Does Massless Spin 2 Imply General Relativity?

[MattRob]

Pretty straightforward title. Looking for recommendations. I've read Kip Thorne's Black Holes and Time Warps, Einstein's Relativity, The Special and General Theory, and a few other titles, just looking for recommendations on where to go next. I've only covered Integral calculus and linear algebra, so I might be able to glean some mathematics though most will be beyond me. I did very much enjoy all the qualitative/descriptive information I could get from a GR textbook I got from my university library, though, but life circumstances kept me from reading past the Sagnac Effect.

Any suggestions?

 

[robphy]

Some books to consider...

https://www.amazon.com/dp/020138423X/?tag=pfamazon01-20

https://www.amazon.com/dp/0198506562/?tag=pfamazon01-20

https://www.amazon.com/dp/1891389823/?tag=pfamazon01-20

 

[bcrowell]

What is your physics background?

 

[ShayanJ]

I suggest you read Hartle's and then Zee's. Then, if you want to learn more deeply, you may take a look at more advanced books like Carroll's or Padmanabhan's.
But you should be careful that learning GR is not just about studying GR. You should have good knowledge of other areas of physics too, the most important of which are classical mechanics(in all three formulations) and classical EM(advanced level).

 

[MattRob]

What is your physics background?

I suggest you read Hartle's and then Zee's. Then, if you want to learn more deeply, you may take a look at more advanced books like Carroll's or Padmanabhan's.
But you should be careful that learning GR is not just about studying GR. You should have good knowledge of other areas of physics too, the most important of which are classical mechanics(in all three formulations) and classical EM(advanced level).

I'm currently an undergraduate in Physics-Astronomy, coming into my Junior year. I've taken physics courses enough that I'm rather well-hearsed in classical Newtonian mechanics (I think?), and just a bit into EM.

Also I'm somewhat familiar with GR due to what I've mentioned in the first post. I did understand them.

 

[MattRob]

Perhaps I should also note that I have formally covered an overview of Special Relativity, as well.

 

[vanhees71]

GR is not as difficult as it looks in the beginning. You only must be willing to learn the true tensor calculus first. The good thing is that most GR books provide it in great detail. I'm not an expert in GR, but I've studied it just for (great) fun for myself and at the moment doing the recitations/exercises for my bosse's lecture on cosmology, so that I glanced through the literature a bit recently. I used Landau/Lifshitz Vol. II which is a great introduction. Recently I also found a very short one by Dirac, which is just great

P. A. M. Dirac, General Theory of Relativity, Wiley (1975)

Very good is also

Adler, R., Bazin, M., Schiffer, M.: Introduction to general relativity, 2 edition, McGraw-Hill Inc., 1975

Its great strength is to provide the details of the (sometimes quite technical) calculations as, e.g., to write down the Christoffel symbols and Ricci tensor for a given ansatz for the metric (in, e.g., deriving the Schwarzschild or Reissner-Nordström metric).

Also of course Weinberg's book is great, but perhaps not so much for a beginner

Weinberg, S.: Gravitation and Kosmologie, Wiley&Sons, Inc., 1972

If you want to learn more advanced and modern techniques like the Cartan calculus of differential forms etc., see

C.W. Misner, K. S. Thorne, J. A. Wheeler, Gravitation, W. H. Freeman & Comp., 1973

For an alternative point of view, deriving the relativistic laws of gravitation as for all other fundamental forces from the point of view of relativistic field theory, see

R. P. Feynman, F. B. Morinigo, W. G. Wagner, Feynman Lectures on Gravitation, Addison-Wesley 1995

Online you find tons of very good manuscript for free (legally!). Among them is the very comprehensive (and has really more the quality of a textbook than simply being lecture notes)

http://www.blau.itp.unibe.ch/GRLecturenotes.html

The only trouble with GR is that there are as many conventions as authors (even Landau Lifshitz changed the GR convention from one edition to the next). It starts with the metric, which can be east (mostly +) or west coast (mostly -) like as in SR. In GR the east-coast convention seems to be more favored, maybe because Einstein used the east-coast convention. Then both the Riemann tensor and the Ricci tensor comes with different overall sign conventions. This one just has to look up in the very beginning by checking how the commutator of covariant derivatives acts on a vector field and then how the so defined Riemann tensor is contracted to the Ricci tensor (either over the 1st and the 3rd or the 1st and the 4th index pair, which just differs in a sign). So thus, I'd be very careful to glance through a lot of books in your university library before starting learn GR to figure out, which one most probably likes best, because to switch from one book to another can be confusing just by the high probability that the sign conventions differ.

