- Planck units
**Planck units**areunits of measurement named after the German physicistMax Planck , who first proposed them in 1899. They are an example ofnatural units , i.e. units of measurement designed so that certain fundamentalphysical constants are normalized to 1. In Planck units, the constants thus normalized are:

*thegravitational constant , "G";

*Dirac's constant or reducedPlanck's constant , $hbar$;

*thespeed of light in a vacuum, "c";

*the Coulomb force constant, $frac\{1\}\{4\; pi\; varepsilon\_0\}\; ext\{;\}$

*Boltzmann's constant , "k"_{B}(or simply "k"). Each of these constants can be associated with at least one fundamental physical theory: "c" withspecial relativity , "G" withgeneral relativity andNewtonian gravity , $hbar$ withquantum physics , "ε"_{0}withelectrostatics , and "k" withstatistical mechanics andthermodynamics . Planck units have profound significance for theoretical physics since they elegantly simplify several recurringalgebraic expression s ofphysical law . They are particularly relevant in research on unified theories such asquantum gravity .**Base Planck units**All systems of measurement feature "base units". (In the SI system, for example, the base unit of length is the

meter .) In the system of Planck units, the Planck base unit of length is known simply as thePlanck length , the base unit of time is thePlanck time , etc. These units are derived from the five fundamental physical constants in Table 1, which are arranged in Table 2 so as to cancel out the unwanted dimensions, leaving only the dimension appropriate to each unit. (Like all systems ofnatural units , Planck units are an instance ofdimensional analysis .)**Table 1: Fundamental physical constants****Planck units simplify the key equations of physics**Ordinarily, 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). However, in theoretical physics this scruple can be set aside in order to simplify calculations. The process by which this is done is called

Nondimensionalization . Table 4 shows how Planck units, by setting the numerical values of the five fundamental constants to unity, simplify many equations of physics and make them nondimensional.**Table 4: Nondimensionalized equations****Alternative normalizations**As already stated in the introduction, 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 constants to normalize is not evident, and the values of the Planck units are sensitive to this choice.

The factor 4π, and multiples of it such as 8π, are ubiquitous in formulas in

theoretical physics because it is the surface area of the unitsphere in three dimensions. For example, gravitational andelectrostatic field s produced by point charges have spherical symmetry. (Barrow 2002: 214-15) The 4π"r"^{2}appearing in the denominator ofCoulomb's law , for example, reflects the fact that theflux of electric field distributes uniformly on the surface of the sphere. If space had more dimensions, the factor corresponding to 4π would be different.In any event, a fundamental choice that has to be made when designing a system of natural units is which, if any, instances of 4"n"π appearing in the equations of physics are to be eliminated via normalization.

*Setting "ε"

_{0}= 1.Planck normalized to 1 theCoulomb force constant 1/(4π"ε"_{0}) (as does thecgs system of units). This sets thePlanck impedance , "Z"_{P}equal to "Z"_{0}/4π, where "Z"_{0}is thecharacteristic impedance of free space . Normalizing thepermittivity of free space "ε"_{0}to 1 not only would make "Z"_{P}equal to "Z"_{0}, but would also eliminate 4π fromMaxwell's equations . On the other hand, the nondimensionalized form of Coulomb's law would now include a factor of 1/(4π).

*Setting 4"n"π"G" = 1.In 1899,general relativity lay some years in the future, so that Newton'slaw of universal gravitation was still seen as fundamental, rather than as a convenient approximation holding for "small" velocities and distances. Hence Planck normalized to 1 thegravitational constant "G" in Newton's law. In theories emerging after 1899, "G" is nearly always multiplied by 4π or multiples.:* 4π"G" appears in the::**Gauss's law for gravity , Φ_{g}= −4π"GM";:**Characteristic impedance ofgravitational radiation in free space, "Z"_{0}= 4π"G"/"c". [*[*] The "c" in the denominator stems from the*http://arxiv.org/abs/0710.1378v4 arXiv:0710.1378v4*]general relativity prediction that gravitational radiation propagates at the same speed as electromagnetic radiation; :**Gravitoelectromagnetic (GEM) equations, which hold in weakgravitational field s or reasonably flat space-time. These equations have the same form as Maxwell's equations (and theLorentz force equation) of electromagnetism, withmass density replacingcharge density , and with 1/(4π"G") replacing ε_{0}.:*8π"G" appears in theEinstein field equations ,Einstein-Hilbert action ,Friedmann equations , and thePoisson equation for gravitation. Planck units modified so that 8π"G" = 1 are known as "reduced Planck units", because the Planck mass is divided by $sqrt\{8pi\}\; ext\{.\}$:*Setting 16π"G" = 1 would eliminate the constant "k" = "c"^{4}/(16π"G") from theEinstein-Hilbert action . TheEinstein field equations withcosmological constant Λ becomes "R_{μν}" − Λ"g_{μν}" = ("Rg_{μν}" − "T_{μν}")/2.Hence a substantial body of physical theory discovered since Planck (1899) suggests normalizing to 1 not "G" but 4"n"π"G", "n" = 1, 2, or 4. However, doing so would introduce a factor of 1/(4"n"π) into the nondimensionalized law of universal gravitation.

