Planck units

In particle physics and physical cosmology, Planck units are a set of units of measurement defined exclusively in terms of five universal physical constants, in such a manner that these five physical constants take on the numerical value of 1 when expressed in terms of these units.

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 nature and not from any human construct. Planck units are only one system of several systems of natural units, but Planck units are not based on properties of any prototype object or particle (that would be arbitrarily chosen), but rather on only the properties of free space. Planck units have significance for theoretical physics since they simplify several recurring algebraic expressions of physical law by nondimensionalization. They are relevant in research on unified theories such as quantum gravity.

The term "Planck scale" refers to the magnitudes of space, time, energy and other units, below which (or beyond which) the predictions of the Standard Model, quantum field theory and general relativity are no longer reconcilable, and quantum effects of gravity are expected to dominate. This region may be characterized by energies around 1.22×1019 GeV (the Planck energy), time intervals around 5.39×10−44 s (the Planck time) and lengths around 1.62×10−35 m (the Planck length). At the Planck scale, current models are not expected to be a useful guide to the cosmos, and physicists no longer have any scientific model whatsoever to suggest how the physical universe behaves. The best known example is represented by the conditions in the first 10−43 seconds of our universe after the Big Bang, approximately 13.8 billion years ago.

The five universal constants that Planck units, by definition, normalize to 1 are:

Each of these constants can be associated with a fundamental physical theory or concept: c with special relativity, G with general relativity, ħ with quantum mechanics, ε0 with electromagnetism, and kB with the notion of temperature/energy (statistical mechanics and thermodynamics).

Base units

Any system of measurement may be assigned a mutually independent set of base quantities and associated base units, from which all other quantities and units may be derived. In the International System of Units, for example, the SI base quantities include length with the associated unit of the metre. In the system of Planck units, a similar set of base quantities may be selected, and the Planck base unit of length is then 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 selected equations of physical law when physical quantities are expressed in terms of Planck units. For example, Newton's law of universal gravitation,

can be expressed as

Both equations are 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 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 explicitly by their corresponding unit:

This last equation (without G) is valid only if F, m1, m2, and r are 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 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."[1]

Table 1: Dimensional universal physical constants normalized with Planck units
Constant Symbol Dimension Value (SI units)[2]
Speed of light in vacuum c L T −1 2.99792458×108 m⋅s−1
(exact by definition of metre)
Gravitational constant G L3 M−1 T −2 6.67430(15)×10−11 m3⋅kg−1⋅s−2[3]
Reduced Planck constant ħ = h/2π
where h is the Planck constant
L2 M T −1 1.054571817...×10−34 J⋅s[4]
(exact by definition of the kilogram since 20 May 2019)
Coulomb constant ke = 1/4πε0
where ε0 is the permittivity of free space
L3 M T −2 Q−2 8.9875517873681764×109 kg⋅m3⋅s−4⋅A−2
(exact by definitions of ampere and metre until 20 May 2019)
Boltzmann constant kB L2 M T −2 Θ−1 1.380649×10−23 J⋅K−1[5]
(exact by definition of the kelvin since 20 May 2019)

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:

Solving the five equations above for the five unknowns results in a unique set of values for the five base Planck units:

Table 2: Base Planck units
Name Dimension Expression Value (SI unit)s[2]
Planck length Length (L) 1.616255(18)×10−35 m[6]
Planck mass Mass (M) 2.176435(24)×10−8 kg[7]
Planck time Time (T) 5.391245(60)×10−44 s[8]
Planck charge Electric charge (Q) 1.875 545 956(41) × 10−18 C[9][4][10]
Planck temperature Temperature (Θ) 1.416785(16)×1032 K[11]

Table 2 clearly defines Planck units in terms of the fundamental constants. Yet relative to other units of measurement such as SI, the values of the Planck units, other than the Planck charge, are only known approximately. This is due to uncertainty in the value of the gravitational constant G as measured relative to SI unit definitions.

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 1/299792458 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 ε0, due to the definition of ampere which sets the vacuum permeability μ0 to 4π × 10−7 H/m and the fact that μ0ε0 = 1/c2. The numerical value of the reduced Planck constant ħ has been determined experimentally to 12 parts per billion, while that of G has been determined experimentally to no better than 1 part in 21300 (or 47000 parts per billion).[2] G appears in the definition of almost every Planck unit in Tables 2 and 3, but not all. 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 ±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 43500, or 23000 parts per billion.)

