# Planck length

Template:Unit of length In physics, the Planck length, denoted P, is a unit of length, equal to Template:Val. 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 ${\displaystyle \ell _{\text{P}}}$ is defined as

${\displaystyle \ell _{\text{P}}={\sqrt {\frac {\hbar G}{c^{3}}}}\approx 1.616\;199(97)\times 10^{-35}{\mbox{ m}}}$

where ${\displaystyle c}$ is the speed of light in a vacuum, ${\displaystyle G}$ is the gravitational constant, and ${\displaystyle \hbar }$ 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, and thus is exceedingly small. At this scale, more Planck lengths would fit inside a grain of sand volumetrically than grains of sand would fit inside the observable Universe.

## Theoretical significance

There is currently no proven physical significance of the Planck length; it is, however, a topic of theoretical research. Since the Planck length is so many orders of magnitude smaller than any current instrument could possibly measure, there is no way of examining it directly. According to the generalized uncertainty principle (a concept from speculative models of quantum gravity), the Planck length is, in principle, within a factor of 10, the shortest measurable length – and no theoretically known improvement in measurement instruments could change that.

In some forms of quantum gravity, the Planck length is the length scale at which the structure of spacetime becomes dominated by quantum effects, and it is impossible to determine the difference between two locations less than one Planck length apart. The precise effects of quantum gravity are unknown; it is often guessed that spacetime might have a discrete or foamy structure at a Planck length scale.{{ safesubst:#invoke:Unsubst||date=__DATE__ |\$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }}

The Planck area, equal to the square of the Planck length, plays a role in black hole entropy. The value of this entropy, in units of the Boltzmann constant, is known to be given by ${\displaystyle A/4\ell _{\text{P}}^{2}}$, where ${\displaystyle A}$ is the area of the event horizon. 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, as was proven by Jacob Bekenstein.[3]

If large extra dimensions exist, the measured strength of gravity may be much smaller than its true (small-scale) value. In this case the Planck length would have no fundamental physical significance, and quantum gravitational effects would appear at other scales.

In string theory, the Planck length is the order of magnitude of the oscillating strings that form elementary particles, and shorter lengths do not make physical sense.[4] The string scale ${\displaystyle l_{s}}$ is related to the Planck scale by ${\displaystyle \ell _{P}=g_{s}^{1/4}l_{s}}$, where ${\displaystyle g_{s}}$ is the string coupling constant. Contrary to what the name suggests, the string coupling constant is not constant, but depends on the value of a scalar field known as the dilaton.

In loop quantum gravity, area is quantized, and the Planck area is, within a factor of 10, the smallest possible area value.

In doubly special relativity, the Planck length is observer-invariant.

The search for the laws of physics valid at the Planck length is a part of the search for the theory of everything.

## Visualization

The size of the Planck length can be visualized as follows: if a particle or dot about 0.1mm in size (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. In other words, the universe is to a visible dot as a visible dot is to Planck length: The diameter of the observable universe is to within less than an order of magnitude, larger than a 0.1 millimeter object, roughly at or near the limits of the unaided human eye, by about the same factor (1031) as that 0.1mm object or dot is larger than the Planck length. More simply – on a logarithmic scale, a dot is halfway between the Planck length and the size of the observable universe.

## Bibliography

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