- #1

elfmotat

- 260

- 2

[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.