discussion of Emmy Noether on BBC

[notFritzArgelander]

I happen to be one of those examples where I seem o be good at mathematics but I do not learn by rote obedience as some schools teach it. Confronted with those teachers and my learning suffered.
I was once actually told to stop asking about the real applications of calculus (grad and curl) as if I was expected to master a tool with no material to work it on.
Many times I struggled until I could grasp a real use of the math being taught.
Eventually I developed ways to teach myself the methods, and remastered differentials that way.

I did much the same process. After the minimum required by my undergrad major I never took a course from a mathematics department again. Much of the maths I learned after that was always in conjunction with physics.

[chasmanian]

thank you nFA.

this book might be of benefit to some of you guys:

https://www.barnesandnoble.com/w/a-most ... 1113785761

Amazon has the Kindle version for $3.99.
I tried posting a link to it, but for some reason, the link does not appear.

[notFritzArgelander]

thank you nFA.

this book might be of benefit to some of you guys:

https://www.barnesandnoble.com/w/a-most ... 1113785761

Amazon has the Kindle version for $3.99.
I tried posting a link to it, but for some reason, the link does not appear.

It looks intriguing.

[notFritzArgelander]

thank you nFA.

this book might be of benefit to some of you guys:

https://www.barnesandnoble.com/w/a-most ... 1113785761

Amazon has the Kindle version for $3.99.
I tried posting a link to it, but for some reason, the link does not appear.

It looks intriguing.

In fact so much so that I got the kindle version and went directly to the discussion of energy conservation in GR.

The author's answer to the question of whether or not energy momentum is conserved in GR is "maybe".

On the "maybe" side he cites John Baez who I mentioned in post #12 of the baby universe thread viewtopic.php?f=74&t=19138

On the "maybe not" side he cites Sean Carroll. He definitely does not represent the "certainly!" POV that I (following Landau and Lifschitz) prefer to defend.

I've skimmed the book and find it highly readable and recommendable. It adopts mostly a geometric POV which I like but there is an abundance of the algebraic treatment so you'll likely never go astro wondering how to compute something. He references the Bernard Schutz books that I like which are also strongly geometric but with enough algebraic content to facilitate computing answers.
If you need more of a maths refresher before tackling Schutz this is a good way to go.

PS I'd like to add that this, like some other books, makes qualms that while the matter-fields stress energy tensor can provide good local definition of conservation laws (differential current conservation) the gravitational energy does not have a good local representation. My view is that's OK, since the energy in the gravitational field (aka spacetime curvature) ISN'T local. Hence the pseudo tensor nature and the need to integrate over volumes to make sure it makes sense. The sum of the two (tensor plus pseudo tensor) DOES have a local conservation current.

[GCoyote]

Thanks for quick review, added to the reading list!

[notFritzArgelander]

Thanks for quick review, added to the reading list!

I've started a thread for discussing the book if there should be a need. It would be nice to keep discussion in one place IMO.

[chasmanian]

thank you very much nFA, for all you posted.and starting a thread for the book.

I'm copying and pasting the Top Review for the book on Amazon, in the event that it is helpful and entertaining and all that jazz.

the reviewer's user name is "I am falling in love with my life". (I love that idea:))

Top reviews from the United States
I am falling in love with my life
4.0 out of 5 stars My first general relativity book
Reviewed in the United States on June 1, 2016
Verified Purchase
(This review is for the second edition.)

When I tried to study general relativity on my own and look for an appropriate book, I realized that I had already three general relativity books.

1. Gravitation by Misner, Thorne, and Wheeler
2. Introducing Einstein's Relativity by D'inverno
3. Introduction to the Theory of Relativity by Bergmann

Although I majored physics in university and have a doctoral degree in mathematics, the above three books were difficult to me. More precisely, in my opinion, any good book should be read thinking "so... so... why? ... so... so... ok. I see. So..." But the above books seemed to force me to accept many things without clear understanding. Relativity is a well-established part of physics, so in principle, it is possible to give reasonable explanations to any statement in relativity books. But for people who learn such a discipline with critical or logical spirit, easy and detailed explanations are essential.

