What Is The Minimum Distance

Units defined only by physical constants

In particle physics and physical cosmology, Planck units are a set of units of measurement divers exclusively in terms of iv universal physical constants, in such a style that these physical constants accept on the numerical value of i when expressed in terms of these units. Originally proposed in 1899 by German physicist Max Planck, these units are a system of natural units because their definition is based on properties of nature, more specifically the properties of free space, rather than a option of prototype object. They are relevant in research on unified theories such as quantum gravity.

The term Planck scale refers to quantities of infinite, time, energy and other units that are similar in magnitude to corresponding Planck units. This region may be characterized by particle energies of around x¹⁹ GeV, time intervals of effectually ten⁻⁴³ s and lengths of around 10⁻³⁵ m (approximately the energy-equivalent of the Planck mass, the Planck time and the Planck length, respectively). At the Planck scale, the predictions of the Standard Model, quantum field theory and full general relativity are non expected to apply, and quantum furnishings of gravity are expected to dominate. The all-time-known example is represented by the weather condition in the first 10⁻⁴³ seconds of our universe later on the Large Bang, approximately 13.8 billion years agone.

The iv universal constants that, past definition, have a numeric value 1 when expressed in these units are:

• the speed of light in vacuum, c,
• the gravitational constant, One thousand,
• the reduced Planck abiding, ħ,
• the Boltzmann constant, k _B.

Planck units do not incorporate an electromagnetic dimension. Some authors choose to extend the system to electromagnetism past, for instance, adding either the electric constant ε ₀ or 4π ε ₀ to this list. Similarly, authors choose to use variants of the organisation that give other numeric values to one or more of the 4 constants higher up.

Introduction

Any organisation 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 instance, the SI base quantities include length with the associated unit of measurement of the metre. In the organization 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 measurement of time is known every bit 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 i), these constants are and so eliminated from equations of physics in which they appear. For example, Newton's law of universal gravitation,

${\displaystyle F=G{\frac {m_{1}m_{two}}{r^{2}}}=\left({\frac {F_{\text{P}}l_{\text{P}}^{ii}}{m_{\text{P}}^{2}}}\right){\frac {m_{1}m_{two}}{r^{ii}}},}$

can be expressed as:

${\displaystyle {\frac {F}{F_{\text{P}}}}={\frac {\left({\dfrac {m_{1}}{m_{\text{P}}}}\correct)\left({\dfrac {m_{two}}{m_{\text{P}}}}\right)}{\left({\dfrac {r}{l_{\text{P}}}}\right)^{2}}}.}$

Both equations are dimensionally consistent and as valid in whatever organization of quantities, only the 2nd equation, with Grand absent, is relating only dimensionless quantities since any ratio of two like-dimensioned quantities is a dimensionless quantity. If, by a autograph convention, it is understood that each concrete quantity is the corresponding ratio with a coherent Planck unit of measurement (or "expressed in Planck units"), the ratios above may be expressed simply with the symbols of physical quantity, without being scaled explicitly by their respective unit:

${\displaystyle F'={\frac {m_{one}'m_{2}'}{r'^{2}}}.}$

This last equation (without G) is valid with F ′, m ₁′, g ₂′, and r ′ being the dimensionless ratio quantities corresponding to the standard quantities, written due east.g. F ′ ≘ F or F ′ = F/F _P , but not as a direct equality of quantities. This may seem to exist "setting the constants c, G, etc., to 1" if the correspondence of the quantities is idea of as equality. For this reason, Planck or other natural units should be employed with intendance. Referring to " G = c = 1", Paul S. Wesson wrote that, "Mathematically information technology is an acceptable pull a fast one on which saves labour. Physically information technology represents a loss of information and can lead to defoliation."^[three]

History and definition

The concept of natural units was introduced in 1874, when George Johnstone Stoney, noting that electric accuse is quantized, derived units of length, time, and mass, now named Stoney units in his honor. Stoney chose his units then that M, c, and the electron charge e would exist numerically equal to 1.^[iv] In 1899, 1 year before the advent of quantum theory, Max Planck introduced what became later on known equally the Planck constant.^{[5] [six]} At the cease of the paper, he proposed the base of operations units later that were named in his honor. The Planck units are based on the quantum of action, now usually known as the Planck abiding, which appeared in the Wien approximation for black-body radiation. Planck underlined the universality of the new unit of measurement arrangement, writing:

... 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 notwendig behalten und welche daher als »natürliche Maßeinheiten« bezeichnet werden können.

