General Relativity: The Essentials

General relativity is wonderfully interesting and extremely important. It is a difficult area to learn well. At one extreme, a lot of popular books avoid all equations and give only imprecise (sometimes incorrect) qualitative notions of what it all means. These are not very good. At another extreme, you might expect to master all of Riemannian geometry and modern manifold theory just to get a start. This is a wonderful little book that manages to find a useful middle ground. It shows you enough of the real mathematics to get you started. That is much more important than it appears because it lets you do further study with more insight concerning what is important. It gives you a very clear notion of what it all means and is certainly the clearest currently available introduction that could lead to useful speculation about quantizing gravity. It is certainly a non-trivial read, but it is clear and concise and, most importantly, correct. Many very useful books are boring to read. You could need lots of coffee. This one is pleasant and will only require as much coffee as you like anyway.

Not for novices, but pretty good for junior-senior physics majors provide they have had a good multivariable calculus course. Years ago, back in the 1960s when I began my quest to "understand" relativity, Pauli's "Theory of Relativity"was recommended as a good read. Rovelli's book is an easier read and very up date with regard to black holes and cosmological models when compared with the Pauli classic. If you are willing to spend some time thinking about theexercises and questions scattered about in the text, I think you will be quite glad you bought the book.

This book is not for those who are not well-versed in the GR formalism. And by "well-versed" I mean graduate-level differential geometry. However, for those who are thoroughly comfortable with the math, this is an excellent overview to consolidate/condense your "big-picture" understanding of GR and its applications. The book is brief, yes, but that is only because it assumes a ton of background knowledge on the reader's part. In that way it's like a friendlier version of Dirac's GR book, which I also recommend to the same audience.

On time

La simpleza chic la que nos muestra un tema tan complejo, como lo es la relatividad general

