Why are General relativity texts so much more formal?

[paralleltransport]

TL;DR Summary

Why are general relativity texts written so much more formally than other physics texts.

Hi all,
What I notice is that there's a significant difference in style between the GR texts and the other textbooks. In particular, GR texts very much try hard to read like a math textbooks, emphasizing theorems and abstract definitions, which I'm not sure are practically useful (though beautiful they are). I have in mind for example something like Carroll's book or Wald's book, compared to equivalent undergraduate/graduate texts in other physics subjects (Jackson for E&M, Goldstein for mechanics, Sakurai/Griffiths for QM).

Some example I've seen:
1. All GR texts I've seen make a big deal about Hausdorff vs. non-Hausdorff or mention it.
2. They all talk about topological space, open/closed set to start defining manifolds.
3. One defines differentiable manifolds using abstract charts and maps and C∞C∞C∞C∞ invertible functions to stitch them together.
4. The descriptions of boundaries etc... often use abstract set notation.
5. Symmetry arguments try to use killing vector fields, orthonormal to foliations of manifold etc...

I've never used the fact about topological sets, and chart compatibility etc... on a single GR problem. Mostly calculus modified to the appropriate setting.

One could say QM's math involves functional analysis (operator theory on hilbert space, subtleties about infinite dimensional operators etc...) but very few QM text make a big deal out of it. Similarly, classical mechanics could involve the study of vector bundles, symplectic forms, and E&M/gauge theories could involve principal-G bundles but Jackson/Goldstein don't try to make a big deal out of it. I don't even know what math QFT would involve but it's highly non-trivial (that one I don't think mathematicians even figured out completely yet).

What's the main motivation behind this dichotomy? Why isn't GR taught the same way other physics subjects are taught (minimal math, and only introduce it if it's necessary)?

 

[PeterDonis]

All GR texts I've seen

Which ones have you seen? You mention Carroll and Wald, but no others. Both of those, I would say, spend much more time on the topics you mention than others. MTW, for example, while it does discuss tensors and differential forms in some detail, does so from a physics viewpoint and does not spend much time at all on the detailed rigorous math of manifolds. (MTW also separates content into "Track 1" and "Track 2", with the former being much more limited in its discussion of rigorous math.)

 

[paralleltransport]

Hi I've seen wald, carroll and schutz, but I don't really think schutz is a graduate text. My impression was wald & carroll are the standard intro to GR texts at a graduate level.

I remember out of a 20 or so GR lecture course we spent 10 solid lectures doing math. We got the physics (Einstein's equation) by lecture 11 or so. The first lecture was just spent talking about manifolds and their charts, and yes there was a 1 hour discussion about properties of open/closed sets, topological spaces etc...

 

[PeterDonis]

My impression was wald & carroll are the standard intro to GR texts at a graduate level.

That may be, but you don't have to be limited by those texts.

MTW (Misner, Thorne, and Wheeler) recently came out in a new edition; perhaps that will lead some graduate programs to consider using it again (it was originally published in 1973 and was based on material that all three of its authors had used in teaching graduate courses in GR back then, and I believe it was used for at least some years after publication as a graduate teaching text).

 

[PeterDonis]

I remember out of a 20 or so GR lecture course we spent 10 solid lectures doing math. We got the physics (Einstein's equation) by lecture 11 or so. The first lecture was just spent talking about manifolds and their charts.

Hm. Sounds like the mathematicians have taken over.

While I'm not a teacher, I have to question whether this is really the best approach pedagogically. The basic physical concepts of GR can be gotten across with much less emphasis on the rigorous mathematical background. I would think a "physics first, math later" approach would make the subject much more approachable.

 

[paralleltransport]

Hm. Sounds like the mathematicians have taken over.

While I'm not a teacher, I have to question whether this is really the best approach pedagogically. The basic physical concepts of GR can be gotten across with much less emphasis on the rigorous mathematical background. I would think a "physics first, math later" approach would make the subject much more approachable.

I see. Glad to hear I'm not alone in this thought. I've never had to think about orientation of a manifold, or point-set topology to think about the schwarzchild metric etc... but I found it weird lecture 1 starts with that.

I have to say, the equivalent of E&M's book like wald, the first chapter would also talk about manifolds and set theory since the arena of gauge theories are bundles with curvature on them, same as GR. I'm glad Jackson doesn't because it would be incomprehensible to me for an introduction.

