- #71
Chris Hillman
Science Advisor
- 2,355
- 10
A stern warning about "on-line books", &c.
Don't be distressed when you get it and see that it is written in the stark typefaces popular for textbooks in dark days of the Cold War, complete with the "duck and cover" cognitive dissonance resulting from a plethora of crude cartoons. If you've ever seen Soviet textbooks from that era, you know that the combatants had an unpleasant tendency to mimic each other's worse characteristics, and not only in textbook publishing. (There was a long and fascinating New Yorker piece on the bizarre saga of cold war textbook propaganda published some 15 years ago.) Anyway, despite this rather gothic appearance, it's actually a great book and very friendly. In fact, so friendly one might easily underestimate the depth of what it offers the reader!
OK, great, probably won't actually do you any HARM to see this. My only concern is that students can get the impression that this is hard subject when they see there is an entire chapter late in the book...
Einstein didn't invent "tensors", or more properly, "multilinear operators" and in particular, bilinear forms. These concepts were used much earlier by generations of mathematicians, including Gauss, Lagrange, Hamilton, Cayley, Sylvester, Frobenius, Riemann, Ricci-Curbastro, and Levi-Civita. The term "tensor" was introduced by Hamilton, but "tensor analysis" in the sense of index gymnastics is due to the Italian school, especially Ricci and his student Levi-Civita. Tullio Levi-Civita was a contemporary of Einstein and together with other leading mathematicians, including Elie Cartan, David Hilbert, and Hermann Weyl, produced most of the first known solutions of the EFE in the years 1915-1925.
In principle, Einstein was exposed to the Riemannian geometry at the Polytechnic in Zurich, but apparently he cut most of his classes and relied on the meticulous notes of his friend Marcel Grossmann at exam time! I would NOT recommend following his example in this respect, by the way, and Einstein himself said pretty much the same thing in his later years. Anyway, it was probably Grossmann who first told Einstein that the mathematical foundation needed for the relativistic classical field theory of gravitation he began searching for circa 1913 was Riemannian geometry, a then arcane subject for which no textbook existed. Grossmann tried to learn it (from Levi-Civita) so that he could teach it to Einstein, but this was not, as they say, his field, and he found it tough sledding, and the lack of good textbooks to study in fact led Einstein and Grossmann to make some very serious errors which blocked their progress. Fortunately, by 1915 Einstein was in close contact with Klein's school at Goettingen, where during several visits he benefited from conversations with Hilbert, Minkowski, and Noether (in particular).
As an aside: even mathematicians may not fully appreciate the extent to which invariant theory and algebraic geometry, as well as differential geometry, played a key role in the final stages of the discovery of gtr, with the input of Hilbert and Noether. A few years later, in the early 1920s, Cartan and Weyl also became involved in the early development of gtr. With the direct involvement of Hilbert, Cartan, and Weyl, Einstein had the assistance of (arguably) the three leading mathematicians in the world, and three of the greatest mathematicians of all time. There just might be a contemporary lesson here: for many decades, the leading mathematicians showed far more interest in gtr than did the leading physicists. I suspect that subjects like string theory and higher dimensional categories may be of greater interest to mathematicians than physicists for many decades, until appropriate applications begin to emerge or physical theories become testable.
Anyway, the point I am somehow trying to express here is that since you haven't yet mastered gtr, you can't possibly appreciate what is most important to learn as background. I am trying to tell you that of all the things you might want to brush up on, "tensor analysis" is the least important topic I can think of. Much more important to read up on linear operators and their matrix representations, plus vector space bases and change of basis, plus algebraic invariants on linear operators like characteristic polynomials and their roots (the "eigenvalues" of the operator), if you want to be systematic--- these things are more related to the algebraic underpinnings of the subject.
No. This would be like saying that "vectors are invariant". You might have meant that "tensor EQUATIONS are invariant under diffeomorphisms" (true), but the components of vectors and tensors are certainly not invariant, not even under rotations (a simple special case of diffeomorphisms).
I'm going to stop yakking about math now, since I am with daverz on a crucial pedagogical point: physical intuition is more important for physics students. You should listen to me when I say that, because I was trained as a mathematician, not a physicist :-/ so this judgement does not reflect narrow-minded professional parochialism.
But I feel I must stop you when you say this:
Sigh... "on-line book", eh? Since you didn't give any other information, I have no idea who wrote this "book" or whether the author has a clue what he is talking about; if so, your description of what the author wrote must have been somewhat mangled.
Aeroboyo, you should always be very careful about what you find on-line (including this forum, although I am confident that you have gotten so far some good advice here).
At least until very recently, textbooks are much MUCH more carefully vetted, in many ways: they are almost always written by tenured faculty at respectable universities, who have pursued a successful research career; this weeds out almost all cranks right there. In addition, the best academic publishers obtain extensive referee reports from third party experts (other professors at other universities) on textbooks, and often hire still more professors to try out a new textbook in their own classrooms, and may hire eagle eyed graduate students to do all the exercises to check for errors. Standard physics textbooks like Taylor and Wheeler, for example, have been studied by generations of smart students who have gone on to successful academic careers, so they have been gone over line by line with extraordinary care.
In contrast, "on-line books" have probably been read by, at best, their author, who probably has not even caught the obvious typographical errors (like misspellings), much less easily overlooked sign errors, much less subtle conceptual errors. Indeed, the author might even be totally clueless, particularly if he has no academic training whatever (although academic training is no guarantee that a given author is credible or even honest).
OK, you probably realized all this, but I think it needs to be said nonetheless.
