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Template:More footnotes In physics, the Planck mass, denoted by m_P, is the unit of mass in the system of natural units known as Planck units. It is defined so that

m_{P} = \sqrt{\frac{ℏ c}{G}}

≈ Template:Val = Template:Val, (or Template:Val),^[1]

where c is the speed of light in a vacuum, G is the gravitational constant, and ħ is the reduced Planck constant.

Particle physicists and cosmologists often use the reduced Planck mass, which is

\sqrt{\frac{ℏ c}{8 π G}}

≈ Template:Val = 2.435 × 10¹⁸ GeV/c².

The factor of $1 / \sqrt{8 π}$ 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's constant, $h = 2 π ℏ$ .

Significance

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.

Unlike all other Planck base units and most Planck derived units, the Planck mass has a scale more or less conceivable to humans. It is traditionally said to be about the mass of a flea, but more accurately it is about the mass of a flea egg.

Derivations

Dimensional analysis

The formula for the Planck mass can be derived by dimensional analysis. In this approach, one starts with the three physical constants ħ, c, and G, and attempt to combine them to get a quantity with units of mass. The expected formula is of the form

m_{P} = c^{n_{1}} G^{n_{2}} ℏ^{n_{3}},

where $n_{1}, n_{2}, n_{3}$ 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:

[c] = L T^{- 1}

[G] = M^{- 1} L^{3} T^{- 2}

[ℏ] = M^{1} L^{2} T^{- 1}

.

Therefore,

[c^{n_{1}} G^{n_{2}} ℏ^{n_{3}}] = M^{- n_{2} + n_{3}} L^{n_{1} + 3 n_{2} + 2 n_{3}} T^{- n_{1} - 2 n_{2} - n_{3}}

If one wants dimensions of mass, the following equations must hold:

- n_{2} + n_{3} = 1

n_{1} + 3 n_{2} + 2 n_{3} = 0

- n_{1} - 2 n_{2} - n_{3} = 0

.

The solution of this system is:

n_{1} = 1 / 2, n_{2} = - 1 / 2, n_{3} = 1 / 2 .

Thus, the Planck mass is:

m_{P} = c^{1 / 2} G^{- 1 / 2} ℏ^{1 / 2} = \sqrt{\frac{c ℏ}{G}} .

Elimination of a coupling constant

Equivalently, the Planck mass is defined such that the gravitational potential energy between two masses m_P 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.

E = \frac{G m_{P}^{2}}{r} = \frac{ℏ c}{r}

Multiplying through,

G m_{P}^{2} = ℏ c

This equation has units of energy times length and equals the value $ℏ c$ , 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:

m_{P} = \sqrt{\frac{ℏ c}{G}}

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.^[2] The Compton wavelength is, loosely speaking, the length-scale where 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

λ_{c} = \frac{h}{m c}

and the Schwarzschild radius is

r_{s} = \frac{2 G m}{c^{2}}

Setting them equal:

m = \sqrt{\frac{h c}{2 G}} = \sqrt{\frac{π c ℏ}{G}}

This is not quite the Planck mass: It is a factor of $\sqrt{π}$ 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.

Notes and references

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Bibliography

Template:Refbegin

Template:Cite arXiv

Template:Refend

External links

The NIST Reference on Constants, Units, and Uncertainty

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↑ CODATA 2010: value in GeV, value in kg
↑ The riddle of gravitation by Peter Gabriel Bergmann, page x

[1] CODATA 2010: value in GeV, value in kg

[2] The riddle of gravitation by Peter Gabriel Bergmann, page x

[1]

[2]

