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A Stress tensor in 3D Anti-De Sitter Space

  1. May 7, 2017 #1
    I am doing some mathematical exercises with 3D anti-de sitter face using the metric

    ds2=-(1+r2)dt2+(1+r2)-1+r22

    I found the three geodesics from the Christoffel symbols, and they seem to look correct to me.

    d2t/dλ2+2(r+1/r)*(dt/dλ)(dr/dλ)=0

    d2r/dλ2+(r+r3)*(dt/dλ)2-r/(r2+1)(dr/dλ)2-(r+r3)(dφ/dλ)2=0

    d2φ/dλ2+2/r*(dφ/dλ)(dr/dλ)=0

    When I started calculating the Riemann and Ricci Tensor however things started to look hairy

    Rφrφr = -(1+r2)-1

    Rtφtφ = -(r+1/r)(r+r3)

    Rtrtr = -2+1/r2-(r+1/r)2

    I found the other components of the Riemann tensor to be 0 which may have been where I went wrong.

    This led me to a messy Ricci Tensor and Ricci Scalar

    Rrr= -2+1/r2-(r+1/r)2-(1+r2)-1

    Rφφ = -(r+1/r)(r+r3)

    R=guvRuv = Rrr(1+r2)+1/r2Rφφ

    R=-r4-4r2-3+1/r2

    This for some reason doesn't look right to me. It leads to a super complicated stress tensor as well.

    What did I do wrong here?
     
  2. jcsd
  3. May 7, 2017 #2

    Paul Colby

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    I'm not sure one considers this "sporting" but the ctensor package of wxmaxima makes short work of this type of problem. I've run your case assuming that the intended metric is

    ##ds^2 = -(1+r^2)dt^2+\frac{1}{1+r^2}dr^2+r^2d\phi^2##​

    Looks based on this that your Christoffel symbols seem to agree but there is a typo in the time component geodesic equation

    ##\Gamma^t_{t r} = r+\frac{1}{r^2}##​

    yours seems to be ##r+\frac{1}{r}## which is likely just in typing your post.

    The Riemann components I get are,

    ##R^r_{t r t} = -(1+r^2)##
    ##R^\phi_{t \phi t} = -(1+r^2)##
    ##R^t_{r r t} = -\frac{1}{1+r^2}##
    ##R^\phi_{r \phi r} = \frac{1}{1+r^2}##
    ##R^t_{\phi \phi t} = -r^2##
    ##R^r_{\phi \phi r} = -r^2##​
     
  4. May 7, 2017 #3

    pervect

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    I recommend using some sort of auotmated package at well. Maxima works and is free, though the ordering conventions are rather strange and it's a bit clunky to use. If there's interest, I could dig up Chris Hillman's file on how to use Maxima.

    Anyway, I get results similar to Pauls, though there seem to be some sign discrepancies, which I haven't tried to track down.

    $$g_{ab} = \left[ \begin {array}
    {ccc} -1-{r}^{2}&0&0\\0& \left( 1+{r}^{2} \right) ^{
    -1}&0\\0&0&{r}^{2}\end {array} \right]$$

    Geodesic equations:

    $$\frac{d^2 t}{d\tau^2} + 2 \frac{r}{1+r^2} \left( \frac{dt}{d\tau} \right) \left( \frac{dr}{d\tau} \right) = 0$$
    $$\frac{d^2 r}{d\tau^2} + r\left(1+r^2\right) \left( \frac{dt}{d\tau} \right)^2 - \frac{r}{1+r^2} \left( \frac{dr}{d\tau} \right)^2 - r\left(1+r^2\right) \left( \frac{d\phi}{d\tau} \right)^2 = 0$$
    $$\frac{d^2 \phi}{d\tau^2} + \frac{2}{r} \left( \frac{dr}{d\tau} \right) \left( \frac{d\phi}{d\tau} \right) = 0$$

    For the Riemann

    $$R^{\phi}{}_{r r \phi} = -R^t{}_{rtr} = \frac{1}{1+r^2} \quad R^t{}_{\phi t \phi} = R^r{}_{\phi r \phi} = -r^2 \quad R^r{}_{ttr} = R^\phi{}_{t t \phi} = -1-r^2 \quad $$

    Note that one needs to use the Bianci identies to get the complete set of nonzero Riemann components, for instance interchanging the last two symbols changes the sign, so that ##R^t{}_{rtr} = -R^t{}_{rrt}## - just one example of many omissions of nonzero components.
     
  5. May 7, 2017 #4

    Paul Colby

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    Agreed, the index order is straight from mars but it has it's charms. wxmaxima is a workbook front end for maxima which I highly recommend. Maxima is a code written in the 60's in lisp (second in age to only fortran).

    I would be very interested in any notes you could dig up.

    I also have Mathematica which for reasons I can't quite fathom I find more obscure to use.
     
  6. May 7, 2017 #5

    Ibix

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    I'd also be interested in notes, @pervect. I tried to get into ctensor in maxima off the back of code in Ben Crowell's GR book, but didn't get very far.
     
  7. May 7, 2017 #6
    This helps immensely thanks! I must have made a typo in Riemann tensor calculation, but going back through it I got the same answers as Pervect.

    Using these tensors, how could I calculate particle path? If I had photon shot from r = 0 at t = 0, moving along a geodesic in the outward radial direction how do I calculate when would it reach r = ∞?
     
  8. May 8, 2017 #7

    pervect

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  9. May 8, 2017 #8
    I was attempting to do this without using Maxima or the like, not to mention my coding is sub-par at best.

    I ended up finding using the metric and null geodesics to find that t(r)=tan-1(r), meaning for a photon leaving r=0 going to r=∞ it will reach ∞ at t=π/2, which is consistent with Anti-De Sitter space.

    Thanks for the help!
     
  10. May 8, 2017 #9

    pervect

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    The geodesic equation will tell you that. In that case ##\tau## is not proper time, but an "affine parameter". Usually people use "s", but it doesn't make any difference. If you need more detail, ask.
     
  11. May 9, 2017 #10

    Ibix

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    Wow, thanks for that link! I just modified his first program to derive the Schwarzschild metric in Schwarzschild coordinates by messing around with the frame field, which was really neat! Something of a twofer - I learned a bit about Maxima and a bit about working in GR.

    If anyone is still in touch with Chris Hillman (I gather he took himself offline), tell him thanks from me.
     
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