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.

Table 3: Derived Planck units
Name Dimension Expression Approximate SI equivalent
Planck area area (L2) 2.6121×10−70 m2
Planck volume volume (L3) 4.2217×10−105 m3
Planck momentum momentum (LMT−1) 6.52485 kg⋅m/s
Planck energy energy (L2MT−2) 1.9561×109 J
Planck force force (LMT−2) 1.21027×1044 N
Planck power power (L2MT−3) 3.62831×1052 W
Planck density density (L−3M) 5.15500×1096 kg/m3
Planck energy density energy density (L−1MT−2) 4.633×10113 J/m3
Planck intensity intensity (MT−3) 1.38893×10122 W/m2
Planck angular frequency angular frequency (T−1) 1.85487×1043 rad/s
Planck pressure pressure (L−1MT−2) 4.633×10113 Pa
Planck current electric current (T−1Q) 3.4789×1025 A
Planck voltage voltage (L2MT−2Q−1) 1.04295×1027 V
Planck impedance resistance (L2MT−1Q−2) 29.9792458 Ω
Planck magnetic inductance magnetic induction (MT−1Q−1) 2.1526×1053 T
Planck electrical inductance inductance (L2MQ−2) 1.616×10−42 H
Planck volumetric flow rate volumetric flow rate (L3T−1) 7.83×10−62 m3/s
Planck viscosity dynamic viscosity (L−1MT−1) 2.498×1070 Pa⋅s
Planck acceleration acceleration (LT−2) 5.560815×1051 m/s2

The charge, as other Planck units, was not originally defined by Planck. It is a unit of charge that is a natural addition to the other units of Planck, and is used in some publications.[12][13] The elementary charge, measured in terms of the Planck charge, is

where is the fine-structure constant

Some Planck units are suitable for measuring quantities that are familiar from daily experience. For example:

However, most Planck units are many orders of magnitude too large or too small to be of 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 physical speed in special relativity;[14] 1 nano-(Planck length per Planck time) is about 1.079 km/h.
  • 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 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.


Planck units are free of anthropocentric arbitrariness. Some physicists argue that communication with extraterrestrial intelligence would have to employ such a system of units in order to be understood.[15] Unlike the metre and second, which exist as base 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:

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)].[16]

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 is comparing apples to oranges, because mass and electric charge are incommensurable quantities. Rather, the disparity of magnitude of force is a manifestation of the fact that the charge on the protons is approximately the unit charge but the mass of the protons is far less than the unit mass.


In Big Bang cosmology, the Planck epoch or Planck era is the earliest stage of the Big Bang, before the time passed was equal to the Planck time, tP, or approximately 10−43 seconds.[17] There is no currently available physical theory to describe such short times, and it is not clear in what sense the concept of time is meaningful for values smaller than the Planck time. It is generally assumed that quantum effects of gravity dominate physical interactions at this time scale. At this scale, the unified force of the Standard Model is assumed to be unified with gravitation. Inconceivably hot and dense, the state of the Planck epoch was succeeded by the Grand unification epoch, where gravitation is separated from the unified force of the Standard Model, in turn followed by the Inflationary epoch, which ended after about 10−32 seconds (or about 1010 tP).[18]

Relative to the Planck epoch, the observable universe today looks extreme when expressed in Planck units, as in this set of approximations:[19][20]

Table 5: Today's universe in Planck units.
Property of
present-day Observable Universe
Approximate number
of Planck units
Age 8.08 × 1060 tP 4.35 × 1017 s, or 13.8 × 109 years
Diameter 5.4 × 1061 lP 8.7 × 1026 m or 9.2 × 1010 light-years
Mass approx. 1060 mP 3 × 1052 kg or 1.5 × 1022 solar masses (only counting stars)
1080 protons (sometimes known as the Eddington number)
Density 1.8 × 10−123 ρP 9.9 × 10−27 kg m−3
Temperature 1.9 × 10−32 TP 2.725 K
temperature of the cosmic microwave background radiation
Cosmological constant 5.6 × 10−122 t−2
1.9 × 10−35 s−2
Hubble constant 1.18 × 10−61 t−1
67.8 (km/s)/Mpc

The recurrence of large numbers close or related to 1060 in the above table is a coincidence that intrigues some theorists. It is an example of the kind of large numbers coincidence that led theorists such as Eddington and Dirac to develop alternative physical theories (e.g. a variable speed of light or Dirac varying-G theory).[21] After the measurement of the cosmological constant in 1998, estimated at 10−122 in Planck units, it was noted that this is suggestively close to the reciprocal of the age of the universe squared.[22] Barrow and Shaw (2011) proposed a modified theory in which Λ is a field evolving in such a way that its value remains Λ ~ T−2 throughout the history of the universe.[23]


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 electron charge, e, to 1.

