Planck length

In physics, the Planck length, denoted , is a unit of length in the system of Planck units that was originally proposed by physicist Max Planck, equal to 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 is defined as:
where formula_2 is the speed of light, is the gravitational constant, and is the reduced Planck constant. Solving the above will show the approximate equivalent value of this unit with respect to the metre:
The two digits enclosed by parentheses are the estimated standard error associated with the reported numerical value.
The Planck length is about 10−20 times the diameter of a proton. It can be defined using the radius of the hypothesized Planck particle.
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 () and the diameter of the observable universe (). 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 ).
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 formula_4, where formula_5 is the gravitational radius, formula_6 is the radial coordinate, formula_7 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: formula_8, where formula_9 is the gravitational constant, formula_10 is body mass, formula_2 is the speed of light, formula_12 is the reduced Planck constant. Reducing identical constants from two sides, we get Heisenberg's uncertainty principle formula_13. The uncertainty principle formula_4 predicts the appearance of virtual black holes and wormholes (quantum foam) on the Planck scale.
Proof: The equation for the invariant interval formula_15 in the Schwarzschild solution has the form 
Substitute according to the uncertainty relations formula_17. We obtain 
It is seen that at the Planck scale formula_19 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 formula_20 fluctuates and generates a quantum foam. These fluctuations formula_21 in the macroworld and in the world of atoms are very small in comparison with formula_22 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 formula_21 agrees with the Bohr-Rosenfeld uncertainty relation formula_24. Due to the smallness of the value formula_25, the formula for the invariant interval in special relativity is always written in the Galilean metric formula_26, 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 formula_27 will result in an increase in formula_28 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 formula_29 of geometry from Euclidean geometry in the gravitational field is determined by the ratio of the gravitational potential formula_30 and the square of the speed of light formula_2: formula_32. When formula_33, the geometry is close to Euclidean geometry; for formula_34, all similarities disappear. The energy of the oscillation of scale formula_35 is equal to formula_36 (where formula_37 is the order of the oscillation frequency). The gravitational potential created by the mass formula_10, at this length is formula_39, where formula_9 is the constant of universal gravitation. Instead of formula_10, we must substitute a mass, which, according to Einstein's formula, corresponds to the energy formula_42 (where formula_43). We get formula_44. Dividing this expression by formula_45, we obtain the value of the deviation formula_46. Equating formula_47, we find the length at which the Euclidean geometry is completely distorted. It is equal to Planck length formula_48.
As noted in Regge (1958) "for the space-time region with dimensions formula_35 the uncertainty of the Christoffel symbols formula_50 be of the order of formula_51, and the uncertainty of the metric tensor formula_52 is of the order of formula_53. If formula_35 is a macroscopic length, the quantum constraints are fantastically small and can be neglected even on atomic scales. If the value formula_35 is comparable to formula_7, 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.