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A Divergence of (covaraint) energymomentum tensor

  1. Dec 25, 2017 #26
    I understand it now. Thank you all very much for the help. You have been excellent in explaining it.
     
  2. Dec 25, 2017 #27

    Orodruin

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    If you struggle with this kind of manipulations I would strongly suggest that you read up on differential geometry using a dedicated text that is more detailed than the summary you would typically find in a GR textbook.
     
  3. Dec 25, 2017 #28
    And again welcome to PF! I hope it works for you now. I'll go back to the chat later (it's still there).

    Note: within 4hrs you can edit your last post (in general) instead of making many consecutive ones, if no one is in between.
     
    Last edited: Dec 25, 2017
  4. Dec 25, 2017 #29

    Dale

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    Younshould not use the lower expression ever. It is syntactically wrong. Never contract a lower index with a lower index.
     
  5. Dec 26, 2017 #30
    Thank you all again.
    I will ask for much today :-)
    I would like to know name of a good reference in introductory Differential Geometry.
    I struggled whole night with the second expression below, I couldn't figure out the positions of the indices

    \[\begin{array}{l}
    {T^{ab}}_{;b} = {T^{ab}}_{,b} + {\Gamma _{bc}}^a{T^{bc}} + {\Gamma _{bd}}^b{T^{ad}} \\
    {T_{ab}}^{;b} = {T_{ab}}^{,b} - {\Gamma _{..}}^{.}{T_{..}} - {\Gamma _{..}}^{.}{T_{..}} \\
    \end{array}\]
    but happily I could write latex expression in PF :-)

    I need a paper which I don't have and I couldn't afford and my institution too! one of these is about non-conserved of energy momentum tensor by Rastall.
     
    Last edited: Dec 26, 2017
  6. Dec 26, 2017 #31
    I will address just the first part (as I am currently ~ in bed and my Diff. Geometry won't be that walken up yet).
    In college, to get ready for General Relativity, I used:
    D. F. Lawden "Introduction to Tensor Calculus, Relativity and Cosmology" (e.g. Dover publ.)
    https://www.amazon.com/Introduction-Calculus-Relativity-Cosmology-Physics/dp/0486425401
    It's kind of educational and affordable.
    I believe you can also find it on line (e.g. pdf)
    https://www.google.gr/search?ei=lvB...67j0i7i30j0i22i10i30j33i160j0i13.ide/B7FO8fE=
     
  7. Dec 26, 2017 #32
    Sorry, there a little mistake in the second expression. The first term in the RHS should be the ordinary partial derivative instead of the covariant derivative.
     
  8. Dec 26, 2017 #33

    Orodruin

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    That would be because you raised the index of the covariant derivative and you simply cannot do that without involving the metric tensor in your expression.
    $$
    T_{ab}^{;b} = \nabla^{b} T_{ab} = g^{bc} \nabla_c T_{ab} = g^{bc} (T_{ab,c} - \Gamma_{ca}^d T_{db} - \Gamma_{cb}^d T_{ad})
    = T_{ab}^{,c} - g^{bc} \Gamma_{ca}^d T_{db} - g^{bc} \Gamma_{cb}^d T_{ad}
    $$
     
  9. Dec 26, 2017 #34
    I thought the metric goes immediately to work into the tensor indices to give a mixed tensor not to the differentiation index itself. I am learning a lot.
    Thank you very much Orodruin.
     
  10. Dec 26, 2017 #35

    Orodruin

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    Of course you can absorb the metric tensor into the energy momentum tensor with lower indices and obtain a mixed tensor, but that did not seem to be what you were after as your template expression included ##T_{\cdot\cdot}##.

    Edit: Note that in the term on the form ##\Gamma_{\cdot\cdot}^\cdot T_{\cdot\cdot}## there is no way to place indices such that there is only one free covariant index left, since there are 4 covariant and only one contravariant index and the left-hand side has only one free covariant index. This should immediately tell you that it is impossible to write the term on that form.
     
  11. Dec 26, 2017 #36
  12. Dec 26, 2017 #37

    Orodruin

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    At least in the sections you are looking at, that reference discusses special relativity with the Euclidean metric using an imaginary time component. It also introduces relativistic mass, which is an essentially deprecated concept. I would dare to say that you will not find any of this in a modern textbook and I suggest that you get a more modern reference (that text is over 50 years old).
     
  13. Dec 26, 2017 #38
    Thank you very much. What about non conserved Energy-momentum in General Relativity because the energy of the gravitational filed in not included in it, and that gravity can be removed locally "free fall"?
     
  14. Dec 26, 2017 #39
    Edit: Note that in the term on the form
    \[\begin{array}{l}
    {\Gamma _{..}}^{.}{T_{..}} \\
    \end{array}\]
    there is no way to place indices such that there is only one free covariant index left, since there are 4 covariant and only one contravariant index and the left-hand side has only one free covariant index. This should immediately tell you that it is impossible to write the term on that form.

    Very clever! helping a lot in understanding and memorizing it. Thank you.
     
  15. Dec 26, 2017 #40

    Orodruin

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    The stress energy tensor is locally conserved in GR ##\nabla_a T^{ba} = 0##. There is generally no good way of defining global energy conservation.
     
  16. Dec 26, 2017 #41
    I quote "no good way of defining global energy conservation", why is that?
     
  17. Dec 26, 2017 #42

    Orodruin

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  18. Dec 26, 2017 #43
    It is very well explained except the paragraph
    "We will not delve into definitions of energy in general relativity such as the hamiltonian (amusingly, the energy of a closed universe always works out to zero according to this definition), various kinds of energy one hopes to obtain by "deparametrizing" Einstein's equations, or "quasilocal energy". There's quite a bit to say about this sort of thing! Indeed, the issue of energy in general relativity has a lot to do with the notorious "problem of time" in quantum gravity... but that's another can of worms."
     
  19. Dec 28, 2017 #44
    (Already shared these references in a conversation with Torg yesterday - they may be useful to others too)

    Take a look at this thread:
    https://www.physicsforums.com/threads/book-on-general-relativity.874853/

    Tensor calculus is generally part of differential geometry. Spivak's book is the one I was trying to recall. Try a google search for:
    Spivak, "Comprehensive Introduction to Differential Geometry"

    For tensors: J.L. Synge, A. Schild, Tensor Calculus (e.g. Dover publ.)
    (Traditional)

    See also
    https://www.physicsforums.com/threads/book-recommendations-in-differential-geometry.917075/

    And
    https://www.physicsforums.com/threads/differential-geometry-book-with-tensor-calculus.880156/

    Or
    https://www.physicsforums.com/threads/a-good-book-on-tensors.914341/

    I hope that helps.
     
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