Inparticle physics andphysical cosmology,Planck units are asystem of units of measurement defined exclusively in terms of four universalphysical constants:c,G,ħ, andkB (described further below). Expressing one of these physical constants in terms of Planck units yields a numerical value of1. They are a system ofnatural units, defined using fundamental properties ofnature (specifically, properties offree space) rather than properties of a chosenprototype object. Originally proposed in 1899 by German physicistMax Planck, they are relevant in research on unified theories such asquantum gravity.
The termPlanck scale refers to quantities of space, time, energy and other units that are similar in magnitude to corresponding Planck units. This region may be characterized by particleenergies of around1019 GeV or109 J,time intervals of around5×10−44 s andlengths of around10−35 m (approximately the energy-equivalent of the Planck mass, the Planck time and the Planck length, respectively). At the Planck scale, the predictions of theStandard Model,quantum field theory andgeneral relativity are not expected to apply, andquantum effects of gravity are expected to dominate. One example is represented by the conditions in thefirst 10−43 seconds of our universe after theBig Bang, approximately 13.8 billion years ago.
The fouruniversal constants that, by definition, have a numeric value 1 when expressed in these units are:
Variants of the basic idea of Planck units exist, such as alternate choices of normalization that give other numeric values to one or more of the four constants above.
Any system of measurement may be assigned a mutually independent set of base quantities and associatedbase units, from which all other quantities and units may be derived. In theInternational System of Units, for example, theSI base quantities include length with the associated unit of themetre. In the system of Planck units, a similar set of base quantities and associated units may be selected, in terms of which other quantities and coherent units may be expressed.[1][2]: 1215 The Planck unit of length has become known as the Planck length, and the Planck unit of time is known as the Planck time, but this nomenclature has not been established as extending to all quantities.
All Planck units are derived from the dimensional universal physical constants that define the system, and in a convention in which these units are omitted (i.e. treated as having the dimensionless value 1), these constants are then eliminated from equations of physics in which they appear. For example,Newton's law of universal gravitation,can be expressed as:Both equations aredimensionally consistent and equally valid inany system of quantities, but the second equation, withG absent, is relating onlydimensionless quantities since any ratio of two like-dimensioned quantities is a dimensionless quantity. If, by a shorthand convention, it is understood that each physical quantity is the corresponding ratio with a coherent Planck unit (or "expressed in Planck units"), the ratios above may be expressed simply with the symbols of physical quantity, without being scaled explicitly by their corresponding unit:This last equation (withoutG) is valid withF′,m1′,m2′, andr′ being the dimensionless ratio quantitiescorresponding to the standard quantities, written e.g.F′ ≘F orF′ =F/FP, but not as a direct equality of quantities. This may seem to be "setting the constantsc,G, etc., to 1" if the correspondence of the quantities is thought of as equality. For this reason, Planck or other natural units should be employed with care. Referring to "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."[3]
The concept ofnatural units was introduced in 1874, whenGeorge Johnstone Stoney, noting that electric charge is quantized, derived units of length, time, and mass, now namedStoney units in his honor. Stoney chose his units so thatG,c, and theelectron chargee would be numerically equal to 1.[4] In 1899, one year before the advent of quantum theory,Max Planck introduced what became later known as the Planck constant.[5][6] At the end of the paper, he proposed the base units that were later named in his honor. The Planck units are based on the quantum ofaction, now usually known as the Planck constant, which appeared in theWien approximation forblack-body radiation. Planck underlined the universality of the new unit system, writing:[5]
... die Möglichkeit gegeben ist, Einheiten für Länge, Masse, Zeit und Temperatur aufzustellen, welche, unabhängig von speciellen Körpern oder Substanzen, ihre Bedeutung für alle Zeiten und für alle, auch ausserirdische und aussermenschliche Culturen nothwendig behalten und welche daher als »natürliche Maasseinheiten« bezeichnet werden können.
