Neat, thank you for posting this. It is nicely organized, and while most HN readers would find §2 pretty steep, the content of §2.1.4 (Massless particles), §2.

Neat, thank you for posting this. It is nicely organized, and while most HN readers would find §2 pretty steep, the content of §2.1.4 (Massless particles), §2.3.2 (Geodesics and affine parameters) and §2.3.4 (Massless particles and the Speed of Light) are concise technical answers to a HN FAQ which is essentially "according to relativity, do photons experience no time?". Really frequently "asked". (tl;dr answer: yes, they certainly do, you just write their "time" down differently from that of massive particles. The difference in their time ("affine time") tells you observable things -- for instance the difference in quantum wavefunction phase, or the difference in redshift, just as does the difference in proper time for electron (anti)neutrinos when e.g. leaving the cores of stars or neutron stars, or traversing cosmological distances). (The missing pieces here are defining a momentum p^\mu = \dot X^\mu where the \dot is the derivative with respect to the affine time, much as the author has actually done in §2.3.3 (Massive Particles), and one of the energy-momentum relations like de Broglie's E = hc / \lambda or \lambda = h / p). §2.3.5 is another good trio of answers to the HN FAQ about the twin paradox, and I agree with the author that the geodesic analysis is more attractive, namely "both Alice and Bob's paths in spacetime begin and end at the same point, but Alice follows a timelike geodesic, while Bob follows a timelike path which is not a geodesic" (emphasis mine). "Since timelike geodesics are exactly the timelike paths which maximize proper time, the proper time experienced by Alice must be larger than the proper time experienced by Bob. Therefore, Alice must be the older twin." That's really all it takes to resolve the apparent paradox. (However, c.f. "Too Many Analyses" in the excellent <https://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_vase.html https://math.ucr.edu/home/baez/physics/Relativity/SR/TwinPar...>). I'm slightly surprised that in skimming the later sections (which I plan to peruse) I did not notice the author discussing the <https://en.wikipedia.org/wiki/Causality_conditions https://en.wikipedia.org/wiki/Causality_conditions> and the causal ladder discussed therein, and only engaged directly with with (in §3.1.3) global hyperbolicity, although some of the many references in §7 might. I suppose that at 60ish pages the author did not want to commit into turning an extensive set of lecture notes into a comprehensive textbook, which is fine. Finally, it has been several years since Bob Geroch's "democracy of causal cones" Faster Than Light? paper <https://arxiv.org/abs/1005.1614 https://arxiv.org/abs/1005.1614>, and exposure to that likely would affect the end of this paper's §4.3 ("Tachyons and Time Travel") which deals specifically with faster-than-light. P.S.: (2019)