*Setting "k" to 2.This would remove the factor of 2 in the nondimensionalized equation for thethermal energy per particle per degree of freedom, and would not affect the value of any base or derived unit other than thePlanck temperature .**Uncertainties in values**Planck units are clearly defined in terms of fundamental constants in Table 2 and yet, relative to other units of measurement such as SI, the values of those 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 nowrap|1/299 792 458 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"_{0}, due to the definition ofampere which sets thevacuum permeability "μ"_{0}to nowrap|4π × 10^{−7}H/m and the fact that "μ"_{0}"ε"_{0}= 1/"c"^{2}. The numerical value of Dirac's constant "ℏ" has been determined experimentally to 50 parts per billion, while that of "G" has been determined experimentally to no better than 1 part in 10000. "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 20,000.)**Discussion**Physicists sometimes humorously refer to Planck units as "God's units", as Planck units are free of arbitrary anthropocentricity. Unlike the

meter andsecond , which exist as fundamental units in theSI system for historical reasons, thePlanck length andPlanck time are conceptually linked at a fundamental physical level.Some Planck units are suitable for measuring quantities that are familiar from daily experience. For example:

*1Planck mass is about 22 micrograms;

*1Planck momentum is about 6.5 kg m/s;

*1Planck energy is about 500 kWh;

*1Planck charge is slightly more than 11elementary charge s;

*1Planck impedance is very nearly 30ohm s.However, most Planck units are many

orders of magnitude too large or too small to be of any empirical and 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 given the current state of physical theory. For example:

*a speed of 1 Planck length per Planck time is thespeed of light in a vacuum, the maximum possible velocity inspecial relativity ;

*our understanding of theBig Bang begins with thePlanck Epoch , when the universe grew older and larger than about 1 Planck time and 1 Planck length, at which time it cooled below about 1 Planck temperature and quantum theory as presently understood becomes applicable. Understanding the universe when it was less than 1 Planck time old requires a theory ofquantum gravity , incorporating quantum effects intogeneral relativity . Such a theory does not yet exist;

*at a Planck temperature of 1, all symmetries are broken since the early Big Bang would be restored, and the four fundamental forces of contemporary physical theory would become one force.Relative to the

Planck Epoch , the universe today looks extreme when expressed in Planck units, as in this set of approximations (see for example): [***] and [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.]Frank J. Tipler , 1986. "The Anthropic Cosmological Principle". Oxford University Press. Harder.**Table 5: Today's universe in Planck units**Natural units can help physicists reframe questions. An example of such reframing is the following passage by

Frank Wilczek :**History**Natural units began in 1881, whenGeorge Johnstone Stoney derived units of length, time, and mass, now namedStoney units in his honor, by normalizing "G", "c", and the electron charge "e" to 1. (Stoney was also the first to hypothesize that electric charge is quantized and hence to see the fundamental character of "e".)Max Planck first set out the base units ("q_{P}" excepted) later named in his honor, in a paper presented to the Prussian Academy of Sciences in May 1899. [*Planck (1899), p. 479.*] [**Tomilin, K. A., 1999, " [*] That paper also includes the first appearance of a constant named "b", and later called "h" and named after him. The paper gave numerical values for the base units, in terms of the metric system of his day, that were remarkably close to those in Table 2. We are not sure just how Planck came to discover these units because his paper gave no algebraic details. But he did explain why he valued these units as follows:*http://dbserv.ihep.su/~pubs/tconf99/ps/tomil.pdf Natural Systems of Units: To the Centenary Anniversary of the Planck System,*] " 287-96.quotation|"...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 units and the invariant scaling of nature**Dirac (1937), and others after him, have conjectured that some physical "constants" might actually change over time, a proposition that introduces many difficult questions such as:

*How would such a change make a noticeable operational difference in physical measurement or, more basically, our perception of reality?

*If some physical constant had changed, would we even notice it?

*How would physical reality be different?

