Relativity #29 - The Lorentz and Poincaré Groups (Very briefly!)

Notes are on my GitHub! github.com/rorg314/WHYBmaths

In this video I will very briefly introduce the Lorentz group, mostly through analogy with the orthogonal rotation groups. These (matrix Lie) groups are very similar, in that group elements satisfy an orthogonality condition, but the Lorentz group is slightly special in that the orthogonality condition contains a matrix which has positive and negative entries - where the orthogonality condition for a 'pseudo-orthogonal' matrix L is slightly modified to be LηL^T = η where eta is a matrix that contains +-1 on the diagonal.

For our case, we have the Minkowski metric η which is (-1, 1, 1, 1) as a diagonal matrix, and thus the pseudo-orthogonal rotation group in 1, 3 dimensions is then the Lorentz group SO(1, 3). In future videos I will talk a lot more about this group and how it can be understood to be the 'symmetry' group of the Minkowski geometry - i.e. it is the group of transformations that leave the metric (squared interval) invariant. I will talk in much more detail about how we should understand this group in future, but for now this is just a short taster!

Lastly, I also comment on how we can extend the Lorentz group to the Poincaré group, by including spatial translations. This is rather trivial, but still useful!