiom, an unchanging dimensionless constant). This hypothetical god-like observer on the outside might observe that light now propagates at half the speed that it previously did (as well as all other observed velocities) but it would still travel [[speed of light|299792458]] of our ''new'' metres in the time elapsed by one of our ''new'' seconds (<math>\scriptstyle \frac{c}{2} \frac{4\sqrt{2}}{2\sqrt{2}}</math> continues to equal 299792458 m/s). ''We'' would not notice any difference.
| |
| | |
| This contradicts what [[George Gamow]] writes in his book ''[[Mr. Tompkins]]''; there, Gamow suggests that if a dimension-dependent universal constant such as ''c'' changed, we ''would'' easily notice the difference. The disagreement is better thought of as the ambiguity in the phrase ''"changing a physical constant"''; what would happen depends on whether (1) all other dimension'''less''' constants were kept the same, or whether (2) all other dimension-'''dependent''' constants are kept the same. The second choice is a somewhat confusing possibility, since most of our units of measurement are defined in relation to the outcomes of physical experiments, and the experimental results depend on the constants. (The only exception is the [[kilogram]].) Gamow does not address this subtlety; the thought experiments he conducts in his popular works assume the second choice for ''"changing a physical constant"''. And Duff or Barrow{{Citation needed|reason=''Who'' said this?|date=July 2013}} would point out that ascribing a change in measurable reality, i.e. [[fine-structure constant|α]], to a specific dimensional component quantity, such as [[speed of light|''c'']], is unjustified. The very same operational difference in measurement or perceived reality could just as well be caused by a change in [[Planck constant|''h'']] or [[elementary charge|''e'']].
| |
| | |
| This unvarying aspect of the Planck-relative scale, or that of any other system of natural units, leads many theorists to conclude that a hypothetical change in dimensionful physical constants can only be manifest as a change in [[dimensionless physical constant]]s. One such dimensionless physical constant is the [[fine-structure constant]]. There are some experimental physicists who assert they have in fact measured a change in the fine structure constant<ref>Webbe, J. K. et al. (1999). [http://arxiv.org/abs/astro-ph/0012539v3 "Further evidence for cosmological evolution of the fine structure constant"]. ''[[Physical Review Letters|Phys. Rev. Lett.]]'' '''82''': 884.</ref> and this has intensified the debate about the measurement of physical constants. According to some theorists<ref>[[Paul C. Davies]], T. M. Davis, and C. H. Lineweaver (2002) "Cosmology: Black Holes Constrain Varying Constants," ''[[Nature]]'' '''418''': 602.</ref> there are some very special circumstances in which changes in the fine-structure constant ''can'' be measured as a change in ''dimensionful'' physical constants. Others however reject the possibility of measuring a change in dimensionful physical constants under any circumstance.<ref name="hep-th0208093"/> The difficulty or even the impossibility of measuring changes in dimensionful physical constants has led some theorists to debate with each other whether or not a dimensionful physical constant has any practical significance at all and that in turn leads to questions about which dimensionful physical constants are meaningful.<ref name="DOV"/>
| |
| | |
| == See also ==
| |
| {{Portal|Physics}}
| |
| * [[Dimensional analysis]]
| |
| * [[Doubly special relativity]]
| |
| * [[Planck scale]]
| |
| * [[Planck particle]]
| |
| * [[Zero-point energy]]
| |
| * [[cGh physics]]
| |
| | |
| ==Notes==
| |
| {{Reflist}}
| |
| | |
| ==References==
| |
| *{{cite book |title=The Constants of Nature; From Alpha to Omega - The Numbers that Encode the Deepest Secrets of the Universe |last=Barrow |first=John D. |authorlink=John D. Barrow |coauthors= |year=2002 |publisher=Pantheon Books |location=New York |isbn=0-375-42221-8 |pages= }} Easier.
| |
| *{{cite book |title=[[Anthropic Principle#The Anthropic Cosmological Principle|The Anthropic Cosmological Principle]] |author=——— |coauthors=[[Frank J. Tipler|Tipler, Frank J.]] |year=1986 |publisher=Claredon Press |location=Oxford |isbn= 0-19-851949-4 |pages= }} Harder.
| |
| *{{cite journal |last=Duff |first=Michael |authorlink=Michael Duff (physicist) |coauthors= |year=2002 |month= |title=Comment on time-variation of fundamental constants |journal=ArΧiv e-prints |volume= |issue= |bibcode=2002hep.th....8093D|pages= 8093 |url= |accessdate= |quote= |arxiv=hep-th/0208093 }}
| |
| *{{cite journal |author=——— |coauthors=Okun, L. B.; [[Gabriele Veneziano|Veneziano, Gabriele]] |year=2002 |month= |title=Trialogue on the number of fundamental constants |journal=Journal of High Energy Physics |volume=03 |issue=|pages=023 |doi=10.1088/1126-6708/2002/03/023 |url= |accessdate= |quote= |arxiv=physics/0110060 |bibcode = 2002JHEP...03..023D }}
| |
| *{{cite book |title=[[The Road to Reality]] |last=Penrose |first=Roger |authorlink=Roger Penrose |coauthors= |year=2005 |publisher=Alfred A. Knopf |location=New York |isbn=0-679-45443-8 |pages=Section 31.1 |nopp=true }}
| |
| *{{cite journal |last=Planck |first=Max |authorlink=Max Planck |coauthors= |year=1899 |month= |title=Über irreversible Strahlungsvorgänge |journal=Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin |volume=5 |issue= |pages=440–480 |id= |url=http://bibliothek.bbaw.de/bibliothek-digital/digitalequellen/schriften/anzeige/index_html?band=10-sitz/1899-1&seite:int=454 |accessdate= |quote= }} pp. 478–80 contain the first appearance of the Planck base units other than the [[Planck charge]], and of [[Planck's constant]], which Planck denoted by ''b''. ''a'' and ''f'' in this paper correspond to ''[[Boltzmann's constant|k]]'' and ''[[gravitation constant|G]]'' in this entry.
| |
| *{{cite journal |last=Tomilin |first=K. A. |author= |authorlink= |coauthors= |title=Natural Systems of Units: To the Centenary Anniversary of the Planck System |version= |pages=287–296 |publisher= |year=1999 |url=http://dbserv.ihep.su/~pubs/tconf99/ps/tomil.pdf |format= |id= |accessdate= }}
| |
| | |
| ==External links==
| |
| *[http://physics.nist.gov/cuu/Constants/index.html Value of the fundamental constants], including the Planck base units, as reported by the [[National Institute of Standards and Technology]] (NIST).
| |
| *Sections C-E of [http://www.planck.com/ collection of resources] bear on Planck units. As of 2011, those pages had been removed from the planck.org web site. Use the [http://web.archive.org/web/*/http://www.planck.com/ Wayback Machine] to access pre-2011 versions of the website. Good discussion of why 8π''G'' should be normalized to 1 when doing [[general relativity]] and [[quantum gravity]]. Many links.
| |
| *[http://www.scientificblogging.com/hammock_physicist/grand_arena The universe and the parameters that describe it in Planck units] Pulls together various physics concepts into one unifying picture.
| |
| | |
| {{Planckunits}}
| |
| | |
| {{Systems of measurement}}
| |
| | |
| {{DEFAULTSORT:Planck Units}}
| |
| [[Category:Natural units| ]]
| |
| [[Category:Max Planck|units]]
| |