... it is possible to gear up units for length, mass, time and temperature, which are independent of special bodies or substances, necessarily retaining their significant for all times and for all civilizations, including extraterrestrial and non-homo ones, which tin can be chosen "natural units of mensurate".

Planck considered merely 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.^[half-dozen] His definitions differ from the mod ones by a factor of ${\displaystyle {\sqrt {ii\pi }}}$ , because the modernistic definitions use ${\displaystyle \hbar }$ rather than ${\displaystyle h}$ .^{[5] [6]}

Tabular array ane: Modernistic values for Planck'south original choice of quantities

Proper noun	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 (K)	${\displaystyle m_{\text{P}}={\sqrt {\frac {\hbar c}{G}}}}$	2.176434(24)×ten−8 kg [eight]
Planck time	time (T)	${\displaystyle t_{\text{P}}={\sqrt {\frac {\hbar K}{c^{5}}}}}$	five.391247(60)×10−44 s [9]
Planck temperature	temperature (Θ)	${\displaystyle T_{\text{P}}={\sqrt {\frac {\hbar c^{five}}{Gk_{\text{B}}^{2}}}}}$	1.416784(16)×1032 1000 [x]

Different the example with the International System of Units, in that location 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 fourth dimension, regarding an additional unit of measurement for temperature to exist redundant.^{[note 1]} Other tabulations add together, in addition to a unit for temperature, a unit for electric charge, and so that the vacuum permittivity ${\displaystyle \epsilon _{0}}$ is also normalized to one.^{[12] [xiii]} Some of these tabulations too replace mass with energy when doing then.^[14] Depending on the author's choice, this accuse unit is given by

${\displaystyle q_{\text{P}}={\sqrt {4\pi \epsilon _{0}\hbar c}}\approx 1.875546\times 10^{-18}{\text{ C}}\approx 11.vii\ e}$

or

${\displaystyle q_{\text{P}}={\sqrt {\epsilon _{0}\hbar c}}\approx v.290818\times 10^{-19}{\text{ C}}\approx three.iii\ eastward.}$

The Planck charge, as well as other electromagnetic units that can be defined like resistance and magnetic flux, are more hard to interpret than Planck's original units and are used less oftentimes.^[15]

In SI units, the values of c, h, due east and g _B are verbal and the values of ε ₀ and G in SI units respectively accept relative uncertainties of one.5×10^{−10 [xvi]} and 2.2×x^−v .^[17] Hence, the uncertainties in the SI values of the Planck units derive nigh entirely from uncertainty in the SI value of Yard.

Derived units

In whatever system of measurement, units for many concrete quantities tin can exist derived from base of operations units. Table two offers a sample of derived Planck units, some of which are seldom used. Equally with the base units, their use is mostly bars to theoretical physics considering nigh of them are too large or too pocket-sized for empirical or practical apply and there are big uncertainties in their values.

