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## The Planck length and the Planck time

In the concluding chapter (Chapter 7) of my book The Cosmic Microwave Background – how it changed our understanding of the Universe (available here) I discuss possible future directions for research in cosmology. One of the things I discuss is the idea of the Planck length and the Planck time. If you haven’t heard of these, here is my attempt to explain them.

My book “The Cosmic Microwave Background – how it changed our understanding of the Universe” is published by Springer. It is available by following this link.

In Chapter 7 of my CMB book I discuss future research, including quantum gravity and the concept of the Planck length and Planck time.

## What is the Planck length?

I will blog in the near future about the Planck constant, which came into Physics when Max Planck realised in 1900 that a blackbody spectrum could be fitted by a mathematical formula if only certain energies were allowed, with the energy of ‘light quanta’ (as he called them, we now call them photons) being given by $E=h \nu$ where $E$ is the energy, $\nu$ is the frequency and $h$ is now known as Planck’s constant. As I will explain more fully in that blog, Planck first of all found a mathematical formula to fit the blackbody curve, and then several months later he figured out a physical explanation for such a formula.

This was the beginning of the quantisation of nature, and culminated in the mid 1920s with the development of Quantum Mechanics by Werner Heisenberg and Erwin Schrodinger. The whole idea behind quantum mechanics is that energy comes in lumps and is not continuous. So, for example, we all know that children come in lumps, you cannot have 2.2 children, even if that is the average per family in some countries! But, the height of those children is not lumpy, a child’s height can be anything, you are not restricted to a height which only comes in e.g. centimetres.

The Planck constant is the constant which gives the lumpiness of energy. But, what about length? Is there a lumpiness to length, or can we just measure smaller and smaller lengths (technology permitting)? Well, it has been suggested that, if we combine three fundamental physical constants $G, h \text{ and } c$ (the universal gravitational constant, Planck’s constant and the speed of light respectively) in the correct way, we can produce something which has the dimensions of length. We call this the Planck length, and it is given by

$\boxed { \ell_{P} = \sqrt{ \frac{ \hbar G }{ c^{3} } } }$

where $\hbar = h/2\pi$. If you plug in the numbers, this comes out to be $\ell_{P} = 1.616 \times 10^{-35} \text{ metres}$.

## The Planck time

The Planck time $t_{P}$ is defined as the time taken by light to travel the Planck length, so given that the speed of light is $c$, the time it takes to travel the Planck length is just going to be $\ell_{P}/c$ or

$\boxed { t_{P} = \sqrt{ \frac{ \hbar G }{ c^{5} } } }$

Again, if you plug in the numbers this comes out to be $t_{P} = 5.391 \times 10^{-44} \text{ seconds}$.

## Do the Planck length and Planck time actually mean anything?

There is no evidence that these quantities mean anything. Many people interpret them as being the smallest distance and smallest time that exist, that space-time becomes quantised at small scale and one cannot distinguish between a length of zero and the Planck length $\ell_{P}$ or between a time of zero and the Planck time $t_{P}$. This is the usual interpretation of these two (linked) quantities.

However, in reality there is no reason why these quantities do actually mean anything. That is to say, there is no evidence that there is something fundamental about them. They are merely a particular combination of three fundamental physical quantities (maybe the three most fundamental), combined in such a way as to give the dimensions of space and time. But, as of yet there is no physical theory or law which argues that there is anything truly fundamental about them. But, that may one day be proved to be the case. Stay tuned!