Planck units

In

.

The term Planck 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 particle

, approximately 13.8 billion years ago.

The four

universal constants

that, by definition, have a numeric value 1 when expressed in these units are:

• c, the speed of light in vacuum,
• G, the gravitational constant,
• ħ, the reduced Planck constant, and
• k_B, the Boltzmann constant.

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.

Introduction

Any system of measurement may be assigned a mutually independent set of base quantities and associated

^: 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 are

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 (without G) is valid with F′, m₁′, m₂′, and r′ being the dimensionless ratio quantities corresponding to the standard quantities, written e.g. F′ ≘ F or F′ = F/F_P, but not as a direct equality of quantities. This may seem to be "setting the constants c, 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]

History and definition

The concept of natural units was introduced in 1874, when George Johnstone Stoney, noting that electric charge is quantized, derived units of length, time, and mass, now named Stoney units in his honor. Stoney chose his units so that G, c, and the electron charge e 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 of action, now usually known as the Planck constant, which appeared in the Wien approximation for black-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 ${\displaystyle G}$, ${\displaystyle h}$, ${\displaystyle c}$, and ${\displaystyle k_{\rm {B}}}$ to arrive at natural units for length, time, mass, and temperature.^[6] His definitions differ from the modern ones by a factor of ${\displaystyle {\sqrt {2\pi }}}$, because the modern definitions use ${\displaystyle \hbar }$ rather than ${\displaystyle h}$.^[5][6]

Table 1: Modern values for Planck's original choice of quantities

Name	Dimension	Expression	Value (SI units)
Planck length	length (L)	${\displaystyle l_{\text{P}}={\sqrt {\frac {\hbar G}{c^{3}}}}}$	1.616255(18)×10−35 m[7]
Planck mass	mass (M)	${\displaystyle m_{\text{P}}={\sqrt {\frac {\hbar c}{G}}}}$	2.176434(24)×10−8 kg[8]
Planck time	time (T)	${\displaystyle t_{\text{P}}={\sqrt {\frac {\hbar G}{c^{5}}}}}$	5.391247(60)×10−44 s[9]
Planck temperature	temperature (Θ)	${\displaystyle T_{\text{P}}={\sqrt {\frac {\hbar c^{5}}{Gk_{\text{B}}^{2}}}}}$	1.416784(16)×1032 K[10]

Unlike the case with the

${\displaystyle k_{\text{e}}}$^[12][13] or the vacuum permittivity ${\displaystyle \epsilon _{0}}$^[14] is normalized to 1. Thus, depending on the author's choice, this charge unit is given by for ${\displaystyle k_{\text{e}}=1}$, or for ${\displaystyle \varepsilon _{0}=1}$. Some of these tabulations also replace mass with energy when doing so.^[15]

In SI units, the values of c, h, e and k_B are exact and the values of ε₀ and G in SI units respectively have relative uncertainties of 1.5×10^−10[16] and 2.2×10⁻⁵.^[17] Hence, the uncertainties in the SI values of the Planck units derive almost entirely from uncertainty in the SI value of G.

Compared to Stoney units, Planck base units are all larger by a factor ${\textstyle {\sqrt {{1}/{\alpha }}}\approx 11.7}$, where ${\displaystyle \alpha }$ is the fine-structure constant.^[18]

Derived units

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.

Table 2: Coherent derived units of Planck units

Derived unit of	Expression	Approximate SI equivalent
area (L2)	${\displaystyle l_{\text{P}}^{2}={\frac {\hbar G}{c^{3}}}}$	2.6121×10−70 m2
volume (L3)	${\displaystyle l_{\text{P}}^{3}=\left({\frac {\hbar G}{c^{3}}}\right)^{\frac {3}{2}}={\sqrt {\frac {(\hbar G)^{3}}{c^{9}}}}}$	4.2217×10−105 m3
momentum (LMT−1)	${\displaystyle m_{\text{P}}c={\frac {\hbar }{l_{\text{P}}}}={\sqrt {\frac {\hbar c^{3}}{G}}}}$	6.5249 kg⋅m/s
energy (L2MT−2)	${\displaystyle E_{\text{P}}=m_{\text{P}}c^{2}={\frac {\hbar }{t_{\text{P}}}}={\sqrt {\frac {\hbar c^{5}}{G}}}}$	1.9561×109 J
force (LMT−2)	${\displaystyle F_{\text{P}}={\frac {E_{\text{P}}}{l_{\text{P}}}}={\frac {\hbar }{l_{\text{P}}t_{\text{P}}}}={\frac {c^{4}}{G}}}$	1.2103×1044 N
density (L−3M)	${\displaystyle \rho _{\text{P}}={\frac {m_{\text{P}}}{l_{\text{P}}^{3}}}={\frac {\hbar t_{\text{P}}}{l_{\text{P}}^{5}}}={\frac {c^{5}}{\hbar G^{2}}}}$	5.1550×1096 kg/m3
acceleration (LT−2)	${\displaystyle a_{\text{P}}={\frac {c}{t_{\text{P}}}}={\sqrt {\frac {c^{7}}{\hbar G}}}}$	5.5608×1051 m/s2

Some Planck units, such as of time and length, are many

. Such a theory does not yet exist.

