Planck length

1 Planck length =

SI units
16.162×10^−36 m	16.162×10^−27 nm
Natural units
11.706 ℓS	305.42×10^−27 a0
US customary units (Imperial units)
53.025×10^−36 ft	636.30×10^−36 in

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 ℓ_P is defined as

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

where $c$ is the speed of light in a vacuum, G is the gravitational constant, and ħ 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.

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.

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 $\frac{A}{4\ell_\mathrm{P}^2}$, where 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 l_s is related to the Planck scale by ℓ_P = g_s^1/4l_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 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 0.1 mm in size (which is approximately 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 0.1 mm dot. In other words, a 0.1 mm dot is halfway between the Planck length and the size of the observable universe on a logarithmic scale.

See also

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

Notes and references

1. John Baez, The Planck Length
2. NIST, "Planck length", NIST's published CODATA constants
3. "Phys. Rev. D 7, 2333 (1973): Black Holes and Entropy". Prd.aps.org. Retrieved 2013-10-21.
4. Cliff Burgess; Fernando Quevedo (November 2007). "The Great Cosmic Roller-Coaster Ride". Scientific American (print) (Scientific American, Inc.). p. 55.

Bibliography

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

External links

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