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utesfan100
2012-Apr-02, 04:00 PM
I have obtained a copy of MTW's Gravitation on loan. I desire to understand the material through chapter 7 far more deeply than the surface knowledge of a first reading. to further develop an ATM idea by expressing it properly in geometric notation.

I plan to work the examples through example 7.1, and desire someone to review my solutions to make sure I am not overlooking subtle nuances of the questions.

Clearly, posting answers to a standard text in a pubic forum is problematic, thus I desire to take this inquiry offline. I would express my answers in LaTeX generated PDF files through e-mail, if I can secure someone to review them.

utesfan100
2012-Apr-16, 07:23 PM
Exercise 2.7 question:

This question asks us to show that \Lambda^T\eta\Lambda=\eta, for the definition of \Lambda given. I am confused because no indices are given, and the transpose operation is not explained in the text.

Why would there need to be a transpose? Should the expression not be: \Lambda^u_{u'}\eta_{uv}\Lambda^v_{v'}=\eta_{u'v'}?

Or should it be read as \Lambda^{u'}_u\eta_{uv}\Lambda^u_{u'}=\eta_{u'v'}, which we are showing have the same mathematical form?

caveman1917
2012-Apr-17, 02:02 PM
Exercise 2.7 question:

This question asks us to show that \Lambda^T\eta\Lambda=\eta, for the definition of \Lambda given. I am confused because no indices are given, and the transpose operation is not explained in the text.

Why would there need to be a transpose?

I don't have Gravitation with me here, but it looks like you're doing a coordinate change in minkowski space. Remember that minkowski space is simply a 4d vector space with the minkowski inner product (which is a bilinear form on that vector space). As such the representation of the inner product (and thus metric) wrt a certain basis will be a 4x4 matrix. Doing a basis change (changing frames) is then nothing more than \Lambda^T \eta \Lambda where \Lambda is the matrix of basis change.

Since you need to show that \Lambda^T \eta \Lambda = \eta it seems like you are being asked to change between inertial frames in minkowski space and show that the metric is invariant under that transformation. You'll just need to calculate out the matrix product.

In general always remember that what is colloquially termed the "metric" (ds^2 = \hdots) is not actually the metric but the first fundamental form of the metric. What we are really doing is working with matrices to represent the metric (or the inner product to be exact).

If you're not familiar with that, i would recommend first reading up on vector spaces, inner products and basis changes in vector spaces, and then look up the minkowski inner product/metric (all have good wikipedia articles that should do the job well enough) before you go further with Gravitation. This is required background knowledge.

grapes
2012-Apr-17, 03:15 PM
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ETA: I just got a message that I reached kilopi with this post. I'm not going to post for a day, just to let it sink in.

utesfan100
2012-Apr-18, 03:05 PM
I don't have Gravitation with me here, but it looks like you're doing a coordinate change in minkowski space. Remember that minkowski space is simply a 4d vector space with the minkowski inner product (which is a bilinear form on that vector space). As such the representation of the inner product (and thus metric) wrt a certain basis will be a 4x4 matrix. Doing a basis change (changing frames) is then nothing more than \Lambda^T \eta \Lambda where \Lambda is the matrix of basis change.
\Lambda is a given expression, and we are showing that this expression satisfies the requirements of a general boost in am arbitrary direction.


Since you need to show that \Lambda^T \eta \Lambda = \eta it seems like you are being asked to change between inertial frames in minkowski space and show that the metric is invariant under that transformation. You'll just need to calculate out the matrix product.
Matrix multiplication seems identical to multiplying two second rank tensors and contracting the fourth rank tensor along a covariant and contravariant component. My question is along which components should I be contracting. How do I determine which indices should be repeated, when no indices are given?

caveman1917
2012-Apr-18, 11:04 PM
\Lambda is a given expression, and we are showing that this expression satisfies the requirements of a general boost in am arbitrary direction.

I have Gravitation with me now. As you see \Lambda is a given expression for a matrix that signifies a coordinate change. The general boost is nothing more than a coordinate change from the first frame to the "boosted" frame, \Lambda is thus a matrix of basis change and should be treated as such.


Matrix multiplication seems identical to multiplying two second rank tensors and contracting the fourth rank tensor along a covariant and contravariant component. My question is along which components should I be contracting. How do I determine which indices should be repeated, when no indices are given?

You're making this way too difficult, and i don't quite understand the problem about repeated indices, the entire matrix of basis change is defined at the start of the exercise.

\Lambda is a 4x4 basis change matrix, from the standard basis to the primed basis. All components are given. For example the first one says
\Lambda^{0'}_0 = \gamma so \Lambda_{00} = \gamma, ie:
\Lambda = \begin{pmatrix} \gamma & \Lambda_{01} & \Lambda_{02} & \Lambda_{03} \\ \Lambda_{10} & \Lambda_{11} & \Lambda_{12} & \Lambda_{13} \\ \Lambda_{20} & \Lambda_{21} & \Lambda_{22} & \Lambda_{23} \\ \Lambda_{30} & \Lambda_{31} & \Lambda_{32} & \Lambda_{33} \end{pmatrix}

Then you fill out the rest of the components as given until you have the full 4x4 matrix.