Accessibility is the best merit of the book, A Most Incomprehensible Thing by Peter Collier. In Preface, the author says,

This book is written for the general reader who wishes to gain a basic but working understanding of the mathematics of Einstein's theory of relativity, one of the cornerstones of modern physics... I never found my ideal volume but instead had to make use of many different sources... The book I had in mind would assume little prior mathematical knowledge (even less than my own patchy sixth-form/high school maths if it were to be suitable for the general reader)...

Some days ago, I finished the book. I was satisfied with the book as a whole, and I felt that I learned somethings, and I am more prepared with more advanced books in general relativity like Relativity, Gravitation, and Cosmology by Chen and Hartle's famous book on introduction to relativity.

But I have to tell you bad news. This book is not easy also. If an instructor specializing in general relativity teaches with this book, then even a freshman could understand it. But as a self-study book, I think this book is difficult. If you can successfully choose topics that you need in the book, focusing important parts, passing unimportant parts, and coming back to necessary parts again and again, finding some typos or errors, not expecting to understand everything in the book, then you may be able to finish the book on your own and be satisfied with the book. But if you want to thoroughly understand the book, you must know a lot of things before reading the book. To do that, you must have taken one year course of freshmen level calculus and physics including somewhat detailed chapters on special relativity. If you are not already familiar with special relativity at least the level of the popular science books, you may have difficulty in understanding the topics related to special relativity in the book. In particular, it would be good to know about time dilation and length contraction in advance.

Besides that, you may know that relativity regards space-time as a differentiable manifold and relativity is formulated with the language of tensors. In the book, there are introductory chapters on differentiable manifold and tensors. But they are insufficient. If you want to comprehensively understand differentiable manifolds and tensors, you must study independently with some sources like the followings.

Regarding tangent vectors,

1. Related sections in Differential Geometry of Curves and Surfaces by do Carmo

2. Related sections in Elementary Differential Geometry by O'Neill

To understand differentiable manifold and tensor, you must know the basics of differential geometry and to do that, you must be familiar with the concept of tangent vector. Differential geometry books deal with tangent vectors in detail. If you have taken one year course of calculus and one-semester course of linear algebra, then you would have no difficulty in reading differential geometry books.

Regarding tensors,

1. Chapter 5- Multiple Integrals, Chapter 6- Vector Analysis, and Chapter 10- Coordinate Transformations in Mathematical Methods in the Physical Sciences by Boas. Try to understand what a line element is and topics related to coordinate transformations.

2. Quick Introduction to Tensor Analysis by Sharipov. You can get this on the internet. If you are familiar only with calculus, linear algebra, then you can read the book. Within only 46 pages, it contains the essentials you must know to understand the tensor formulations in relativity. Exercises are really helpful.

Regarding covariant derivatives (Connection),

1. Differential Geometry of Curves and Surfaces by do Carmo

General relativity says that energy makes the space-time curved. What is the meaning of 'a differentiable manifold is curved'? If a covariant derivative is given to the space, then we can say about how much the space is curved. In my opinion, do Carmo's book mentioned above is the best to learn covariant derivative among differential geometry books that I've ever seen.

2. Chapter 2 - Riemannian Geometry in Morse Theory by Milnor

Do Carmo's book deals with covariant derivative of a 2-dimensional manifold embedded in R^3. But to understand general relativity, you must go further. We should know how to define covariant derivative on an abstract differentiable manifold, not only on a 2-dimensional manifold embedded in R^3. The mentioned chapter of Milnor's book is about only 20 pages, but contains all the basic essentials of covariant derivative in the generalized sense, also with a concise introduction to Riemann curvature tensor that is also needed to understand general relativity.

