General Relativity: Advice please about the textbook by Misner, Thorne and Wheeler

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You might want to reconsider that since very few prominent specialists in GR have been professional mathematicians. The only one I can think of off the top of my head is Roger Penrose.
And maybe Hermann Weyl. It's as usual: His math is brillant, also from a didactic point of view when referring to his very famous textbook "Raum, Zeit, Materie" ("Space, Time, Matter"). The mathematicians of his time, however seem to have thought not so positively about this book, because in Heisenberg's book "Der Teil und das Ganze" you can read about his experience with the famous mathematician Ferdinand Lindemann, whom he consulted concerning the choice of his subject of study at Munich university. When he told Lindemann that he has already read Weyl's book, Lindemann told him that he is already spoiled for a serious study of mathematics ;-)).

Weyl's physics is not that brilliant, because the idea to gauge scale invariance of the matter-free gravitational field and taking the corresponding gauge field as the electromagnetic field was immediately considered wrong by Einstein and also Pauli, because indeed the measures of rods doesn't depend on their "electromagnetic history". In any case this idea of "gauging of symmetries" was ingenious in its own write. Weyl simply gauged the wrong symmetry in this case, and the entire thing gave the name associated with this idea till today: "gauge theory", "gauging a symmetry", etc.

Demystifier
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I think that MTW is a wonderful book for learning General Relativity, if you have 20 years to spare.
Gee, it's been sitting on my bookshelf since 1988. I could have learned it by now!

kith, ohwilleke and Demystifier
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And maybe Hermann Weyl.
Yes, he's another example. I remember reading an English translation of "Space, Time, Matter" back when I was an undergraduate, and I couldn't make head or tail of it. Then, years later, after I had read through MTW and was more familiar with GR and tensors and so on, I suddenly realized what he was talking about.

vanhees71 and bob012345
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Yes, he's another example. I remember reading an English translation of "Space, Time, Matter" back when I was an undergraduate, and I couldn't make head or tail of it. Then, years later, after I had read through MTW and was more familiar with GR and tensors and so on, I suddenly realized what he was talking about.
Thanks for the reference. I found it on Project Gutenberg. It is astonishing that this book is over 100 years old. Still, I want it.

https://www.gutenberg.org/files/43006/43006-pdf.pdf

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Yes, he's another example. I remember reading an English translation of "Space, Time, Matter" back when I was an undergraduate, and I couldn't make head or tail of it. Then, years later, after I had read through MTW and was more familiar with GR and tensors and so on, I suddenly realized what he was talking about.
I read the book also early in my undergraduate studies, and I found it pretty intuitive, at least the parts where he talks about the mathematical foundations. I had only trouble to understand the physics part. This I learned then from Landau and Lifshitz vol 2 :-).

ohwilleke
ergospherical
this is probably one of the worst textbooks I've ever seen
not even trying to be edgy, it's just a complete mess and I don't know how anybody has learned anything from it

ohwilleke
Gold Member
this is probably one of the worst textbooks I've ever seen
not even trying to be edgy, it's just a complete mess and I don't know how anybody has learned anything from it
Are you referring to MTW?

Gold Member
this is probably one of the worst textbooks I've ever seen
not even trying to be edgy, it's just a complete mess and I don't know how anybody has learned anything from it
Carroll, in his Lecture Notes on General Relativity, said abot MTW the following:
"A heavy book, in various senses. Most things you want to know are in here, although you might have to work hard to get to them (perhaps learning something unexpected in the process)."

ohwilleke
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this is probably one of the worst textbooks I've ever seen
not even trying to be edgy, it's just a complete mess and I don't know how anybody has learned anything from it
You mean Landau Lifshitz vol 2? For me it's the most straightforward introduction to GR I've ever read. Of course you can critisize that it sticks to the Ricci calculus only, but that's good for the beginner. Without it, I'd not have had the chance to understand MTW when I learned the subject as a student. MTW is great, but you get too easily lost in all the details before you have an overview about GR from a more introductory level. Of course for the purpose to really get a deeper understanding also for more modern math (Cartan calculus) and more subtle discussions of the physics it's a gem.

Demystifier and TeethWhitener
this is probably one of the worst textbooks I've ever seen
not even trying to be edgy, it's just a complete mess and I don't know how anybody has learned anything from it
Which one?

ergospherical
gravitation by charles misner, kip thorne and john wheeler

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gravitation by charles misner, kip thorne and john wheeler
No textbook will be suitable for everybody, but many, many people have learned a great deal from this one, so I think you might want to mute your criticism a bit.

robphy
ergospherical
No textbook will be suitable for everybody, but many, many people have learned a great deal from this one, so I think you might want to mute your criticism a bit.
no doubt there's some really unique content, I just wish the explanations and the formatting weren't so convoluted and dragged-out so as to make much of it unreadable (to me at least)

maybe it'll be more useful later in my studies, but for now a terser book like Hawking and Ellis is proving much more enjoyable

ohwilleke
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2022 Award
MTW has a special style, but I simply love it, because it's so different from nearly all other books on the topic. The same holds for Kip Thornes newest textbook, which I like even more:

K. S. Thorne, R. D. Blanford, Classical Modern Physics, PUP (2017)

It contains really all "classical physics" (i.e., all non-quantum physics) treating it from a coherent conceptual point of view, emphasizing the geometric aspects of all physics. It treats the classical topics (mechanics, optics, stat. phys.) within both Newtonian and special+general relativistic physics in a really clear way. Maybe it's a bit sparse in showing every detail of all calculations, but emphasizes the general concepts. E.g., the explanation, why the phase-space distribution function in statistical physics is a relativistic scalar, makes this issue (which confuses even practitioners in the field sometimes) very clear.