In 1898, Max Planck discovered that action is quantized, and published the result in a paper presented to the Prussian Academy of Sciences in May 1899.[24][25] 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 (or ħ) is now common. Planck underlined the universality of the new unit system, writing:

...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 kB to arrive at natural units for length, time, mass, and temperature.[25] Planck did not adopt any electromagnetic units. However, since the non-rationalized gravitational constant, G, is set to 1, a natural extension of Planck units to a unit of electric charge is to also set the non-rationalized Coulomb constant, ke, to 1 as well.[26] Another convention is to use the elementary charge as the basic unit of electric charge in the Planck system.[27] Such a system is convenient for black hole physics. The two conventions for unit charge differ by a factor of the square root of the fine-structure constant. Planck's paper also gave numerical values for the base units that were close to modern values.

List 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 6 shows how the use of Planck units simplifies many fundamental equations of physics, because this gives each of the five fundamental constants, and products of them, a simple numeric value of 1. In the SI form, the units should be accounted for. In the nondimensionalized form, the units, which are now Planck units, need not be written if their use is understood.

Table 6: How Planck units simplify the key equations of physics
SI form Nondimensionalized form
Newton's law of universal gravitation
Einstein field equations in general relativity
Mass–energy equivalence in special relativity
Energy–momentum relation
Thermal energy per particle per degree of freedom
Boltzmann's entropy formula
Planck–Einstein relation for energy and angular frequency
Planck's law (surface intensity per unit solid angle per unit angular frequency) for black body at temperature T.
Stefan–Boltzmann constant σ defined
BekensteinHawking black hole entropy[28]
Schrödinger's equation
Hamiltonian form of Schrödinger's equation
Covariant form of the Dirac equation
Coulomb's law
Maxwell's equations

Ideal gas law

Alternative choices of normalization

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 in theoretical physics because the surface area of a sphere of radius r is 4πr2. This, along with the concept of flux, are the basis for the inverse-square law, Gauss's law, and the divergence operator applied to flux density. For example, gravitational and electrostatic fields produced by point charges have spherical symmetry (Barrow 2002: 214–15). The 4πr2 appearing in the denominator of Coulomb's law in rationalized form, for example, follows from the flux of an electrostatic field being distributed uniformly on the surface of a sphere. Likewise for Newton's law of universal gravitation. (If space had more than three spatial dimensions, the factor 4π would have to be changed according to the geometry of the sphere in higher dimensions.)

Hence a substantial body of physical theory developed since Planck (1899) suggests normalizing not G but either 4πG (or 8πG or 16πG) to 1. Doing so would introduce a factor of 1/4π (or 1/8π or 1/16π) 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 of 1/4π in the nondimensionalized form of Coulomb's law as well, so that the nondimensionalized Maxwell's equations for electromagnetism and gravitoelectromagnetism 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" Lorentz–Heaviside units. When applied additionally to gravitation and Planck units, these are called rationalized Planck units[29] and are seen in high-energy physics.

The rationalized Planck units are defined so that . These are the Planck units based on Lorentz–Heaviside units (instead of on the more conventional Gaussian units) as depicted above.

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 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 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.

  • Normalizing 4πG to 1:
  • Setting 8πG = 1. This would eliminate 8πG from the Einstein field equations, Einstein–Hilbert action, and the Friedmann equations, for gravitation. Planck units modified so that 8πG = 1 are known as reduced Planck units, because the Planck mass is divided by 8π. Also, the Bekenstein–Hawking formula for the entropy of a black hole simplifies to SBH = (mBH)2/2 = 2πABH.
  • Setting 16πG = 1. This would eliminate the constant c4/16πG from the Einstein–Hilbert action. The form of the Einstein field equations with cosmological constant Λ becomes Rμν + Λgμν = 1/2(Rgμν + Tμν).