Chris Hillman
aeroboyo said:
Don't be distressed when you get it and see that it is written in the stark typefaces popular for textbooks in dark days of the Cold War, complete with the "duck and cover" cognitive dissonance resulting from a plethora of crude cartoons. If you've ever seen Soviet textbooks from that era, you know that the combatants had an unpleasant tendency to mimic each other's worse characteristics, and not only in textbook publishing. (There was a long and fascinating New Yorker piece on the bizarre saga of cold war textbook propaganda published some 15 years ago.) Anyway, despite this rather gothic appearance, it's actually a great book and very friendly. In fact, so friendly one might easily underestimate the depth of what it offers the reader!
aeroboyo said:
OK, great, probably won't actually do you any HARM to see this. My only concern is that students can get the impression that this is hard subject when they see there is an entire chapter late in the book...
aeroboyo said:
Einstein didn't invent "tensors", or more properly, "multilinear operators" and in particular, bilinear forms. These concepts were used much earlier by generations of mathematicians, including Gauss, Lagrange, Hamilton, Cayley, Sylvester, Frobenius, Riemann, Ricci-Curbastro, and Levi-Civita. The term "tensor" was introduced by Hamilton, but "tensor analysis" in the sense of index gymnastics is due to the Italian school, especially Ricci and his student Levi-Civita. Tullio Levi-Civita was a contemporary of Einstein and together with other leading mathematicians, including Elie Cartan, David Hilbert, and Hermann Weyl, produced most of the first known solutions of the EFE in the years 1915-1925.
In principle, Einstein was exposed to the Riemannian geometry at the Polytechnic in Zurich, but apparently he cut most of his classes and relied on the meticulous notes of his friend Marcel Grossmann at exam time! I would NOT recommend following his example in this respect, by the way, and Einstein himself said pretty much the same thing in his later years. Anyway, it was probably Grossmann who first told Einstein that the mathematical foundation needed for the relativistic classical field theory of gravitation he began searching for circa 1913 was Riemannian geometry, a then arcane subject for which no textbook existed. Grossmann tried to learn it (from Levi-Civita) so that he could teach it to Einstein, but this was not, as they say, his field, and he found it tough sledding, and the lack of good textbooks to study in fact led Einstein and Grossmann to make some very serious errors which blocked their progress. Fortunately, by 1915 Einstein was in close contact with Klein's school at Goettingen, where during several visits he benefited from conversations with Hilbert, Minkowski, and Noether (in particular).
As an aside: even mathematicians may not fully appreciate the extent to which invariant theory and algebraic geometry, as well as differential geometry, played a key role in the final stages of the discovery of gtr, with the input of Hilbert and Noether. A few years later, in the early 1920s, Cartan and Weyl also became involved in the early development of gtr. With the direct involvement of Hilbert, Cartan, and Weyl, Einstein had the assistance of (arguably) the three leading mathematicians in the world, and three of the greatest mathematicians of all time. There just might be a contemporary lesson here: for many decades, the leading mathematicians showed far more interest in gtr than did the leading physicists. I suspect that subjects like string theory and higher dimensional categories may be of greater interest to mathematicians than physicists for many decades, until appropriate applications begin to emerge or physical theories become testable.
Anyway, the point I am somehow trying to express here is that since you haven't yet mastered gtr, you can't possibly appreciate what is most important to learn as background. I am trying to tell you that of all the things you might want to brush up on, "tensor analysis" is the least important topic I can think of. Much more important to read up on linear operators and their matrix representations, plus vector space bases and change of basis, plus algebraic invariants on linear operators like characteristic polynomials and their roots (the "eigenvalues" of the operator), if you want to be systematic--- these things are more related to the algebraic underpinnings of the subject.
aeroboyo said:
No. This would be like saying that "vectors are invariant". You might have meant that "tensor EQUATIONS are invariant under diffeomorphisms" (true), but the components of vectors and tensors are certainly not invariant, not even under rotations (a simple special case of diffeomorphisms).
I'm going to stop yakking about math now, since I am with daverz on a crucial pedagogical point: physical intuition is more important for physics students. You should listen to me when I say that, because I was trained as a mathematician, not a physicist :-/ so this judgement does not reflect narrow-minded professional parochialism.
But I feel I must stop you when you say this:
aeroboyo said:
Sigh... "on-line book", eh? Since you didn't give any other information, I have no idea who wrote this "book" or whether the author has a clue what he is talking about; if so, your description of what the author wrote must have been somewhat mangled.
Aeroboyo, you should always be very careful about what you find on-line (including this forum, although I am confident that you have gotten so far some good advice here).
At least until very recently, textbooks are much MUCH more carefully vetted, in many ways: they are almost always written by tenured faculty at respectable universities, who have pursued a successful research career; this weeds out almost all cranks right there. In addition, the best academic publishers obtain extensive referee reports from third party experts (other professors at other universities) on textbooks, and often hire still more professors to try out a new textbook in their own classrooms, and may hire eagle eyed graduate students to do all the exercises to check for errors. Standard physics textbooks like Taylor and Wheeler, for example, have been studied by generations of smart students who have gone on to successful academic careers, so they have been gone over line by line with extraordinary care.
In contrast, "on-line books" have probably been read by, at best, their author, who probably has not even caught the obvious typographical errors (like misspellings), much less easily overlooked sign errors, much less subtle conceptual errors. Indeed, the author might even be totally clueless, particularly if he has no academic training whatever (although academic training is no guarantee that a given author is credible or even honest).
OK, you probably realized all this, but I think it needs to be said nonetheless.
Chris Hillman
Last edited:
