The Planck length, denoted by $l\_P\; \backslash$, is the unit of length in the system of units known as Planck units. The Planck length is deemed "natural" because it can be defined from three fundamental physical constants: the speed of light, Planck's constant, and the gravitational constant.

Value

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The Planck length equals:

$l\_P\; =\backslash sqrt\{\backslash frac\{\backslash hbar\; G\}\{c^3\}\}\; \backslash cong\; 1.616\; 24\; (12)\; \backslash times\; 10^\{-35\}$ metre

where:

• $\backslash hbar\; \backslash$ is Dirac's constant, Planck's constant divided by 2π ;
• G is the gravitational constant;
• c is the speed of light in vacuum.

The two digits between the parentheses denote the uncertainty (standard deviation) in the last two digits of the value.

In SI units, the Planck length is approximately 10^-35 meters. The estimated radius of the observable Universe (7.4 × 10²⁶ m) is 1.2 × 10⁶² Planck lengths.

Significance

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Ignoring a factor of $\backslash pi$, the Planck mass is roughly the mass of a black hole with a Schwarzschild radius equal to its Compton wavelength. The radius of such a black hole would be, roughly, the Planck length.

The following thought experiment illuminates this fact. The task is to measure an object's position by bouncing electromagnetic radiation, namely photons, off of it. The shorter the wavelength of the photons, and hence the higher their energy, the more accurate the measurement. If the photons are sufficiently energetic to make possible a measurement more precise than a Planck length, their collision with the object would, in principle, create a minuscule black hole. This black hole would "swallow" the photon and thereby make it impossible to obtain a measurement. A simple calculation using dimensional analysis suggests that this problem arises if we attempt to measure an object's position with a precision of less than a Planck length.

This thought experiment draws on both general relativity and the Heisenberg uncertainty principle of quantum mechanics. Combined, these two theories imply that it is impossible to measure position to a precision less than the Planck length, or duration to a precision greater than the time a photon traveling at C would take to travel a Planck length. Hence in any theory of quantum gravity combining general relativity and quantum mechanics, traditional notions of space and time will break down at distances shorter than the Planck length or times shorter than the Planck time.

History

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Max Planck was the first to propose the Planck length, a base unit in a system of measurement he called "natural units." By design, the Planck length, Planck time, and Planck mass are such that the physical constants c, G, and $\backslash hbar\; \backslash$ all equal 1 and thus disappear from the equations of physics. Although quantum mechanics and general relativity were unknown when Planck proposed his natural units, it later became clear that at a distance equal to the Planck length, gravity begins to display quantum effects, whose understanding would require a theory of quantum gravity.

See also

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• Planck units

External links

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• John Baez, "Length Scales in Physics: The Planck length."
• John Baez, "Higher-Dimensional Algebra and Planck-Scale Physics: The Planck Length."

Units of length | Natural units

Longitud de Planck | Longitud de Planck | Longueur de Planck | Lunghezza di Planck | Planck-Länge | Longitudo planckiana | Planck-hossz | Plancklengte | プランク長 | Długość Plancka | Comprimento de Planck | Độ dài Planck | 蒲朗克長度