Planck length

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 metre:

${\displaystyle 1\ \ell _{\mathrm {P} }\approx 1.616\;255(18)\times 10^{-35}\ \mathrm {m} }$

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.[5][6]

The Planck length is about 10⁻²⁰ times the diameter of a proton.[7] 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.[8][9] 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 size of the Planck length can be visualized as follows: if a particle or dot about 0.1mm in size (the diameter of human hair, which is at or near the smallest the unaided human eye can see) 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 0.1mm dot. Alternatively: There are approximately 62 orders of magnitude between the Planck Length (1.616e-35 m.) and the diameter of the Observable Universe (1e27 m.). Right in the middle, 31 orders of magnitude (Ten million trillion trillion) from either end, is the human hair (diameter ~100 micrometers, or 1e-4 m).

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. This scale is known as Quantum foam.[10] 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.[11] 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.[12] The Planck length may represent the diameter of the smallest possible black hole.[5]

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.[13][14]

Proof: The equation for the invariant interval ${\displaystyle dS}$ in the Schwarzschild solution has the form

${\displaystyle dS^{2}=\left(1-{\frac {r_{s}}{r}}\right)c^{2}dt^{2}-{\frac {dr^{2}}{1-{r_{s}}/{r}}}-r^{2}(d\Omega ^{2}+\sin ^{2}\Omega d\varphi ^{2})}$

Substitute according to the uncertainty relations ${\displaystyle r_{s}\approx \ell _{P}^{2}/r}$. We obtain

${\displaystyle dS^{2}\approx \left(1-{\frac {\ell _{P}^{2}}{r^{2}}}\right)c^{2}dt^{2}-{\frac {dr^{2}}{1-{\ell _{P}^{2}}/{r^{2}}}}-r^{2}(d\Omega ^{2}+\sin ^{2}\Omega d\varphi ^{2})}$

It is seen that at the Planck scale ${\displaystyle r=\ell _{P}}$ spacetime metric in special and general relativity is bounded below by the Planck length (division by zero appears), and on this scale, there are real and virtual black holes.

The spacetime metric ${\displaystyle g_{00}=1-\Delta g\approx 1-\ell _{P}^{2}/(\Delta r)^{2}}$ fluctuates and generates a quantum foam. These fluctuations ${\displaystyle \Delta g\sim \ell _{P}^{2}/(\Delta r)^{2}}$ in the macroworld and in the world of atoms are very small in comparison with ${\displaystyle 1}$ and become noticeable only on the Planck scale. Lorentz-invariance is violated at the Planck scale. The formula for the fluctuations of the gravitational potential ${\displaystyle \Delta g\sim \ell _{P}^{2}/(\Delta r)^{2}}$ agrees with the Bohr-Rosenfeld uncertainty relation ${\displaystyle \Delta g\,(\Delta r)^{2}\gtrsim \ell _{P}^{2}}$.[15] Quantum fluctuations in geometry are superimposed on the large-scale slowly changing curvature predicted by the classical deterministic general relativity. Classical curvature and quantum fluctuations coexist with each other.[13]

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.[16] A decrease in ${\displaystyle \Delta r}$ will result in an increase in ${\displaystyle \Delta r_{s}}$ and vice versa. A subsequent increase of the energy will end up with larger black holes that have a worse resolution, not better. So, the Planck length is the minimum distance one may probe.

The Planck length refers to the internal architecture of particles and objects. Many other quantities that have units of length may be much shorter than the Planck length. For example, the photon's wavelength may be arbitrarily short: any photon may be boosted, as special relativity guarantees, so that its wavelength gets even shorter.[17]

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.[10] 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,[10] 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.[10][18] 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. 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 ${\textstyle \ell _{P}={\sqrt {G\hbar /c^{3}}}\approx 10^{-35}\mathrm {m} }$.[19]

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".[20] Conjecturally, this could imply that space-time becomes a quantum foam at the Planck scale.[21]

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