 

[robphy]

Possibly interesting:
https://physicstoday.scitation.org/doi/10.1063/PT.3.1605 “Teaching general relativity to undergraduates” by Christensen & Moore

https://arxiv.org/abs/gr-qc/0511073
“Teaching General Relativity” by Wald

https://www.aapt.org/doorway/TGRU/
AAPT TOPICAL WORKSHOP:
TEACHING GENERAL RELATIVITY TO UNDERGRADUATES (2006).
The website is old, and a little buggy… but the content is there.
[ I presented a poster here]

 

[paralleltransport]

I see. I read Wald's article. It's quite interesting that both GR and QM have similar hurdle:
- in QM one needs to unlearn naive conception of vector spaces as arrows and accept bra-ket notation (abstract linear vector spaces, where "vectors" could be functions and matrices could be bounded operators on said space).
- in GR one needs to unlearn naive conception of talking about manifolds embedded in higher dimensional space, and vectors as pure arrows.

It just seems the GR people emphasize that part more than QM folks.

 

[martinbn]

My speculation is that after Einstein's papers appeared (even for them mathematicians were involeved), GR was developed also (even mostly) by mathematicians and mathematically oriented physicists. I am not sure but I think that the first courses in GR were taught in the Maths departments.

 

[paralleltransport]

One idea I have is that GR is such a "complete" or self-contained theory the same way analytical mechanics is. A lot of the 20th century research for it could be well posed as math problems with rigorous proofs. This attracts people like Penrose etc... who come from a pure math background and instill this culture.

This is unlike other areas like QM and stat mech where most areas of research, the questions could not be well posed as math problems (a lot of the research in modern quantum field theory involves even asking the right question! think the renormalization group).

 

[PeterDonis]

It just seems the GR people emphasize that part more than QM folks.

I think this is at least partly because in QM, state vectors are already understood to be vectors in a different space (Hilbert space) from ordinary 3-space, or ordinary 4-dimensional spacetime. Hilbert space is already abstract, so additional abstraction doesn't seem like as big a hurdle.

In relativity, however, the basic thing being studied is still ordinary 4-dimensional spacetime, which seems to be concrete--it's the actual spacetime we live in, not some abstract thing. So having to accept that vectors are not actually arrows in this concrete spacetime we live in is more of an issue and has to be dealt with explicitly.

 

[PeterDonis]

in GR one needs to unlearn naive conception of talking about manifolds embedded in higher dimensional space

For manifolds, I think the key element in GR that is not present in QM is curvature, more precisely intrinsic curvature, which has to be carefully distinguished from extrinsic curvature. That's why intuitions involving embedding have to be unlearned--because an embedding focuses attention on extrinsic curvature instead of intrinsic curvature.

 

[robphy]

Of course, an important feature of relativity is to free one's self from Euclidean space and learn to generalize to vectors spaces with a Lorentz-signature metric and to pseudo-Riemannian spaces. One needs to relax and generalize what is known about Euclidean and Riemannian spaces.

In addition, one should not try to get too tied to "coordinate systems"... but instead try to think geometrically (e.g. vectors, not components).

Of course, different authors have different levels of rigor and mathematical detail
depending on the intended audience of the textbook.

By the way, here is Moore's book:
https://uscibooks.aip.org/books/a-general-relativity-workbook/

 

[jbergman]

Which ones have you seen? You mention Carroll and Wald, but no others. Both of those, I would say, spend much more time on the topics you mention than others. MTW, for example, while it does discuss tensors and differential forms in some detail, does so from a physics viewpoint and does not spend much time at all on the detailed rigorous math of manifolds. (MTW also separates content into "Track 1" and "Track 2", with the former being much more limited in its discussion of rigorous math.)

I guess I'm alone in preferring the more mathematical treatments. Personally, I find MTW abysmal.

Maybe one of the reasons for more mathematical treatments is that it's almost a requirement to get into related fields like Quantum Gravity and String Theory.

