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{{more footnotes|date=December 2010}}
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.
In [[physics]], the '''Planck mass''', denoted by ''m''<sub>P</sub>, is the unit of [[mass]] in the system of [[natural units]] known as [[Planck units]]. It is defined so that
:<math>m_\text{P}=\sqrt{\frac{\hbar c}{G}}</math>≈ {{val|1.2209|e=19|u=[[GeV]]/c<sup>2</sup>}} = {{val|2.17651|(13)|e=-8|u=kg}}, (or {{val|21.7651|u=μg}}),<ref>CODATA 2010: [http://physics.nist.gov/cgi-bin/cuu/Value?plkmc2gev value in GeV], [http://physics.nist.gov/cgi-bin/cuu/Value?plkm value in kg]</ref>


where ''c'' is the [[speed of light]] in a vacuum, ''G'' is the [[gravitational constant]], and ''ħ'' is the [[reduced Planck constant]].
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
* Only registered users will be able to execute this rendering mode.
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[[Particle physics|Particle physicists]] and [[physical cosmology|cosmologists]] often use the '''reduced Planck mass''', which is
Registered users will be able to choose between the following three rendering modes:  
:<math>\sqrt\frac{\hbar{}c}{8\pi G}</math> ≈ {{val|4.341|e=-9|u=kg}} = 2.435 × 10<sup>18</sup> [[GeV]]/c<sup>2</sup>.
The factor of  <math>1/\sqrt{8\pi}</math>  simplifies a number of equations in [[general relativity]].


The name honors [[Max Planck]] because the unit measures the approximate scale at which quantum effects, here in the case of gravity, become important.  Quantum effects are typified by the magnitude of [[Planck constant|Planck's constant]], <math>h = 2\pi\hbar</math>.
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


==Significance==
<!--'''PNG'''  (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


The Planck mass is nature’s maximum allowed mass for point-masses ([[quanta]]). If two quanta of the Planck mass or greater met, they could spontaneously form a [[black hole]] whose [[Schwarzschild radius]] equals their [[de Broglie wavelength]]. Once such a hole formed, other particles would fall in, and the black hole would experience runaway, explosive growth (assuming it did not evaporate via [[Hawking Radiation]]). Nature’s stable point-mass particles, such as [[electrons]] and [[quarks]], are many, many orders of magnitude smaller than the Planck mass and cannot form black holes in this manner. On the other hand, extended objects (as opposed to point-masses) can have any mass.
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


Unlike all other [[Planck units|Planck base units]] and most Planck derived units, the Planck mass has a scale more or less conceivable to [[human]]s. It is traditionally said to be about the mass of a [[flea]], but more accurately it is about the mass of a flea egg.
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].


==Derivations==
==Demos==
===Dimensional analysis===
The formula for the Planck mass can be derived by [[dimensional analysis]]. In this approach, one starts with the three [[physical constant]]s ħ, c, and G, and attempt to combine them to get a quantity with units of mass. The expected formula is of the form
:<math>m_\text{P} = c^{n_1} G^{n_2} \hbar^{n_3},</math>
where <math>n_1,n_2,n_3</math> are constants to be determined by matching the dimensions of both sides. Using the symbol L for length, T for time, M for mass, and writing "[x]" for the dimensions of some physical quantity x, we have the following:
:<math>[c] = LT^{-1} \ </math>
:<math>[G] = M^{-1}L^3T^{-2} \ </math>
:<math>[\hbar] = M^1L^2T^{-1} \ </math>.
Therefore,
:<math>[c^{n_1} G^{n_2} \hbar^{n_3}] = M^{-n_2+n_3} L^{n_1+3n_2+2n_3} T^{-n_1-2n_2-n_3}</math>
If one wants dimensions of mass, the following equations must hold:
:<math>-n_2 + n_3 = 1 \ </math>
:<math>n_1 + 3n_2 + 2n_3 = 0 \ </math>
:<math>-n_1 - 2n_2 - n_3 = 0 \ </math>.
The solution of this system is:
:<math>n_1 = 1/2, n_2 = -1/2, n_3 = 1/2. \ </math>
Thus, the Planck mass is:
:<math>m_\text{P} = c^{1/2}G^{-1/2}\hbar^{1/2} = \sqrt{\frac{c\hbar}{G}}. </math>


