Calculating the Riemann Tensor for a 4D Sphere

[Mr-R]

Dear All,

I am trying to calculate the Riemann tensor for a 4D sphere. In D'inverno's book, I have this equation $R^{a}_{bcd}=\partial_{c}\Gamma^{a}_{bd}-\partial_{d}\Gamma^{a}_{bc}+\Gamma^{e}_{bd}\Gamma^{a}_{ec}-\Gamma^{e}_{bc}\Gamma^{a}_{ed}$

But the exercise asks me to calculate $R_{abcd}$. Do I lower it with a metric? if so, a metric with what indices?

An lastly, in the above equation, there is an extra index e. Is it another dummy variable that I have to sum over with?

too much work

 

[haushofer]

Yes, e is dummy. You should check your concention, but your metric has lower indices,

R_{abcd}= g_{af}R^{f}_{bcd}

The index f is again dummy.

 

[Mr-R]

Yes, e is dummy. You should check your concention, but your metric has lower indices,

$R_{abcd}= g_{af}R^{f}_{bcd}$

The index f is again dummy.

Thank you for your reply haushofer. Man this is going to take me some time. How many Riemann tensors should I get from this calculation? Or do I choose any order of indices for the tensor and work it out?

Sorry If I am mixing up stuff.

 

[haushofer]

Just one, with the indices shown. Work it out explicitly taking your time, and things hopefully get more clear :)

 

[haushofer]

If you like long calculations, try supergravity next. It makes you longing for ordinary GR really soon :P

 

[Mr-R]

maybe when I grow up

Thanks for your help

 

[haushofer]

Good luck, and take your time!

 

[Mr-R]

So I am back and my Riemann tensor is zero for some reason :(

Basically I have my 4D sphere (t,r,Θ,ø). And all of the Christoffle symbols I have calculate are zero (the ones that have the index t). Therefore, the Riemann tensor is zero and it should not be zero. Problem is that in the above formula for the Riemann tensor, the index a=t. And all of the Christoffle symbols contain a=t. My page contains a forest of zeros. What Am I doing wrong?

help

 

[pervect]

too much work

I highly recommend a symbolic algebra package of any sort. Maxima is one of the free ones.

 

[Mr-R]

Thanks pervect. I will look for it.

 

[Mr-R]

As an addition to post #8, I think I know where my mistake is, hopefully.

$ds^{2}=e^{\nu}dt^{2}-e^{\lambda}dr^{2}-r^{2}d\theta^{2}-r^{2}sin^{2}(\theta)d\varphi^{2}$.

I have been putting the $dt$ and $dr$ terms as 1 and -1 respectively in the metric tensor. So when I took the derivatives when calculating the Christoffle symbols I got mostly zeros. Can this be the mistake?

$\nu$ and $\lambda$ are functions of $(t,r)$

Should not the dt terms have c (m/s) multiplied with it? and what am I supposed to expect to get from the calculation? I have got no sources or refrences to check from. Just me and D'Inverno.

Thanks all

 

[ChrisVer]

As your metric is, you are supposed to get some nice algebra, but I don't think you should look for a reasonable answer. The reasonable answer appears by the time you can identify those $\nu,\lambda$ in some way. Why don't you try to warm yourself up by doing a calculation in smaller dimensions? Like fore example the 2 sphere...
As for your questions:
Obviously what you said you did, can be a mistake- partially. However you wouldn't have found the C.Symbols 0 even in this case.
For example:
$\Gamma^{1}_{33}$ I think shouldn't be zero (by setting what is in front of dr equal to -1)...

$\Gamma^{1}_{33} = \frac{1}{2} g^{11} (g_{31,3} + g_{31,3} - g_{33,1}) = \frac{1}{2} g_{33,1} = -\frac{1}{2}2r \sin^{2}(\theta)=-r \sin^{2}(\theta)$

You don't really need to care about $c$ since you can always choose your units such that $c=1$. In those units, length and time are equivalent.

 

[Mr-R]

Thanks ChrisVer for the reply .

By doing what I did (putting the dt term as 1 and dr term as -1 in the metric), I got the Christoffle symbols for a 3D sphere (6 unique symbols). But I have a 4D sphere line element in the problem. I will try again by putting the "e"'s in the metric instead of 1 and -1. This should give me another function when derived w.r.t r or t instead of zeros.

This is an excercise in D'inverno's book page 90.