@@ Line 1: / Line 1: @@
-{{Unreferenced|date=December 2009}}
+{{more footnotes|date=December 2010}}
-{{incomplete|date=January 2011}}
+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
-'''Cyclotron radiation''' is [[electromagnetic radiation]] emitted by moving [[electric charge|charge]]d particles deflected by a [[magnetic field]].  The [[Lorentz force]] on the particles acts perpendicular to both the magnetic field lines and the particles' motion through them, creating an acceleration of charged particles that causes them to emit radiation as a result of the acceleration they undergo as they spiral around the lines of the magnetic field.
+:<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>
-The name of this radiation derives from the [[cyclotron]], a type of [[particle accelerator]] used since the 1930s to create highly energetic particles for study. The cyclotron makes use of the circular orbits that charged particles exhibit in a uniform magnetic field. Furthermore, the period of the orbit is independent of the energy of the particles, allowing the cyclotron to operate at a set [[frequency]]. Cyclotron radiation is emitted by all charged particles travelling through magnetic fields, not just those in cyclotrons. Cyclotron radiation from [[Plasma (physics)|plasma]] in the [[interstellar medium]] or around [[black hole]]s and other astronomical phenomena is an important source of information about distant magnetic fields. The [[Power (physics)|power]] (energy per unit time) of the emission of each electron can be calculated:
+where ''c'' is the [[speed of light]] in a vacuum, ''G'' is the [[gravitational constant]], and ''ħ'' is the [[reduced Planck constant]].
-<math>{-dE \over dt}={\sigma_t B^2 V^2 \over c \mu_o} </math>
+[[Particle physics|Particle physicists]] and [[physical cosmology|cosmologists]] often use the '''reduced Planck mass''', which is
+:<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]].
-where ''E'' is energy, ''t'' is time, <math> \sigma_t </math> is the [[Thomson cross section]] (total, not differential), ''B'' is the magnetic field strength, ''V'' is the velocity perpendicular to the magnetic field, ''c'' is the speed of light and <math> \mu_o </math> is the [[permeability of free space]]. In the context of [[magnetic fusion energy]], cyclotron radiation losses translate into a requirement for a minimum plasma energy density in relation to the magnetic field energy density (see [[Aneutronic fusion#Power density and energy balance]]).
+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>.
-Cyclotron radiation would likely be produced in a high altitude nuclear explosion. [[Gamma ray]]s produced by the explosion would [[ionization|ionize]] [[atom]]s in the upper atmosphere and those free electrons would interact with the Earth's magnetic field to produce cyclotron radiation in the form of an [[electromagnetic pulse]] (EMP).  This phenomenon is of concern to the military as the EMP may damage solid state electronic equipment.
+==Significance==
-Cyclotron radiation has a spectrum with its main spike at the
+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.
-same fundamental frequency as the particle's orbit, and [[harmonic]]s at higher integral factors. Harmonics are the result of imperfections in the actual emission environment, which also create a broadening of the [[spectral line]]s. The most obvious source of line broadening is non-uniformities in the magnetic field; as an electron passes
-from one area of the field to another, its emission frequency will change with the strength of the field. Other sources of broadening include collisional broadening as the electron will invariably fail to follow a perfect orbit, distortions of the emission caused by interactions with the surrounding plasma, and [[special relativity|relativistic]] effects if the charged particles are sufficiently energetic.  When the electrons are moving at relativistic speeds, cyclotron radiation is known as [[synchrotron radiation]].
-The recoil experienced by a particle emitting cyclotron radiation is called [[radiation reaction]]. Radiation reaction acts as a resistance to motion in a cyclotron; and the work necessary to overcome it is the main energetic cost of accelerating a particle in a cyclotron. Cyclotrons are prime examples of systems which experience radiation reaction.
+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.
+==Derivations==
+===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===
+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
+:<math>\lambda_c = \frac{h}{mc}</math>
+and the Schwarzschild radius is
+:<math>r_s = \frac{2Gm}{c^2}</math>
+Setting them equal:
+:<math>m = \sqrt{\frac{hc}{2G}} = \sqrt{\frac{\pi c \hbar}{G}}</math>
+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.
 ==See also==
-*[[Auroral kilometric radiation]] (AKR)
+* [[Micro black hole]]
-*[[Bremsstrahlung]]
+* [[Orders of magnitude (mass)]]
-*[[Synchrotron radiation]]
+* [[Planck length]]
-*[[Free electron laser]]
+* [[Planck particle]]
-*[[Larmor formula]]
+== Notes and references==
+{{Reflist|2}}
+==Bibliography==
+{{Refbegin}}
+*{{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}}
+{{Refend}}
+==External links==
+* [http://physics.nist.gov/cuu/Constants/index.html The NIST Reference on Constants, Units, and Uncertainty]
-{{DEFAULTSORT:Cyclotron Radiation}}
+{{Planck's natural units}}
-[[Category:Electromagnetic radiation]]
+{{Portal bar|Physics}}
-[[Category:Plasma physics]]
-[[Category:Experimental particle physics]]
-[[ar:إشعاع سيكلوتروني]]
+{{DEFAULTSORT:Planck Mass}}
-[[ca:Radiació ciclotró]]
+[[Category:Physical constants]]
-[[fr:Rayonnement cyclotron]]
+[[Category:Natural units|Mass]]
-[[is:Hringhraðlageislun]]
+[[Category:Units of mass]]
-[[it:Radiazione di ciclotrone]]
-[[nl:Cyclotronstraling]]
-[[pt:Radiação ciclotrônica]]
-[[ru:Циклотронное излучение]]
-[[uk:Циклотронне випромінювання]]
-[[zh:回旋辐射]]

Main Page: Difference between revisions

Revision as of 08:41, 10 August 2014

Contents

Significance

Derivations

Dimensional analysis

Elimination of a coupling constant

Compton wavelength and Schwarzschild radius

See also

Notes and references

Bibliography

External links

Navigation menu

Main Page: Difference between revisions

Revision as of 08:41, 10 August 2014

Significance

Derivations

Dimensional analysis

Elimination of a coupling constant

Compton wavelength and Schwarzschild radius

See also

Notes and references

Bibliography

External links

Navigation menu

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