Planck length

Template:Unit of length In physics, the Planck length, denoted ℓ_P, is a unit of length, equal to 1.616199(97)×10⁻³⁵ metres. 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 $\ell _{\text{P}}$ is defined as

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

where $c$ is the speed of light in a vacuum, $G$ is the gravitational constant, and $\hbar$ is the reduced Planck constant. 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, and thus is exceedingly small.

Simple dimensional analysis shows that the measurement of the position of physical objects with precision to the Planck length is problematic. Consider the following thought experiment. Suppose we want to determine the position of an object using electromagnetic radiation, i.e., photons. The greater the energy of photons, the shorter their wavelength and the more accurate the measurement. If the photon has enough energy to measure objects the size of the Planck length, it would collapse into a black hole and the measurement would be impossible. Thus, the Planck length sets the fundamental limits on the accuracy of length measurement.

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 order unity, the shortest measurable length – and no theoretically known improvement in measurement instruments could change that.

Physical meaning of the Planck length can be determined as follows:

A particle of mass $m$ has a reduced Compton wavelength

${\overline {\lambda }}_{C}={\frac {\lambda _{C}}{2\pi }}={\frac {\hbar }{mc}}$

Schwarzschild radius of the particle is

$r_{s}={\frac {2Gm}{c^{2}}}$

The product of these values is always constant and equal to

$r_{s}{\overline {\lambda }}_{C}={\frac {2G\hbar }{c^{3}}}=2\ell _{P}^{2}$

Accordingly, the uncertainty relation between the Schwarzschild radius of the particle and Compton wavelength of the particle will have the form

$\Delta r_{s}\Delta {\overline {\lambda }}_{C}\geq {\frac {G\hbar }{c^{3}}}=\ell _{P}^{2}$

which is another form of Heisenberg's uncertainty principle at the Planck scale. Indeed, substituting the expression for the Schwarzschild radius, we obtain

$\Delta \left({\frac {2Gm}{c^{2}}}\right)\Delta {\overline {\lambda }}_{C}\geq {\frac {G\hbar }{c^{3}}}$

Reducing the same characters, we come to the Heisenberg uncertainty relation

$\Delta \left(mc\right)\Delta {\overline {\lambda }}_{C}\geq {\frac {\hbar }{2}}$

Uncertainty relation between the gravitational radius and the Compton wavelength of the particle is a special case of the general Heisenberg's uncertainty principle at the Planck scale

$\Delta R_{\mu }\Delta x_{\mu }\geq \ell _{P}^{2}$

where $R_{\mu }$ is the radius of curvature of space-time small domain; $x_{\mu }$ is the coordinate small domain.

This implies that the Planck scale is the limit below which the very notions of space and length cease to exist. Any attempt to investigate the possible existence of shorter distances (less than 1.6 ×10⁻³⁵ m), 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. Reduction of the Compton wavelength of the particle increases the Schwarzschild radius. The resulting uncertainty relation generates at the Planck scale virtual black holes.

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.

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 $A/4\ell _{\text{P}}^{2}$, where $A$ is the area of the event horizon. The Planck area is the area by which a spherical black hole increases when the black hole swallows one bit of information, as was proven by Jacob Bekenstein.

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. The string scale $l_{s}$ is related to the Planck scale by $\ell _{P}=g_{s}^{1/4}l_{s}$, where $g_{s}$ 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 order unity, 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 a Planck dot: 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.

See also

• Fock–Lorentz symmetry
• Orders of magnitude (length)
• Planck energy
• Planck mass
• Planck epoch
• Planck scale
• Planck temperature
• Planck time

Bibliography

• Garay, Luis J. (January 1995). "Quantum gravity and minimum length". International Journal of Modern Physics A. 10 (2): 145 ff. arXiv:arXiv:gr-qc/9403008v2. Bibcode:1995IJMPA..10..145G. doi:10.1142/S0217751X95000085. {{cite journal}}: Check |arxiv= value (help)

External links

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