... it is possible to set up units for length, mass, time and temperature, which are independent of special bodies or substances, necessarily retaining their meaning for all times and for all civilizations, including extraterrestrial and non-human ones, which can be called "natural units of measure".
Planck considered only the units based on the universal constants,,, and to arrive at natural units forlength,time,mass, andtemperature.[6] His definitions differ from the modern ones by a factor of, because the modern definitions use rather than.[5][6]
| Name | Dimension | Expression | Value (SI units) |
|---|---|---|---|
| Planck length | length (L) | 1.616255(18)×10−35 m[7] | |
| Planck mass | mass (M) | 2.176434(24)×10−8 kg[8] | |
| Planck time | time (T) | 5.391247(60)×10−44 s[9] | |
| Planck temperature | temperature (Θ) | 1.416784(16)×1032 K[10] |
Unlike the case with theInternational System of Units, there is no official entity that establishes a definition of a Planck unit system. Some authors define the base Planck units to be those of mass, length and time, regarding an additional unit for temperature to be redundant.[note 1] Other tabulations add, in addition to a unit for temperature, a unit for electric charge, so that either theCoulomb constant[12][13][14] or thevacuum permittivity[15] is normalized to 1. Thus, depending on the author's choice, this charge unit is given byfor, orfor. Some of these tabulations also replace mass with energy when doing so.[15]In SI units, the values ofc,h,e andkB are exact and the values ofε0 andG in SI units respectively have relative uncertainties of1.6×10−10[16] and2.2×10−5.[17] Hence, the uncertainties in the SI values of the Planck units derive almost entirely from uncertainty in the SI value ofG.
Compared toStoney units, Planck base units are all larger by a factor, where is thefine-structure constant.[18]
In any system of measurement, units for many physical quantities can be derived from base units. Table 2 offers a sample of derived Planck units, some of which 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.
| Derived unit of | Expression | ApproximateSI equivalent |
|---|---|---|
| area (L2) | 2.6121×10−70 m2 | |
| volume (L3) | 4.2217×10−105 m3 | |
| momentum (LMT−1) | 6.5249 kg⋅m/s | |
| energy (L2MT−2) | 1.9561×109 J | |
| force (LMT−2) | 1.2103×1044 N | |
| density (L−3M) | 5.1550×1096 kg/m3 | |
| acceleration (LT−2) | 5.5608×1051 m/s2 |
Some Planck units, such as of time and length, are manyorders of magnitude too large or too small to be of practical use, so that Planck units as a system are typically only relevant to theoretical physics. In some cases, a Planck unit may suggest a limit to a range of a physical quantity where present-day theories of physics apply.[19] For example, our understanding of theBig Bang does not extend to thePlanck epoch, i.e., when the universe was less than one Planck time old. Describing the universe during the Planck epoch requires a theory ofquantum gravity that would incorporate quantum effects intogeneral relativity. Such a theory does not yet exist.
Several quantities are not "extreme" in magnitude, such as the Planck mass, which is about22 micrograms: very large in comparison with subatomic particles, and within the mass range of living organisms.[20]: 872 Similarly, the related units of energy and of momentum are in the range of some everyday phenomena.
Planck units have littleanthropocentric arbitrariness, but do still involve some arbitrary choices in terms of the defining constants. Unlike themetre andsecond, which exist asbase units in theSI system for historical reasons, thePlanck length and Planck time are conceptually linked at a fundamental physical level. Consequently, natural units help physicists to reframe questions.Frank Wilczek puts it succinctly:
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 1/13quintillion.[21]
While 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, this is not about the relative strengths of the two fundamental forces. From the point of view of Planck units, this iscomparing apples with oranges, becausemass andelectric charge areincommensurable quantities. Rather, the disparity of magnitude of force is a manifestation of that theproton charge is approximately the unit charge but theproton mass is far less than the unit mass in a system that treats both forces as having the same form.
