Understanding Tensors: Is Misner Thorne and Wheeler Enough?

[bob012345]

Summary:: Does the textbook Misner Thorne and Wheeler have all I need to understand tensors in order to learn GR?

Does the textbook Misner Thorne and Wheeler have all I need to understand tensors in order to learn GR? I have that textbook but never went through it. Tensors greatly intimidate me with all the indexes and symbols and summing this way and that and all the terminology. Is there a better source? Thanks.

P.S. I always liked that the book MTW is it's own pun...

 

[PeterDonis]

Does the textbook Misner Thorne and Wheeler have all I need to understand tensors in order to learn GR?

Yes. However:

Tensors greatly intimidate me with all the indexes and symbols and summing this way and that and all the terminology.

As you seem to realize, MTW is probably not the source you want to learn tensor calculus from if you want to avoid this problem.

I would suggest Sean Carroll's online lecture notes on GR as a gentler introduction to the subject.

 

[bob012345]

Yes. However:As you seem to realize, MTW is probably not the source you want to learn tensor calculus from if you want to avoid this problem.

I would suggest Sean Carroll's online lecture notes on GR as a gentler introduction to the subject.

Ok, thanks.

 

[Orodruin]

Summary:: Does the textbook Misner Thorne and Wheeler have all I need to understand tensors in order to learn GR?

Tensors greatly intimidate me with all the indexes and symbols and summing this way and that and all the terminology.

Do not break the first commandment https://www.physicsforums.com/insights/the-10-commandments-of-index-expressions-and-tensor-calculus/ ;)

Do not despair. It does look daunting in the beginning but once you get the hang of it there are mainly a few basic things to keep in mind.

 

[vanhees71]

The greatest obstacle for me is to obey Commandment 2 ;-)). Another important commandment is to also obey carefully the horizontal order of indices not only the vertical one! I've seen many documents providing in principle a good approach to the topic but are completely useless in not clearly writing the indices in a well-defined horizontal ordering, where it is needed.

 

[Orodruin]

The greatest obstacle for me is to obey Commandment 2 ;-))

Yet it has been the bane of many a student calculations

 

[Demystifier]

The greatest obstacle for me is to obey Commandment 2 ;-)). Another important commandment is to also obey carefully the horizontal order of indices not only the vertical one! I've seen many documents providing in principle a good approach to the topic but are completely useless in not clearly writing the indices in a well-defined horizontal ordering, where it is needed.

Especially when one needs to be careful about both at the same time. For instance, the Lorentz-transformation tensor obeys $(\Lambda^{-1})^{\alpha}_{\;\,\beta}=\Lambda^{\;\,\alpha}_{\beta}$.

 

[dyn]

Especially when one needs to be careful about both at the same time. For instance, the Lorentz-transformation tensor obeys $(\Lambda^{-1})^{\alpha}_{\;\,\beta}=\Lambda^{\;\,\alpha}_{\beta}$.

Can you recommend a textbook that covers this clearly and explains what the α and β refer to in terms of rows and columns on either side of the equation. I have always found this confusing

 

[Demystifier]

Can you recommend a textbook that covers this clearly and explains what the α and β refer to in terms of rows and columns on either side of the equation. I have always found this confusing

I don't know a textbook, but it's easy. Lorentz group is an orthogonal group SO(1,3). Orthogonal matrix, by definition, obeys $\Lambda^{-1}=\Lambda^T$, where $T$ denotes the transpose. The rest should be easy.

 

[Orodruin]

I don't know a textbook, but it's easy. Lorentz group is an orthogonal group SO(1,3). Orthogonal matrix, by definition, obeys $\Lambda^{-1}=\Lambda^T$, where $T$ denotes the transpose. The rest should be easy.

This is true for SO(n). It is not true for SO(1,n). In particular, for the standard Lorentz boost in the x-direction, it is not true as $\Lambda = \Lambda^T$.

 

[vanhees71]

I don't know a textbook, but it's easy. Lorentz group is an orthogonal group SO(1,3). Orthogonal matrix, by definition, obeys $\Lambda^{-1}=\Lambda^T$, where $T$ denotes the transpose. The rest should be easy.

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

 

[Demystifier]

@Orodruin and @vanhees71 thanks for the correction!

 

[dyn]

My confusion is over the following -
If Λ^μ_ν represents the entry in the Λ matrix in row μ and column ν what does Λ_ν^μ represent in terms of rows and columns ?
In other words in Λ^μ_ν the row is indicated by μ ; but is that because it is the top index or the 1st index going from left to right ?

 

[Demystifier]

In other words in Λ^μ_ν the row is indicated by μ ; but is that because it is the top index or the 1st index going from left to right ?

It is because it's first from the left. In terms of a matrix, the difference between upper and lower indices doesn't make sense. For example, for the metric tensor $g$ with matrix entries $g_{\mu\nu}$, the quantities called $g^{\mu\nu}$ are really the matrix entries of $g^{-1}$. So in matrix language, $g_{\mu\nu}$ and $g^{\mu\nu}$ are entries of different matrices, one being the inverse of another.

 

[vanhees71]

The difference between upper and lower indices makes a lot of sense. It's a mnemonic notation for knowing whether the corresponding tensor components have to be transformed covariantly (lower indices) or contravariantly.

