Planck length

From Exampleproblems

The Planck length is the natural unit of length, denoted by $l_P \$ .

1 History
2 Value
3 Significance
4 See also
5 External links

History

This unit was first developed by Max Planck who created a system of measurement based on natural units by using universal physical constants. The Planck length, Planck time, and Planck mass are units judiciously chosen so that c, G, and $\hbar \$ are all equal to 1 and thus disappear from equations of physical law that use those scaling factors. Although quantum mechanics and general relativity were unknown at the time that the units were proposed, it later became clear that at distances of the Planck length, gravity would begin to display quantum mechanical effects, requiring a theory of quantum gravity to explain the effects.

Value

Ignoring a factor of $\pi \$ , the Planck mass is a mass whose Schwarzschild radius and Compton length are equal distances. This length, called the Planck length, is equal to:

$l_P =\sqrt{\frac{\hbar G}{c^3}} \cong 1.616 24 (12) \times 10^{-35}$ metre

where:

$\hbar \$ is the reduced Planck Constant (or Dirac's constant)

G is the gravitational constant

c is the speed of light in vacuum

The estimated size of the observable Universe (7.4 × 10²⁶ m) is 1.2 × 10⁶² Planck lengths.

Significance

Ignoring a factor of $π$ , the Planck mass is roughly the mass of a black hole whose Schwarzschild radius equals its Compton wavelength. The radius of such a black hole is roughly the Planck length.

The meaning of this can be illustrated by a thought experiment. Assume the task is to measure an object's position by bouncing light off of it. To measure its position to high accuracy, photons of high energy and short wavelength are needed. If their energy is high enough to measure more precisely than a Planck length, they would in principle create a black hole when they collide with the object. The black hole would "swallow" the photon and make it impossible to obtain a measurement, and therefore defeat the experiment. A simple calculation using dimensional analysis suggests that this problem occurs when the object's position is measured at a length more precise than Planck Length.

Note that this thought experiment involves both general relativity and quantum mechanics (namely the Heisenberg uncertainty principle). Together, these theories state it is impossible to measure positions more accurately than the Planck length. So, it suggests that in any theory of quantum gravity combining general relativity and quantum mechanics, traditional notions of space and time will break down at distances shorter than the Planck length or times shorter than the Planck time.