Contrary to the (very) popular belief, the Planck length has not been proven to be the smallest possible unit of space.

The Planck length is part of a series of units called the Planck units, which were, unsurprisingly, developed by the famous physicist Max Planck

To develop these units, you begin with 5 fundamental constants:

• The speed of light, [math]c = 299792458[/math]ms[math]^{-1}[/math]
  [2]
• The gravitation constant, [math]G = 6.674 08 \times 10^{-11} [/math]m[math]^3[/math] kg[math]^{-1}[/math] s[math]^{-2}[/math]
  [3]
• The reduced Planck’s constant, [math]\hbar =1.054 571 800 \times 10^{-34} [/math]kg m[math]^2[/math] s[math]^{-1} [/math]
  [4]
• The electric constant, [math]\frac{1}{4 \pi \epsilon_0} = [/math][math]8.9875517873681764\times10^9 [/math]kg m[math]^3 [/math]s[math]^{−4}[/math] A[math]^{−2}[/math]
  [5]
• The Boltzman constant, [math]k_B = [/math][math]1.38064852 \times 10^{−23} [/math]kg m[math]^2[/math]s[math]^{-1} [/math]K[math]^{−1}[/math]
  [6]

To produce a Planck unit, you then simply need to work out what combination of these 5 constants you need.

Let’s say we want to define the Planck time, [math]t_p[/math].

Obviously, [math]t_p[/math] has units of time — so [math][t_p] = T[/math], by dimensional analysis

We now construct the following:

[math]t_p = c^\alpha G^\beta \hbar^\gamma \left(\frac{1}{4 \pi \epsilon_0} \right)^\delta k_B^\eta[/math]

Where the Greek letters are unknown constants.

Looking at the units in the list above, we can then write:

[math][t_p] = [c]^\alpha [G]^\beta [\hbar]^\gamma \left[\frac{1}{4 \pi \epsilon_0} \right]^\delta [k_B]^\eta[/math]

[math]T =[/math] [math]\left(LT^{-1} \right)^\alpha \times[/math] [math]\left(L^3 M^{-1} T^{-2} \right)^\beta \times[/math] [math]\left(M L^2 T^{-1} \right)^\gamma \times[/math] [math]\left( M L^3 T^{-4} Q^{-2} \right)^\delta \times[/math] [math]\left( M L^2 T^{-1} \Theta^{-1} \right)^\eta[/math]

By matching all of our terms, we then get:

[math]T^1 = L^{\left(\alpha + 3\beta + 2\gamma + 3\delta + 2\eta\right)} T^{\left(-\alpha -2\beta - \gamma -4 \delta -\eta\right)} M^{\left(-\beta +\gamma + \delta + \eta\right)} Q^{-2\delta} \Theta^{-\eta}[/math]

By inspection, we can immediately see that since [math]Q[/math] and [math]\Theta[/math] do not appear on the left hand side, that [math]\delta = \eta = 0.[/math]

We are left with three equations:

[math]T: \quad 1 = - \alpha - 2\beta - \gamma[/math]

[math]L: \quad 0 = \alpha + 3 \beta + 2 \gamma[/math]

[math]M: \quad 0 = - \beta + \gamma[/math]

From (M), we see that [math]\beta = \gamma[/math]

From (L), we see that [math]\alpha = - 5 \beta = - 5 \gamma[/math]

From T, we see that [math]1 = \left(5 -2 -1\right) \beta[/math]

Putting this all together:

[math]\alpha = -\frac{5}{2} \quad \quad \beta = \frac{1}{2} \quad \quad \gamma = \frac{1}{2}[/math]

Therefore:

[math]\large \boxed{t_p = \sqrt{\frac{\hbar G}{c^5}}}[/math]

It is then a simple matter to see that if we want the Planck length, since [math]v = \frac{d}{t}[/math], we expect [math]l_p = v_p \times t_p[/math] — but the Planck speed is the speed of light!

Therefore:

[math]l_p = \sqrt{\frac{\hbar G}{c^3}}[/math]

That is how you derive the Planck length.

Nothing to do with anything fundamental about the nature of space!

We are literally just multiplying units together, and seeing what combination gives us the units we need.