Planck units normalize to 1 the Coulomb force constant ke = 1/4πε0 (as does the cgs system of units). This sets the Planck impedance, ZP equal to Z0/4π, where Z0 is the characteristic impedance of free space.

where is the fine-structure constant. This convention is seen in high-energy physics.


Planck normalized to 1 the Boltzmann constant kB.

  • Normalizing 1/2kB to 1:
    • Removes the factor of 1/2 in the nondimensionalized equation for the thermal energy per particle per degree of freedom.
    • Introduces a factor of 2 into the nondimensionalized form of Boltzmann's entropy formula.
    • Does not affect the value of any of the base or derived Planck units listed in Tables 3 and 4 other than the Planck temperature, which it doubles.

Planck units and the invariant scaling of nature

Some theorists (such as Dirac and Milne) have proposed cosmologies that conjecture that physical "constants" might actually change over time (e.g. a variable speed of light or 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, or 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 in his paper "Comment on time-variation of fundamental constants".[30][31]

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:

[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 mP ] you cannot tell because all the pure numbers defined by the ratios of any pair of masses are unchanged.

— Barrow 2002[19]

Referring to Duff's "Comment on time-variation of fundamental constants"[30] and Duff, Okun, and Veneziano's paper "Trialogue on the number of fundamental constants",[32] 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, mp/me, changes (atomic structures would change) but if all dimensionless physical quantities remained unchanged (this includes all possible ratios of identically dimensioned physical quantity), we cannot 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 has changed, even drastically.

If the speed of light c, were somehow suddenly cut in half and changed to 1/2c (but with the axiom that all dimensionless physical quantities remain the same), then the Planck length would increase by a factor of 22 from the point of view of some unaffected observer on the outside. Measured by "mortal" observers in terms of Planck units, the new speed of light would 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:

Then atoms would be bigger (in one dimension) by 22, each of us would be taller by 22, and so would our metre sticks be taller (and wider and thicker) by a factor of 22. 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 42 (from the point of view of this unaffected observer on the outside) because the Planck time has increased by 42 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 unaffected 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 299792458 of our new metres in the time elapsed by one of our new seconds (1/2c × 42 ÷ 22 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 significantly, 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 dimensionless 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 would point out that ascribing a change in measurable reality, i.e. α, to a specific dimensional component quantity, such as c, is unjustified. The very same operational difference in measurement or perceived reality could just as well be caused by a change in h or e if α is changed and no other dimensionless constants are changed. It is only the dimensionless physical constants that ultimately matter in the definition of worlds.[30][33]

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 constants. 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[34] and this has intensified the debate about the measurement of physical constants. According to some theorists[35] 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.[30] 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.[32]

See also


  1. ^ General relativity predicts that gravitational radiation propagates at the same speed as electromagnetic radiation.



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  12. ^ [Theory of Quantized Space – Date of registration 21/9/1994 N. 344146 protocol 4646 Presidency of the Council of Ministers – Italy – Dep. Information and Publishing, literary, artistic and scientific property]
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  23. ^ Barrow, John D.; Shaw, Douglas J. (2011). "The value of the cosmological constant". General Relativity and Gravitation. 43 (10): 2555–2560. arXiv:1105.3105. Bibcode:2011GReGr..43.2555B. doi:10.1007/s10714-011-1199-1.
  24. ^ Planck (1899), p. 479.
  25. ^ a b *Tomilin, K. A., 1999, "Natural Systems of Units: To the Centenary Anniversary of the Planck System", 287–296.
  26. ^ Pavšic, Matej (2001). The Landscape of Theoretical Physics: A Global View. Fundamental Theories of Physics. Dordrecht: Kluwer Academic. pp. 347–352. ISBN 978-0-7923-7006-2.
  27. ^ Tomilin, K. (1999). "Fine-structure constant and dimension analysis". Eur. J. Phys. 20 (5): L39–L40. Bibcode:1999EJPh...20L..39T. doi:10.1088/0143-0807/20/5/404.
  28. ^ Also see Roger Penrose (1989) The Road to Reality. Oxford Univ. Press: 714-17. Knopf.
  29. ^ Sorkin, Rafael (1983). "Kaluza-Klein Monopole". Physical Review Letters. 51 (2): 87–90. Bibcode:1983PhRvL..51...87S. doi:10.1103/PhysRevLett.51.87.
  30. ^ a b c d Michael Duff (2002). "Comment on time-variation of fundamental constants". arXiv:hep-th/0208093.
  31. ^ Michael Duff (2014). "How fundamental are fundamental constants?". arXiv:1412.2040. doi:10.1080/00107514.2014.980093 (inactive 28 May 2019).
  32. ^ a b Duff, Michael; Okun, Lev; Veneziano, Gabriele (2002). "Trialogue on the number of fundamental constants". Journal of High Energy Physics. 2002 (3): 023. arXiv:physics/0110060. Bibcode:2002JHEP...03..023D. doi:10.1088/1126-6708/2002/03/023.
  33. ^ John Baez How Many Fundamental Constants Are There?
  34. ^ Webb, J. K.; et al. (2001). "Further evidence for cosmological evolution of the fine structure constant". Phys. Rev. Lett. 87 (9): 884. arXiv:astro-ph/0012539v3. Bibcode:2001PhRvL..87i1301W. doi:10.1103/PhysRevLett.87.091301. PMID 11531558.
  35. ^ Davies, Paul C.; Davis, T. M.; Lineweaver, C. H. (2002). "Cosmology: Black Holes Constrain Varying Constants". Nature. 418 (6898): 602–3. Bibcode:2002Natur.418..602D. doi:10.1038/418602a. PMID 12167848.


External links

Atomic units

Atomic units (au or a.u.) form a system of natural units which is especially convenient for atomic physics calculations. There are two different kinds of atomic units, Hartree atomic units and Rydberg atomic units, which differ in the choice of the unit of mass and charge. This article deals with Hartree atomic units, where the numerical values of the following four fundamental physical constants are all unity by definition:

In Hartree units, the speed of light is approximately . Atomic units are often abbreviated "a.u." or "au", not to be confused with the same abbreviation used also for astronomical units, arbitrary units, and absorbance units in different contexts.

Boltzmann constant

The Boltzmann constant (kB or k) is a physical constant named after its discoverer, Ludwig Boltzmann, which relates the average relative kinetic energy of particles in a gas with the temperature of the gas and occurs in Planck's law of black-body radiation and in Boltzmann's entropy formula.

It is the gas constant R divided by the Avogadro constant NA:

The Boltzmann constant has the dimension energy divided by temperature, the same as entropy. As of 2017, its value in SI units is a measured quantity. The recommended value (as of 2015, with standard uncertainty in parentheses) is 1.38064852(79)×10−23 J/K.

Historically, measurements of the Boltzmann constant depended on the definition of the kelvin in terms of the triple point of water. However, in the redefinition of SI base units adopted at the 26th General Conference on Weights and Measures (CGPM) on 16 November 2018, the definition of the kelvin was changed to one based on a fixed, exact numerical value of the Boltzmann constant, similar to the way that the speed of light was given an exact numerical value at the 17th CGPM in 1983. The final value (based on the 2017 CODATA adjusted value of 1.38064903(51)×10−23 J/K) is 1.380649×10−23 J/K.

CGh physics

cGh physics refers to the mainstream attempts in physics to unify relativity, gravitation and quantum mechanics, in particular following the ideas of Matvei Petrovich Bronstein and George Gamow. The letters are the standard symbols for the speed of light (c), the gravitational constant (G), and Planck's constant (h).

If one considers these three universal constants as the basis for a 3-D coordinate system and envisions a cube, then this pedagogic construction provides a framework, which is referred to as the cGh cube, or physics cube, or cube of theoretical physics (CTP). This cube can used for organizing major subjects within physics as occupying each of the eight corners. The eight corners of the cGh physics cube are:

Classical mechanics (_,_,_)

Special relativity (c,_,_), Gravitation (_,G,_), Quantum mechanics (_,_,h)

General relativity (c,G,_), Quantum field theory (c,_,h), Non-relativistic quantum theory with gravity (_,G,h)

Theory of everything, or relativistic quantum gravity (c,G,h)Other cGh subjects include Planck units, Hawking radiation and black hole thermodynamics.

While there are several other physical constants, these three are given special consideration, because they can be used to define all Planck units and thus all physical quantities. The three constants are therefore used sometimes as a framework for philosophical study and as one of pedagogical patterns.

