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, Planck's 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. It is the smallest length possible, also see quantum foam.

Physical significance

There is currently no directly proven physical significance of the Planck length; it is, however, a topic of research. Because the Planck length is so many orders of magnitude smaller than any current instrument could possibly measure, there is currently no way of probing this length scale directly. Research on the Planck length is therefore mostly theoretical. According to the generalized uncertainty principle, the Planck length is in principle, within a factor of order unity, the shortest measurable length – and no improvements in measurement instruments could change that.

The Planck length is the square root of the Planck area, which is the area by which a spherical black hole increases when the black hole swallows one bit of information. The proof is relatively simple and was first set out by Jacob Bekenstein.

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 would become impossible to determine the difference between two locations less than one Planck length apart. The precise effects of quantum gravity are unknown; often it is suggested that spacetime might have a discrete or foamy structure at 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.

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.

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, that is, about the size of smallest object the naked human eye can see. In other words, 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 (10^31) 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 universe.

See also

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

Further reading

• Garay, Luis J. (1995). "Quantum gravity and miminum 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); Unknown parameter |month= ignored (help)

External links

• Bowley, Roger (2010). "Planck Length". Sixty Symbols. Brady Haran for the University of Nottingham. {{cite web}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)