===Elimination of a coupling constant===
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:
Equivalently, the Planck mass is defined such that the [[gravitational potential energy]] between two masses ''m''<sub>P</sub> of separation ''r'' is equal to the energy of a photon (or graviton) of angular wavelength ''r'' (see the [[Planck relation]]), or that their ratio equals one.
:<math>E=\frac{G m_\text{P}^2}{r}=\frac{\hbar c}{r}</math>
Multiplying through,
:<math>G m_\text{P}^2=\hbar c</math>
This equation has units of energy times length and equals the value <math>\hbar c</math>, a ubiquitous quantity when deriving the Planck units. Since the two quantities are equal their ratio equals one. From here, it is easy to isolate the mass that would satisfy this equation in our system of units:
:<math>m_\text{P}=\sqrt{\frac{\hbar c}{G}}</math>
Note in the second equation that if instead of planck masses the electron mass were used, the equation would no longer be unitary and instead equal a [[gravitational coupling constant]], analogous to how the equation of the [[fine-structure constant]] operates with respect to the [[elementary charge]] and the [[Planck charge]]. Thus, the planck mass is an attempt to absorb the gravitational coupling constant into the unit of mass (and those of distance/time as well), as the planck charge does for the fine-structure constant; naturally it is impossible to truly set either of these dimensionless numbers to zero.


===Compton wavelength and Schwarzschild radius===
The Planck mass can be derived approximately by setting it as the mass whose [[Compton wavelength]] and [[Schwarzschild radius]] are equal.<ref>[http://books.google.com/books?id=WYxkrwMidp0C&pg=PR10 The riddle of gravitation] by Peter Gabriel Bergmann, page x</ref> The Compton wavelength is, loosely speaking, the length-scale where [[quantum mechanics|quantum effects]] start to become important for a particle; the heavier the particle, the smaller the Compton wavelength. The Schwarzschild radius is the radius in which a mass, if confined, would become a [[black hole]]; the heavier the particle, the larger the Schwarzschild radius. If a particle were massive enough that its Compton wavelength and Schwarzschild radius were approximately equal, its dynamics would be strongly affected by [[quantum gravity]]. This mass is (approximately) the Planck mass.


The Compton wavelength is
* accessibility:
:<math>\lambda_c = \frac{h}{mc}</math>
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
and the Schwarzschild radius is
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
:<math>r_s = \frac{2Gm}{c^2}</math>
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
Setting them equal:
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
:<math>m = \sqrt{\frac{hc}{2G}} = \sqrt{\frac{\pi c \hbar}{G}}</math>
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
This is not quite the Planck mass: It is a factor of <math>\sqrt{\pi}</math> larger. However, this is a heuristic derivation, only intended to get the right order of magnitude. On the other hand, the previous "derivation" of the Planck mass should have had a proportional sign in the initial expression rather than an equal sign. Therefore, the extra factor might be the correct one.
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


==See also==
==Test pages ==
* [[Micro black hole]]
* [[Orders of magnitude (mass)]]
* [[Planck length]]
* [[Planck particle]]


== Notes and references==
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
{{Reflist|2}}
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]


==Bibliography==
*[[Inputtypes|Inputtypes (private Wikis only)]]
{{Refbegin}}
*[[Url2Image|Url2Image (private Wikis only)]]
*{{cite arXiv |last=Sivaram |first=C. |authorlink= |eprint=0707.0058  |title=What is Special About the Planck Mass? |class=gr-qc |year=2007 |version=v1 |accessdate=13 November 2013}}
==Bug reporting==
{{Refend}}
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
 
==External links==
* [http://physics.nist.gov/cuu/Constants/index.html The NIST Reference on Constants, Units, and Uncertainty]
 
{{Planck's natural units}}
{{Portal bar|Physics}}
 
{{DEFAULTSORT:Planck Mass}}
[[Category:Physical constants]]
[[Category:Natural units|Mass]]
[[Category:Units of mass]]

Latest revision as of 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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