> what am i missing ? Comma and semicolon typographical notation for partial resp. covariant derivatives. The comma is used to indicate a partial derivative:

> what am i missing ? Comma and semicolon typographical notation for partial resp. covariant derivatives. The comma is used to indicate a partial derivative: V^{\alpha}_{\comma\beta} = \partial_{\beta}V_{\alpha}. The semicolon V^{\alpha}_{\semicolon\beta} turns somewhat complicatedly (via Christoffel symbols) into \nabla{\beta}V^{\alpha}. There’s more detail at https://en.wikipedia.org/wiki/Covariant_derivative#Notation https://en.wikipedia.org/wiki/Covariant_derivative#Notation These notations are reflected in “comma-goes-to-semicolon” which is discussed in MTW chapter 20 & 22 (esp. §22.4, Electrodynamics in Curved Spacetime) and other textbooks. Carroll’s online lecture notes at https://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll4.html https://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll... discusses it briefly starting with the paragraph beginning “Our next task is to show how the remaining laws of physics…” and in particular (starting with) that paragraph’s penultimate sentence. Even more briefly, we need generally covariant physics to escape the flat spacetime background into general curved spacetime and still get sensible electromagnetism and so on. Even more briefly: the semicolon indicates a valid tensor equation, showing that we are fine when the basis vectors are not constant. Switching , to ; when it works hides away a bunch of work with Christoffel symbols, and what makes it work is the Equivalence Principle. Semicolon-goes-to-comma always works by gauge fixing, but unfortunately V^{a}_{,b} -> V^{a}_{;b} sometimes doesn’t; doing the full expansion by hand (well, or with modern automation) is one way to see if your physical theory is really generally covariant/background independent. > superdense Yes! As an example, and I haven’t really done more than scrub through it, and he really doesn’t go into any details beyond what you’d get in a standard textbook: https://www.youtube.com/watch?v=DfHln-nzpv0 https://www.youtube.com/watch?v=DfHln-nzpv0 (note LOCAL equations only in this video and its predecessor https://www.youtube.com/watch?v=XqNVxFY-lCs https://www.youtube.com/watch?v=XqNVxFY-lCs) Which leans heavily on https://en.wikipedia.org/wiki/Einstein_notation https://en.wikipedia.org/wiki/Einstein_notation Putting the extra typographical dot on top of the , giving a ; can literally represent several pages of dense equations. > jargon Yes! A lot of writing that turns out to generally be really mechanical has been squashed out of General Relativity through the years. Unfortunately the notation itself — as typography — tends to “leak out”, so we talk about indices being in the basement or upstairs, Greek vs Roman, etc etc.