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I can't see how stress-energy tensor meets the minumum tensor requirement

  1. Jun 17, 2004 #1
    Gentlemen,

    I am sorry. I did a few typing errors here in order to put latex in and I even use 12 minute = 1 hour. This might confuse you.

    Let me try to correct this.

    Said , I use the simple dust model with 216,000 grams in a volume of 1 light-hour^3. So, [tex] T^\mu\nu [/tex] = diag(216000 , 0, 0, 0) in a coordinates using light-hour as the unit.

    If now, I change to the unit of light-minute. The energy density shall be 216000 gram/ 60^3 = 1 gram/light-minute^3. So, in this coordinate
    [tex] T^{\mu\nu} [/tex] = diag(1,0,0,0).

    Now, if I use the standard tensor translation:
    [tex] T^{\mu' \nu'} [/tex] = [tex] T^{\mu\nu} * \partial x^\mu' / \partial x^\mu * \partial x^\nu' / \partial x^\nu [/tex]

    I will never get it right.
    Rather, [tex] \partial x^\mu' / \partial x^\mu [/tex] = diag (60, 60, 60, 60).

    Every 60 light-minute equals to 1 light-hour. For a point as (1,1,1,1) in the [tex] \mu [/tex] coordinate, its coordinates will be (60,60,60,60) for the light-minute coordinates.

    I will have 216000*3600 for the energy density for the stress-energy tensor in the coordiante of light-minute then.

    Did I do something wrong here?

    If not, how do you reconcile this?

    Thanks
     
    Last edited: Jun 18, 2004
  2. jcsd
  3. Jun 17, 2004 #2
    Gentlemen,

    Sorry,

    I intend for upper [tex] \mu [/tex] , but can't get it working.
     
  4. Jun 17, 2004 #3

    selfAdjoint

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    Use braces {} around your mu and nu thus: T^{\mu\nu}.
     
  5. Jun 17, 2004 #4
    Thanks.

    Any way, I figured out what's going on. It's a convenient way for you to write it and describe [tex] T^{00} [/tex] as mass density.

    Actually, it is probably more correct to write a tensor as
    [tex] T^\mu _{abc} [/tex]
    in away such that
    [tex] T^0 _{abc} = (1/6)*Mass*Time/Volume [/tex] for (a,b,c) = perm(1,2,3)
    and
    [tex] T^i _{abc} = (1/6)*Mass*Length/(Area*Time) [/tex] for i not= 0 and (a,b,c) = perm(0,1,2) or perm(0,1,3) or perm(0,2,3)

    The above tensor could be used in integration.

    And the stress-energy tensor in most article will be:

    [tex] T^{\mu\nu} = T^\mu _{abc} \epsilon^{abc\nu} [/tex]

    Any way a definition taking out the mass density part as a coefficient probablly will do too.

    Regards
     
  6. Jun 24, 2004 #5
    Robphy has introduced this article in the thread about energy:

    http://relativity.livingreviews.org...04-4/index.html

    In its page 11, EQ (6) has shown how the symmetric stress tensor needs to be transformed to a 3-form for integration, so as in EQ (7). .

    In its page 26, a tensor of rank(0,4) as 'Bel-Robinson" momentum was shown how it can be integrated to become a quantity as energy.

    In its page 25, one of the approach to a conserved quantity could be integration over a 4-dimensional domain. This of course will better be a tensor of either rank (0,4) or (1,4). I think.

    Most of current approaches mentioned treat energy as a quantity pertained to a spacelike 3-dimensional hypersurface.

    I wonder it's possible to look for a physic quantity pertained to a 4-dimensional domain.
     
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