Table 2: Coherent derived units of Planck units

Derived unit of	Expression	Guess SI equivalent
area (L2)	${\displaystyle l_{\text{P}}^{2}={\frac {\hbar G}{c^{3}}}}$	2.6121×10−70 chiliad2
volume (L3)	${\displaystyle l_{\text{P}}^{3}=\left({\frac {\hbar G}{c^{3}}}\right)^{\frac {three}{two}}={\sqrt {\frac {(\hbar K)^{3}}{c^{9}}}}}$	iv.2217×10−105 m3
momentum (LMT−1)	${\displaystyle m_{\text{P}}c={\frac {\hbar }{l_{\text{P}}}}={\sqrt {\frac {\hbar c^{iii}}{G}}}}$	6.5249 kg⋅m/due south
energy (L2MT−two)	${\displaystyle E_{\text{P}}=m_{\text{P}}c^{2}={\frac {\hbar }{t_{\text{P}}}}={\sqrt {\frac {\hbar c^{5}}{G}}}}$	1.9561×x9 J
strength (LMT−two)	${\displaystyle F_{\text{P}}={\frac {E_{\text{P}}}{l_{\text{P}}}}={\frac {\hbar }{l_{\text{P}}t_{\text{P}}}}={\frac {c^{4}}{G}}}$	1.2103×1044 Due north
density (Fifty−3G)	${\displaystyle \rho _{\text{P}}={\frac {m_{\text{P}}}{l_{\text{P}}^{3}}}={\frac {\hbar t_{\text{P}}}{l_{\text{P}}^{v}}}={\frac {c^{5}}{\hbar Grand^{2}}}}$	five.1550×1096 kg/miii
acceleration (LT−2)	${\displaystyle a_{\text{P}}={\frac {c}{t_{\text{P}}}}={\sqrt {\frac {c^{7}}{\hbar G}}}}$	five.5608×ten51 1000/s2

Some Planck units, such as of time and length, are many orders of magnitude too large or too pocket-size to be of practical utilize, and so that Planck units as a system are typically but relevant to theoretical physics. In some cases, a Planck unit may suggest a limit to a range of a physical quantity where nowadays-twenty-four hour period theories of physics apply.^[18] For example, our understanding of the Big Blindside does not extend to the Planck epoch, i.east., when the universe was less than ane Planck fourth dimension old. Describing the universe during the Planck epoch requires a theory of quantum gravity that would contain breakthrough furnishings into general relativity. Such a theory does not however be.

Several quantities are non "extreme" in magnitude, such as the Planck mass, which is about 22 micrograms: very large in comparison with subatomic particles, and within the mass range of living organisms.^{[19] : 872} Similarly, the related units of free energy and of momentum are in the range of some everyday phenomena.

Significance

Planck units have trivial anthropocentric arbitrariness, but do still involve some arbitrary choices in terms of the defining constants. Different the metre and second, which be as base of operations units in the SI system for historical reasons, the Planck length and Planck fourth dimension are conceptually linked at a central physical level. Consequently, natural units assistance physicists to reframe questions. Frank Wilczek puts it succinctly:

Nosotros see that the question [posed] is not, "Why is gravity then feeble?" but rather, "Why is the proton's mass and then pocket-size?" 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 [ane/(13 quintillion)].^[20]

While it is true that the electrostatic repulsive force betwixt two protons (alone in free space) profoundly exceeds the gravitational attractive force between the aforementioned ii protons, this is not well-nigh the relative strengths of the 2 fundamental forces. From the point of view of Planck units, this is comparison apples with 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 of measurement charge just the mass of the protons is far less than the unit mass.

Planck calibration

In particle physics and physical cosmology, the Planck scale is an energy scale effectually 1.22×x¹⁹ GeV (the Planck energy, corresponding to the energy equivalent of the Planck mass, is 2.17645 ×10⁻⁸ kg) at which quantum effects of gravity become significant. At this scale, present descriptions and theories of sub-atomic particle interactions in terms of quantum field theory interruption down and go inadequate, due to the impact of the apparent non-renormalizability of gravity within current theories.

Relationship to gravity

At the Planck length calibration, the strength of gravity is expected to become comparable with the other forces, and it is theorized that all the fundamental forces are unified at that calibration, only the exact mechanism of this unification remains unknown. The Planck scale is therefore the bespeak at which the effects of quantum gravity can no longer be ignored in other fundamental interactions, where current calculations and approaches begin to intermission downwards, and a means to take account of its affect is necessary.^[21] On these grounds, information technology has been speculated that information technology may be an estimate lower limit at which a black hole could be formed by plummet.^[22]

While physicists have a adequately skilful understanding of the other primal interactions of forces on the breakthrough level, gravity is problematic, and cannot be integrated with quantum mechanics at very high energies using the usual framework of breakthrough field theory. At lesser free energy levels information technology is usually ignored, while for energies approaching or exceeding the Planck scale, a new theory of breakthrough gravity is necessary. Approaches to this trouble include cord theory and M-theory, loop quantum gravity, noncommutative geometry, and causal set theory.

