Planck length

Planck length
Unit system	Planck units
Unit of	length
Symbol	ℓP
Conversions
1 ℓP in ...	... is equal to ...
SI units	1.616255(18)×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 in the system of Planck units that was originally proposed by physicist Max Planck, equal to 1.616255(18)×10⁻³⁵ m. The Planck length can be defined from three fundamental physical constants: the speed of light, the Planck constant, and the gravitational constant. It is exceedingly small. It is also the reduced Compton wavelength of a particle with Planck mass. Regardless of whether it represents some fundamental limit to the universe, it is a useful unit in theoretical physics.

Value

The Planck length ℓ_P is defined as:

$\ell _{\mathrm {P} }={\sqrt {\frac {\hbar G}{c^{3}}}}\approx 1.616\;255(18)\times 10^{-35}\ \mathrm {m} ,$

where $c$ is the speed of light, G is the gravitational constant, and ħ is the reduced Planck constant.

The two digits enclosed by parentheses are the standard uncertainty of the reported numerical value.

The Planck length is about 10⁻²⁰ times the diameter of a proton. It can be defined as the reduced Compton wavelength of a black hole for which this equals its Schwarzschild radius.

History

In 1899, Max Planck suggested that there existed some fundamental natural units for length, mass, time and energy. He derived these using dimensional analysis, using only the Newton gravitational constant, the speed of light and the Planck constant (though it was not yet called this). The modern convention is to use the reduced Planck constant in place of the Planck constant in the definition of the resulting units. The derived natural units became known as the "Planck length", the "Planck mass", the "Planck time" and the "Planck energy".

Visualisation

The size of the Planck length can be visualized as follows: if a particle or dot about 0.1  mm 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 0.1 mm dot. There are approximately 62 orders of magnitude between the Planck length (1.616×10⁻³⁵ m) and the diameter of the observable universe (10²⁷ m). At the geometric mean of these extremes, 31 orders of magnitude (ten million trillion trillion) from either end, is the human hair (diameter ~100 μm, or 10⁻⁴ m).

Theoretical significance

The Planck length is approximately the size of a black hole where quantum and gravitational effects are at the same scale: where its Compton wavelength and Schwarzschild radius are approximately the same.

The main role in quantum gravity will be played by the uncertainty principle $\Delta r_{s}\Delta r\geq \ell _{P}^{2}$, where $r_{s}$ is the gravitational radius, $r$ is the radial coordinate, $\ell _{P}$ 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: $\Delta (2Gm/c^{2})\Delta r\geq G\hbar /c^{3}$, where $G$ is the gravitational constant, $m$ is body mass, $c$ is the speed of light, $\hbar$ is the reduced Planck constant. Reducing identical constants from two sides, we get Heisenberg's uncertainty principle $\Delta p\,\Delta r\geq \hbar /2$. The uncertainty principle $\Delta r_{s}\Delta r\geq \ell _{P}^{2}$ predicts the appearance of virtual black holes and wormholes (quantum foam) on the Planck scale.

Proof: The equation for the invariant interval $dS$ in the Schwarzschild solution has the form

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

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

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

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

The spacetime metric $g_{00}=1-\Delta g\approx 1-\ell _{P}^{2}/(\Delta r)^{2}$ fluctuates and generates a quantum foam. These fluctuations $\Delta g\sim \ell _{P}^{2}/(\Delta r)^{2}$ in the macroworld and in the world of atoms are very small in comparison with $1$ 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 $\Delta g\sim \ell _{P}^{2}/(\Delta r)^{2}$ agrees with the Bohr-Rosenfeld uncertainty relation $\Delta g\,(\Delta r)^{2}\gtrsim \ell _{P}^{2}$. Due to the smallness of the value $\ell _{P}^{\,2}/(\Delta r)^{2}$, the formula for the invariant interval in special relativity is always written in the Galilean metric $(+1,-1,-1,-1)$, which actually does not correspond to reality. The correct formula must take into account the fluctuations in the spacetime metric and the presence of virtual black holes and wormholes (quantum foam) at Planck scale distances. Ignoring this circumstance leads to ultraviolet divergences in quantum field theory. 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.

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. A decrease in $\Delta r$ will result in an increase in $\Delta r_{s}$ and vice versa. A subsequent increase of the energy will end up with larger black holes that have a worse resolution, not better. Therefore, the Planck length is the minimum distance that can be explored.

Implications

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. The Planck length does however provide practical limits on current physics. To measure a Planck length distance would require another particle with the Planck energy, approximately four quadrillion times greater than the Large Hadron Collider is capable of.

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

Planck length and Euclidean geometry

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 $\zeta$ of geometry from Euclidean geometry in the gravitational field is determined by the ratio of the gravitational potential $\varphi$ and the square of the speed of light $c$: $\zeta =\varphi /c^{2}$. When $\zeta \ll 1$, the geometry is close to Euclidean geometry; for $\zeta \sim 1$, all similarities disappear. The energy of the oscillation of scale $l$ is equal to $E=\hbar \nu \sim \hbar c/l$ (where $c/l$ is the order of the oscillation frequency). The gravitational potential created by the mass $m$, at this length is $\varphi =Gm/l$, where $G$ is the constant of universal gravitation. Instead of $m$, we must substitute a mass, which, according to Einstein's formula, corresponds to the energy $E$ (where $m=E/c^{2}$). We get $\varphi =GE/l\,c^{2}=G\hbar /l^{2}c$. Dividing this expression by $c^{2}$, we obtain the value of the deviation $\zeta =G\hbar /c^{3}l^{2}=\ell _{P}^{2}/l^{2}$. Equating $\zeta =1$, we find the length at which the Euclidean geometry is completely distorted. It is equal to Planck length ${\textstyle \ell _{P}={\sqrt {G\hbar /c^{3}}}\approx 10^{-35}\mathrm {m} }$.

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

See also

• Fock–Lorentz symmetry
• Immirzi parameter
• Orders of magnitude (length)
• Planck epoch

References

Bibliography

• Garay, Luis J. (January 1995). "Quantum gravity and minimum length". International Journal of Modern Physics A. 10 (2): 145–165. arXiv:gr-qc/9403008v2. Bibcode:1995IJMPA..10..145G. doi:10.1142/S0217751X95000085. S2CID 119520606.

External links

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