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

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  • #26
bob012345
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It seems hopeless as highly educated intelligent people can't even agree on the math. Maybe GR is best left to professional mathematicians.
 
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PeterDonis
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highly educated intelligent people can't even agree on the math
I don't think there's any disagreement on the math. The disagreement in this thread has been over terminology--whether, for example, the Lorentz transformation can be properly described as a "tensor". Nobody is disagreeing on how to use the Lorentz transformation mathematically, or any other mathematical object.
 
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  • #28
PeterDonis
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Maybe GR is best left to professional mathematicians.
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.
 
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  • #29
caz
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You might want to reconsider that since very few prominent specialists in GR have been professional mathematicians.
Maybe that’s why the Earth is still blocking your view of Venus … :-p
 
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PeterDonis
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Maybe that’s why the Earth is still blocking your view of Venus … :-p
Yes, since de-modulation isn't working, perhaps I need to consider finding a black hole somewhere and using that to swallow the Earth...
 
  • #31
bob012345
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I don't think there's any disagreement on the math. The disagreement in this thread has been over terminology--whether, for example, the Lorentz transformation can be properly described as a "tensor". Nobody is disagreeing on how to use the Lorentz transformation mathematically, or any other mathematical object.
Well, maybe it's just an apparent disagreement then because I am so unfamiliar with it I can't even make the distinction yet. I'm starting with Sean Carroll's notes BTW and he even has an abridged version which is still not so clear. I'm really more interested in understanding the physics than the mathematics of GR.
 
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PeterDonis
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I'm really more interested in understanding the physics than the mathematics of GR.
Then you probably don't want to focus on the nuts and bolts of tensors, coordinate transformations, etc, since those are more for detailed calculations than for conceptual understanding.

Edwin Taylor's Exploring Black Holes might be helpful; it only treats a particular class of spacetimes (basically the ones we use to model things like the solar system and black holes), but it brings the math down to the level of ordinary algebra and calculus (by making use of well chosen coordinates and the symmetries of this class of spacetimes), and seems to me to focus more on physical understanding than on brute force calculation. It is available online here:

https://www.eftaylor.com/general.html
 
  • #33
bob012345
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Then you probably don't want to focus on the nuts and bolts of tensors, coordinate transformations, etc, since those are more for detailed calculations than for conceptual understanding.

Edwin Taylor's Exploring Black Holes might be helpful; it only treats a particular class of spacetimes (basically the ones we use to model things like the solar system and black holes), but it brings the math down to the level of ordinary algebra and calculus (by making use of well chosen coordinates and the symmetries of this class of spacetimes), and seems to me to focus more on physical understanding than on brute force calculation. It is available online here:

https://www.eftaylor.com/general.html
Thanks. I'm not totally giving up ever following the math by the way..
 
  • #34
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It's not that easy! You have ##(\eta^{\mu \nu})=\mathrm{diag}(1,-1,-1,-1)## (I'm in the west-coast camp, but there's no big difference when using the east-coast convention). An ##\mathbb{R}^{4 \times 4}##-matrix is called a Lorentz-transformation matrix if,
$${\Lambda^{\mu}}_{\rho} {\Lambda^{\nu}}_{\sigma} \eta_{\mu \nu}=\eta_{\mu \nu}.$$
In matrix notation (note that here the index positioning gets lost, so you have to keep in mind that the matrix ##\hat{\Lambda}## has a first upper and a second lower index while the matrix ##\hat{\eta}## as two lower indices) this reads
$$\hat{\Lambda}^{\text{T}} \hat{\eta} \hat{\Lambda}=\hat{\eta}.$$
Since ##\hat{\eta}^2=\hat{1}## we have
$$\hat{\eta} \hat{\Lambda}^{\text{T}} \hat{\eta}=\hat{\Lambda}^{-1}.$$
In index notation that reads restoring the correct index placement (note that also ##\hat{\eta}^{-1}=(\eta^{\mu \nu})=\hat{\eta}=(\eta_{\mu \nu})##)
$${(\hat{\Lambda}^{-1})^{\mu}}_{\nu} = \eta_{\nu \sigma} {\Lambda^{\sigma}}_{\rho} \eta^{\rho \mu}={\Lambda_{\nu}}^{\mu}.$$
Just wanted to clarify as this is still confusing to me. If you write in matrix vector notation, everything flows clearly, and if you write in index notation everything flows clearly too. The connection between the two is what's confusing. For example,