Cutoff (physics)

In theoretical physics, cutoff is an arbitrary maximal or minimal value of energy, momentum, or length, used in order that objects with larger or smaller values than these physical quantities are ignored in some calculation. It is usually represented within a particular energy or length scale, such as Planck units.

When used in this context, the traditional terms "infrared" and "ultraviolet" are not literal references to specific regions of the spectrum.

Dyson series

In scattering theory, a part of mathematical physics, the Dyson series, formulated by Freeman Dyson, is a perturbative series where each term is represented by Feynman diagrams. This series diverges asymptotically, but in quantum electrodynamics (QED) at the second order the difference from experimental data is in the order of 10−10. This close agreement holds because the coupling constant (also known as the fine structure constant) of QED is much less than 1. Notice that in this article Planck units are used, so that ħ = 1 (where ħ is the reduced Planck constant).

Geometrized unit system

A geometrized unit system or geometric unit system is a system of natural units in which the base physical units are chosen so that the speed of light in vacuum, c, and the gravitational constant, G, are set equal to unity.

The geometrized unit system is not a completely defined or unique system: latitude is left to also set other constants to unity. We may, for example, also set Coulomb's constant, ke, and the electric charge, , to unity.

The reduced Planck constant, ħ, is not equal to 1 in this system (Stoney units), in contrast to Planck units.

This system is useful in physics, especially in the special and general theories of relativity. All physical quantities are identified with geometric quantities such as areas, lengths, dimensionless numbers, path curvatures, or sectional curvatures.

Many equations in relativistic physics appear simpler when expressed in geometric units, because all occurrences of G and of c drop out. For example, the Schwarzschild radius of a nonrotating uncharged black hole with mass m becomes r = 2m. For this reason, many books and papers on relativistic physics use geometric units. An alternative system of geometrized units is often used in particle physics and cosmology, in which G = 1 instead. This introduces an additional factor of 8π into Newton's law of universal gravitation but simplifies Einstein's equations, the Einstein–Hilbert action, the Friedmann equations and the Newtonian Poisson equation by removing the corresponding factor.

Practical measurements and computations are usually done in SI units, but conversions are generally quite straightforward[citation needed].


Gravitoelectromagnetism, abbreviated GEM, refers to a set of formal analogies between the equations for electromagnetism and relativistic gravitation; specifically: between Maxwell's field equations and an approximation, valid under certain conditions, to the Einstein field equations for general relativity. Gravitomagnetism is a widely used term referring specifically to the kinetic effects of gravity, in analogy to the magnetic effects of moving electric charge. The most common version of GEM is valid only far from isolated sources, and for slowly moving test particles.

The analogy and equations differing only by some small factors were first published in 1893, before general relativity, by Oliver Heaviside as a separate theory expanding Newton's law.

Immirzi parameter

The Immirzi parameter (also known as the Barbero–Immirzi parameter) is a numerical coefficient appearing in loop quantum gravity (LQG), a nonperturbative theory of quantum gravity. The Immirzi parameter measures the size of the quantum of area in Planck units. As a result, its value is currently fixed by matching the semiclassical black hole entropy, as calculated by Stephen Hawking, and the counting of microstates in loop quantum gravity.

Natural units

In physics, natural units are physical units of measurement based only on universal physical constants. For example, the elementary charge e is a natural unit of electric charge, and the speed of light c is a natural unit of speed. A purely natural system of units has all of its units defined in this way, and usually such that the numerical values of the selected physical constants in terms of these units are exactly 1. These constants are then typically omitted from mathematical expressions of physical laws, and while this has the apparent advantage of simplicity, it may entail a loss of clarity due to the loss of information for dimensional analysis. It precludes the interpretation of an expression in terms of fundamental physical constants, such as e and c, unless it is known which units (in dimensionful units) the expression is supposed to have. In this case, the reinsertion of the correct powers of e, c, etc., can be uniquely determined.

Planck acceleration

Planck acceleration is the acceleration from zero speed to the speed of light during one Planck time. It is a derived unit in the Planck system of natural units.

Planck charge

In physics, the Planck charge, denoted by , is one of the base units in the system of natural units called Planck units. It is a quantity of electric charge defined in terms of fundamental physical constants.