Then you take \eta

\eta = \begin{pmatrix} -1&0&0&0 \\ 0&1&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \end{pmatrix}

, which is nothing more than the matrix representation of a bilinear form (the minkowski metric in this case) wrt the first basis, and do a basis change on that, ie \Lambda^T \eta \Lambda.
This gets you the matrix representation of that bilinear form wrt the primed basis, and you need to show that it is invariant. So just calculate out the matrix product.

utesfan100
2012-Apr-19, 02:19 PM
I have Gravitation with me now. As you see \Lambda is a given expression for a matrix that signifies a coordinate change. The general boost is nothing more than a coordinate change from the first frame to the "boosted" frame, \Lambda is thus a matrix of basis change and should be treated as such.



You're making this way too difficult, and i don't quite understand the problem about repeated indices, the entire matrix of basis change is defined at the start of the exercise.

\Lambda is a 4x4 basis change matrix, from the standard basis to the primed basis. All components are given. For example the first one says
\Lambda^{0'}_0 = \gamma so \Lambda_{00} = \gamma, ie:
\Lambda = \begin{pmatrix} \gamma & \Lambda_{01} & \Lambda_{02} & \Lambda_{03} \\ \Lambda_{10} & \Lambda_{11} & \Lambda_{12} & \Lambda_{13} \\ \Lambda_{20} & \Lambda_{21} & \Lambda_{22} & \Lambda_{23} \\ \Lambda_{30} & \Lambda_{31} & \Lambda_{32} & \Lambda_{33} \end{pmatrix}

Then you fill out the rest of the components as given until you have the full 4x4 matrix.

Then you take \eta

\eta = \begin{pmatrix} -1&0&0&0 \\ 0&1&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \end{pmatrix}

, which is nothing more than the matrix representation of a bilinear form (the minkowski metric in this case) wrt the first basis, and do a basis change on that, ie \Lambda^T \eta \Lambda.
This gets you the matrix representation of that bilinear form wrt the primed basis, and you need to show that it is invariant. So just calculate out the matrix product.
Thank you. It was clear before that the mechanics that you outlined above work out. I was thinking the transpose was referring to the tensor object, but rather it appears to be a consequence of the matrix representation of the tensor.

If I am now understanding the notation, the transpose in the problem is similar to the transpose in the following:

u^i\cdot v^i=u^i\eta_{ij}v^j=u^T\eta v

Where the last expression uses column vectors and matrix representations, not tensor notation.

caveman1917
2012-Apr-19, 09:22 PM
If I am now understanding the notation, the transpose in the problem is similar to the transpose in the following:

u^i\cdot v^i=u^i\eta_{ij}v^j=u^T\eta v

Where the last expression uses column vectors and matrix representations, not tensor notation.

Yes that's indeed the case.

Cougar
2012-Apr-21, 01:26 AM
...and do a basis change on that, ie \Lambda^T \eta \Lambda....

I thought I was following along, in a general sort of way. But then why is \Lambda transposed for this basis change?

caveman1917
2012-Apr-21, 01:42 AM
I thought I was following along, in a general sort of way. But then why is \Lambda transposed for this basis change?

The short answer: because that's how you transform (the Gram matrix of) a bilinear form to a different basis.

The longer answer: because the statement that two matrices represent the same bilinear form in different bases can be shown to be the same as the statement that the two matrices are congruent. Suppose \eta_1 and \eta_2 are two matrix representations in two different bases of a bilinear form, then they represent the same bilinear form iff there exists an invertible matrix P such that \eta_2 = P^T \eta_1 P. In such case this matrix P is called the matrix of basis change, here represented by the symbol \Lambda.

utesfan100
2012-Apr-25, 02:35 PM
I thought I was following along, in a general sort of way. But then why is \Lambda transposed for this basis change?

We are representing \eta_{ij} as a i\times j matrix, and likewise for \Lambda^k_{k'}.

To get \Lambda^i_{i'}\eta_{ij}\Lambda^j_{j'}, without the transpose we would have (i\times i')(i\times j)(j\times j'). The first matrix needs to be transposed so that we are summing along i, not mixing it with i'.

We are transposing the matrix to conform to the Tensor summation convention.

ngc3314
2012-Apr-25, 04:35 PM
I desire to understand the material through chapter 7 far more deeply than the surface knowledge of a first reading. to further develop an ATM idea by expressing it properly in geometric notation.


May I asked the assembled BAUTisti to join me in a round of applause for taking the time to Do It Right?