I have been out of the field of physics for over 5 years, didn't get to make use of my training in quantitative analysis too much in my previous job as a postdoctoral researcher at UC Davis, and make even less use of my training in quantitative analysis in my current job as a transportation planner at Cambridge Systematics. When I was doing work in my PhD in physics and when I was a student in school and college before graduate school, I enjoyed solving problems in math & physics and growing & applying my toolbox of quantitative skills, so since leaving the field, and especially more recently, I have felt a slight itch to recover some of those skills purely for my own personal satisfaction. To that end, I've resolved to learn or relearn some parts of math that I did not learn at all or to the full extent that I should have in college or graduate school (because I could ultimately manage my work in my PhD without knowing those things to that extent). Currently, I'm interested in learning differential geometry as well as complex analysis. It remains to be seen whether there are other topics in physics-relevant math that I become interested in; if they are, then I will certainly make an effort to learn at least a little bit about them. I've enjoyed learning these topics in math to broaden my knowledge & skills, and I've particularly enjoyed pondering definitions & rules of these concepts in math almost like a lawyer (which also makes this useful for my current job in a very indirect way, because my current job in part involves analyzing laws & regulations and creating intuitive explanations of them for public sector clients).
I can't guarantee that I'll write a blog post about every topic in math that I learn about. However, I am writing this post specifically about differential geometry in parts because I feel that I have learned the basic ideas in it in the context of general relativity to my satisfaction (which was my original goal) and because I have some questions/concerns that I have not been able to satisfactorily resolve based on what I have read in lecture notes or textbooks. This post is a way to further flesh out those questions/concerns. Follow the jump to see more.
There are a few conventions & assumptions to note throughout this post.
- The dimension of the manifold will generically be denoted \( N \). For most nonrelativistic physics, \( N = 3 \), while for most relativistic physics (including general relativity), \( N = 4 \); exceptions include constrained low-dimensional systems or different physical models that have different dimensions analogous to differential geometry.
- I will consistently use Einstein summation unless otherwise specified. This means that indices that are repeated with one as an upper index and one as a lower index will be summed; indices should never be present more than twice at all and more than once in the same (upper versus lower) position.
- Although the convention in general relativity is to use lowercase Greek letters for indices, I will keep things easier to read by using lowercase English letters for indices.
- The motivation of general relativity means that I will only consider differentiable manifolds and specifically smooth (infinitely differentiable) scalar functions or tensor components on them.
- The motivation of general relativity also means that I will only consider torsion-free connection coefficients that can be expressed in terms of the metric.
