Planck length

The Planck length ℓ_P is defined as:

${\displaystyle \ell _{\mathrm {P} }={\sqrt {\frac {4\pi \hbar G}{c^{3}}}}}$ (Lorentz–Heaviside version)

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

Solving the above two formulas will show the approximate equivalent value of this unit with respect to the metre:

${\displaystyle 1\ \ell _{\mathrm {P} }\approx 5.279(38)\times 10^{-35}\ \mathrm {m} }$ (Lorentz–Heaviside version)

${\displaystyle 1\ \ell _{\mathrm {P} }\approx 1.616(23)\times 10^{-35}\ \mathrm {m} }$ (Gaussian version)

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.[3][4]

The Planck length is about 10⁻²⁰ times the diameter of a proton.[5] It can be defined using the radius of the hypothesized Planck particle.

In 1899 Max Planck suggested that there existed some fundamental natural units for length, mass, time and energy.[6][7] These he derived using dimensional analysis, using only the Newton gravitational constant, the speed of light and the "unit of action", which later became the Planck constant. The natural units he further derived became known as the "Planck length", the "Planck mass", the "Planck time" and the "Planck energy".

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.[8] 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.[9] To measure anything the size of Planck length, the photon momentum needs to be very large due to Heisenberg's uncertainty principle and so much energy in such a small space would create a tiny black hole with the diameter of its event horizon equal to a Planck length.[10] The Planck length may represent the diameter of the smallest possible black hole.[3]

The main role in quantum gravity will be played by the uncertainty principle ${\displaystyle \Delta r_{s}\Delta r\geq \ell _{P}^{2}}$, where ${\displaystyle r_{s}}$ is the gravitational radius, ${\displaystyle r}$ is the radial coordinate, ${\displaystyle \ell _{P}}$ is the Planck length. This uncertainty principle is another form of Heisenberg's uncertainty principle between momentum and coordinate as applied to the Planck scale. Indeed, this ratio can be written as follows: ${\displaystyle \Delta (2Gm/c^{2})\Delta r\geq G\hbar /c^{3}}$, where ${\displaystyle G}$ is the gravitational constant, ${\displaystyle m}$ is body mass, ${\displaystyle c}$ is the speed of light, ${\displaystyle \hbar }$ is the reduced Planck constant. Reducing identical constants from two sides, we get Heisenberg's uncertainty principle . The uncertainty principle ${\displaystyle \Delta r_{s}\Delta r\geq \ell _{P}^{2}}$ predicts the appearance of virtual black holes and wormholes (quantum foam) on the Planck scale.[11][12]

Any attempt to investigate the possible existence of shorter distances, by performing higher-energy collisions, would inevitably result in black hole production. Higher-energy collisions, rather than splitting matter into finer pieces, would simply produce bigger black holes.[14] A decrease in ${\displaystyle \Delta r}$ will result in an increase in ${\displaystyle \Delta r_{s}}$ and vice versa.

The Planck length is sometimes misconceived as the minimum length of space-time, but this is not accepted by conventional physics, as this would require violation or modification of Lorentz symmetry.[8] 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,[8] or attempt to establish the Planck length as observer-invariant, known as doubly special relativity.

The strings of string theory are modeled to be on the order of the Planck length.[8][15] In theories of large extra dimensions, the Planck length has no fundamental physical significance, and quantum gravitational effects appear at other scales.

The Planck length is the length at which quantum zero oscillations of the gravitational field completely distort Euclidean geometry.[16] 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 ${\textstyle \ell _{P}={\sqrt {G\hbar /c^{3}}}\approx 10^{-35}\mathrm {m} }$.[17]

As noted in Regge (1958) "for the space-time region with dimensions ${\displaystyle l}$ the uncertainty of the Christoffel symbols ${\displaystyle \Delta \Gamma }$ be of the order of ${\displaystyle \ell _{P}^{2}/l^{3}}$, and the uncertainty of the metric tensor ${\displaystyle \Delta g}$ is of the order of ${\displaystyle \ell _{P}^{2}/l^{2}}$. If ${\displaystyle l}$ is a macroscopic length, the quantum constraints are fantastically small and can be neglected even on atomic scales. If the value ${\displaystyle l}$ is comparable to ${\displaystyle \ell _{P}}$, then the maintenance of the former (usual) concept of space becomes more and more difficult and the influence of micro curvature becomes obvious".[18] Conjecturally, this could imply that space-time becomes a quantum foam at the Planck scale.[19]

• Bowley, Roger; Eaves, Laurence (2010). "Planck Length". Sixty Symbols. Brady Haran for the University of Nottingham.