You can also generate the Planck mass ([math]m_p = \sqrt{\frac{\hbar c}{G}})[/math], planck charge ( [math]q_p = \frac{e}{\sqrt{\alpha}}[/math]) and the Planck temperature ([math]\Theta = \sqrt{\frac{\hbar c^5}{G k_b^2}}[/math]) — from which you can derive everything from acceleration, to power, to voltage!

The Planck units were established because they simplify many of the more fundamental equations — if you write down your equations in Planck units, you can do away with many physical constants and not have to worry about dimensions.

Newton’s law of gravitation becomes:

[math]F = \frac{G m_1 m_2}{r^2} \mapsto F = \frac{m_1 m_2}{r^2}[/math]

Mass-Energy equivalence becomes:

[math]E = mc^2 \mapsto E = m[/math]

And so on and so forth. This is a process called nondimensionalization

So — why does this myth persist?

It is true that several people have estimated that the scale of the Planck length is roughly the order of magnitude around which the structure of spacetime becomes dominated by quantum effects — or roughly the scale of the “quantum foam”.

Please note the phrase “roughly the scale of”.

Human body height is roughly of the scale of [math]1[/math]m. But if someone told you that all humans were 1m tall, you would look at them like they were madmen!

There’s also a certain amount of arbitrarity in the choice of the base units we used to construct the Planck units — note that we use the reduced Planck constant, [math]\hbar = \frac{h}{2 \pi}[/math]. There’s no reason not to use [math]h[/math] — the difference is a (non-dimensional) factor of [math]2 \pi[/math]. The same can be said of the factor of [math]4 \pi[/math] in the electric constant.

It’s difficult to ascribe such fundamental importance to a series of units where you can multiply by [math]2\pi[/math] and leave the result unchanged…

So — what is true?

Some people

Somebody then went “huh - that’s funny — remember those units that guy came up with 100 years ago? The length in that unit is approximately [math]10^{-35}[/math]metres.”

That somehow got morphed into “OMG the Planck unit is the smallest possible unit of space!”

I’ve then also seen this then applied to the Planck time — people then claim that the Planck time is the smallest possible unit of time….which is nonsense. There’s no reason to assert that.

Also I’ve seen the same assertion made about the Planck charge — that it is the smallest possible unit of charge. Except, [math]q_p = \frac{e}{\sqrt{\alpha}}[/math], where [math]\alpha \approx \frac{1}{137}[/math] …. so the electron charge ([math]-e[/math]) is several times smaller than the Planck charge! It’s trivial to show that this statement is utter nonsense.

Bonus points go to the person (who shall remain nameless) who then asserted that since [math]l_p[/math] and [math]t_p[/math] are the smallest time and length scales, since the speed of light is [math]c = \frac{l_p}{t_p}[/math], that’s why the speed of light is an invariant….I mean….wow.

It’s true that these time and length scales are so tiny not to be physically meaningful to us as humans— but at the moment, there is no evidence to attribute to them any special significance other than that which is seemingly coincidental.

So to the question — is anything smaller than a Planck length?

Well - I can give you a physical result which has a value smaller than the Planck length.

In 1973, Jacob Bekenstein published a paper where he showed that the surface area of a black hole increases by [math]1 A_p[/math] for every bit of information which crosses the event horizon

The surface area of a sphere is given by [math]4 \pi R^2[/math] .

Therefore, the new surface area is given by [math]4 \pi R_{new}^2 = 4 \pi R_{old}^2 + l_p^2[/math]

Then:

[math]R_{new} = \sqrt{R_{old}^2 + \frac{l_p^2}{4 \pi}}[/math]

Therefore the change in radius is given by [math]\delta R = R_{new} - R_{old}[/math]

[math]\delta R = R_{old} \sqrt{1 + \frac{l_p^2}{4 \pi R_{old}^2}} - R_{old}[/math]

We then expand the brackets (since [math]R_{old} \ggg l_p[/math] for any black hole), such that:

[math]\delta R \approx R_{old} \left( 1 + \frac{l_p^2}{8 \pi R_{old}^2}\right) - R_{old}[/math]

Therefore:

[math]\delta R \approx \frac{l_p^2}{8 \pi R_{old}} \ll l_p[/math]

Therefore, via Bekenstein’s result, the change in radius of a black hole when 1 bit of information is added is much smaller than 1 Planck length (given that [math]R[/math] for a typical black hole will be on the orders of thousands of km’s — approx [math]10^{40} l_p[/math])

The Academic Space — A blog for academic-level & evidence-based answers.

Footnotes