 

[paralleltransport]

I think a similar reasoning could be said about QFT. QFT is a pre-requisite to get into string theory. But most textbook on QFT just use heuristic math (if they didn't, they'd be either unreadable or 2000 pages+ monographs). For most problems we only care about the math in sofar as it gives the rules to compute things. Hence distinction like hausdorff/non-hausdorff, C∞C∞C∞C∞ differentiability do not contribute much to the physics discussion. Relativistic QFT also occur in the 4-d manifold as an area, and there the fact the number of points are literally infinite cause serious non-trivial issue in the path integral measure, leading to things like renormalization. Yet for a introduction to the subject, no text try to formalize this point for good reasons.

 

[PeterDonis]

"Maybe one of the reasons for more mathematical treatments is that it's almost a requirement to get into related fields like Quantum Gravity and String Theory.

Source https://www.physicsforums.com/threa...xts-so-much-more-formal.1009155/#post-6564811"

@paralleltransport you can use the "Quote" feature of PF to quote things you want to respond to. Highlight the portion of a post that you want to quote and then click "Reply" in what pops up. The quote will appear in the post editing window, then you can just enter your reply.

 

[ergospherical]

A book to consider may well be General Relativity: An Introduction for Physicists; Hobson, Efstathiou, Lasenby. The authors are members of the Physics/Astrophysics departments as opposed to the Applied Maths department, so there is a little more emphasis on physical principles rather than mathematical formalism. On the flip side it is designed for a third year course, so isn't ridiculously advanced (although some of the problems are tricky!).

 

[Nugatory]

I would think a "physics first, math later" approach would make the subject much more approachable.

That would be Hartle.

 

[robphy]

. I would think a "physics first, math later" approach would make the subject much more approachable.

That would be Hartle.

Even Schutz would agree
https://www.physicsforums.com/threa...-as-a-challenge-for-physics-education.966053/

Here are Hartle's slides from that AAPT workshop ( https://www.aapt.org/doorway/TGRU/ )
https://www.aapt.org/doorway/TGRUTalks/Hartle/HartleTalk1of5.htm

 

[vanhees71]

That may be, but you don't have to be limited by those texts.

MTW (Misner, Thorne, and Wheeler) recently came out in a new edition; perhaps that will lead some graduate programs to consider using it again (it was originally published in 1973 and was based on material that all three of its authors had used in teaching graduate courses in GR back then, and I believe it was used for at least some years after publication as a graduate teaching text).

AFAIK there's nothing new compared to the original edition except for some preface. I'd not recommend it for a first encounter with GR, but if you have some knowledge, what GR is about, it's just great.

For an introduction I still think Landau Lifshitz vol. 2 is a gem, because it precisely avoids all the mathematical finesses you stated in #1 ;-)).

 

[Nullstein]

I think the difference is that GR is only rarely being applied in engineering. Most people who study it, intend to do foundational research. But in order to do foundational research, you really need to know all those details. On the other hand, most people who study QM, ED or CM don't worry much about foundational research and pursue more application-oriented careers, where you can get quite far without knowing a lot of math. However, if you want to get into cutting-edge research in these fields, you don't get around learning all these intricate details as well.

 

[PeterDonis]

I'd not recommend it for a first encounter with GR

Outside the context of a course with instruction, I agree. I could see it being a useful text in a course with instruction, though, because the instructors can zero in on the key points and help the students to navigate the text.

 

[paralleltransport]

I think the difference is that GR is only rarely being applied in engineering. Most people who study it, intend to do foundational research. But in order to do foundational research, you really need to know all those details. On the other hand, most people who study QM, ED or CM don't worry much about foundational research and pursue more application-oriented careers, where you can get quite far without knowing a lot of math. However, if you want to get into cutting-edge research in these fields, you don't get around learning all these intricate details as well.

I'm not sure that's quite true. QFT is almost never applied in engineering but is not taught formally the first time either. Similarly QM is very rarely used in real life work. Even in semiconductor research it's semi-classical treatments rarely the full formalism. Yet QM is not taught formally either.

Finally I have a hard time believing GR research involves thinking about manifold point set topology. We all know that the physics breaks down at the Planck scale, so there's no point even debating the smoothness of the manifold.

 

[martinbn]

Finally I have a hard time believing GR research involves thinking about manifold point set topology. We all know that the physics breaks down at the Planck scale, so there's no point even debating the smoothness of the manifold.