 

[ShayanJ]

For an alternative point of view, deriving the relativistic laws of gravitation as for all other fundamental forces from the point of view of relativistic field theory, see

R. P. Feynman, F. B. Morinigo, W. G. Wagner, Feynman Lectures on Gravitation, Addison-Wesley 1995

That's not completely reliable. See this paper!

 

[bcrowell]

If you've had single-variable calculus, SR, and mechanics, but not E&M or differential equations, then the perfect book for you should be Exploring Black Holes, by Taylor and Wheeler.

 

[MattRob]

GR is not as difficult as it looks in the beginning. You only must be willing to learn the true tensor calculus first. The good thing is that most GR books provide it in great detail. I'm not an expert in GR, but I've studied it just for (great) fun for myself and at the moment doing the recitations/exercises for my bosse's lecture on cosmology, so that I glanced through the literature a bit recently. I used Landau/Lifshitz Vol. II which is a great introduction. Recently I also found a very short one by Dirac, which is just great

P. A. M. Dirac, General Theory of Relativity, Wiley (1975)

Very good is also

Adler, R., Bazin, M., Schiffer, M.: Introduction to general relativity, 2 edition, McGraw-Hill Inc., 1975

Its great strength is to provide the details of the (sometimes quite technical) calculations as, e.g., to write down the Christoffel symbols and Ricci tensor for a given ansatz for the metric (in, e.g., deriving the Schwarzschild or Reissner-Nordström metric).

Also of course Weinberg's book is great, but perhaps not so much for a beginner

Weinberg, S.: Gravitation and Kosmologie, Wiley&Sons, Inc., 1972

If you want to learn more advanced and modern techniques like the Cartan calculus of differential forms etc., see

C.W. Misner, K. S. Thorne, J. A. Wheeler, Gravitation, W. H. Freeman & Comp., 1973

For an alternative point of view, deriving the relativistic laws of gravitation as for all other fundamental forces from the point of view of relativistic field theory, see

R. P. Feynman, F. B. Morinigo, W. G. Wagner, Feynman Lectures on Gravitation, Addison-Wesley 1995

Online you find tons of very good manuscript for free (legally!). Among them is the very comprehensive (and has really more the quality of a textbook than simply being lecture notes)

http://www.blau.itp.unibe.ch/GRLecturenotes.html

The only trouble with GR is that there are as many conventions as authors (even Landau Lifshitz changed the GR convention from one edition to the next). It starts with the metric, which can be east (mostly +) or west coast (mostly -) like as in SR. In GR the east-coast convention seems to be more favored, maybe because Einstein used the east-coast convention. Then both the Riemann tensor and the Ricci tensor comes with different overall sign conventions. This one just has to look up in the very beginning by checking how the commutator of covariant derivatives acts on a vector field and then how the so defined Riemann tensor is contracted to the Ricci tensor (either over the 1st and the 3rd or the 1st and the 4th index pair, which just differs in a sign). So thus, I'd be very careful to glance through a lot of books in your university library before starting learn GR to figure out, which one most probably likes best, because to switch from one book to another can be confusing just by the high probability that the sign conventions differ.

You've put a lot of work into compiling all this, just to be nice, thanks! Given that I lack diff eq, multivariable calculus and advanced EM, though, I wonder if this other recommendation:

If you've had single-variable calculus, SR, and mechanics, but not E&M or differential equations, then the perfect book for you should be Exploring Black Holes, by Taylor and Wheeler.

Might be the best place to start, especially given I've seen it mentioned twice, now:

Some books to consider...

https://www.amazon.com/dp/020138423X/?tag=pfamazon01-20

https://www.amazon.com/dp/0198506562/?tag=pfamazon01-20

https://www.amazon.com/dp/1891389823/?tag=pfamazon01-20

So on that note, I think I'll start there, with Exploring Black Holes: Introduction to General Relativity, then look over all the suggestions here and see where to go next. In the meantime, though, while the book is on its way, I'll probably look around those lecture notes since they're just right here/immediately/easily accessible.

