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Conservation laws in a curve spacetime

  1. Feb 22, 2007 #1
    1. The problem statement, all variables and given/known data

    Given the energy-momentum tensor for a perfect fluid, what is the conservation laws that I can compute from

    [tex]
    \nabla_b T^{ab}=0
    [/tex]

    in a curve space-time.

    2. Relevant equations

    [tex]
    T^{ab}=(\rho + p) u^a u^b - p g^{ab}
    [/tex]

    where p is the pressure and [itex]\rho[/itex] is the density.

    3. The attempt at a solution

    I have already compute the geodesic equation, the equivalent to the equation of motion in a flat space-time.
    I would like to know if there is any other conservation law to get. I supose that might be another one equivalent to the Navie-Strokes in flat space-time...

    Thanks in advance
     
  2. jcsd
  3. Feb 22, 2007 #2

    Dick

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    Science Advisor
    Homework Helper

    Just write [tex]\nabla_b T^{ab}=0[/tex] explicitly in components (in terms of density, pressure and scale factor). You will get a conservation type eqn.
     
  4. Feb 22, 2007 #3
    In terms of scale factor? The problem is that they dont give me a metric. I dont know what to with the [itex]g^{ab}[/itex] term...
     
  5. Feb 22, 2007 #4

    Dick

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    I guess I was thinking of the problem in FRW background. Can you simplify anything w/o a metric???? Not sure...
     
  6. Feb 22, 2007 #5
    Well. I have already compute the equation of motion, in a curve space, from [itex]\nabla_b T^{ab}=0[/itex]. So I suppose that there may be another one...
     
  7. Feb 24, 2007 #6
    In general, there isn't any other conservation law, besides the covariant divergence of the energy-momentum tensor. However, if you have a space-time with a continuous symetry, there's a Killing field defined on the space-time, associated to that symettry. It can then serves to define another conservation law, in a similar way as a Noether current.
     
  8. Feb 24, 2007 #7
    What about this:

    [tex]

    \nabla_b T^{ab}=\nabla_b((\rho + p) u^a u^b) - g^{ab} \nabla_b p = 0

    [/tex]

    where I have used [itex]\nabla_b g^{ab} = 0[/itex]

    [tex]
    u^a \nabla_b((\rho + p) u^b) + (\rho + p) u^b \nabla_b u^a -g^{ab} \nabla_b p = 0
    [/tex]

    [tex]
    u_a u^a \nabla_b((\rho + p) u^b) + (\rho + p) u^b u_a \nabla_bu^a -g^{ab} u_a \nabla_b p = 0
    [/tex]

    using the fact that [itex]\nabla_b (u_a u^a) = \nabla_b 1 \Leftrightarrow 2 u_a \nabla_b u^a = 0[/itex], I get

    [tex]
    \nabla_b((\rho + p) u^b) = g^{ab} u_a \nabla_b p
    [/tex]

    using this in the second equation I get

    [tex]
    (\rho + p) u^b \nabla_b u^a = 0
    [/tex]

    or

    [tex]
    \nabla_u u^a = 0
    [/tex]

    So I can saw that the equation of motion (a conservation law) can be derived from [itex]\nabla_b T^{ab}=0[/itex].
    Am I wrong?

    Just has I said above, they dont give me any metric, so I cant compute the Killing vectors.
     
    Last edited: Feb 25, 2007
  9. Feb 25, 2007 #8
    From equation

    [tex]

    \nabla_b T^{ab}=\nabla_b((\rho + p) u^a u^b) - g^{ab} \nabla_b p = 0,

    [/tex]

    you get the following :

    (1) [tex]
    \nabla_a ((\rho + p) u^a) = u^a \nabla_a p,
    [/tex]

    AND :

    (2) [tex]
    (\rho + p) u^a \nabla_a ( u^b) = (g^{ab} - u^a u^b) \nabla_a p.
    [/tex]

    In the case of a pressureless gaz, p = 0, and you get the conservation of "matter" AND the geodesics equation :

    (3) [tex]
    \nabla_a (\rho u^a) = 0,
    [/tex]

    AND :

    (4) [tex]
    u^a \nabla_a ( u^b) = 0.
    [/tex]


    There's no need of any particular metric to define conserved quantites with the Killing fields.
     
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