In matrix vector notation,

$$\eta = \Lambda^T \eta \Lambda \quad \rightarrow \quad 1 = \eta \Lambda^T \eta \Lambda \quad \rightarrow \quad \Lambda^{-1} = \eta \Lambda^T \eta \Lambda \Lambda^{-1} \quad \rightarrow \quad \Lambda^{-1} = \eta \Lambda^T \eta$$

In index notation,

$$\eta_{\rho\sigma} = (\Lambda^T)^{\mu\;}_{\;\rho} \Lambda^{\nu\;}_{\;\sigma} \eta_{\mu\nu} \quad \rightarrow \quad \delta^\alpha_\rho = (\Lambda^T)^{\mu\;}_{\;\rho} \Lambda^{\nu\;}_{\;\sigma} \eta_{\mu\nu} \eta^{\sigma\alpha} \quad \rightarrow \quad \delta^\alpha_\rho ((\Lambda^T)^{-1})^{\mu\;}_{\;\rho} = \Lambda^{\nu\;}_{\;\sigma} \eta_{\mu\nu} \eta^{\sigma\alpha} \quad \rightarrow \quad \delta^\alpha_\rho ((\Lambda^{-1})^T)^{\mu\;}_{\;\rho} = \Lambda^{\nu\;}_{\;\sigma} \eta_{\mu\nu} \eta^{\sigma\alpha} \quad \rightarrow \quad ((\Lambda^{-1})^T)^{\mu\;}_{\;\alpha} = \Lambda_{\mu\;}^{\;\alpha} \quad \rightarrow \quad (\Lambda^{-1})^{\alpha\;}_{\;\mu} = \Lambda_{\mu\;}^{\;\alpha}$$

The possible misconceptions here are,
1. Is it correct in index notation to write ##(\Lambda^T)^{\mu\;}_{\;\rho}## which is just the counterpart for the matrix vector notation? So that ##(\Lambda^T)^{\mu\;}_{\;\rho} = \Lambda^{\rho\;}_{\;\mu}##. This makes a lot of sense actually.
2. In the expression ##\;\delta^\alpha_\rho = (\Lambda^T)^{\mu\;}_{\;\rho} \Lambda^{\nu\;}_{\;\sigma} \eta_{\mu\nu} \eta^{\sigma\alpha}\;##, it is correct to just multiply both sides by the inverse of ##(\Lambda^T)^{\mu\;}_{\;\rho}## even there are indices present such that ##\; \delta^\alpha_\rho ((\Lambda^T)^{-1})^{\mu\;}_{\;\rho} = \Lambda^{\nu\;}_{\;\sigma} \eta_{\mu\nu} \eta^{\sigma\alpha} \;## right?
*For #1 I have not seen anybody write the transpose explicitly like I did above so I think we just follow the index placement obeying the transformation rules and make sure we have the corresponding matrix vector version in our head and just make sense of the resulting index version equation and say, "hey, we should transpose this matrix to have the correct row-column operation...". For #2 I believe even in index notations the symbol we use can also follow the matrix version as to not confuse the correspondence between the two versions right?
3. In @vanhees71 SR notes (Special Relativity by van Hees) in Appendix A.6, the steps are missing so maybe someone could fill in all the steps so as to lift this long-time confusion already. For example, in eq. A.6.3 it has a term ##\Lambda^{\mu\;}_{\;\rho}## which is the term when written in matrix vector notation, the term with the transpose ##\Lambda^T##, but then when he multiplied by the inverse both sides, the indices in the resulting equation in eq. A.6.4 just suddenly flipped (with consideration to ##\delta^\alpha_\rho##), i.e. ##(\Lambda^{-1})^{\alpha\;}_{\;\mu}##. However there are no step which indicated why that is so.

I hope these may be clarified.
 
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  • #35
stevendaryl
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I think that MTW is a wonderful book for learning General Relativity, if you have 20 years to spare.
 
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  • #36
vanhees71
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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.
 
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  • #37
bob012345
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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!
 
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  • #38
PeterDonis
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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.
 
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  • #39
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
 
  • #40
vanhees71
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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 learnt then from Landau and Lifshitz vol 2 :-).
 

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