The drawback of the matrix notation is that this information gets hidden, i.e., you have to always keep in mind which of the indices in the matrix is an upper or lower index. Another drawback is that you can't handle tensors with rank 3 and higher. The advantage is a somewhat shorter notation.

 

[Orodruin]

It is because it's first from the left. In terms of a matrix, the difference between upper and lower indices doesn't make sense. For example, for the metric tensor $g$ with matrix entries $g_{\mu\nu}$, the quantities called $g^{\mu\nu}$ are really the matrix entries of $g^{-1}$. So in matrix language, $g_{\mu\nu}$ and $g^{\mu\nu}$ are entries of different matrices, one being the inverse of another.

It should be pointed out that this is true particularly for the metric tensor. It is not generally true that $T^{\mu\nu}$ are the components of the inverse of $T_{\mu\nu}$. However, for the metric tensor, it holds that
$$
g^{\mu\nu}g_{\nu\sigma} v^\sigma = g^{\mu\nu} v_\nu = v^\mu
$$
for all $v$ and so $g^{\mu\nu}g_{\nu\sigma} = \delta^\mu_\sigma$.

 

[dyn]

If Λ^μ_ν represents the entry in the Λ matrix in row μ and column ν what does Λ_ν^μ represent in terms of rows and columns ?
In other words in Λ^μ_ν the row is indicated by μ ; but is that because it is the top index or the 1st index going from left to right ?

If Λ is a 4x4 matrix and Λ^μ_ν represents the entry in row μ and column ν what does Λ_v^u represent ?

I am looking for a textbook that explains in the clearest sense this kind of tensor/index notation and what it means

 

[ohwilleke]

I have a copy of MTW and it is not at all my favorite one to use for independent studying of something like tensor calculus from scratch.

 

[PeterDonis]

If Λ is a 4x4 matrix and Λ^μ_ν represents the entry in row μ and column ν what does Λ_v^u represent ?

The entry in row $\nu$ and column $\mu$.

However, $\Lambda$ is not a tensor, it's a coordinate transformation represented as a matrix. They're not the same thing.

I am looking for a textbook that explains in the clearest sense this kind of tensor/index notation and what it means

First you need to understand the distinction between tensors and coordinate transformations. You have been talking about $\Lambda$, meaning the Lorentz transformation, as though it were a tensor; but as noted above, it isn't.

 

[PeterDonis]

the Lorentz-transformation tensor

A coordinate transformation isn't a tensor. They're different kinds of objects, even though similar notation is used for both.

 

[Demystifier]

A coordinate transformation isn't a tensor.

Lorentz transformation is a Lorentz tensor. Under Lorentz coordinate transformations, components of $\Lambda^{\mu}_{\;\nu}$ transform as components of a tensor. But of course, it's not a tensor under general coordinate transformations.

 

[Orodruin]

A coordinate transformation isn't a tensor. They're different kinds of objects, even though similar notation is used for both.

I was about to write something similar, but this depends on whether you consider an active or passive Lorentz transformation. For a passive transformation, the coordinate transformation certainly is not a tensor as it is just a relabelling of components. However, for an active transformation, a Lorentz transformation is indeed a linear map from vectors in Minkowski space to vectors in Minkowski space - which is the very definition of a (1,1) tensor. However ...

Lorentz transformation is a Lorentz tensor. Under Lorentz coordinate transformations, components of $\Lambda^{\mu}_{\;\nu}$ transform as components of a tensor. But of course, it's not a tensor under general coordinate transformations.

... I generally dislike the very common introduction of tensors as being "components transforming in a particular way" as I have seen it lead to several misunderstandings. In the case of a passive Lorentz transformation (which is what people will generally think about), it certainly does not satisfy the requirements for being a tensor in the sense of being a map from tangent vectors to tangent vectors as it is just a relabelling of the coordinates used to describe the vector.

 

[Demystifier]

Then can we at least agree that there are several inequvalent definitions of a "tensor"?

 

[dextercioby]

Then can we at least agree that there are several inequvalent definitions of a "tensor"?

In mathematics, there is only one definition as a multilinear map. There is no such thing as "inequivalent definitions", this would mean different notions.

 

[vanhees71]

If Λ is a 4x4 matrix and Λ^μ_ν represents the entry in row μ and column ν what does Λ_v^u represent ?

I am looking for a textbook that explains in the clearest sense this kind of tensor/index notation and what it means

The horizontal position of the indices always tells you what's counted: the left index counts the row, the right index the column. In the matrix notation the vertical positioning of the indices is simply lost. You always have to remember what a given matrix represents, including the vertical positioning of the indices. For this reason I tend to avoid the matrix notation when it comes to relativity when doing calculations.

 

[bob012345]

It seems hopeless as highly educated intelligent people can't even agree on the math. Maybe GR is best left to professional mathematicians.

 

[PeterDonis]

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.

 

[PeterDonis]

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.

 

[Frabjous]

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 …

 

[PeterDonis]

Maybe that’s why the Earth is still blocking your view of Venus …

Yes, since de-modulation isn't working, perhaps I need to consider finding a black hole somewhere and using that to swallow the Earth...

 

[bob012345]

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.

 

[PeterDonis]

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

 

[bob012345]

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..

 

[shinobi20]

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.

 

[stevendaryl]

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