When Planck proposed his units, the goal was only that of establishing a universal ("natural") way of measuring objects, without giving any special meaning to quantities that measured one single unit. In 1918,Arthur Eddington suggested that the Planck length could have a special significance for understanding gravitation, but this suggestion was not influential.[22][23] During the 1950s, multiple authors includingLev Landau andOskar Klein argued that quantities on the order of the Planck scale indicated the limits of the validity of quantum field theory.John Archibald Wheeler proposed in 1955 that quantum fluctuations of spacetime become significant at the Planck scale, though at the time he was unaware of Planck's unit system.[22][24] In 1959, C. A. Mead showed that distances that measured of the order of one Planck length, or, similarly, times that measured of the order of Planck time, did carry special implications related toHeisenberg'suncertainty principle:[25]
An analysis of the effect of gravitation on hypothetical experiments indicates that it is impossible to measure the position of a particle with error less than𝛥𝑥 ≳ √𝐺 = 1.6 × 10−33 cm, where 𝐺 is the gravitational constant in natural units. A similar limitation applies to the precise synchronization of clocks.
Inparticle physics andphysical cosmology, the Planck scale is anenergy scale around1.22×1028 eV (the Planck energy, corresponding to theenergy equivalent of the Planck mass,2.17645×10−8 kg) at whichquantum effects ofgravity become significant. At this scale, present descriptions and theories of sub-atomic particle interactions in terms ofquantum field theory break down and become inadequate, due to the impact of the apparentnon-renormalizability of gravity within current theories.[19]
At the Planck length scale, the strength of gravity is expected to become comparable with the other forces, and it has been theorized that all the fundamental forces are unified at that scale, but the exact mechanism of this unification remains unknown.[26] The Planck scale is therefore the point at which the effects of quantum gravity can no longer be ignored in otherfundamental interactions, where current calculations and approaches begin to break down, and a means to take account of its impact is necessary.[27] On these grounds, it has been speculated that it may be anapproximate lower limit at which a black hole could be formed by collapse.[28]
While physicists have a fairly good understanding of the other fundamental interactions of forces on the quantum level,gravity is problematic, and cannot be integrated withquantum mechanics at very high energies using the usual framework of quantum field theory. At lesser energy levels it is usually ignored, while for energies approaching or exceeding the Planck scale, a new theory ofquantum gravity is necessary. Approaches to this problem includestring theory andM-theory,loop quantum gravity,noncommutative geometry, andcausal set theory.[29]
InBig Bang cosmology, thePlanck epoch orPlanck era is the earliest stage of theBig Bang, before thetime passed was equal to the Planck time,tP, or approximately 10−43 seconds.[30] There is no currently available physical theory to describe such short times, and it is not clear in what sense the concept oftime is meaningful for values smaller than the Planck time. It is generally assumed thatquantum effects of gravity dominate physical interactions at this time scale. At this scale, theunified force of theStandard Model is assumed to beunified with gravitation. Immeasurably hot and dense, the state of the Planck epoch was succeeded by thegrand unification epoch, where gravitation is separated from the unified force of the Standard Model, in turn followed by theinflationary epoch, which ended after about 10−32 seconds (or about 1011 tP).[31]
Table 3 lists properties of the observable universe today expressed in Planck units.[32][33]
| Property of present-dayobservable universe | Approximate number of Planck units | Equivalents |
|---|---|---|
| Age | 8.08 × 1060tP | 4.35 × 1017 s or 1.38 × 1010 years |
| Diameter | 5.4 × 1061lP | 8.7 × 1026 m or 9.2 × 1010light-years |
| Mass | approx. 1060mP | 3 × 1052 kg or 1.5 × 1022solar masses (only counting stars) 1080 protons (sometimes known as theEddington number) |
| Density | 1.8 × 10−123mP⋅lP−3 | 9.9 × 10−27 kg⋅m−3 |
| Temperature | 1.9 × 10−32TP | 2.725 K temperature of thecosmic microwave background radiation |
| Cosmological constant | ≈ 10−122l −2 P | ≈ 10−52 m−2 |
| Hubble constant | ≈ 10−61t −1 P | ≈ 10−18 s−1 ≈ 102 (km/s)/Mpc |