Thanks for all the suggestions, everyone! I'm excited to get those pages in my hands and start reading. Summer's giving me lots of time and I hope to capitalize on that as much as possible.

That's not completely reliable. See this paper!

Thanks for the heads up.

 

[vanhees71]

Well, if you didn't have multivariable calculus, you must start with more basic topics first. You should have a good understanding of special relativity and the manifest covariant formulation of electrodynamics first. Also an understanding of Hamilton's principle for both relativistic point-particle mechanics and field theory (electromagnetics) is very helpful. Also the first steps towards relativistic (ideal) hydrodynamics is of use for GR (and particularly cosmology) is helpful. Most important is, as I said before, that you have a good understanding of tensor analysis, which is multivariable calculus + a formulation making the formalism covariant under certain transformations (Lorentz invariance for special relativity, general diffeomorphism invariance for general relativity).

 

[RyanH42]

But you should be careful that learning GR is not just about studying GR. You should have good knowledge of other areas of physics too, the most important of which are classical mechanics(in all three formulations) and classical EM(advanced level).

I asked the same question as MattRob https://www.physicsforums.com/threads/book-recommend-for-gr.820002

https://www.physicsforums.com/threads/book-recommend-for-gr.820002/).I am in high school and my only classical mechanic knowladge is Leonard Susskind Lectures. I don't want to learn GR immediately and deeply.So do I need really learn Classical Mechanics ?

Note:My GR book is Lieber https://www.amazon.com/dp/1589880447/?tag=pfamazon01-20
and calculus book is https://www.amazon.com/dp/0071615458/?tag=pfamazon01-20

 

[ShayanJ]

I asked the same question as MattRob https://www.physicsforums.com/threads/book-recommend-for-gr.820002

https://www.physicsforums.com/threads/book-recommend-for-gr.820002/).I am in high school and my only classical mechanic knowladge is Leonard Susskind Lectures. I don't want to learn GR immediately and deeply.So do I need really learn Classical Mechanics ?

Note:My GR book is Lieber https://www.amazon.com/dp/1589880447/?tag=pfamazon01-20
and calculus book is https://www.amazon.com/dp/0071615458/?tag=pfamazon01-20

You're not going to learn GR in the level that MattRob is going to learn it. Lieber is a much more elementary book than the ones suggested to MattRob.
So for now, I don't think my post applies to you. But after some time, when you want to read more advanced books, you will need to know advanced classical mechanics and EM.

 

[MattRob]

Just wanted to come back here and drop a thanks to those who recommended Exploring Black Holes: Introduction to General Relativity now that I can speak from having dug into it a little; It's been really great so far and it's thrilling to have GR problems on a level that I can work them! Roughly 2/5th the way through and I'm thoroughly enjoying it and also very much looking forward to the later chapters on the Kerr solution! Exploring the Schwarzschild has been extremely fascinating!

 

[atyy]

That's not completely reliable. See this paper!

I'm not sure either. How about http://arxiv.org/abs/hep-th/0007220 ?

"We investigate, in any spacetime dimension >=3, the problem of consistent couplings for a finite collection of massless, spin-2 fields described, in the free limit, by a sum of Pauli-Fierz actions. We show that there is no consistent (ghost-free) coupling, with at most two derivatives of the fields, that can mix the various "gravitons". In other words, there are no Yang-Mills-like spin-2 theories. The only possible deformations are given by a sum of individual Einstein-Hilbert actions."

 

[ShayanJ]

I'm not sure either. How about http://arxiv.org/abs/hep-th/0007220 ?

"We investigate, in any spacetime dimension >=3, the problem of consistent couplings for a finite collection of massless, spin-2 fields described, in the free limit, by a sum of Pauli-Fierz actions. We show that there is no consistent (ghost-free) coupling, with at most two derivatives of the fields, that can mix the various "gravitons". In other words, there are no Yang-Mills-like spin-2 theories. The only possible deformations are given by a sum of individual Einstein-Hilbert actions."

I'm just going to begin learning QFT, so I'm not sure about this. But as far as I know, ghosts arise when we try to quantize some classical theories. So this paper is actually about naive quantum gravity. Is this true?
But the paper by Padmanabhan is about classical gravity. So these two papers investigate the issue at different levels.
So it seems not only the non-geometrical view of classical gravity doesn't work as well as for other interactions, but also it makes things worse for quantum gravity.