Several quantities are not "extreme" in magnitude, such as the Planck mass, which is about

^: 872 Similarly, the related units of energy and of momentum are in the range of some everyday phenomena.

Significance

Planck units have little

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.^[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 is

is far less than the unit mass.

Planck scale

In

Relationship to gravity

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.

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 with quantum 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 of quantum gravity is necessary. Approaches to this problem include string theory and M-theory, loop quantum gravity, noncommutative geometry, and causal set theory.^[25]

In cosmology

In

Table 3 lists properties of the observable universe today expressed in Planck units.[28]^[29]

Table 3: Today's universe in Planck units

Property of present-day observable universe	Approximate number of Planck units	Equivalents
Age	8.08 × 1060 tP	4.35 × 1017 s or 1.38 × 1010 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 mP⋅lP−3	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	≈ 10−122 l −2 P	≈ 10−52 m−2
Hubble constant	≈ 10−61 t −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 the age of the universe (T) squared. Barrow and Shaw 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.^[30]

Analysis of the units

Planck length

The Planck length, denoted ℓ_P, is a unit of length defined as:

It is equal to 1.616255(18)×10⁻³⁵ m

)

The Planck length is a distance scale of interest in speculations about quantum gravity. The Bekenstein–Hawking entropy of a black hole is one-fourth the area of its event 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.^[35][36][37] This is sometimes expressed by saying that "spacetime becomes a foam at the Planck scale".^[38] 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.^[39]

The strings of

, the Planck length calculated from the observed value of ${\displaystyle G}$ can be smaller than the true, fundamental Planck length.^{[11]: 61 [42]}

Planck time

The Planck time t_P is the time required for light to travel a distance of 1 Planck length in vacuum, which is a time interval of approximately 5.39×10⁻⁴⁴ s. No current physical theory can describe timescales shorter than the Planck time, such as the earliest events after the Big Bang.^[26] Some conjectures state that the structure of time need not remain smooth on intervals comparable to the Planck time.^[43]

Planck energy

The Planck energy E_P 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). The ultra-high-energy cosmic ray observed in 1991 had a measured energy of about 50 J, equivalent to about 2.5×10⁻⁸ E_P.^[44][45]

Proposals for theories of doubly 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.^[46][47]

Planck unit of force

The Planck unit of force may be thought of as the derived unit of force 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.^[48]

Some authors have argued that the Planck force is on the order of the maximum force that can occur between two bodies.^[49][50] However, the validity of these conjectures has been disputed.^[51][52]

Planck temperature

The Planck temperature T_P is 1.416784(16)×10³² K.^[10] At this temperature, the wavelength of light emitted by thermal radiation reaches the Planck length. There are no known physical models able to describe temperatures greater than T_P; a quantum theory of gravity would be required to model the extreme energies attained.^[53] Hypothetically, a system in thermal equilibrium at the Planck temperature might contain Planck-scale black holes, constantly being formed from thermal radiation and decaying via Hawking evaporation. Adding energy to such a system might decrease its temperature by creating larger black holes, whose Hawking temperature is lower.^[54]

Nondimensionalized equations

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 called nondimensionalization. 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 the energy–momentum relation E² = (mc²)² + (pc)², which becomes E² = m² + p², and the Dirac equation (iħγ^μ∂_μ − mc)ψ = 0, which becomes (iγ^μ∂_μ − m)ψ = 0.

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

^: 56

Hence a substantial body of physical theory developed since Planck's 1899 paper suggests normalizing not G but 4πG (or 8πG) to 1. Doing so would introduce a factor of 1/4π (or 1/8π) 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 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". When applied additionally to gravitation and Planck units, these are called rationalized Planck units^[55] and are seen in high-energy physics.^[56]

The rationalized Planck units are defined so that c = 4πG = ħ = ε₀ = k_B = 1.

There are several possible alternative normalizations.

Gravitational constant

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 (and therefore setting G = 1/4π):
  • Gauss's law for gravity becomes Φ_g = −M (rather than Φ_g = −4πM in Planck units).
  • Eliminates 4πG from the
    Poisson equation
    .
  • Eliminates 4πG in the
    mass density replacing charge density
    , and with 1/4πG replacing ε₀.
  • Normalizes the
    gravitational radiation in free space to 1 (normally expressed as 4πG/c).^{[note 2]}
  • Eliminates 4πG from the Bekenstein–Hawking formula (for the entropy of a black hole in terms of its mass m_BH and the area of its event horizon A_BH) which is simplified to S_BH = πA_BH = (m_BH)².
• Setting 8πG = 1 (and therefore setting G = 1/8π). 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 S_BH = (m_BH)²/2 = 2πA_BH.

See also

• cGh physics
• Dimensional analysis
• Doubly special relativity
• Trans-Planckian problem
• Zero-point energy

Explanatory notes

References

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External links

• Value of the fundamental constants, including the Planck units, as reported by the National Institute of Standards and Technology (NIST).
• The Planck scale: relativity meets quantum mechanics meets gravity from 'Einstein Light' at UNSW