Recently, I had the pleasure of studying Sidney Coleman's Lectures on Relativity (Cambridge University Press). Recently, I studied Rovelli's "essentials" (reprint, 2021, published by Cambridge University Press). Take note: If you want to learn relativity, study Sidney Coleman. If you have already learned relativity, then Rovelli is a source of ideas. However, I do not recommend Rovelli as a first introduction into any of the topics he expounds, as this exposition of "essentials" seems merely an afterthought. Note, too, there is little in his "essentials" that you won't find elsewhere. Some of this "essential" material appears in his Quantum Gravity (2004, Cambridge University Press), where you read "the structural similarity between Maxwell and Einstein's theories is evident," page 52. These "essentials" neglects to include Misner, Thorne & Wheeler's Gravitation as part of the reading list (page 157: "classic manuals"). That could be explained due to his belief that "the geometrical interpretation of the gravitational field…is going to be largely lost in the quantum theory." (Quantum Gravity, page 96). Let us dig deeper:(1) Cursory, that is an apt description of this book. Rovelli hits the high notes (say, equivalence principle, page 59) without ever covering anything in sufficient depth. Why bother with such a cursory description of differential forms ? (pages 47-50; Study Needham's Visual Differential Geometry and Forms). "Action," here, is compressed to two pages (65-67), a worthy topic for elaboration (he does use it on page 110). For that, see Zee, Padmanabhan, or Dirac. An exercise (page 2, dimensional analysis of Planck length)is presented better in Schutz's First Course in General Relativity (exercise 2d, page 210). Not surprisingly, chapter 6 of "essentials" reinforces all that need be said: "another radical consequence." (pages 68-74).(2) Not all typos are created equal, but the typo on page 43 (line 26) is puzzling for a student. Among the more serious ones, line 27, then equation 3.80 (both page 47) are simply un-decipherable.Also, what is generally spelled contravariant elsewhere, Rovelli spells 'controvariant' (page 42).Herein, general covariance means diffeomorphism invariance. (page 69). Here is a statement of dubious value: "If you want to do great science, read philosophy." (page 13).(3) Analogies between electromagnetic field and gravitational field are emphasized. Read: "we can take inspiration from the electromagnetic case." (page 77) and " theory of electromagnetic waves is a good model for the gravitational waves." (page 83). For more: "On the analogies between gravitational and electromagnetic radiative energy" (Gomes and Rovelli, arXiv:2303.14064). Now, I urge the reader to make comparisons between Rovelli's exposition (page 87) and Schutz's exposition (page 218).(4) As customary, there is reliance on phrase "to first order." ( "Up to first order in distance," page23).That statement implies something useful (approximations, local versus global, linear versus nonlinear). So, be on guard as to what "first order" entails each time it is used, both from a mathematics and physics standpoint (page 109). Second-order will make an appearance for frame-dragging (page 121).(5) Pithy, that is another way to describe this exposition. Being concise is not equivalent to being lucid.(Of course, bigger books are not necessarily more lucid). Here, time is equated to be a clock (page 107). Statements, such as: "atoms and moving parts of the clock interact with the gravitational field" (page 14) and the reason "things fall towards a mass is because a mass slows down time" (page 81) are not made intuitive to the uninitiated (if time equates to clock, the first statement makes little sense).(6) Is there any consensus on Rovelli's discussion regards "dark energy" ? (that is, pages 62, 119, 120).(7) Is there pedagogic value displaying the compact notation of formula 4.11 (page 61) ? Compressing an equation to sweep away all detail does no one any favors. Stating that "The interaction with the Dirac field cannot be written in terms of the metric, it requires the tetrad" provides little guidance for understanding equation 4.11 ! Now, page 61 reminds us to switch Lorentz metric to general metric, and to switch partial to covariant derivatives when gravity is involved (and, so it appears as if gravity 'arises' from a purely magical mathematical substitution). Read: Misner, Thorne and Wheeler (page 387) !(8) Let us talk coordinates: Find a multitude of coordinate transformations (pages: 36, 53-55, 125, 128, 134-137). Read: "proper time is to be distinguished from coordinate time." (page 60) and "Coordinates--by itself-- carry no information at all." (page 68). Now, read page 115: "This is the speed of light near the Sun in these coordinates: light slows down near the star." If it all sounds confusing, turn to Cheng's Relativity, Gravitation and Cosmology, where he actually explains it beautifully (pages 89-93).(9) Akin to Misner, Thorne & Wheeler's Gravitation, Rovelli places in his final chapter his speculations: White Holes (time reversed Black Holes), space discreteness, superposition of geometries (but, argument for superposition of geometries is unconvincing, pages 148-150). More about coordinates and "time" in Rovelli's 2004 book Quantum Gravity (pages 82-97).(10) Who is the audience for this book ? Certainly not laymen. It is also not an introduction (although it tries to be). The imbedded exercises ( "do it" ) in some cases are trivial, in some cases not. You will learn "ideas and key results," but I do not believe you get "additional clarity." (preface). In contrast to James Hartle's beautiful text, Rovelli's book is simply not offering insight that you can't get elsewhere. Rovelli's terseness recalls Dirac's 1975 monograph and (in a perverse sense) I do suggest studying the two books in tandem ! Then, work your way through "Problem Book in Relativity and Gravitation" (by Lightman, Press & Price). Happy is a footnote (page ix) recommending Steven Weinberg's book, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity !Addendum: Physics Today has a review of this book (2023, Vol 76, No 9). Read: "The book’s thorough grounding in examples and simple cases is somewhat rare among general relativity textbooks." (Now, in my view, that is unfounded). Read: "his treatment of the mathematical content is so terse that it is a bit opaque." (And, in my view, his treatment... is more than "a bit" opaque !).

I am interested in the philosophy of reality since I started with Niels Bohr Atomic Physics and Human Knowledge when it was written in Danish, decades ago. Mathematic? I know 2 + 2 makes 4. but this book very readable and gives me a deeper understanding of existence in the front of understanding