The Planck charge is defined as:


From a classical calculation, the electric potential energy of one Planck charge on the surface of a sphere that is one Planck length in diameter is one Planck energy.

The Gaussian cgs units are defined so that , in which case has the following simple form,

It is customary in theoretical physics to adopt the Lorentz–Heaviside units (also known as rationalized cgs). When made natural (, ), they are like the SI system with . Therefore, it is more appropriate to instead define the Planck charge as

When charges are measured in units of , which is commonly used in quantum field theory, we have

Planck force

Planck force is the derived unit of force resulting from the definition of the base Planck units for time, length, and mass. It is equal to the natural unit of momentum divided by the natural unit of time.

Planck length

In physics, the Planck length, denoted ℓP, is a unit of length that is the distance light travels in one unit of Planck time. It is equal to 1.616255(18)×10−35 m. It is a base unit in the system of Planck units, developed by physicist Max Planck. The Planck length can be defined from three fundamental physical constants: the speed of light in a vacuum, the Planck constant, and the gravitational constant.

Planck mass

In physics, the Planck mass, denoted by mP, is the unit of mass in the system of natural units known as Planck units. It is approximately 0.02 milligrams. Unlike some other Planck units, such as Planck length, Planck mass is not a fundamental lower or upper bound; instead, Planck mass is a unit of mass defined using only what Max Planck considered fundamental and universal units. One Planck mass is roughly the mass of a flea egg. For comparison, this value is of the order of 1015 (a quadrillion) times larger than the highest energy available to contemporary particle accelerators.

It is defined as:

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

Substituting values for the various components in this definition gives the approximate equivalent value of this unit in terms of other units of mass:

For the Planck mass , the Schwarzschild radius () and the Compton wavelength () are of the same order as the Planck length .

Particle physicists and cosmologists often use an alternative normalization with the reduced Planck mass, which is

The factor of simplifies a number of equations in general relativity.

Planck momentum

Planck momentum is the unit of momentum in the system of natural units known as Planck units. It has no commonly used symbol of its own, but can be denoted by , where is the Planck mass and is the speed of light in a vacuum. Then


is the reduced Planck's constant,
is the Planck length,
is the gravitational constant.

In SI units Planck momentum is approximately 6.5 kg·m/s. It is equal to the Planck mass multiplied by the speed of light, usually associated with the momentum of primordial photons in some prevailing Big Bang models. Unlike most of the other Planck units, Planck momentum occurs on a human scale. By comparison, running with a five-pound object (108×Planck mass) at an average running speed (10−8×speed of light in a vacuum) would give the object Planck momentum. A 70 kg human moving at an average walking speed of 1.4 m/s (5.0 km/h; 3.1 mph) would have a momentum of about 15 . A baseball, which has mass 0.145 kg, travelling at 45 m/s (160 km/h; 100 mph) would have a Planck momentum.

Planck temperature

Planck temperature, denoted by TP, is the unit of temperature in the system of natural units known as Planck units.

It serves as the defining unit of the Planck temperature scale. In this scale the magnitude of the Planck temperature is equal to 1, while that of absolute zero is 0.

Other temperatures can be converted to Planck temperature units. For example, 0 °C = 273.15 K = 1.9279×10−30 TP.

In SI units, the Planck temperature is about 1.417×1032 kelvin (equivalently, degrees Celsius, since the difference is trivially small at this scale), or 2.55×1032 degrees Fahrenheit or Rankine.

Planck time

In quantum mechanics, the Planck time (tP) is the unit of time in the system of natural units known as Planck units. A Planck time unit is the time required for light to travel a distance of 1 Planck length in a vacuum, which is a time interval of approximately 5.39 × 10 −44 s. The unit is named after Max Planck, who was the first to propose it.

The Planck time is defined as:


ħ = ​h2π is the reduced Planck constant (sometimes h is used instead of ħ in the definition)
G = gravitational constant
c = speed of light in vacuum

Using the known values of the constants, the approximate equivalent value in terms of the SI unit, the second, is

where the two digits between parentheses denote the standard error of the approximated value.

Stoney units

In physics the Stoney units form a system of units named after the Irish physicist George Johnstone Stoney, who first proposed them in 1881. They are the first historical example of natural units, i.e. units of measurement designed so that certain fundamental physical constants serve as base units.

Base Planck units
Derived Planck units
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