Question from Schutz, A First Course in GR

[Jimmy Snyder]

Can someone help me understand something on page 220 of the book 'A First Course in General Relativity' by Bernard Schutz?
Near the middle of the page, the line element is given as

$ds^2 = -dudv + f^2(u)dx^2 + h^2(u)dy^2$

(I changed g to h so I can talk about the metric tensor) which I think is:

$$\Large \[ \mathbf{g}_{\alpha\beta} = \left( \begin{array}{cccc} 0 & -\frac{1}{2} & 0 & 0 \\ -\frac{1}{2} & 0 & 0 & 0 \\ 0 & 0 & f^2(u) & 0 \\ 0 & 0 & 0 & h^2(u) \end{array} \right) \]$$

Is this correct? If so, then:

$$\Large \[ \mathbf{g}^{\alpha\beta} = \left( \begin{array}{cccc} 0 & -2 & 0 & 0 \\ -2 & 0 & 0 & 0 \\ 0 & 0 & \frac{1}{f^2(u)} & 0 \\ 0 & 0 & 0 & \frac{1}{h^2(u)} \end{array} \right) \]$$

How am I doing so far?

Then using equation 5.75 on page 143

$\Large \Gamma^{\gamma}_{\beta\mu} = \frac{1}{2}\mathbf{g}^{\alpha\gamma}(g_{\alpha\beta,\mu} + g_{\alpha\mu,\beta} - g_{\beta\mu,\alpha})$

we have

$\Large \Gamma^{v}_{xx} = \frac{1}{2}\mathbf{g}^{\alpha v}(g_{\alpha x,x} + g_{\alpha x, x} - g_{xx,\alpha})$

but the only value of $\alpha$ for which $\mathbf{g}^{\alpha v}$ is not zero is u. so

$\Large \Gamma^{v}_{xx} = \frac{1}{2}\mathbf{g}^{uv}(g_{ux,x} + g_{ux, x} - g_{xx,u}) = \frac{1}{2}(-2)(-2f\dot{f}) = 2f\dot{f}$

but the book has:

$\Large \Gamma^{v}_{xx} = 2\dot{f}/f$

Is the book wrong or am I?

 

[pervect]

I don't have the book, so I can't tell if they might be doing something different than what you describe, but everything you typed here looks OK to me.

 

[Jimmy Snyder]

I don't have the book, so I can't tell if they might be doing something different than what you describe, but everything you typed here looks OK to me.

Thank you for these encouraging words. However, I wouldn't have posted if I didn't think what I typed looked OK. I find it hard to believe that Schutz would have made such a mistake. I hope someone who has a copy will take a look and see if I haven't made some error.

 

[pervect]

Sorry I can't be of more help. If it helps your confidence any, I did put your line element into GRTensorII and had it verify the value of $\Gamma^v{}_{xx}$.

The only issue that I'm aware of is that the equations you are using do require you to be using a coordinate basis. Unless Schutz was talking about a basis of one forms somewhere in the text (possibly an orthonormal basis) in addition to giving you the line element, it seems unlikely that you would be using other than a coordinate basis.

 

[Jimmy Snyder]

It finally dawned on me too.

 

[pervect]

It finally dawned on me too.

LOL, I see you read my .sig

 

[George Jones]

Yesterday, I looked at the library's copy of Shutz, and I checked your calculation. Like pervect, I ended up with your result. Today, I looked at the library's copy of d'Inverno, which, on page 280, also confirms your result. d'Inverno also disagrees with Schutz's last connection coefficient, and with the the curvature components given by Schutz.

Shutz and d'Inverno do agee on the vacuum field equation, though.

My copies of Shutz and d'Inverno (and my other books) come out of storage in 2 or 3 weeks.

Regards,
George

 

[Jimmy Snyder]

Thank both of you for spending so much time and effort on this. Emboldened by the knowledge that Schutz has the wrong value for the Riemann tensor components, here is my calculation using eqn. 6.67 on page 169.

$$R^{\alpha}_{\beta \mu \nu} = \frac{1}{2}g^{\alpha \sigma}(g_{\sigma \nu,\beta \mu} - g_{\sigma \mu,\beta \nu} + g_{\beta \mu,\sigma \nu} - g_{\beta \nu,\sigma \mu})$$

so

$$R^{x}_{uxu} = \frac{1}{2}g^{x \sigma}(g_{\sigma u,ux} - g_{\sigma x,uu} + g_{ux,\sigma u} - g_{uu,\sigma x})$$

Again, since the only value of $\sigma$ for which $g^{x \sigma}$ is not zero is x, so

$$R^{x}_{uxu} = \frac{1}{2}g^{xx}(g_{xu,ux} - g_{xx,uu} + g_{ux,xu} - g_{uu,xx}) = \frac{1}{2f^2}(-2(\dot{f}^2 + f\ddot{f})) = -\frac{1}{f^2}(\dot{f}^2 + f\ddot{f}))$$
$$= -(\dot{f}/f)^2 - \ddot{f}/f$$

whereas the book has:

$$R^{x}_{uxu} = -\ddot{f}/f$$

 

[pervect]

Sorry, but the computer says your textbook is right on this one

$$R^x{}_{uxu}{} = -{\frac {{\frac {d^{2}}{d{u}^{2}}}f \left( u \right) }{f \left( u \right) }}$$

I'm not familiar with the particular formula you quoted, the formula I've seen (for a coordinate basis) is considerably more complicated:

$$R^u{}_{vab} = \partial_a \Gamma^u{}_{vb} - \partial_b\Gamma^u{}_{va}+ \Gamma^u{}_{pa}\Gamma^p{}_{vb} - \Gamma^u{}_{pb}\Gamma^p{}_{va}$$

 

[George Jones]

Thank both of you for spending so much time and effort on this.

Well, it's interesting stuff. Also, I have to be honest - I already had Schutz, d'Inverno, and a number of other relativity books out from the library for my own personal use. I hope I'll be able to take a closer look tomorrow.

Sorry, but the computer says your textbook is right on this one

Very interesting! d'Inverno, using a different sign convention and interchanging x and z, and gets

$$R_{0303} = ff''.$$

I'm not familiar with the particular formula you quoted

Schutz say that it is for a local inertial coodinate system about a point p, i.e., at p, g takes its special relativistic form, and the first derivatives of g are all zero. Unfortunately, it doesn't apply here, and (Schutz's version of) the formula Pervect gives must be used.

BTW, I was able to use GRTensorII to do some calculations in 2+1 gravity. Thanks for the help Pervect.

Regards,
George

 

[Jimmy Snyder]

This is a fine kettle of fish pervect. I used the eqn you provided (with different sub/superscripts and slightly different notation)

$$R^{\alpha}_{\beta \gamma \delta} = \Gamma^{\alpha}_{\beta \delta,\gamma} - \Gamma^{\alpha}_{\beta \gamma,\delta} + \Gamma^{\alpha}_{\sigma \gamma}\Gamma^{\sigma}_{\beta \delta} - \Gamma^{\alpha}_{\sigma \delta}\Gamma^{\sigma}_{\beta \gamma}$$

and got:

$$R^{x}_{uxu} = \Gamma^{x}_{uu,x} - \Gamma^{x}_{ux,u} + \Gamma^{x}_{\sigma x}\Gamma^{\sigma}_{uu} - \Gamma^{x}_{\sigma u}\Gamma^{\sigma}_{ux}$$

The first term on the right is zero, as is the second factor of the third term for all $\sigma$. As for the fourth term, the only value of $\sigma$ for which this is not zero is x. So we have:

$$R^{x}_{uxu} = - \Gamma^{x}_{ux,u} - \Gamma^{x}_{xu}\Gamma^{x}_{ux} = (\dot{f}/f),u - (\dot{f}/f)^2 = -\ddot{f}/f + (\dot{f}/f)^2 - (\dot{f}/f)^2 = -\ddot{f}/f$$

This is the same as Schutz, and the same as your computer (I assume you mean you ran GRTensorII again). But according to George, D'Inverno has something else. Can't anyone in this gang shoot straight?

 

[pervect]

I'll just include GRTensorII's output for all the non-zero Christoffel symbols while I'm at it:

$$\Gamma^x{}_{xu}=\Gamma^x{}_{ux}= {\frac {{\frac {d}{du}}f \left( u \right) }{f \left( u \right) }}\hspace{.25 in} \Gamma^y{}_{yu}=\Gamma^y{}_{uy}= {\frac {{\frac {d}{du}}h \left( u \right) }{h \left( u \right) }}\hspace{.25 in} \Gamma^v{}_{xx}= 2\,f \left( u \right) {\frac {d}{du}}f \left( u \right) \hspace{.25 in} \Gamma^v{}_{yy}= 2\,h \left( u \right) {\frac {d}{du}}h \left( u \right) \hspace{.25 in}$$

 

[Jimmy Snyder]

In a previous post, I used eqn. 6.67 from page 169 of Schutz. As George indicates, if I had looked closer at the text on that and the following page I would have realized that this is not coordinate invariant. Actually, the commas are a dead giveaway. The formulae that I have derived and now believe to be correct are the following:

$$\Gamma^{x}_{xu} = \dot{f}/f$$
$$\Gamma^{y}_{yu} = \dot{h}/h$$
$$\Gamma^{v}_{xx} = 2f\dot{f}$$
$$\Gamma^{v}_{yy} = 2h\dot{h}$$
$$R^{x}_{uxu} = -\ddot{f}/f$$
$$R^{y}_{uyu} = -\ddot{h}/h$$

I went to the library to look at a copy of D'Inverno. Like Schutz, he does not have these equations exactly. In fact, as George points out, it looks like he has an equation for

$$R_{uxux}$$

which may not be of interest here. I have not looked at his notational conventions yet.

It seems that both Schutz and D'Inverno have errors (typos?), different ones, in their calculations. I have found typos elsewhere in both books.

I am ready to forge on to calculate the vacuum field equation. I get the same answer as Schutz and D'Inverno as follows:

$$R_{uu} = R^{\mu}_{u \mu u} = -(\ddot{f}/f + \ddot{h}/h)$$
$$R_{\alpha \beta} = 0$$ if $$\alpha \ne u$$ or $$\beta \ne u$$

$$R^{vv} = g^{\mu v}g^{\nu v}R_{\mu \nu} = g^{uv}g^{uv}R_{uu} = -4(\ddot{f}/f + \ddot{h}/h)$$
$$R^{\alpha \beta} = 0$$ if $$\alpha \ne v$$ or $$\beta \ne v$$

$$R = g^{\mu \nu}R_{\mu \nu} = g^{uu}R_{uu} = 0$$

$$G^{\alpha \beta} = R^{\alpha \beta} - \frac{1}{2}g^{\alpha \beta}R = 0$$

or

$$\ddot{f}/f + \ddot{h}/h = 0$$

 

[pervect]

I went to the library to look at a copy of D'Inverno. Like Schutz, he does not have these equations exactly. In fact, as George points out, it looks like he has an equation for

$$R_{uxux}$$

which may not be of interest here. I have not looked at his notational conventions yet

The symmetries of the Riemann make
$$R_{abcd} = - R_{abdc}$$ always, and
$$R_{abcd} = -R_{bacd}$$ whenever you have a metric (which is always in GR, perhaps not in some other applications).

Thus $$R_{uxux} = R_{xuxu}$$. Now $$R^x{}_{uxu} = g^{xo} R_{ouxu}$$, where you have to sum over o, but only $g^{xx}$ is nonzero.

Thus $$R^x{}_{uxu} = g^{xx} R_{uxux}$$.

It seems that both Schutz and D'Inverno have errors (typos?), different ones, in their calculations. I have found typos elsewhere in both books.

I am ready to forge on to calculate the vacuum field equation. I get the same answer as Schutz and D'Inverno as follows:

$$R_{uu} = R^{\mu}_{u \mu u} = -(\ddot{f}/f + \ddot{h}/h)$$
$$R_{\alpha \beta} = 0$$ if $$\alpha \ne u$$ or $$\beta \ne u$$

$$R^{vv} = g^{\mu v}g^{\nu v}R_{\mu \nu} = g^{uv}g^{uv}R_{uu} = -4(\ddot{f}/f + \ddot{h}/h)$$
$$R^{\alpha \beta} = 0$$ if $$\alpha \ne v$$ or $$\beta \ne v$$

$$R = g^{\mu \nu}R_{\mu \nu} = g^{uu}R_{uu} = 0$$

$$G^{\alpha \beta} = R^{\alpha \beta} - \frac{1}{2}g^{\alpha \beta}R = 0$$

or

$$\ddot{f}/f + \ddot{h}/h = 0$$

I get the same results from GrtensorII.

 

[Jimmy Snyder]

Thanks for your generous help in this matter. I forgot to apply the symmetries. Thanks also for confirming my calculations. I'm new at this and don't feel comfortable with it yet, but with your help I am gaining strength.

 

[George Jones]

Thus $$R^x{}_{uxu} = g^{xx} R_{uxux}$$.

Which means that I was too hasty when I said "d'Inverno also disagrees with ..., and with the the curvature components given by Schutz.", i.e., Schutz and d'Inverno agree (up to a sign convention) about curvature since

$$R^x{}_{uxu} = g^{xx} R_{uxux} = \frac{1}{-f^2} \left( ff'' \right).$$

There is a minus sign in the denominator since Schutz and dInverno use metrics with different signatures.

It looks like Jimmy and d'Inverno have things right. Schutz has a typo/transcription error. Easy to make and easy to miss in the proofreadinding process.

Regards,
George

 

[Jimmy Snyder]

Easy to make and easy to miss in the proofreadinding process.

Indeed!

Extra text added to satisfy an unnecessary criterion.

 

[George Jones]

Extra text added to satisfy an unnecessary criterion.

LOL!

I, too, was puzzled by this criterion when I first ran afoul of it.

Regards,
George

 

[pmb_phy]

jimmysnyder - Have you ever considered simply e-mailing Schutz? His e-mail address is - schutz@aei-potsdam.mpg.de

pervect - I can scan that part of Schutz that jimmy is talking about and e-mail them to you if you'd like?

Pete

 

[pervect]

I suppose it would be good if someone told Schutz about his apparent typo, but since I don't own the book it's not going to be me.

Did I just see a volunteer, Pete? :-).

 

[pmb_phy]

I suppose it would be good if someone told Schutz about his apparent typo, but since I don't own the book it's not going to be me.

Did I just see a volunteer, Pete? :-).

I haven't read that section of the book. I've avoided gravitational waves for as long as I could.

It was never something I wanted to learn. I guess I could take a look though and finally read about g-waves. Sheesh! See what you folks have me doing now? Ya got me learning! Yipes

I'll scan and e-mail that section to you pervect. I'd like to discuss it with you since its a debatable section.

Pete

 

[Jimmy Snyder]

Have you ever considered simply e-mailing Schutz?

I just sent him an e-mail concerning this typo. Normally, I do contact the authors of books when I find these kinds of errors. My name appears in the third printing of the 4th edition of Professor Liboff's book "Introductory Quantum Mechanics", Professor Zweibach's web page of errata for his book "A First Course In String Theory" and on Professor Kubelsky's web page for errata in his book "Elements of Operator Theory"

In this case I held back because I wasn't sure if the error was his or mine. However, there are other typos in the book. If Professor Schutz responds possitively, I will send him about 10 or so. None of them are as glaring as this one, but once I get started, I pick every nit. That's how I get my name in.

 

[pmb_phy]

I just sent him an e-mail concerning this typo. Normally, I do contact the authors of books when I find these kinds of errors. My name appears in the third printing of the 4th edition of Professor Liboff's book "Introductory Quantum Mechanics", Professor Zweibach's web page of errata for his book "A First Course In String Theory" and on Professor Kubelsky's web page for errata in his book "Elements of Operator Theory"

In this case I held back because I wasn't sure if the error was his or mine. However, there are other typos in the book. If Professor Schutz responds possitively, I will send him about 10 or so. None of them are as glaring as this one, but once I get started, I pick every nit. That's how I get my name in.

I'm the same way. I have my name in Exploring Black Holes by Taylor and Wheeler and Classical Mechanics - 3rd Ed, Goldstein, Safko and Poole.

Pete

 

[Jimmy Snyder]

As I might have said, I got a copy of d'Inverno from the library and looked at page 280. It might have helped me more if you guys hadn't already told me what I need to know. I skimmed through the book and decided that it would not be a good idea for me to read this book along side of Schutz. However, as long as I have it out, I will consult it when I get stuck. I especially liked his paragraph on page 15 that begins with the following:

The very success of the activity of modelling has, throughout the history of science, turned out to be counterproductive. Time and again, the successful model has been confused with the ultimate reality, and this in turn has stultified progress.

Also, I was surprised to see on page 11 a reference to the book "Einstein's Theory of Relativity" by Lillian Lieber. I had borrowed a copy of this from the county library and so liked it that I purchased a copy from a used book dealer over the internet. This book is addressed to the general public, and yet has a rather detailed mathematical description of tensors, geodesics, and the like. It has to be seen to be believed.

 

[selfAdjoint]

Also, I was surprised to see on page 11 a reference to the book "Einstein's Theory of Relativity" by Lillian Lieber. I had borrowed a copy of this from the county library and so liked it that I purchased a copy from a used book dealer over the internet. This book is addressed to the general public, and yet has a rather detailed mathematical description of tensors, geodesics, and the like. It has to be seen to be believed

I LOVE that book! I first read it as a teenager (like, 1949?) and have owned it now for a number of years. I just recently reread the GR portion. It confirmed again my early impression; she takes you on a very narrow road^*, but a true one, to the Schwartschild solution and the famous tests of GR. She wants you to derive the math for these tests yourself, or at least understand clearly when she derives it.

In my recent reading I was able to appreciate some of her pedagogical strategies, for example her derivation of the Riemann-Christoffel tensor from the difference of second covariant derivatives in opposite orders of an arbitrary contravariant vector. Not the most motivating of derivations but the one that quickly clears away the lumber for the big show ahead.

^*Her brother's witty drawings invoking a hike threough the mountains, emphasize this feature. You can't help but wonder what's off behind that mountain on the left.