After the measurement of the cosmological constant (Λ) in 1998, estimated at 10−122 in Planck units, it was noted that this is suggestively close to the reciprocal of theage of the universe (T) squared. Barrow and Shaw proposed a modified theory in whichΛ is a field evolving in such a way that its value remainsΛ ~T−2 throughout the history of the universe.[34]
The Planck length, denotedℓP, is a unit oflength defined as:It is equal to1.616255(18)×10−35 m[7] (the two digits enclosed by parentheses are the estimatedstandard error associated with the reported numerical value) or about10−20 times the diameter of aproton.[35] It can be motivated in various ways, such as considering a particle whosereduced Compton wavelength is comparable to itsSchwarzschild radius,[35][36][37] though whether those concepts are in fact simultaneously applicable is open to debate.[38] (The same heuristic argument simultaneously motivates the Planck mass.[36])
The Planck length is a distance scale of interest in speculations about quantum gravity. TheBekenstein–Hawking entropy of a black hole is one-fourth the area of itsevent horizon in units of Planck length squared.[11]: 370 Since the 1950s, it has been conjectured that quantum fluctuations of the spacetime metric might make the familiar notion of distance inapplicable below the Planck length.[24][39][22] This is sometimes expressed by saying that "spacetime becomes afoam at the Planck scale".[40] It is possible that the Planck length is the shortest physically measurable distance, since any attempt to investigate the possible existence of shorter distances, by performing higher-energy collisions, would result in black hole production. Higher-energy collisions, rather than splitting matter into finer pieces, would simply produce bigger black holes.[41]
The strings ofstring theory are modeled to be on the order of the Planck length.[42][43] In theories withlarge extra dimensions, the Planck length calculated from the observed value of can be smaller than the true, fundamental Planck length.[11]: 61 [44]
The Planck time, denotedtP, is defined as:This is thetime required forlight to travel a distance of 1 Planck length invacuum, which is a time interval of approximately5.39×10−44 s. No current physical theory can describe timescales shorter than the Planck time, such as the earliest events after the Big Bang.[30] Some conjectures state that the structure of time need not remain smooth on intervals comparable to the Planck time.[45]
The Planck energyEP is approximately equal to the energy released in the combustion of the fuel in an automobile fuel tank (57.2 L at 34.2 MJ/L of chemical energy). Theultra-high-energy cosmic rayobserved in 1991 had a measured energy of about 50 J, equivalent to about2.5×10−8 EP.[46][47]
Proposals for theories ofdoubly special relativity posit that, in addition to the speed of light, an energy scale is also invariant for all inertial observers. Typically, this energy scale is chosen to be the Planck energy.[48][49]
The Planck unit of force may be thought of as the derived unit offorce in the Planck system if the Planck units of time, length, and mass are considered to be base units.It is the gravitational attractive force of two bodies of 1 Planck mass each that are held 1 Planck length apart. One convention for the Planck charge is to choose it so that the electrostatic repulsion of two objects with Planck charge and mass that are held 1 Planck length apart balances the Newtonian attraction between them.[50]
Some authors have argued that the Planck force is on the order of the maximum force that can occur between two bodies.[51][52] However, the validity of these conjectures has been disputed.[53][54]
The Planck temperatureTP is1.416784(16)×1032 K.[10] At this temperature, the wavelength of light emitted bythermal radiation reaches the Planck length. There are no known physical models able to describe temperatures greater thanTP; aquantum theory of gravity would be required to model the extreme energies attained.[55] Hypothetically, a system inthermal equilibrium at the Planck temperature might contain Planck-scale black holes, constantly being formed from thermal radiation and decaying viaHawking evaporation. Adding energy to such a system mightdecrease its temperature by creating larger black holes, whose Hawking temperature is lower.[56]
Physical quantities that have different dimensions (such as time and length) cannot be equated even if they are numerically equal (e.g., 1 second is not the same as 1 metre). In theoretical physics, however, this scruple may be set aside, by a process callednondimensionalization. The effective result is that many fundamental equations of physics, which often include some of the constants used to define Planck units, become equations where these constants are replaced by a 1.
