Thread by @WKCosmo

So what exactly is the Planck length? A thread. 1/

https://twitter.com/elonmusk/status/1650658418816950273

Let's start with the Heisenberg uncertainty principle, which we can write like this: 2/

What this means that is if we probe a system with a spatial resolution of Δr, then that introduces an uncertainty in the momentum associated with that system of 3/

Here ℏ is Planck's constant h divided by (2 π). 4/

What this means physically is if we probe any system (even empty space!) with on shorter and shorter distance scales, there is a correspondingly larger and larger uncertainty in the momentum of that system. 5/

We can also write the uncertainty in the momentum as an uncertainty ΔE in the energy, as 6/

We can also think of this uncertainty in the energy as an uncertainty Δm in the mass by using Einstein's E = m c^2, so that 7/

Or 8/

The Heisenberg uncertainty relation then means that if we probe space with a resolution Δr, there is an associated uncertainty in the mass contained within the radius Δr, 9/

The smaller the radius Δr, the larger the uncertainty in the mass Δm contained within that radius! 10/

Now we can ask: how small must Δr be for the associated mass Δm to be large enough to make a black hole? 11/

The radius Δr of a black hole with a mass Δm is given by the Schwarzschild radius, 12/

Combining this with the uncertainty in the mass from the Heisenberg relation gives. 13/

The Heisenberg uncertainty relation tells us that if we probe space with a resolution smaller than this distance, called the Planck length, we will start making tiny black holes as a result of the measurement process. 14/

This means that we cannot describe systems of this size or smaller without taking into account the quantum properties of spacetime itself, which requires a theory of quantum gravity. 15/

Which we don't have. 16/

So we do not actually know whether or not it is possible to divide space up smaller than this, because we do not have a theory which would describe space on scales this small. 17/

Answering that question requires an understanding of quantum gravity. 18/

So ... 19/

Maybe the Planck length represents a minimum possible distance, some discretization scale for spacetime, and maybe it doesn't. We don't really know. 20/

This is one of many reasons why theorists are so interested in finding a working theory of quantum gravity, because it would enable us to answer questions of this sort. 21/

In the meantime, this rough calculation of the Planck length gives us a guidline for when we should stop trusting the theories we have. Beyond that ... here there be dragons. 22/22

@elonmusk think of how cool threads like this could be if Twitter supported Mathjax for equations!