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B Mean of G

  1. Oct 12, 2016 #1
    Is it possible to take the "Eistein Scalar" from the Einstein Tensor, like one can take the Ricci Scalar from the Ricci Tensor? If so, is the G of Newton's law the same as this "Einstein Scalar" or is it just the same symbol used in very different things.

    (Sorry for my bad English. I think Meaning of G would be a better title for the thread.)
     
    Last edited: Oct 12, 2016
  2. jcsd
  3. Oct 12, 2016 #2

    pervect

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    The term isn't used much, but given the Ricci scalar ##R = R^a{}_a## , we can write the equivalent for the Einstein tensor, ##G^a{}_a = -R^a{}_a##. See for instance http://math.ucr.edu/home/baez/gr/outline2.html.

    In particular, note that ##R = R^a{}_a = -T^a{}_a##, while ##G^a{}_a = T^a{}_a##, thus ##G^a{}_a = -R^a{}_a##

    Because one is just the negative of the other, people usually talk about the Ricci scalar ##R^a{}_a## and there's no real need to talk about ##G^a{}_a##.

    Neither one is related to the G in Newton's law, which can be regarded as a unit conversion factor. Physically, the significance of R would be that if we choose a local orthonormal frame, ##R = \rho - P_{xx} - P_{yy} - P_{zz}## or in the case where the pressure is isotropic, ##R = \rho - 3P##. Here ##\rho## is the density (energy per unit volume), and P is the pressure (or the x, y, and z components of the pressure if they're not all the same).
     
    Last edited: Oct 12, 2016
  4. Oct 12, 2016 #3
    Many thanks.
     
  5. Oct 12, 2016 #4

    pervect

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    You're welcome - I should probably add that I followed Baez's lead in glossing over some constant conversions factors when I wrote ##G_{aa} = T_{aa}## rather than ##G_{aa} = 8 \pi T_{aa}## or even ##G_{aa} = (8 \pi G / c^4) T_{aa}##.
     
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