Template:Unit of length The Planck length, denoted by ${\displaystyle \scriptstyle \ell _{P}\ }$, is the unit of length approximately 1.6 × 10⁻³⁵ metres, 6.3 × 10⁻³⁴ inches, or about 10⁻²⁰ times the diameter of a proton. It is in the system of units known as Planck units. The Planck length is deemed "natural" because it can be defined from three fundamental physical constants: the speed of light, Planck's constant, and the gravitational constant.

Value

The Planck length equals ^[1][2]

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

where:

• ${\displaystyle c}$ is the speed of light in vacuum;
• ${\displaystyle G}$ is the gravitational constant;
• ${\displaystyle \hbar }$ (pronounced h-bar) is Dirac's constant, Planck's constant divided by 2π .

The two digits between the parentheses denote the uncertainty in the last two digits of the value.

In SI units, the Planck length is approximately 1.6 × 10⁻³⁵ metres. The estimated radius of the observable universe (4.4 × 10²⁶ m or 46 billion light-years) is 2.7 × 10⁶¹ Planck lengths.

Physical significance

The physical significance of the Planck length is somewhat abstract. Because it is the only length (up to a constant factor) obtainable from the constants c, G, and ${\displaystyle \hbar }$, 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, but in other theories of quantum gravity there are no such effects predicted. 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 more precise than a Planck length, their collision with the object 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 to within a Planck length.

This thought experiment draws on both general relativity and the Heisenberg uncertainty principle of quantum mechanics. Combined, these two theories imply that it is impossible to measure position to a precision shorter than the Planck length, or duration to a precision to a shorter time interval than a Planck time. These limits may apply to a theory of quantum gravity as well.^[3][4]

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 ${\displaystyle \hbar \ }$ 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 begins to intrude on one's ability to make any useful statements about what is actually happening.

References in science fiction

• In the novel Hyperion by Dan Simmons, Planck length is referred to as "the shortest length possible".
• Many objects in the video game series Xenosaga are compressed to a "planck-scale size".
• A modified Planck length is used as a wish-fulfillment device in Rudy Rucker's novel Master of Space and Time.