Planck length

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Planck length
Unit system	Planck units
Unit of	length
Symbol	ℓP
Unit conversions
1 ℓP in ...	... is equal to ...
SI units	1.616229(38)×10−35 m
natural units	11.706 ℓS  3.0542×10−25 a0
imperial/US units	6.3631×10−34 in

In physics, the Planck length, denoted ℓ_P, is a unit of length, equal to 1.616229(38)×10⁻³⁵ metres. 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.

Value

The Planck length ℓ_P is defined as:

${\displaystyle \ell _{\mathrm {P} }={\sqrt {\frac {\hbar G}{c^{3}}}}}$

Solving the above will show the approximate equivalent value of this unit with respect to the meter:
${\displaystyle 1\ \ell _{\mathrm {P} }\approx 1.616\;229(38)\times 10^{-35}\ \mathrm {m} }$, where ${\displaystyle c}$ is the speed of light in a vacuum, G is the gravitational constant, and ħ is the reduced Planck constant. The two digits enclosed by parentheses are the estimated standard error associated with the reported numerical value.^[1][2]

The Planck length is about 10⁻²⁰ times the diameter of a proton. It can be defined using the radius of Planck particle.

Measuring the Planck length

In 2017 it was suggested by E. Haug^[3] that the Planck length can be indirectly measured independent of any knowledge of Newton's gravitational constant with for example the use of a Cavendish apparatus. Further, it seems like the error in the Planck length measures must be exactly half of that in the measurement errors of the Newton's gravitational constant. That is the error as measured in percentage term, also known as the relative standard uncertainty. This is in line with the relative standard uncertainty reported by NIST, which for the gravitational constant is ${\displaystyle 4.7\times 10^{-5}}$ and for the Planck length is ${\displaystyle 2.3\times 10^{-5}}$.

History

In 1899 Max Planck^[4] suggested that there existed some fundamental natural units for length, mass, time and energy. These he derived using dimensional analysis only using the Newton gravitational constant, the speed of light and the Planck constant. The natural units he derived has later been known as: the Planck length, the Planck mass, the Planck time and the Planck energy.

Theoretical significance

The Planck length is the scale at which quantum gravitational effects are believed to begin to be apparent, where interactions require a working theory of quantum gravity to be analyzed.^[5] The Planck area is the area by which the surface of a spherical black hole increases when the black hole swallows one bit of information.^[6]

The Planck length is sometimes misconceived as the minimum length of spacetime, but this is not accepted by conventional physics, as this would require violation or modification of Lorentz symmetry.^[5] However, certain theories of loop quantum gravity do attempt to establish a minimum length on the scale of the Planck length, though not necessarily the Planck length itself^[5], or attempt to establish the Planck length as observer-invariant, known as doubly special relativity.^{[citation needed]}

The strings of string theory are modelled to be on the order of the Planck length.^[5][7] In theories of large extra dimensions, the Planck length has no fundamental physical significance, and quantum gravitational effects appear at other scales.^{[citation needed]}

Planck length and Euclidean geometry

The gravitational field performs zero-point oscillations, and the geometry associated with it also oscillates. The ratio of the circumference to the radius varies near the Euclidean value. The smaller the scale, the greater the deviations from the Euclidean geometry. Let us estimate the order of the wavelength of zero gravitational oscillations, at which the geometry becomes completely unlike the Euclidean geometry. The degree of deviation ${\displaystyle \zeta }$ of geometry from Euclidean geometry in the gravitational field is determined by the ratio of the gravitational potential ${\displaystyle \varphi }$ and the square of the speed of light ${\displaystyle c}$: ${\displaystyle \zeta =\varphi /c^{2}}$. When ${\displaystyle \zeta \ll 1}$, the geometry is close to Euclidean geometry; for ${\displaystyle \zeta \sim 1}$, all similarities disappear. The energy of the oscillation of scale ${\displaystyle l}$ is equal to ${\displaystyle E=\hbar \nu \sim \hbar c/l}$ (where ${\displaystyle c/l}$ is the order of the oscillation frequency). The gravitational potential created by the mass ${\displaystyle m}$, at this length is ${\displaystyle \varphi =Gm/l}$, where ${\displaystyle G}$ is the constant of universal gravitation. Instead of ${\displaystyle m}$, we must substitute a mass, which, according to Einstein's formula, corresponds to the energy ${\displaystyle E}$ (where ${\displaystyle m=E/c^{2}}$). We get ${\displaystyle \varphi =GE/l\,c^{2}=G\hbar /l^{2}c}$. Dividing this expression by ${\displaystyle c^{2}}$, we obtain the value of the deviation ${\displaystyle \zeta =G\hbar /c^{3}l^{2}=\ell _{P}^{2}/l^{2}}$. Equating ${\displaystyle \zeta =1}$, we find the length at which the Euclidean geometry is completely distorted. It is equal to Planck length ${\displaystyle \ell _{P}={\sqrt {G\hbar /c^{3}}}\approx 10^{-35}}$m. Here there is a quantum foam.

Visualization

The size of the Planck length can be visualized as follows: if a particle or dot about 0.005 mm in size (which is the same size as a small grain of silt) were magnified in size to be as large as the observable universe, then inside that universe-sized "dot", the Planck length would be roughly the size of an actual 0.005 mm dot. In other words, a 0.005 mm dot is halfway between the Planck length and the size of the observable universe on a logarithmic scale.^[8] All said, the attempt to visualize to an arbitrary scale of a 0.005 mm dot is only for a hinge point. With no fixed frame of reference for time or space, where the spatial units shrink toward infinitesimally small spatial sections and time stretches toward infinity, scale breaks down. Inverted, where space is stretched and time is shrunk, the scale adjusts the other way according to the ratio V-squared/C-squared (Lorentz transformation).^{[clarification needed]}