3. Riemannian Geometry by do Carmo

If you can be satisfied with the two books mentioned, it would be good. But if you want to understand what a differentiable manifold is and the concepts like metrics, geodesics, more on covariant derivatives, Riemann curvature tensor, the relationship between the Riemann curvature tensor and the Gaussian curvature (this would be the curvature you might think when you hear that a surface is curved), this book is recommended.

4. An Introduction to Differentiable Manifolds and Riemannian Geometry by Boothby

If you can be satisfied with the three books mentioned, it would be good. But if you think that do Carmo's Riemannian Geometry is not logically so reasonable in some parts, then you can consult the classic Boothby's book. For example, covariant derivative is a global concept. But to prove properties of covariant derivative, the author uses local concepts. It's possible, but there is no enough explanation of doing that. Boothby explains that in detail. Besides that, if you want to more comprehensively understand the important concepts stated above, that is, tangent vectors, metrics, geodesics, covariant derivatives, Riemann curvature tensor, the relationship between the Riemann curvature tensor and the Gaussian curvature, then Boothby's book would be perfectly helpful.

I'd like to reemphasize the best merit of the book. There were huge level-gaps between the level of special relativity in the books for freshmen and of general relativity in the existing books like D'inverno's or Ohanian's for undergraduate students. The author exactly pointed out the gap and succeeded at large.

Here are detailed points and my questions.

1. If a real beginner would read the book, then he wouldn't understand the equation at page 96 regarding the divergence of a gravitational field. The appearance of the mass density was abrupt. He would also had difficulty in understanding Section 3.3.3 dealing with introducing the second observer's space-time diagram in the first observer's space-time diagram not because the content is hard to understand, but because he had no motivation to follow the not-so much interesting long arguments. And he would fail to understand the Ricci tensor (and so the Ricci scalar) because of typos at several places. If he already knows Ricci tensor, these typos would be trivial things. But if he is really a novice in relativity and differential geometry, this kind of typos can be a big obstacle to read the book.

2. If a real beginner would read the book, then he would fail to understand the meaning that Schwarzschild metric is a solution of the Einstein field equation. It would be better if the author had explained more explicitly that a solution of the Einstein field equation is a metric and the Schwarzschild metric is a solution under some conditions. As for the latter, the book is good, but as for the former, not so good. The derivation of Einstein field equation was good. But the derivation of the Lorentz transformation was not good. In my opinion, it would be better if the author postulated them as axioms. The explanation of the Einstein field equation was appropriate, but as for the Lorentz transformation, it was too lengthy.

3. About the Friedmann equations, I feel the lack of a detailed explanation of how the Friedmann equations are derived from the Einstein field equations. I don't want to see and follow every mathematical detail, but want to see the logic.

4. In page 151, it says that the scalar product of two four vectors is invariant under Lorentz transformations. It's true, but I think that detailed explanations are needed.

5. In page 152, it is said that relativistic momentum is conserved in all inertial frames. What's the meaning of this?

6. In page 207, it says that small regions of spacetime are locally flat. Does locally flat mean that the Riemann curvature tensor is 0? So it is not curved? But Einstein field equation says that the energy makes spacetime curved. It's absurd.

7. In page 221, it says "We mentioned earlier that, using dummy index, the covariant derivative of a tensor field is equivalent to the divergence of that field." But I cannot find any relevant remarks about it in earlier pages.

8. In page 249, it says "We've actually slipped an assumption here: that the coordinate time difference between two events in Schwarzschild spacetime is the same as that recorded by a distant observer." But I cannot understand where the assumption is used.

9. In page 301, the density of radiation is proportional to the inverse of the fourth power of the scale factor. But the argument seems insufficient.

As my first general relativity book, I am satisfied with the book as a whole, and I thank the author. Now I am reading my (possibly) second general relativity book.

[notFritzArgelander]

Oh, well, there's no pleasing everybody. Looking at what he does and does not like.... I think we have different tastes. In particular it comes as somewhat of a shock that a maths PhD would find Misner, Thorne and Wheeler difficult.