Demystifier and dextercioby
Homework Helper
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this is probably one of the worst textbooks I've ever seen
not even trying to be edgy, it's just a complete mess and I don't know how anybody has learned anything from it
Are you referring to MTW?

gravitation by charles misner, kip thorne and john wheeler

I think that these students (from the 60s and 70s) were able to learn something from them...
(from https://www.genealogy.math.ndsu.nodak.edu/id.php?id=31332 )

Plesset, Milton Yale University 1932 29 Feynman, Richard Princeton University 1942 48 Wightman, Arthur Princeton University 1949 801 Tiomno, Jayme Princeton University 1950 5 Everett III, Hugh Princeton University 1957 Misner, Charles Princeton University 1957 58 Ernst, Frederick University of Wisconsin-Madison 1958 1 Brill, Dieter Princeton University 1959 34 Klauder, John Princeton University 1959 24 Shepley, Lawrence Princeton University 1965 2 Thorne, Kip Princeton University 1965 274 Geroch, Robert Princeton University 1967 30 Fischer, Arthur Princeton University 1969 Christodoulou, Demetrios Princeton University 1971 31 Unruh, William Princeton University 1971 5 Hu, Bei-Lok Princeton University 1972 38 Wald, Robert Princeton University 1972 5 Ford, Lawrence Princeton University 1974 10 Hojman, Sergio Princeton University 1975 2 Kheyfets, Arkady University of Texas at Austin 1986 1 Miller, Warner University of Texas at Austin 1986 1

Possibly interesting:
http://www.oobject.com/john-wheeler-and-his-elaborate-blackboard-presentations/

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dextercioby and vanhees71
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E.g., the explanation, why the phase-space distribution function in statistical physics is a relativistic scalar, makes this issue (which confuses even practitioners in the field sometimes) very clear.
Thanks for the tip! But I looked into it and found their derivation, in terms of Lorentz contraction, rather clumsy. Do you know a reference with more elegant derivation? Perhaps a derivation that works even in curved coordinates?

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It works in curved spacetime, because it's a local concept. I don't know, whether this is the same as in MTW, my approach is this: There's the "lab frame" with spacetime coordinates ##x=(x^{\mu})## and on-shell momenta ##p=p^{\mu}=(E_p,\vec{p})##. Now we consider the particles with momentum ##\vec{p}## and define ##\mathrm{d} N^*## as the number of particles at time ##t^*## with ##\Sigma^*## the reference frame, where these particles are at rest and write
$$\mathrm{d} N^*=f^*(x^*,\vec{p}^*=0) \mathrm{d}^3 x^* \mathrm{d}^3 p^*.$$
Now we express this very same particles in terms of the quantities in our observational frame ##\Sigma##. Because we measure our volume element at fixed time ##t## we have length contraction and thus
$$\mathrm{d}^3 x=\frac{1}{\gamma} \mathrm{d}^3 x=\frac{E}{m} \mathrm{d}^3 x.$$
For the on-shell momenta volume elements you have
$$\mathrm{d}^3 p^*/E^*=\mathrm{d}^3 p^*/m=\mathrm{d}^3 p/E$$
and thus
$$\mathrm{d}N^*=\mathrm{d} N = f(x,\vec{p}) \mathrm{d}^3 x \mathrm{d}^3 p = f(x,\vec{p}) \frac{m}{E} \mathrm{d}^3 x^* \frac{E}{m} \mathrm{d}^3 p^*=f(x,\vec{p}) \mathrm{d}^3 x^* \mathrm{d} p^* \' \Rightarrow \; f^*(x^*,\vec{p}^*=0)=f(x,\vec{p}).$$
In this way it's clear that ##f(x,\vec{p})## is a Lorentz-scalar one-particle phase-space distribution function.

Another argument is that the particle-number four-current,
$$N^{\mu}(x)=\int_{\mathbb{R}^3} \mathrm{d}^3 p \frac{p^{\mu}}{E} f(x,\vec{p})$$
must be a vector field, and thus since for on-shell particles ##\mathrm{d}^3 p/E## is a Lorentz scalar and ##p^{\mu}## is a four-vector ##f## must be a scalar.

For GR everything goes through analogously. You only have to write everything in a general covariant way with the corresponding tensor densities.

Demystifier
andresB

That's a dedicated teacher.

Homework Helper
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vanhees71, bob012345, PeterDonis and 1 other person
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vanhees71
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That's a dedicated teacher.
FWIW, what makes sense for a course taught by one of the textbook's authors in a classroom setting and what makes sense for someone studying independently without the backup of lectures, office hours, TAs and study groups from people in the same class, can be quite different.

yucheng
Why is MTW a pun though? I seriously do not get it.

Gold Member
Why is MTW a pun though? I seriously do not get it.
It’s so massive that it exerts its own gravitational field or it demonstrates the topic by its weight.

vanhees71 and yucheng
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It’s so massive that it exerts its own gravitational field or it demonstrates the topic by its weight.
The old joke was that Wheeler wanted the book to undergo gravitational collapse and form a black hole.

ohwilleke, vanhees71 and Frabjous
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The old joke was that Wheeler wanted the book to undergo gravitational collapse and form a black hole.
I hadn’t heard that one.

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