Planck length

Template:Unit of length In physics, the Planck length, denoted $\scriptstyle \ell _{P}\$, is unit of length, equal to about 1.616 252 × 10^-35 meters. It is a base unit in the system of Planck units, the most widely used system of natural units. 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. Current theory suggests that 1 Planck length is the smallest distance or size about which anything can be known.

Value

The Planck length equals:

$\ell _{P}={\sqrt {\frac {\hbar G}{c^{3}}}}\thickapprox 1.616252(81)\times 10^{-35}{\mbox{ meters}}$

(about 0.000 000 000 000 000 000 000 000 000 000 000 016 2 meters), where:

• c is the speed of light in a vacuum;
• G is the gravitational constant;
• $\hbar$ (pronounced "h-bar") $\ =h/2\pi$ is the reduced Planck constant.

The two digits enclosed by parentheses are the estimated standard error associated with the reported numerical value.

Physical significance

Some facts from physics:

• The diameter of a proton is about 10²⁰ Planck lengths;
• 1 Planck length is the Schwarzschild radius corresponding to 1 Planck mass;
• The radius of the observable universe is about 46 billion light-years, equivalent to 4.4 × 10²⁶ meters or 2.7 × 10⁶¹ Planck lengths.

Because the Planck length is the only length (up to a constant factor) obtainable from the constants c, G, and $\hbar$, it is expected to play some role in a theory of quantum gravity. In some theories or forms of quantum gravity, it is the length scale at which the structure of spacetime becomes dominated by quantum effects, giving it a discrete or foamy structure. Other theories of quantum gravity predict no such effects. If there are large extra dimensions, such as those implied by string theory, the measured strength of gravity may be much smaller than its true (small-scale) value. In this case the Planck length would have no physical significance, and quantum gravitational effects would appear at much larger scales.

The Planck mass is the mass for which the Schwarzschild radius is equal to the Compton length divided by π. 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 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 whose precision is less than 1 Planck length, their collision with the object under study would, in theory, 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 equal to 1 Planck length. However, this black hole could, at least in theory, produce Hawking Radiation by "swallowing" other photons. Therefore, this radiation could be measured.

This thought experiment draws on both general relativity and the Heisenberg uncertainty principle of quantum mechanics. These two theories combined imply that it is impossible to measure position with a precision smaller than the Planck length, or duration with a precision smaller than the Planck time. These limits may also apply to a theory of quantum gravity.

History

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 $\hbar \$ 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 seem to require a theory of quantum gravity. Note that at such a distance scale, the uncertainty principle materially impairs the ability to make any useful statements about what is actually happening.

Speculative connection to "holographic noise" at gravitational wave detectors

The Fermilab physicist Craig Hogan has claimed that the holographic principle may imply a particular type of quantum fluctuations in spatial position characterized by the Planck length. Such fluctuations would lead to apparent background noise or "holographic noise" measurable at gravitational wave detectors, in particular the GEO 600 detector near Hannover.

See also

• Doubly special relativity
• Orders of magnitude (length)
• Planck scale
• Planck units
• Quantum gravity
• Planck epoch
• Fock-Lorentz symmetry