- #26

bob012345

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*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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- Thread starter bob012345
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- #26

bob012345

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- #27

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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.highly educated intelligent people can't even agree on the math

- #28

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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.Maybe GR is best left to professional mathematicians.

- #29

caz

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Maybe that’s why the Earth is still blocking your view of Venus …You might want to reconsider that since very few prominent specialists in GR have been professional mathematicians.

- #30

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Yes, since de-modulation isn't working, perhaps I need to consider finding a black hole somewhere and using that to swallow the Earth...Maybe that’s why the Earth is still blocking your view of Venus …

- #31

bob012345

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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.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.

- #32

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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.I'm really more interested in understanding the physics than the mathematics of GR.

Edwin Taylor's

https://www.eftaylor.com/general.html

- #33

bob012345

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Thanks. I'm not totally giving up ever following the math by the way..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'sExploring Black Holesmight 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

- #34

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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,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})##)

$${(\hat{\Lambda}^{-1})^{\mu}}_{\nu} = \eta_{\nu \sigma} {\Lambda^{\sigma}}_{\rho} \eta^{\rho \mu}={\Lambda_{\nu}}^{\mu}.$$

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

I hope these may be clarified.

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- #35

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I think that MTW is a wonderful book for learning General Relativity, if you have 20 years to spare.

- #36

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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 ;-)).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.

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.

- #37

bob012345

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Gee, it's been sitting on my bookshelf since 1988. I could have learned it by now!I think that MTW is a wonderful book for learning General Relativity, if you have 20 years to spare.

- #38

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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.And maybe Hermann Weyl.

- #39

bob012345

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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.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.

https://www.gutenberg.org/files/43006/43006-pdf.pdf

- #40

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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 :-).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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