[chasmanian]

thank you nFA.

hmmm, the exception to not pleasing everybody,
perhaps:

[notFritzArgelander]

thank you nFA.

hmmm, the exception to not pleasing everybody,
perhaps:

Sara Lee is too sweet for me.

[GCoyote]

He's not wrong in one regard.
"If he already knows Ricci tensor, these typos would be trivial things. But if he is really a novice in relativity and differential geometry, this kind of typos can be a big obstacle to read the book."

I find a good deal of material for the popular audience is no longer given the scrutiny it really needs. Publishers apparently expect new submissios to be ready to print. They don't want to spend the money for a subject matter expert to check and edit a manuscript resulting in more errors slipping through.

[Gmetric]

thank you very much nFA, for all you posted.and starting a thread for the book.

I'm copying and pasting the Top Review for the book on Amazon, in the event that it is helpful and entertaining and all that jazz.

the reviewer's user name is "I am falling in love with my life". (I love that idea:))

Top reviews from the United States
I am falling in love with my life
4.0 out of 5 stars My first general relativity book
Reviewed in the United States on June 1, 2016
Verified Purchase
(This review is for the second edition.)

When I tried to study general relativity on my own and look for an appropriate book, I realized that I had already three general relativity books.

1. Gravitation by Misner, Thorne, and Wheeler
2. Introducing Einstein's Relativity by D'inverno
3. Introduction to the Theory of Relativity by Bergmann

Although I majored physics in university and have a doctoral degree in mathematics, the above three books were difficult to me. More precisely, in my opinion, any good book should be read thinking "so... so... why? ... so... so... ok. I see. So..." But the above books seemed to force me to accept many things without clear understanding. Relativity is a well-established part of physics, so in principle, it is possible to give reasonable explanations to any statement in relativity books. But for people who learn such a discipline with critical or logical spirit, easy and detailed explanations are essential.

Accessibility is the best merit of the book, A Most Incomprehensible Thing by Peter Collier. In Preface, the author says,

This book is written for the general reader who wishes to gain a basic but working understanding of the mathematics of Einstein's theory of relativity, one of the cornerstones of modern physics... I never found my ideal volume but instead had to make use of many different sources... The book I had in mind would assume little prior mathematical knowledge (even less than my own patchy sixth-form/high school maths if it were to be suitable for the general reader)...

Some days ago, I finished the book. I was satisfied with the book as a whole, and I felt that I learned somethings, and I am more prepared with more advanced books in general relativity like Relativity, Gravitation, and Cosmology by Chen and Hartle's famous book on introduction to relativity.

But I have to tell you bad news. This book is not easy also. If an instructor specializing in general relativity teaches with this book, then even a freshman could understand it. But as a self-study book, I think this book is difficult. If you can successfully choose topics that you need in the book, focusing important parts, passing unimportant parts, and coming back to necessary parts again and again, finding some typos or errors, not expecting to understand everything in the book, then you may be able to finish the book on your own and be satisfied with the book. But if you want to thoroughly understand the book, you must know a lot of things before reading the book. To do that, you must have taken one year course of freshmen level calculus and physics including somewhat detailed chapters on special relativity. If you are not already familiar with special relativity at least the level of the popular science books, you may have difficulty in understanding the topics related to special relativity in the book. In particular, it would be good to know about time dilation and length contraction in advance.

Besides that, you may know that relativity regards space-time as a differentiable manifold and relativity is formulated with the language of tensors. In the book, there are introductory chapters on differentiable manifold and tensors. But they are insufficient. If you want to comprehensively understand differentiable manifolds and tensors, you must study independently with some sources like the followings.

Regarding tangent vectors,

1. Related sections in Differential Geometry of Curves and Surfaces by do Carmo

2. Related sections in Elementary Differential Geometry by O'Neill

To understand differentiable manifold and tensor, you must know the basics of differential geometry and to do that, you must be familiar with the concept of tangent vector. Differential geometry books deal with tangent vectors in detail. If you have taken one year course of calculus and one-semester course of linear algebra, then you would have no difficulty in reading differential geometry books.

