[…]

This is Einstein’s final contribution to physics… and unfortunately it was left unfinished.

Also, a huge thanks to @ChrisPattisonCosmo for working with me on this video. You need to go check out his channel now if you haven’t seen it already. Also, he did a video looking at Einstein’s blackboard in more detail. Check it out here:

• Before He Died, E…

A photographer recorded forever the final work of Albert Einstein when he took a photo of Einstein’s office on the day of his death. This photo was printed in Time magazine, and is hugely famous for obvious reasons. In this video, we try to understand what exactly Einstein was working on.

By the types of markings and mathematics found on the board, we can understand that the mathematics is that of General Relativity, Einstein’s theory that best describes gravity and the universe on a large scale. We try and understand a few specific elements seen on the blackboard.

Firstly, we see that the metric tensor (or just “metric” for short) features quite a lot on the blackboard. The metric is used to describe the curvature of any region of spacetime that we are studying. Basically, it helps us to calculate distances between two points within spacetime depending on how the fabric of reality is warped. The general metric is described by the letter “g”, and it has two subscripts because it is a “rank 2” tensor. We also see the greek letter “eta” used to represent the metric, but this is only in the special case where we are studying a “flat” (or curvature-free) spacetime.

Next, we understand that Einstein was trying to write the metric tensor for a generic spacetime in terms of “tetrads”. Tetrads are simply vectors that can be defined at each point in spacetime, which point in the direction that each coordinate increases. In flat spacetime, they always point in the same directions, but in curved spacetimes their directions may change. Tetrads are useful for any observers that are actually within the spacetime being studied, whereas the metric is good for taking an overview or general look at the spacetime as a whole.

On the surface, it may seem useful to write a rank 2 tensor (metric) in terms of vectors or rank 1 tensors (tetrads), since tetrads are more simple entities. However, Einstein also showed on his blackboard that the number of degrees of freedom are exactly the same regardless of how we write general relativity (but just packaged up in different places).

Degrees of freedom refer to how many variables (or coordinates) can change in any system we’re studying. For example, for a ball moving through the air we need 3 coordinates – usually x, y, z. Hence, there are 3 degrees of freedom. But for a ball moving on a table, we only need x and y, since its height does not change. Hence this system of a ball on a table only has 2 degrees of freedom.

Finally, we understand that Einstein designed a convention for writing sums in a short way when working with general relativity. This is known as the “Einstein Summation Convention”. It helps us not have to write long and boring sigma (sum) signs, and just get on with the maths instead. However, we see on the blackboard that Einstein did not use the summation convention (nor always put in the required number of subscripts for tensors). This may have been because he was explaining his ideas and theories to someone else, and hence erasing parts of his explanations that were no longer relevant.

[…]