I don't buy this. Physics doesn't break at the plank scale. You probably mean the current physics theories not physics. But even that we don't know for sure. Even if we did know it, it is no reason not to study the smooth models. We defenitely know that they are very good at other scales. So there is a good reason to study all the differential geometry, topology and so on. Here is an analogy. In fluid mechanics, we definitely know that the continuous models are not accurate at all scales. That doesn't stop people from studying them rigorously. It is consider important enough to have a million dollar award for resloving rigorously a specific problem.

 

[paralleltransport]

I don't buy this. Physics doesn't break at the plank scale. You probably mean the current physics theories not physics. But even that we don't know for sure. Even if we did know it, it is no reason not to study the smooth models. We defenitely know that they are very good at other scales. So there is a good reason to study all the differential geometry, topology and so on. Here is an analogy. In fluid mechanics, we definitely know that the continuous models are not accurate at all scales. That doesn't stop people from studying them rigorously. It is consider important enough to have a million dollar award for resloving rigorously a specific problem.

Yes I mean the current physical theory. Are there examples in GR research that need fine mathematical distinction on the smoothness/construction of manifold structures? I don't know of a single example. For example let's say distinction on hausdorff vs. non-hausdorff etc... It seems we can always assume C∞C∞ and not even bother with fine distinctions that are of mathematical interests.

The problem you're referring to is the Navier Stokes equation and the Clay Award right? I think that's a mathematical problem, which is distinct from solving the statistical distribution turbulent flow which is a physics problem.

One analogy would be like saying one needs to know measure theory to do use integrals... not in practice it seems.

 

[vanhees71]

Outside the context of a course with instruction, I agree. I could see it being a useful text in a course with instruction, though, because the instructors can zero in on the key points and help the students to navigate the text.

Don't get me wrong: I think MTW is still one of the best books on GR and a mile stone in a new paradigm of textbook writing. I think it's just not so good as a first textbook, because you get too easily lost in all the details. When I first read it I found myself with all ten fingers in the book and jumping back and forth between different sections in "Track 1 and 2", etc. So first I found it a bit confusing, but after I had systematically studied Landau&Lifhitz vol. 2, I found MTW the ideal follow-up reading, including the introduction to the more modern coordinate free Cartan formulation of tensor calculus.

A more modern book in the same style is Thorne, Modern Classical Physics, which is a big review not only about general relativity but "general non-quantum physics", including Newtonian as well as general-relativistic treatment.

Another gem, but also rather for a second read, is Weinberg, Gravitation and Cosmology (1971). For me that's the book with right balance between the physical and geometric approach to the subject.

As a first book now I'd also recommend the new book by Adler, General Relativity and Cosmolgy, Springer. His older book (together with Bazin and and Schiffer) is also a good 2nd read.

 

[Nullstein]

Yes I mean the current physical theory. Are there examples in GR research that need fine mathematical distinction on the smoothness/construction of manifold structures?

Yes. E.g. the no-hair theorem has been proven only under the assumption that the manifold is analytic for example, but this is not a physical assumption, so it is an open question whether this also holds for manifolds that are merely smooth or $C^2$. The no-hair theorem plays a central role in the justification of more advanced concepts such as holography, so it is very important to study this in detail.

 

[jbergman]

Don't get me wrong: I think MTW is still one of the best books on GR and a mile stone in a new paradigm of textbook writing. I think it's just not so good as a first textbook, because you get too easily lost in all the details. When I first read it I found myself with all ten fingers in the book and jumping back and forth between different sections in "Track 1 and 2", etc. So first I found it a bit confusing, but after I had systematically studied Landau&Lifhitz vol. 2, I found MTW the ideal follow-up reading, including the introduction to the more modern coordinate free Cartan formulation of tensor calculus.

Professor Hughes says the same thing in his MIT OCW General Relativity lectures and Kip Thorne was his doctoral advisor! In short, he states that MTW is a great reference and covers certain topics better than other books, but that it isn't a good first book for GR. They use Carroll's book.

https://ocw.mit.edu/courses/physics/8-962-general-relativity-spring-2020/video-lectures/index.htm

 

[Demystifier]

TL;DR Summary: Why are general relativity texts written so much more formally than other physics texts.

Carroll's book ...

Some example I've seen:
1. All GR texts I've seen make a big deal about Hausdorff vs. non-Hausdorff or mention it.
2. They all talk about topological space, open/closed set to start defining manifolds.

The Carroll's book actually does not talk about such things.