*Which changed constants would result in a meaningful and measurable difference?John Barrow has spoken to these questions as follows:

When measuring a length with a ruler or tape measure, one is actually counting tick marks on a given standard, i.e., measuring the length relative to that given standard; the result is a dimensionless value. It is no different for physical experiments, as all physical quantities are measured relative to some other like-dimensioned values. 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 we would measure when observing nature or conducting experiments would be dimensionless numbers. See Duff (2004) and section III.5 (by Duff alone) of Duff, Okun, and Veneziano (2002).

We can notice a difference if some dimensionless physical quantity such as α or the proton/electron mass ratio changes; either change would alter atomic structures. But if all dimensionless physical quantities remained constant (this includes all possible ratios of identically dimensioned physical quantities), we could not tell if a dimensionful quantity, such as the speed of light, "c", had changed. And, indeed, the Tompkins concept becomes meaningless in our existence if a dimensional quantity such as "c" has changed, even drastically.

If the speed of light "c", were somehow suddenly cut in half and changed to frac|"c"|2, (but with "all" dimensionless physical quantities continuing to remain constant), then the

Planck length would "increase" by a factor of √(8) from the point-of-view of some unaffected "god-like" observer on the outside. But then the size of atoms (approximately theBohr radius ) are related to the Planck length by an unchanging dimensionless constant::$a\_0\; =$4pivarepsilon_0hbar^2}over{m_e e^2= m_P}over{m_e alpha l_P

Atoms would then be bigger (in one dimension) by √(8), each of us would be taller by √(8), and so would our meter sticks be taller (and wider and thicker) by a factor of √(8) and we would not know the difference. Our perception of distance and lengths relative to the Planck length is logically an unchanging dimensionless constant.

Moreover, our clocks would tick slower by a factor of √(32) (from the point-of-view of this unaffected "god-like" observer) because the Planck time has increased by √(32) but we would not know the difference. (Our perception of durations of time relative to the Planck time is, axiomatically, an unchanging dimensionless constant.) This hypothetical god-like outside observer might observe that light now travels at half the speed that it used to (as well as all other observed velocities) but it would still travel 299792458 of our "new" meters in the time elapsed by one of our "new" seconds (frac|"c"|2 frac|√(32)|√(8) continues to equal 299792458 m/s). "We" would not notice any difference.

This contradicts what

George Gamow wrote in his book "Mr. Tompkins in Wonderland "; where he suggested that if a dimension-dependent universal constant such as "c" changed, we "would" easily notice the difference. The disagreement stems from the ambiguity in the phrase "changing a physical constant"; what would happen depends on whether we hold constant all other (1) dimension**less**constants, or (2) dimension-**dependent**constants. (2) is a somewhat confusing alternative, 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 thekilogram .) Gamow does not address this subtlety; the thought experiments he conducts in his popular works tacitly assume that (2) defines a "changing physical constant."According to

doubly special relativity (a recent and unconventional development of relativity theory) the Planck length is an invariant, minimum length in the same way that the speed of light is an invariant, maximum velocity.**See also***

Dimensional analysis

*Physical constant s

*Natural units

*zero-point energy

*Planck scale

*Planck particle

*Planck epoch

*Planck length

*Planck time

*Planck force

*doubly special relativity **Footnotes****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=0375422218 |pages= Easier.

*cite book |title=The Anthropic Cosmological Principle |author=——— |coauthors=Tipler, Frank J. |year=1986 |publisher=Claredon Press |location=Oxford |isbn= 0198519494 |pages= Harder.

*cite journal |last=Duff |first=Michael |authorlink=Michael Duff |coauthors= |year=2002 |month= |title=Comment on time-variation of fundamental constants |journal=ArΧiv e-prints |volume= |issue= |pages= |id=arXiv|hep-th|0208093 |url= |accessdate= |quote=

*cite journal |author=——— |coauthors=Okun, L. B.; Veneziano, Gabriele |year=2002 |month= |title=Trialogue on the number of fundamental constants |journal=Journal of High Energy Physics |volume=3 |issue= |pages=023 |id=arXiv|physics|0110060 |doi=10.1088/1126-6708/2002/03/023 |url= |accessdate= |quote=

*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 thePlanck charge , and ofPlanck's constant , which Planck denoted by "b". "a" and "f" in this paper correspond to "k" and "G" in this entry.

*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=0679454438 |pages=Section 31.1

*cite paper |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= |date=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 theNational Institute of Standards and Technology (NIST).

*Sections C-E of [*http://www.planck.com/ collection of resources*] bear on Planck units. Good discussion of why 8π"G" should be normalized to 1 when doinggeneral relativity andquantum gravity . Many links.

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