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About the ADM Formalism?

  1. Jun 18, 2011 #1
    Hi! I am novice in the Quantum Gravity field, so it was logical for me to start with the ADM formalism, but I am very confused about the Shift vector, specially in the Peldan paper http://arxiv.org/abs/gr-qc/9305011.

    In the first equation of (2.29): why the contraction of N^a with VaI does not vanish as it was the case for the third equation of (2.29), the N^a and N^I aren't the same vector but viewed by two different coordinate systems?

    I am already apologizing for my bad english :)
     
  2. jcsd
  3. Jun 18, 2011 #2

    atyy

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    I'm not sure, but the equations look different to me. The superscript for N matches the first and second subscripts of V in the first and third equations respectively.
     
  4. Jun 18, 2011 #3
    I mean the right hand side of the first equation of (2.29): N^a contracted with V^aI, isn't supposed to vanish as the third equation on (2.29) where the projection of N^I with V^aI is zero?

    :S
     
  5. Jun 18, 2011 #4

    atyy

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    It looks like in one case the contraction is with "a" and in the other case with "I", which are indices that have different positions on VaI.

    Also, the lower case Roman indices "a" appear to take possible values {1,2,3} (space), whereas upper case indices "I,J,K.." appear to take values {0,1,2,3} (local Minkowski basis). He also uses lower case Greek indices "α" which take values {0,1,2,3} (spacetime coordinates).
     
    Last edited: Jun 18, 2011
  6. Jun 18, 2011 #5
    - Thank you :), I must confess that i have some difficulties with "this" Shif Vector and his behavior.
     
  7. Jun 18, 2011 #6
    Another question :)

    Is there any relation between N^I and N^a, are they the same vector but viewed by different coordinate systems?

    If Yes, which map allows us to pass from one coordinate systems to another?
     
  8. Jun 18, 2011 #7

    atyy

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    I don't think so. It looks like if we take Peldan's tetrad basis vectors to be coordinate basis vectors, then http://arxiv.org/abs/gr-qc/9305011" [Broken] Eq 4.31.

    So Peldan's and Gourgoulhon's N are the same. Peldan's NI is Gourgoulhon's n. Peldan's Na are the components of Gourgoulhon's β.
     
    Last edited by a moderator: May 5, 2017
  9. Jun 18, 2011 #8

    marcus

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    Just as a general (historical?) reference you might be interested in looking at the original ADM paper:
    http://arxiv.org/abs/gr-qc/0405109

    This dates back to around 1962. I think it was a chapter in a book compiled by Louis
    Witten and published in 1962. You are probably familiar with arxiv. If not just click where it says "pdf" for a free download.

    Another free online source that might be useful as context is the draft version of Rovelli's book "Quantum Gravity". It is not the final version that was published by Cambridge U. Press in 2004, but it is pretty close to final as you might expect in the early chapters coverning standard material.

    The link is posted here
    http://www.cpt.univ-mrs.fr/~quantumgravity/
    The pdf link, for download, is this
    http://www.cpt.univ-mrs.fr/~rovelli/book.pdf
     
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