Regarding tensors,

1. Chapter 5- Multiple Integrals, Chapter 6- Vector Analysis, and Chapter 10- Coordinate Transformations in Mathematical Methods in the Physical Sciences by Boas. Try to understand what a line element is and topics related to coordinate transformations.

2. Quick Introduction to Tensor Analysis by Sharipov. You can get this on the internet. If you are familiar only with calculus, linear algebra, then you can read the book. Within only 46 pages, it contains the essentials you must know to understand the tensor formulations in relativity. Exercises are really helpful.

Regarding covariant derivatives (Connection),

1. Differential Geometry of Curves and Surfaces by do Carmo

General relativity says that energy makes the space-time curved. What is the meaning of 'a differentiable manifold is curved'? If a covariant derivative is given to the space, then we can say about how much the space is curved. In my opinion, do Carmo's book mentioned above is the best to learn covariant derivative among differential geometry books that I've ever seen.

2. Chapter 2 - Riemannian Geometry in Morse Theory by Milnor

Do Carmo's book deals with covariant derivative of a 2-dimensional manifold embedded in R^3. But to understand general relativity, you must go further. We should know how to define covariant derivative on an abstract differentiable manifold, not only on a 2-dimensional manifold embedded in R^3. The mentioned chapter of Milnor's book is about only 20 pages, but contains all the basic essentials of covariant derivative in the generalized sense, also with a concise introduction to Riemann curvature tensor that is also needed to understand general relativity.

3. Riemannian Geometry by do Carmo

If you can be satisfied with the two books mentioned, it would be good. But if you want to understand what a differentiable manifold is and the concepts like metrics, geodesics, more on covariant derivatives, Riemann curvature tensor, the relationship between the Riemann curvature tensor and the Gaussian curvature (this would be the curvature you might think when you hear that a surface is curved), this book is recommended.

4. An Introduction to Differentiable Manifolds and Riemannian Geometry by Boothby

If you can be satisfied with the three books mentioned, it would be good. But if you think that do Carmo's Riemannian Geometry is not logically so reasonable in some parts, then you can consult the classic Boothby's book. For example, covariant derivative is a global concept. But to prove properties of covariant derivative, the author uses local concepts. It's possible, but there is no enough explanation of doing that. Boothby explains that in detail. Besides that, if you want to more comprehensively understand the important concepts stated above, that is, tangent vectors, metrics, geodesics, covariant derivatives, Riemann curvature tensor, the relationship between the Riemann curvature tensor and the Gaussian curvature, then Boothby's book would be perfectly helpful.

I'd like to reemphasize the best merit of the book. There were huge level-gaps between the level of special relativity in the books for freshmen and of general relativity in the existing books like D'inverno's or Ohanian's for undergraduate students. The author exactly pointed out the gap and succeeded at large.

Here are detailed points and my questions.

1. If a real beginner would read the book, then he wouldn't understand the equation at page 96 regarding the divergence of a gravitational field. The appearance of the mass density was abrupt. He would also had difficulty in understanding Section 3.3.3 dealing with introducing the second observer's space-time diagram in the first observer's space-time diagram not because the content is hard to understand, but because he had no motivation to follow the not-so much interesting long arguments. And he would fail to understand the Ricci tensor (and so the Ricci scalar) because of typos at several places. If he already knows Ricci tensor, these typos would be trivial things. But if he is really a novice in relativity and differential geometry, this kind of typos can be a big obstacle to read the book.

2. If a real beginner would read the book, then he would fail to understand the meaning that Schwarzschild metric is a solution of the Einstein field equation. It would be better if the author had explained more explicitly that a solution of the Einstein field equation is a metric and the Schwarzschild metric is a solution under some conditions. As for the latter, the book is good, but as for the former, not so good. The derivation of Einstein field equation was good. But the derivation of the Lorentz transformation was not good. In my opinion, it would be better if the author postulated them as axioms. The explanation of the Einstein field equation was appropriate, but as for the Lorentz transformation, it was too lengthy.