In cosmology

In Large Blindside 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, t _P, or approximately ten⁻⁴³ seconds.^[23] There is no currently available concrete 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 causeless that quantum effects of gravity dominate physical interactions at this time scale. At this scale, the unified strength of the Standard Model is assumed to be unified with gravitation. Immeasurably hot and dense, the state of the Planck epoch was succeeded by the grand unification epoch, where gravitation is separated from the unified strength of the Standard Model, in plow followed by the inflationary epoch, which concluded after nigh 10⁻³² seconds (or about 10¹¹t _P).^[24]

Tabular array three lists properties of the observable universe today expressed in Planck units.^{[25] [26]}

Table 3: Today'due south universe in Planck units

Holding of nowadays-day observable universe	Approximate number of Planck units	Equivalents
Age	8.08 × 1060 t P	four.35 × 1017 due south or 1.38 × x10 years
Bore	v.4 × ten61 l P	8.7 × ten26 thou or nine.2 × x10 light-years
Mass	approx. ten60 yard P	3 × 1052 kg or 1.5 × 1022 solar masses (only counting stars) 10fourscore protons (sometimes known as the Eddington number)
Density	1.viii × 10−123 m P⋅l P −three	9.9 × 10−27 kg⋅thou−3
Temperature	1.nine × 10−32 T P	2.725 K temperature of the catholic microwave background radiation
Cosmological constant	≈ 10−122 50  −ii P	≈ ten−52 m−two
Hubble constant	≈ 10−61 t  −1 P	≈ 10−eighteen s−1 ≈ 10ii (km/south)/Mpc

Afterwards 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 mode that its value remains Λ ~ T ⁻² throughout the history of the universe.^[27]

Assay of the units

Planck length

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

$${\displaystyle \ell _{\mathrm {P} }={\sqrt {\frac {\hbar G}{c^{iii}}}}}$$

It is equal to 1.616255(18)×10⁻³⁵ thousand,^[7] where the two digits enclosed past parentheses are the estimated standard mistake associated with the reported numerical value, or most 10⁻²⁰ times the bore of a proton.^[28] It tin can be motivated in diverse means, such every bit considering a particle whose reduced Compton wavelength is comparable to its Schwarzschild radius,^{[28] [29] [30]} though whether those concepts are in fact simultaneously applicable is open to debate.^[31] (The same heuristic argument simultaneously motivates the Planck mass.^[29])

The Planck length is a distance scale of interest in speculations most quantum gravity. The Bekenstein–Hawking entropy of a black hole is i-quaternary the area of its upshot horizon in units of Planck length squared.^{[11] : 370} Since the 1950s, information technology has been conjectured that quantum fluctuations of the spacetime metric might make the familiar notion of distance inapplicable below the Planck length.^{[32] [33] [34]} This is sometimes expressed by saying that "spacetime becomes a foam at the Planck scale".^[35] It is possible that the Planck length is the shortest physically measurable distance, since any effort to investigate the possible existence of shorter distances, past performing higher-energy collisions, would result in blackness hole product. Higher-energy collisions, rather than splitting matter into finer pieces, would simply produce bigger black holes.^[36]

The strings of string theory are modeled to exist on the social club of the Planck length.^{[37] [38]} In theories with large extra dimensions, the Planck length calculated from the observed value of ${\displaystyle G}$ can be smaller than the true, fundamental Planck length.^{[11] : 61 [39]}

Planck fourth dimension

The Planck time t _P is the time required for light to travel a distance of 1 Planck length in vacuum, which is a fourth dimension interval of approximately 5.39×10⁻⁴⁴ s. No current concrete theory can describe timescales shorter than the Planck time, such as the primeval events after the Large Bang,^[23] and it is conjectured that the structure of time breaks down on intervals comparable to the Planck time.^[40]

