Faster than light, smaller than a black hole, or half a photon?

Such nonsense could be straight from a science-fiction story. But think about this question:

Is space made of points?

If you answered yes, then you are also answering yes to at least one of the title questions! Those three ideas are in contrast with observations. And so is the question about points. Here is why.

Take the maximum speed v ≤ c, the minimum action W ≥ ℏ, and your favourite black hole limit, such as, for example, the maximum force F ≤ c⁴/4G or, if you prefer, the maximum mass per length ratio m/d ≤ c²/4G.

Insert the limits into the definition for action W (the quantity that measures the change of a system and that appears in the principle of least action) which is given by

W = F d t = F d² / v,

where F is force, d is distance, t is time, and v is speed. Equivalently, action is also given by

W = E t = m c² t = (m/d) c d²,

where E is energy and m is mass. After inserting the limits, in both cases the result is

d ≥ √ (4Gℏ/c³) ≈ 3 × 10⁻³⁵ m.

In other words, there is a smallest length in nature, given by twice the Planck length. Let that sink in. Smaller lengths cannot be measured, because they do not exist.

A smaller length does not exist, in the same way that nothing faster than light exists, nothing smaller than a black hole exists, and no change smaller than Planck’s quantum of action exists.

Points would be smaller than the minimum length. If you believe in points, you contradict at least one of the three limits.

If you believe in points, you believe in faster than light, or smaller than black holes, or half photons — or several together.

Modern physics — relativity and quantum theory — shows that points do not exist. Modern physics contradicts what we have been told in school since 2500 years.