1. Value

The Planck length ℓ_P is defined as:

[math]\displaystyle{ \ell_\mathrm{P} =\sqrt\frac{\hbar G}{c^3} \approx 1.616\;255 (18) \times 10^{-35}\ \mathrm{m} , }[/math]

where [math]\displaystyle{ c }[/math] is the speed of light, G is the gravitational constant, and ħ is the reduced Planck constant.^[1][2]

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.^[3] It can be defined as the reduced Compton wavelength of a black hole for which this equals its Schwarzschild radius.^[4]

2. History

In 1899, Max Planck suggested that there existed some fundamental natural units for length, mass, time and energy.^[5][6] 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".

3. 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.^[1] Some proposals for a theory of quantum gravity predict quantum foam appearing at the Planck scale due to fluctuations in the spacetime metric.^[7]

The Planck length is expected to be the shortest measurable distance, since 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.^[8]

The strings of string theory are modeled to be on the order of the Planck length.^[9] In theories of large extra dimensions, the Planck length has no fundamental, physical significance, and quantum gravitational effects appear at other scales.^[10]

4. 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 [math]\displaystyle{ \zeta }[/math] of geometry from Euclidean geometry in the gravitational field is determined by the ratio of the gravitational potential [math]\displaystyle{ \varphi }[/math] and the square of the speed of light [math]\displaystyle{ c }[/math]: [math]\displaystyle{ \zeta=\varphi/c^2 }[/math]. When [math]\displaystyle{ \zeta\ll 1 }[/math], the geometry is close to Euclidean geometry; for [math]\displaystyle{ \zeta\sim 1 }[/math], all similarities disappear. The energy of the oscillation of scale [math]\displaystyle{ l }[/math] is equal to [math]\displaystyle{ E=h\nu \sim \hbar c/l }[/math] (where [math]\displaystyle{ c/l }[/math] is the order of the oscillation frequency). The gravitational potential created by the mass [math]\displaystyle{ m }[/math], at this length is [math]\displaystyle{ \varphi=Gm/l }[/math], where [math]\displaystyle{ G }[/math] is the constant of universal gravitation. Instead of [math]\displaystyle{ m }[/math], we must substitute a mass, which, according to Einstein's formula, corresponds to the energy [math]\displaystyle{ E }[/math] (where [math]\displaystyle{ m=E/c^2 }[/math]). We get [math]\displaystyle{ \varphi=GE/l\,c^2=G\hbar/l^2c }[/math]. Dividing this expression by [math]\displaystyle{ c^2 }[/math], we obtain the value of the deviation [math]\displaystyle{ \zeta=G\hbar/c^3l^2=\ell^2_P/l^2 }[/math]. Equating [math]\displaystyle{ \zeta=1 }[/math], we find the length at which the Euclidean geometry is completely distorted. It is equal to Planck length [math]\displaystyle{ \ell_P=\sqrt{G\hbar/c^3}\approx 10^{-35}\mathrm{m} }[/math].^[11]

As noted in Regge (1958) "for the space-time region with dimensions [math]\displaystyle{ l }[/math] the uncertainty of the Christoffel symbols [math]\displaystyle{ \Delta\Gamma }[/math] be of the order of [math]\displaystyle{ \ell^2_P/l^3 }[/math], and the uncertainty of the metric tensor [math]\displaystyle{ \Delta g }[/math] is of the order of [math]\displaystyle{ \ell^2_P/l^2 }[/math]. If [math]\displaystyle{ l }[/math] is a macroscopic length, the quantum constraints are fantastically small and can be neglected even on atomic scales. If the value [math]\displaystyle{ l }[/math] is comparable to [math]\displaystyle{ \ell_P }[/math], then the maintenance of the former (usual) concept of space becomes more and more difficult and the influence of micro curvature becomes obvious".^[12] Conjecturally, this could imply that space-time becomes a quantum foam at the Planck scale.^[13]