Planck energy

About Planck units are extremely small, as in the example of Planck length or Planck time, or extremely large, as in the example of Planck temperature or Planck density. For comparison, the Planck energy E _P is approximately equal to the free energy stored in an automobile gas tank (57.two L of gasoline at 34.2 MJ/L of chemic energy). The ultra-high-energy catholic ray observed in 1991 had a measured energy of about 50 J, equivalent to about 2.v×10⁻⁸E _P .^{[41] [42]}

Proposals for theories of doubly special relativity posit that, in addition to the speed of low-cal, an energy scale is also invariant for all inertial observers. Typically, this energy scale is called to be the Planck free energy.^{[43] [44]}

Planck unit of measurement of force

The Planck unit of strength may be thought of as the derived unit of strength in the Planck system if the Planck units of time, length, and mass are considered to be base units.

${\displaystyle F_{\text{P}}={\frac {m_{\text{P}}c}{t_{\text{P}}}}={\frac {c^{iv}}{G}}\approx \mathrm {1.2103\times 10^{44}~N} }$

Information technology is the gravitational attractive forcefulness of two bodies of i 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 ii objects with Planck accuse and mass that are held 1 Planck length apart exactly balances the Newtonian allure between them.^[45]

Various authors accept argued that the Planck force is on the order of the maximum force that can exist observed in nature.^{[46] [47]} However, the validity of these conjectures has been disputed.^{[48] [49]}

Planck temperature

The Planck temperature T _P is 1.416784(16)×10³² M.^[10] At this temperature, the wavelength of light emitted past thermal radiations reaches the Planck length. In that location are no known concrete models able to describe temperatures greater than T _P; a quantum theory of gravity would be required to model the extreme energies attained.^[50] Hypothetically, a system in thermal equilibrium at the Planck temperature might incorporate Planck-scale black holes, constantly being formed from thermal radiation and decomposable via Hawking evaporation. Calculation energy to such a system might decrease its temperature by creating larger black holes, whose Hawking temperature is lower.^[51]

Nondimensionalized equations

Physical quantities that have unlike dimensions (such equally time and length) cannot be equated fifty-fifty if they are numerically equal (eastward.g., ane 2nd is non the same every bit 1 metre). In theoretical physics, nonetheless, this scruple may be set aside, by a procedure chosen nondimensionalization. The constructive result is that many central equations of physics, which ofttimes include some of the constants used to define Planck units, become equations where these constants are replaced past a 1.

Examples include the energy–momentum relation ${\displaystyle E^{two}=(mc^{2})^{two}+(pc)^{two}}$ , which becomes ${\displaystyle Eastward^{ii}=chiliad^{2}+p^{ii}}$ , and the Dirac equation ${\displaystyle \ (i\hbar \gamma ^{\mu }\partial _{\mu }-mc)\psi =0}$ , which becomes ${\displaystyle \ (i\gamma ^{\mu }\partial _{\mu }-yard)\psi =0}$ .

Culling choices of normalization

As already stated to a higher place, Planck units are derived past "normalizing" the numerical values of certain fundamental constants to 1. These normalizations are neither the only ones possible nor necessarily the all-time. Moreover, the pick 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 in iii-dimensional infinite, the surface area of a sphere of radius r is ivπ r ² . This, forth with the concept of flux, are the basis for the inverse-square police, Gauss's police, and the divergence operator practical to flux density. For example, gravitational and electrostatic fields produced by point objects have spherical symmetry, then the electric flux through a sphere of radius r around a point charge volition exist distributed uniformly over that sphere. From this, it follows that a factor of 4π r ² will appear in the denominator of Coulomb's law in rationalized class.^{[25] : 214–15} (Both the numerical gene and the power of the dependence on r would modify if infinite were college-dimensional; the right expressions can be deduced from the geometry of higher-dimensional spheres.^{[11] : 51} ) Too for Newton'southward constabulary of universal gravitation: a cistron of fourπ naturally appears in Poisson's equation when relating the gravitational potential to the distribution of matter.^{[11] : 56}

