Finding Constant \alpha_M in SET Definition

[elfmotat]

So I was looking through Wald when I noticed his definition of the stress-energy for an arbitrary matter field:

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

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

\mathcal{L}=R\sqrt{-g}+\alpha_M \mathcal{L}_M

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

I feel like I'm missing something painfully obvious.

 

[Bill_K]

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

The Einstein Equations are G_μν = 8πG T_μν. To get this equation we use an action I = I_G + I_M where I_G = (1/16πG) ∫√-g R d⁴x and T_μν = (2/√-g) δI_M/δg_μν.

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