(just saw you latex sorry, it won't affect this Christoffle symbol. It will effect the Christoffle symbols that have the index 0 or t)

 

[ChrisVer]

I fixed what I wrote above... It's not exactly a 3D sphere, not until you reparametrize your dr.
Nevertheless, make sure you get the Christoffel Symbols right, since I showed above that even in your case, there existed a non-zero christoffel symbol...
The formula is:

$\Gamma^{\alpha}_{\mu \nu} = \frac{1}{2} g^{\alpha \rho} (\partial_{\nu} g_{\mu \rho}+\partial_{\mu} g_{\nu \rho}- \partial_{\rho} g_{\mu \nu})$

$\rho$ is summed over... and be careful that $g^{\mu \nu}$ are the elements the inverse of the matrix with elements $g_{\mu \nu}$... they are not the same matrices.
in the case with the exponentials for example, $g^{11} = -e^{-\lambda}$

 

[Mr-R]

Could you please see the attached picture Chris?

As you can see I got these symbols. Is it okay to derive the exponential to the power of lambda like in the picture?
I am okay with calculating the symbols and such. My problem was getting zeros for the symbols that contain the index 0 or t. Edit: sorry this is how I derived e^lambda: $y=e^{λ}, y^{'}=λ^{'}e^{λ}$

Again, thank you for taking your time to help me. I will try again as soon as crohns stops bothering me.

 

[ChrisVer]

To avoid as many calculations as possible you just can use the symmetry of the Christoffel symbols. $\Gamma^{\rho}_{\mu \nu}= \Gamma^{\rho}_{\nu \mu}$.
For someone one has to find the inverse of the metric, and start looking at:
$\Gamma^{0}_{\mu \nu}$
$\Gamma^{1}_{\mu \nu}$
$\Gamma^{2}_{\mu \nu}$
$\Gamma^{3}_{\mu \nu}$
Keeping in mind that the metric is diagonal (so maybe you can find terms that are zero without needed to calculate them). Eg derivatives of $g_{11}, g_{00}$ with respect to 2,3 don't contribute anything. In fact the diagonal metric is what makes the above handy to minimize as possible the work needed.

So for example:
$\Gamma^{3}_{\mu \nu}= \frac{1}{2} g^{33} (g_{\mu 3, \nu} + g_{\nu 3, \mu} - g_{\mu \nu,3} )$
you don't have to care about $\mu, \nu = 0,1,2$ they are identically zero... the first 2 terms because the metric is diagonal, and the last term because the elements $g_{00}, g_{11},g_{22}$ which could exist don't depend on $\phi$. So you only need to take in account the:
$\Gamma^{3}_{3 \nu}$ case ($\mu=3$)... and once you find it, you also have the $\Gamma^{3}_{\mu 3}$ because of symmetry- which would be the case to take $\nu=3$. So all $\Gamma^{3}_{...}$'s found.

 

[ChrisVer]

In the same way as above, I can see that there exist a $\Gamma^{0}$ which is not zero.

$\Gamma^{0}_{11} \ne 0$

$\Gamma^{0}_{11} = -\frac{1}{2} g^{00} g_{11,0}$ and $g_{11}$ depends on t.

 

[ChrisVer]

yes that's the derivative of exp[lambda]

 

[Mr-R]

That is if you substituted the e's in the metric which I foolishly, did not do.

Thanks for your help and time. Hopefully I will be able to do it now.

 

[ChrisVer]

Your
$\Gamma^{r}_{\theta \theta} = \frac{1}{2} g^{rr} ( 2 g_{\theta r, \theta} - g_{\theta \theta,r})= -\frac{1}{2} g^{rr} g_{\theta \theta,r} = - e^{-\lambda} r$

Well if you didn't have the exponentials, to which metric are you asking me to check the Christoffel Symbols you calculated? :P plus to this in general one with a program which can evaluate christoffel symbols can be more helpful - I can only check some of the C.S. by hand, not all that can arise (thanks the GR exam is passed for me).
But if you are a fish in GR, you must do those calculations at least once by hand...

 

[Mr-R]

haha don't worry I will do it by myself :tongue2:. pervect recommended me a program to do the calculation, but I prefer to do it manually first (I am a single atom, not even a fish :P) before using programs. I will not be taking GR until my last year at Uni. I am just learning it cause I like the theory :!)

 

[ChrisVer]

You should spend your time with more basic stuff rather than going to 4D problems from the beginning ... Since it's not part of your course, you should try to get the 2 sphere:
$ds^{2}= r^{2} d \theta^{2} + r^{2} \sin^{2}(\theta) d \phi^{2}$
and try to find everything up to the Ricci Scalar... having only two indices can give you more confidence after going to higher dimensional problems. You don't start from the 11 dimensional supergravity (as someone mentioned) to learn something...

