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Einstein-Hilbert term and mass, gravity

  1. Dec 25, 2012 #1
    It is said that the Einstein-Hilbert term in the following couple mass and gravity.

    Lagrangian = R/(16*pi*GN) [Einstein-Hilbert term] + L (nongravitational)

    How is the above derived? Is it enough to prove that there should be a theory of quantum gravity since mass and gravity are coupled together in that term?
  2. jcsd
  3. Dec 25, 2012 #2


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    Gold Member

    That Lagrangian does not have the form L = Lgrav + Lmatter + Linteraction which is usually necessary. If there was such a term it would be something like θabTab where θ is the field and T is the matter SET as in field theory gravity (FTG). I understand that this term gives rise to the spin-2 carrier bosons. I'm no expert though.
  4. Dec 25, 2012 #3


    Staff: Mentor

    Can you give a specific reference where this is said? The usual interpretation that I'm familiar with is that the Einstein-Hilbert term only contains spacetime curvature (i.e., "gravity"), and the other term, which you call L(nongravitational), only contains the "matter", which means all non-gravitational fields. The only coupling between the two, as far as classical GR is concerned, comes from the Einstein Field Equation, which is obtained by varying the Lagrangian with respect to the metric.
  5. Dec 26, 2012 #4
    What I heard is just like what you described, that the coupling between them comes from the Einstein Field Equation, which as you said is "obtained by varying the Lagrangian with respect to the metric". Can you please describe what it means to "vary the Lagrangian with respect to the metric"? Is varying the Lagrangian natural or kinda forced?
  6. Dec 26, 2012 #5


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    That's the whole idea behind the action principle; you regard the metric as fundamental field. The action is then a functional of the metric, and its extremum gives the EOM.

    You can look at e.g. Carroll's notes for a nice motivation for this action: it is "simple" and contains up to second order derivatives of the metric.

    Btw, it's not just mass, but energy in general which couples to gravity. An electromagnetic field curves spacetime, deflecting even electrically neutral particles.
  7. Dec 26, 2012 #6


    Staff: Mentor

    That's not the same as saying "the Einstein-Hilbert term couples mass and gravity", which is what you said in the OP. That's why I wasn't sure what you were referring to.

    As haushofer said, you find the extremum of the action (the action is just the integral of the Lagrangian over all of spacetime), and that gives you the equation of motion (which is what the Einstein Field Equation is: it's the "equation of motion" for gravity).

    Finding the extremum just means taking the derivative (of the action) and finding where it is zero. Varying "with respect to the metric" just means the metric is what you take the derivative of the action with respect to.

    I'm not sure what you would consider "natural" vs. "forced". Finding the extremum of a function by finding where its derivative is zero is an elementary operation in calculus.
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