Hi everyone, hope you’re all doing well :) In this short post, I’ll be sharing the slides from my recent presentation at SASMS (Short Attention Span Math Seminars), hosted by the Pure Math Club at the University of Waterloo.
The presentation was on an informal construction of the ideas developed in some posts on this blog:
- Scalar Field Lagrangian From Symmetry Considerations
- Gauge Invariance in Classical Field Theory
- Scalar Field Lagrangian From Symmetry Considerations: Part 2 (Gauge Invariance)
- Deriving the Lagrangian Density for Newtonian Gravitation
In the talk, I gave an overview of the upgradation of classical mechanics to classical field theory. The motivation for this was to examine the striking analogies between gravitation and electromagnetism, by putting them on a somewhat equal footing in terms of the language being used to describe them, i.e. field theory.
It turns out that the more we delve into the structure of classical field theory, the more do the analogies between the said theories deepen. For instance, we later found in the exposition that Newtonian gravitation is a non-relativistic, 3-dimensional application of the Klein-Gordon theory, which in the relativistic, 4-dimensional setting, describes electrons (and couples interestingly with electromagnetism in scalar electrodynamics and quantum field theory).
This, as was discussed in the talk, is not a coincidence; in classical field theory, the condition of the field-theoretic energy-momentum tensor being symmetric is a severe one, which ends up making any classical field theory obeying the condition, a Klein-Gordon theory!
Armed with this understanding, we found a field-theoretic formulation for Newtonian gravitation by treating its equation of motion, the Poisson equation, to resemble the Klein-Gordon equation. Finally, we incorporated matter fields into the model by considering a Lagrangian density for matter and imposing that when varied with respect to the gravitational field, it must yield the mass density field characterizing the matter, as we would expect in a theory respecting both gravitation, matter and their interaction.
Without further ado, here are the slides: 1
Overall, it was super fun preparing the document and being able to present it to an involved audience. Working on this topic also meant getting a fresh perspective on variational principles, Klein-Gordon analysis, gauge theories and so on, which will be implemented in this blog in the future.
That brings us to a little note I’d like to conclude this post with. It’s that we’ll be having plenty of related posts on this blog as soon as possible! They’ll be to do with special relativity, classical field theory, analysis and, surprise, topology. It might still take a couple of months to get them published given that they’re taking time to read up and write on. Until then, current progress and future ideas for this blog can be seen on its new Notion database!
And with that, folks, have a good one! :)