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Constant in SET Definition

  1. Jan 30, 2013 #1
    So I was looking through Wald when I noticed his definition of the stress-energy for an arbitrary matter field:

    [tex]T_{ab}=-\frac{\alpha_M}{8\pi} \frac{1}{ \sqrt{-g}} \frac{\delta S_M}{\delta g^{ab}}[/tex]

    where [itex]S_M[/itex] is the action for the particular type of matter field being considered, and [itex]\alpha_M[/itex] is some constant that determines the form of the Lagrangian for the coupled Einstein-matter field equations:

    [tex]\mathcal{L}=R\sqrt{-g}+\alpha_M \mathcal{L}_M[/tex]

    For example, for a Klein-Gordon field we take [itex]\alpha_{KG}=16\pi[/itex], and for an EM field we take [itex]\alpha_{EM}=4[/itex]. Now, my question is whether or not there is some prescription for finding the value of [itex]\alpha_M[/itex]. How could I go about finding [itex]\alpha_M[/itex] for an arbitrary [itex]\mathcal{L}_M[/itex]?

    I feel like I'm missing something painfully obvious.
  2. jcsd
  3. Jan 31, 2013 #2


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    Science Advisor

    I guess everybody has his own conventions. The usual ones follow.

    The Einstein Equations are Gμν = 8πG Tμν. To get this equation we use an action I = IG + IM where IG = (1/16πG) ∫√-g R d4x and Tμν = (2/√-g) δIM/δgμν.

    For electromagnetism, L = (-1/4)FμνFμν. This is in Heaviside units where e2/4πħc = 1/137. In Gaussian units where e2/ħc = 1/137, the Lagrangian would instead be IM = (-1/16π)FμνFμν.
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