3. About the Friedmann equations, I feel the lack of a detailed explanation of how the Friedmann equations are derived from the Einstein field equations. I don't want to see and follow every mathematical detail, but want to see the logic.

4. In page 151, it says that the scalar product of two four vectors is invariant under Lorentz transformations. It's true, but I think that detailed explanations are needed.

5. In page 152, it is said that relativistic momentum is conserved in all inertial frames. What's the meaning of this?

6. In page 207, it says that small regions of spacetime are locally flat. Does locally flat mean that the Riemann curvature tensor is 0? So it is not curved? But Einstein field equation says that the energy makes spacetime curved. It's absurd.

7. In page 221, it says "We mentioned earlier that, using dummy index, the covariant derivative of a tensor field is equivalent to the divergence of that field." But I cannot find any relevant remarks about it in earlier pages.

8. In page 249, it says "We've actually slipped an assumption here: that the coordinate time difference between two events in Schwarzschild spacetime is the same as that recorded by a distant observer." But I cannot understand where the assumption is used.

9. In page 301, the density of radiation is proportional to the inverse of the fourth power of the scale factor. But the argument seems insufficient.

As my first general relativity book, I am satisfied with the book as a whole, and I thank the author. Now I am reading my (possibly) second general relativity book.

Nice one Chas, some important points that you raise here but I have a question. I've read many of your previous posts and questions to nFA but did I miss the post where you mentioned that you were a physics major and a maths PhD?

[notFritzArgelander]

Oh, well, there's no pleasing everybody. Looking at what he does and does not like.... I think we have different tastes. In particular it comes as somewhat of a shock that a maths PhD would find Misner, Thorne and Wheeler difficult.

Which shows how much taste variation there is: more than we, specifically I, can imagine. In this case it is likely the reviewer is accustomed to the straight and narrow path and is uncomfortable with the gestures in the direction of geometry that MTW provides.

[notFritzArgelander]

Somewhat off topic it happened that this article on Noether's life, emphasizing her contributions to algebra, appeared in my feed today.

https://theconversation.com/emmy-noethe ... 1626441662

[chasmanian]

thank you nFA.

hmmm, the exception to not pleasing everybody,
perhaps:

Sara Lee is too sweet for me.

probably me too!!

[chasmanian]

"Nice one Chas, some important points that you raise here but I have a question. I've read many of your previous posts and questions to nFA but did I miss the post where you mentioned that you were a physics major and a maths PhD?"

I wish!! hahaha

although someone did call me a genius today.
ok, it was my Mom.
boy, did I laugh at that one!! har har har de har har and all that.

that post was a review of the book by someone on Amazon, user name
I am falling in love with my life (indeed a splendiferous paradisiacal thought)

but this is truly an extra special day for me,
to be assumed to be a Maths PhD, physics major and genius.

and now that I think of it, to boot, someone compared me to Paul McCartney.

oh, if they could only see me now. chortle guffaw chuckle etc

[Gmetric]

"Nice one Chas, some important points that you raise here but I have a question. I've read many of your previous posts and questions to nFA but did I miss the post where you mentioned that you were a physics major and a maths PhD?"

I wish!! hahaha

although someone did call me a genius today.
ok, it was my Mom.
boy, did I laugh at that one!! har har har de har har and all that.

that post was a review of the book by someone on Amazon, user name
I am falling in love with my life (indeed a splendiferous paradisiacal thought)

but this is truly an extra special day for me,
to be assumed to be a Maths PhD, physics major and genius.

and now that I think of it, to boot, someone compared me to Paul McCartney.

oh, if they could only see me now. chortle guffaw chuckle etc

I'm sure your mama was right

[chasmanian]

you're very kind.
thank you.