Examples include theenergy–momentum relation (which becomes) and theDirac equation (which becomes).
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.
The factor 4π is ubiquitous intheoretical physics because in three-dimensional space, the surface area of asphere of radiusr is 4πr2. This, along with the concept offlux, are the basis for theinverse-square law,Gauss's law, and thedivergence operator applied toflux density. For example,gravitational andelectrostatic fields produced by point objects have spherical symmetry, and so the electric flux through a sphere of radiusr around a point charge will be distributed uniformly over that sphere. From this, it follows that a factor of 4πr2 will appear in the denominator of Coulomb's law inrationalized form.[32]: 214–15 (Both the numerical factor and the power of the dependence onr would change if space were higher-dimensional; the correct expressions can be deduced from the geometry ofhigher-dimensional spheres.[11]: 51 ) Likewise for Newton's law of universal gravitation: a factor of 4π naturally appears inPoisson's equation when relating the gravitational potential to the distribution of matter.[11]: 56
Hence a substantial body of physical theory developed since Planck's 1899 paper suggests normalizing notG but 4πG (or 8πG) to 1. Doing so would introduce a factor of1/4π (or1/8π) into the nondimensionalized form of the law of universal gravitation, consistent with the modern rationalized formulation of Coulomb's law in terms of the vacuum permittivity. In fact, alternative normalizations frequently preserve the factor of1/4π in the nondimensionalized form of Coulomb's law as well, so that the nondimensionalized Maxwell's equations for electromagnetism andgravitoelectromagnetism both take the same form as those for electromagnetism in SI, which do not have any factors of 4π. When this is applied to electromagnetic constants,ε0, this unit system is called "rationalized". When applied additionally to gravitation and Planck units, these are calledrationalized Planck units[57] and are seen in high-energy physics.[58]
The rationalized Planck units are defined so thatc = 4πG =ħ =ε0 =kB = 1.
There are several possible alternative normalizations.
In 1899, Newton's law of universal gravitation was still seen as exact, rather than as a convenient approximation holding for "small" velocities and masses (the approximate nature of Newton's law was shown following the development ofgeneral relativity in 1915). Hence Planck normalized to 1 thegravitational constantG in Newton's law. In theories emerging after 1899,G nearly always appears in formulae multiplied by 4π or a small integer multiple thereof. Hence, a choice 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.
{{cite book}}: CS1 maint: location missing publisher (link)Just as in our discussion of the Fermi theory, the nonrenormalizability of quantum gravity tells us that at the Planck energy scale ... new physics must appear. Fermi's theory cried out, and the new physics turned out to be the electroweak theory. Einstein's theory is now crying out.
Like all orders of magnitude estimates, this procedure is not rigorous since it extrapolates the concepts of black hole and of Compton wavelength to a new regime in which both concepts would probably lose their accepted meanings and would, strictly speaking, cease being valid. However, this is how one gains intuition into a new physical regime.
However, for most experimental physicists, approaching energies comparable to Planck energy is little more than a distant fantasy. The most powerful accelerators on Earth miss Planck energy of 15 orders of magnitude, while ultra high energy cosmic rays are still 9 orders of magnitude short ofMp.
The point at which our physical theories run into most serious difficulties is that where matter reaches a temperature of approximately 1032 degrees, also known as Planck's temperature. The extreme density of radiation emitted at this temperature creates a disproportionately intense field of gravity. To go even farther back, aquantum theory of gravity would be necessary, but such a theory has yet to be written.
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