 

[vanhees71]

Hm, from the abstract, this only excludes theories with more than one massless spin-2 fields, but not such with only one of such fields. Maybe, Feynman's approach is not rigorous, but I still think it's a valid heuristics to get convinced that the reinterpretation of the graviton field in terms of space-time geometry is pretty natural. Otherwise the honest way to introduce Einstein's ideas on gravity is to just state that space-time is described by a pseudo-Riemannian manifold with a pseudo-metric of signature (1,3) or (3,1), with the additional assumption that the equations of motion for the gravitational field should be generally covariant and of 2nd order. Then you are necessarily forced to make the kinetic part of the action proportional to the Ricci scalar $R$. This argument of course also implies the possibility of a non-zero cosmological constant, but you have no more additional freedom, but I think this is a pretty unconvincing way to introduce GR, because how doe you justify the assumption of this specific space-time structure?

You can also use the point of view that GR is derived from "gauging" Poincare symmetry (leading to the somewhat problematic non-compact gauge group GL(4)). This approach can be found in the textbook

P. Ramond, Quantum Field Theory: A modern Primer, 2nd edition, Westview Press

 

[ShayanJ]

Maybe, Feynman's approach is not rigorous, but I still think it's a valid heuristics to get convinced that the reinterpretation of the graviton field in terms of space-time geometry is pretty natural.

How can it be natural when its not correct?
Remember that Feynman claims the infinite series resulting from the procedure will give the EH action. But the paper I mentioned proves that this is impossible.
For me, its more natural to think gravity is a consequence of the spacetime structure.

but I think this is a pretty unconvincing way to introduce GR, because how doe you justify the assumption of this specific space-time structure?

Its not a problem. Its the same in all of physics, we're convinced this is how it is because it works so well.
In fact, even if the Feynman's approach was correct, it wouldn't be a more convincing approach. Because, after all, the fact he chose spin-2 is because other spins don't give satisfactory theories. And how GR developed is exactly the same, other approaches didn't work as well. This is how all physical theories are developed. "Guess $\rightarrow$ figure out the consequences of the guess $\rightarrow$ compare with observations".

 

[ShayanJ]

Hm, from the abstract, this only excludes theories with more than one massless spin-2 fields, but not such with only one of such fields. Maybe, Feynman's approach is not rigorous, but I still think it's a valid heuristics to get convinced that the reinterpretation of the graviton field in terms of space-time geometry is pretty natural. Otherwise the honest way to introduce Einstein's ideas on gravity is to just state that space-time is described by a pseudo-Riemannian manifold with a pseudo-metric of signature (1,3) or (3,1), with the additional assumption that the equations of motion for the gravitational field should be generally covariant and of 2nd order. Then you are necessarily forced to make the kinetic part of the action proportional to the Ricci scalar $R$. This argument of course also implies the possibility of a non-zero cosmological constant, but you have no more additional freedom, but I think this is a pretty unconvincing way to introduce GR, because how doe you justify the assumption of this specific space-time structure?

You can also use the point of view that GR is derived from "gauging" Poincare symmetry (leading to the somewhat problematic non-compact gauge group GL(4)). This approach can be found in the textbook

P. Ramond, Quantum Field Theory: A modern Primer, 2nd edition, Westview Press

But I think I should explain something here. Feynman's procedure doesn't reproduce the EH action but the part that it misses is exactly the part the we always get rid of when we want to vary the action(the boundary term) and find out the equations of motion. Also you can get the equations of motion directly with such an approach. So if you don't care about the whole Lagrangian and that boundary term and only want the equations of motion, then Feynman's procedure works(But there are still some issues. The paper explains them extensively.).
But the point is, that boundary term is important in some considerations like the Thermodynamics of horizons and so is essential to emergent gravity/spacetime approaches(which is Padmanabhan's favourite approach).

 

[atyy]

Hm, from the abstract, this only excludes theories with more than one massless spin-2 fields, but not such with only one of such fields.

Yes, I thought the point of the article was that it is in agreement with the idea that massless spin 2 implies GR. So perhaps Feynman's and other people's derivations are sloppy, but this article seems to say massless spin 2 (plus certain conditions which they state) implies GR or multiple non-interacting copies of GR, which is basically massless spin 2 implies GR.