The biggest obstacle of my graduate studies right now is to find myself learning things again. And this turns out to be surprisingly hard to do. I suspected that somewhere in my third year of undergraduate, I was “burnt out”. Physics and mathematics are nothing more than things which I happen to (hopefully) do better than most other things. Is doing something you are good at instead of what you “want” (if you know what that is) necessarily bad? I don’t know.
I found out when I realized that I could not motivate myself to learn new things. I used to be able to take a book – including advanced textbooks – and read them. I may not have read them carefully enough to “understand” them, but I did go through them. Now? I do not even have the slightest desire most of the time. It is ironic considering I have already signed up for Master’s degree in Waterloo.
This inertia is dangerous and I know it. I am aware of it. However, it is like a chronic disease which you usually do not know how to deal with it.
Paradoxically, I can possibly enumerate the things I do want to learn – even if I cannot get myself to commit to them right now yet (hopefully it is just a “yet” than never able to commit at all). To see this, let’s try.
I have worked on this for some time now, though really it is only a little over half a year. I got the basic intuitions, standard introductory methods like using Christoffel symbols to compute Riemann tensor and solve the field equations. I know what parallel transport and covariant derivatives are. I know some standard black hole solutions in general relativity and how to get them. I know enough basic differential geometry to know the standard properties of e.g. null hypersurfaces, and also basic symmetry properties to know what Killing tensor fields are. I know basic calculus on manifolds and how to work with simple differential forms and tensor fields as (multi)linear maps. I know basic framework of linearized gravity and hence gravitational waves. I have solved for Einstein field equations using variational approach. I know how to read basic Penrose diagram (at least for asymptotically flat spacetimes). I even know less well-known stuff like how to compute for black hole temperature and entropy from geometric arguments a la Hawking and Wald respectively.
These are far from enough. I want to understand more about null geodesic congruences – how light propagates as a bundle. After all, it tells the difference – via Raychaudhuri equation, for instance – that Kerr black hole null congruences are not twist-free while Schwarzschild ones are. These also appear in a form of tetrad formalism used as an alternative to Einstein field equations, known as the Newman-Penrose formalism. This formalism is powerful to study gravitational radiation. I have seen how this is used to study e.g. mass loss due to gravitational radiation and also the “peeling property”, but never actually do them myself. The spinor formalism seems attractive too, since it allows you to possibly attach spinors to spacetime. I want to know more about general geometric and topological structures of spacetimes: what do the asymptotic symmetries say? What exactly is the difference between the symmetry group for flat spacetimes (BMS group) compared to the full diffeomorphism group? Decomposing Riemann tensor into Ricci scalar, a semi-traceless part (“Ricci tensor part”) and fully traceless Weyl tensor, one can seem to understand certain things using parts of Riemann tensor: Gravitational waves have no trace and hence can be purely studied using Weyl tensor. Last but not least, conformal structure of spacetime: I have learnt that in a 4-spacetime with cosmological constant, imposing conformal flatness on null infinity implies no gravitational radiation. Why? How much physics can you extract from understanding the conformal structure alone?
And really, these are not even all.
Geometry and Topology
This is another topic. I would want to (hell, wishful thinking) read John Lee’s Introduction to Smooth Manifolds back to back if I have the time. It is both rigorous but well-written (I only skimmed part of it). And it is complete – I surmised that I would gain most of what I need for say a decade from such a book. Once done, I should be able to get to the basics of Riemannian geometry and all its standard objects – Riemann tensor, affine connection, etc.
Two main objects of interests are connection and holonomy. If you are a physicist, you most likely need not understand connection beyond the fact that we are using a “metric-compatible” Levi-Civita connection and that it is unique. No one really discusses how the connection – which is a geometric object – is related to the Christoffel symbols. In fact I think it must be the case that a connection is defined by both the Christoffel symbols and “torsion”, and this is something I possibly won’t see unless I study connection as geometric entity properly. Holonomy is supposed to be another way to characterize curvature, and I know next to nothing at the moment. It appeared twice in my life: once in algebraic topology though it was told as “Holonomy functor”, and another one in loop quantum gravity. I understood neither, but I really want to. I feel that I would understand something deep and significant once I get past this. Interestingly, because of the functor thing, I am somehow interested in it (if I have free time). In fact, in loop quantum gravity, holonomy was used to quantify something when one “drags” elements of a Lie algebra around a loop (like SU(2) or others) – something I am completely lost of.
And hell, fiber and vector bundles, symplectic geometry, etc. I think the list goes on.
Even a while back, I was interested in logic – incompleteness theorem, method of forcing, continuum hypothesis, second-order logic; scientific computing like Python and Mathematica‘s tensor manipulation; Lie groups and Lie algebras; conformal field theory and quantum field theory; quantum information and standard quantum mechanics; statistical mechanics and renormalization group; condensed matter physics and topological insulators; complex analysis and complex manifolds; and so on. Even violin and Japanese.
Really, forget about anything above. If I cannot find the motivation to do any of these, even just one of them, I can forget about learning anything at all. Then my life will be as wasted as it has always been.