Hence a substantial body of physical theory adult since Planck'south 1899 paper suggests normalizing not G just ivπ 1000 (or 8π G) to one. Doing so would introduce a gene of 1 / 4π (or 1 / 8π ) into the nondimensionalized course of the police force of universal gravitation, consistent with the modern rationalized conception of Coulomb's police force in terms of the vacuum permittivity. In fact, culling normalizations frequently preserve the factor of 1 / ivπ in the nondimensionalized form of Coulomb's police as well, so that the nondimensionalized Maxwell's equations for electromagnetism and gravitoelectromagnetism both accept the same grade every bit those for electromagnetism in SI, which practise not have whatsoever factors of fourπ. When this is practical to electromagnetic constants, ε ₀, this unit of measurement organisation is called "rationalized ". When applied additionally to gravitation and Planck units, these are called rationalized Planck units ^[52] and are seen in high-energy physics.^[53]

The rationalized Planck units are divers so that ${\displaystyle c=4\pi Thou=\hbar =\varepsilon _{0}=k_{\text{B}}=ane}$ .

At that place are several possible alternative normalizations.

Gravitational constant

In 1899, Newton'south law of universal gravitation was still seen as exact, rather than as a convenient approximation holding for "pocket-sized" velocities and masses (the approximate nature of Newton'south constabulary was shown following the development of full general relativity in 1915). Hence Planck normalized to 1 the gravitational constant Thousand in Newton's law. In theories emerging after 1899, Yard about always appears in formulae multiplied past 4π or a small integer multiple thereof. Hence, a choice to exist made when designing a organization of natural units is which, if whatsoever, instances of ivπ appearing in the equations of physics are to be eliminated via the normalization.

• Normalizing ivπ Chiliad to 1 (and therefore setting G = 1 / 4π ):
  • Gauss's constabulary for gravity becomes Φ _g = −Chiliad (rather than Φ _g = −4π Thou in Planck units).
  • Eliminates 4π G from the Poisson equation.
  • Eliminates 4π One thousand in the gravitoelectromagnetic (Gem) equations, which hold in weak gravitational fields or locally flat spacetime. These equations have the same course every bit Maxwell's equations (and the Lorentz force equation) of electromagnetism, with mass density replacing charge density, and with i / fourπ G replacing ε ₀.
  • Normalizes the characteristic impedance Z _g of gravitational radiations in gratis space to 1 (ordinarily expressed as 4π G / c ).^{[note 2]}
  • Eliminates ivπ G from the Bekenstein–Hawking formula (for the entropy of a black hole in terms of its mass chiliad _BH and the area of its event horizon A _BH) which is simplified to Due south _BH = π A _BH = (k _BH)² .
• Setting 8π Yard = 1 (and therefore setting G = 1 / 8π ). This would eliminate 8π G from the Einstein field equations, Einstein–Hilbert action, and the Friedmann equations, for gravitation. Planck units modified then that 8π G = 1 are known as reduced Planck units, because the Planck mass is divided past √8π . Also, the Bekenstein–Hawking formula for the entropy of a black hole simplifies to S _BH = (m _BH)²/2 = 2π A _BH .

See as well

• cGh physics
• Dimensional analysis
• Doubly special relativity
• Trans-Planckian trouble
• Aught-betoken energy

Explanatory notes

1. ^ For example, both Frank Wilczek and Barton Zwiebach exercise so,^{[1] [xi] : 54} as does the textbook Gravitation.^{[two] : 1215}
2. ^ General relativity predicts that gravitational radiations propagates at the same speed as electromagnetic radiation.^{[54] : 60 [55] : 158}

References

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2. ^ ^{a b} Misner, Charles W.; Thorne, Kip S.; Wheeler, John A. (1973). Gravitation. New York. ISBN0-7167-0334-3. OCLC 585119.
3. ^ Wesson, P. S. (1980). "The awarding of dimensional analysis to cosmology". Space Scientific discipline Reviews. 27 (two): 117. Bibcode:1980SSRv...27..109W. doi:ten.1007/bf00212237. S2CID 120784299.
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External links

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

What Is The Minimum Distance,

Source: https://en.wikipedia.org/wiki/Planck_units

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