Roadmap for General Relativity
The best thing about general relativity is that it can be learned concurrently to the standard quantum route (quantum mechanics, statistical mechanics, quantum field theory, supersymmetry). I personally found general relativity to be easier to learn than quantum field theory, at least at a ~4th year undergraduate level. This was probably due to my prior interest in mathematics: general relativity requires a much, much broader maths background than QM, but once you acquire an intuitive visualisation of differential geometry concepts, it aids the learning process significantly.
Definite prerequisites
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Special relativity: Unlike for QFT, here you have to know special relativity in and out, 100%. SR is a very problem-solvey kind of subject, so it helps to do all the practice you can get. I also recommend reading a bit about the ontological differences between SR and GR prior to diving in head first - I paired this with a historical account of Einstein’s development of GR (modern sources present it in a much cleaner way mathematically, but this slightly conceals what Einstein’s motivations and thought process were)
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Newtonian Gravity: Goes without saying. Make sure to have worked with the potential formalism, and find out some of its conflicts with general relativity right away.
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Multivariable calculus: When a theory operates in 4 dimensions, you know you’re gonna need at least multivariable calculus. In addition, you need to be solid with differential equations and basic PDE (that’s quite literally what the Einstein field equations are)
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Tensor calculus: Constructing and analysing Lorentz covariant objects is critical, as is seeing how quantities change after coordinate transformations. This does require a background in abstract (multi) linear algebra, particularly vector spaces, their duals, changing basis, multilinear maps, tensor products, etc. Tensor manipulations in GR are actually simple to do in practice (once you get the hang of the notation) but they encode quite a bit of information, so it is important to know what you’re doing. I recommend you master this before even opening a GR textbook.
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Real Analysis: The quintessential tedious but necessary precursor to “actual maths”. Actually thinking about spacetime topology is something you won’t have to do right from the beginning, but of course metric spaces, differentiable maps, etc. are the basis for introducing differential geometry. My advice? Don’t sweat it, a basic understanding is all that’s required, you certainly don’t need to have solved every problem in Munkres.
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Differential Geometry: My recommendation will stray from the conventional wisdom. I recommend that if you have even a remote mathematical inclination, you become proficient in basic differential geometry before commencing with any serious GR. I usually recommend people learn mathematics in the context of physics if they can, since that physical context provides intuition, but I don’t think the tenets of GR have any significant insight to provide in this case. By “basic”, I mean manifolds, frames, parallel transport, covariant derivatives, curvature. A lot of this has quite a bit of overlap with what others call “tensor calculus”.
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Electrodynamics Theory: Maxwell’s equations in Lorentz-covariant form, stress-energy tensor, EM Lagrangian. These are useful for illustrating analogous constructions in GR. Note that I’ve qualified this with the word “theory” to emphasise that “only” a mathematical background in E&M (as opposed to augmented with physics-style tricks and techniques) is sufficient.
Learn it along the way (or before, for a better understanding):
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Electromagnetic waves: The clear-cut cousin of gravitational waves, clearly not a prerequisite but it is useful to know about vectorial polarisation before upgrading to tensorial gravitational polarisation.
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Classical field theory: Action principles, symmetries, conservation laws - all very helpful in understanding GR as a field theory. Formulating GR as a gauge theory is a very nice rabbit hole to pursue, and tools like the Hilbert action and Palatini formalism are powerful to have your disposal. You will need predominantly Lagrangian mechanics here (the features of a diffeomorphism-invariant theory are really intriguing on this front), but experts in the Hamiltonian formalism can take a look at the ADM formalism, which feeds right into canonical quantum gravity.
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Gauge theory: For those people who heard “covariant derivative” and immediately assumed GR was a Yang-Mills theory. There are lots of nice analogies, and many key differences between the two. Principal and vector bundles find great use in this regard.
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Basic cosmology: Why not? Even if you’re a diehard spacetime structure theorist and hate our actual universe, cosmology ties together E&M, quantum mechanics and gravity in a wonderfully tangible, physics-y manner.
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Anything on the quantum track - there is zero interplay between curved spacetime and quantum effects for a long, long part of the learning curve, so you can dip your toes in both simultaneously guilt-free.