 

[WannabeNewton]

You should really use Mathematica for calculations like these. There isn't much use in doing them by hand really because it's extremely straightforward but there is easy room for silly mistakes. James B. Hartle has a publicly available Mathematica notebook that executes commands for connection coefficient calculations, Riemann tensor calculations etc. in coordinate bases. You can also easily write your own notebook for calculations in non-coordinate bases (tetrads). As an undergraduate I get a huge discount through my university for Mathematica 10 yearly subscription so maybe you could look into whether or not your university offers this as well. It's good to get used to writing Mathematica notebooks now rather than later because if you plan on going to graduate school Mathematica will soon become your best friend.

If you want to calculate connection coefficients and Riemann tensor components by hand then it would be much more instructive for you to work in a non-coordinate basis (tetrad) and use the Cartan calculus to calculate the Riemann tensor and Ricci rotation coefficients (analogous to Christoffel symbols). These are less straightforward to calculate since they involve some guess work so you can be challenged more and they are far more efficient and elegant computationally. d'Inverno doesn't discuss this method but almost all advanced GR texts will; see e.g. section 3.4b of Wald.

 

[Mr-R]

You should really use Mathematica for calculations like these. James B. Hartle has a publicly available Mathematica notebook that executes commands for connection coefficient calculations, Riemann tensor calculations etc. in coordinate bases. You can also easily write your own notebook for calculations in non-coordinate bases (tetrads). As an undergraduate I get a huge discount through my university for Mathematica 10 yearly subscription so maybe you could look into whether or not your university offers this as well. It's good to get used to writing Mathematica notebooks now rather than later because if you plan on going to graduate school Mathematica will soon become your best friend.

Thanks for the advice WannabeNewton, I already know how to use Mathematica (maybe not as you or graduate students, but I can do all kinds of integration, Diff. equ., matrices, etc, Plotting..). I will check the discount thing.

Thanks again :thumbs:

 

[WannabeNewton]

...I already know how to use Mathematica (maybe not as you or graduate students, but I can do all kinds of integration, Diff. equ., matrices, etc, Plotting..).

That's awesome. If you're already used to Mathematica then Hartle's prewritten notebook for GR tensor calculations will be very, very easy for you to use. Here's the notebook: http://wps.aw.com/aw_hartle_gravity_1/0,6533,512496-,00.html

To be honest I basically only use Mathematica for the NDSolve and Series commands haha.

 

[Mr-R]

That's awesome. If you're already used to Mathematica then Hartle's prewritten notebook for GR tensor calculations will be very, very easy for you to use. Here's the notebook: http://wps.aw.com/aw_hartle_gravity_1/0,6533,512496-,00.html

To be honest I basically only use Mathematica for the NDSolve and Series commands haha.

Will check Hartle's thanks. Haha yes I used DSolve a lot to check my coursework before submitting. It promotes self learning to be honest, nice.

 

[gnnmartin]

I'm pleased that you are working with Ray's book: I was taught by him, and his is the book I know best. He was a wonderful teacher.

I expect you have cracked the problem now, but in case you haven't I'll make a couple of comments. I assume you are doing exercise 6.31 in Ray's book "Introducing Einstein's Relativity', in which case you can check your calculated connection against the answers given at the back of the book. In particular, the picture you included in post 15 above had a couple of mistakes in it. You will see what these are when you compare the values you wrote with the answers Ray gives.

Ray uses the result of this exercise in chapter 14.5 to calculate the Schwarzschild metric, and it might help to read that.

 

[Mr-R]

I'm pleased that you are working with Ray's book: I was taught by him, and his is the book I know best. He was a wonderful teacher.

I expect you have cracked the problem now, but in case you haven't I'll make a couple of comments. I assume you are doing exercise 6.31 in Ray's book "Introducing Einstein's Relativity', in which case you can check your calculated connection against the answers given at the back of the book. In particular, the picture you included in post 15 above had a couple of mistakes in it. You will see what these are when you compare the values you wrote with the answers Ray gives.

Ray uses the result of this exercise in chapter 14.5 to calculate the Schwarzschild metric, and it might help to read that.

gnnmartin, thank you for your comments. Yes it has been some time since I have asked that question. I did not a have a good foundation or grasp of Tensor calculus and differential geometry which made it hard for me to understand Ray's book which was my first book on GR. I switch to Schutz since then and now I can hold Ray's book in my hands and understand it :)

Moderators: You may close the thread.

 

[Nugatory]

This has turned out more happily than most attempts at thread necromancy... But yes, time to close.