> I'm just wondering if I'm being lazy by not trying hard enough or efficient. It's definitely not for lack of curiosity, but I also don't like to fool myself into thinking I understand something that I don't.
I don't think you're lazy or not trying hard enough. The truth is just that… it's a hard and long road. Even more so when you're studying by yourself.
I took about the straightest path you can take to learning Quantum Mechanics and General Relativity (and all the math you need for it), attending classes and sitting down on my butt every day for 3 to 4 semesters. There are ways to shorten it but I'm not sure how rewarding that would be.
It also depends on what you mean by "understanding". If you really want to understand things at the deepest level possible, there's no way around a university-level education. I'm saying "university-level", not "university", because one could certainly learn these things on one's own. But to be honest with you I think chances of pulling this off are very small, mostly because physics outsiders don't have access to the same resources or social networks as enrolled students, which makes studying even more frustrating.
Anyway, FWIW here's a roadmap for learning General Relativity at the deepest-possible level along with the minimum timeframe needed (IMO) and the most important topics / keywords you really need to understand well:
Linear Algebra (1-2 semesters): vector spaces, linear maps, dual maps, matrices, symmetric bilinear forms – These things lay the foundation for pretty much anything in math and are needed to understand differentiation in several variables (see below) as well as pretty much anything in Differential Geometry and Relativity.
Real Analysis (1-2 semesters): Mostly differentiation of functions of one to several variables + a bit of integration theory – needed for pretty much anything in Differential Geometry and Relativity (especially coordinate changes and manifolds) but also in mechanics.
Differential Geometry aka (Semi-)Riemannian Geometry (1 semester): manifolds, tensors, metric, connections, geodesics, curvature – These concepts are at the heart of General Relativity
Mechanics (1-2 semesters): Newtonian Mechanics, Lagrangian Mechanics, Special Relativity (all with a focus on both theoretical and experimental physics) – without these there's no hope of understanding (or appreciating) the physics content of GR
General Relativity (1 semester)
(Side note: Don't let anyone tell you that you don't really need Differential Geometry and all the other math to understand Relativity. They're lying and probably don't really understand Relativity, either.)
For quantum mechanics it's a bit shorter because the math is not as involved (at least to understand the basic concepts:
Linear Algebra: (1-2 semesters): (finite-dimensional) vector spaces, linear maps, dual maps, matrices, symmetric bilinear forms – No way around this. 99% of quantum mechanics is encoding fuzzy concepts in linear algebra.
Mechanics (1 semester): Newtonian Mechanics, Lagrangian Mechanics
Quantum Mechanics
Bonus: Hilbert space theory, basics in functional analysis
I don't think you're lazy or not trying hard enough. The truth is just that… it's a hard and long road. Even more so when you're studying by yourself.
I took about the straightest path you can take to learning Quantum Mechanics and General Relativity (and all the math you need for it), attending classes and sitting down on my butt every day for 3 to 4 semesters. There are ways to shorten it but I'm not sure how rewarding that would be.
It also depends on what you mean by "understanding". If you really want to understand things at the deepest level possible, there's no way around a university-level education. I'm saying "university-level", not "university", because one could certainly learn these things on one's own. But to be honest with you I think chances of pulling this off are very small, mostly because physics outsiders don't have access to the same resources or social networks as enrolled students, which makes studying even more frustrating.
Anyway, FWIW here's a roadmap for learning General Relativity at the deepest-possible level along with the minimum timeframe needed (IMO) and the most important topics / keywords you really need to understand well:
(Side note: Don't let anyone tell you that you don't really need Differential Geometry and all the other math to understand Relativity. They're lying and probably don't really understand Relativity, either.)For quantum mechanics it's a bit shorter because the math is not as involved (at least to understand the basic concepts: