Definition of the stress-energy tensor in terms of functional derivatives in G.R

Question

I have found confusing definitions in various places regarding the stress-energy tensor, in particular when used to derive Einstein GR equations from the principle of stationary action. Some of these various definitions are $${T}_{\mu\nu}=-\frac{2}{\sqrt{-g}} \frac{\delta{\mathcal{L}}_M}{\delta{g}^{\mu\nu}}, \tag{1}$$ $${T}_{\mu\nu}=-\frac{2}{\sqrt{-g}} \frac{\delta{\mathcal{S}}_M}{\delta{g}^{\mu\nu}}, \tag{2}$$ or even $${T}_{\mu\nu}=-\frac{2}{\sqrt{-g}} \frac{\delta(\mathcal{L}_M\sqrt{-g})}{\delta{g}^{\mu\nu}}. \tag{3}$$

I have been able to follow the derivation leading to the G.R. equations using the definition $(1)$, which I have also seen in these questions. But then I found the other definitions here which really confused me. Is $(1)$ the correct one? Otherwise, which one is correct?

[Here I'm using the Minkowski sign convention $(-,+,+,+)$.]

Answer 1

All definitions (1)-(3) are in principle the same. However, the various notations$^1$ may warrant some explanation:

• Eq. (2) uses the standard/traditional definition of a functional/variational derivative (FD) of the action functional $S=\int\!d^dx ~{\cal L}$ in $d$ spacetime dimensions.

• Eq. (1) uses a 'same-spacetime' FD $$ \frac{\delta {\cal L}(x)}{\delta\phi^{\alpha} (x)}~:=~ \frac{\partial{\cal L}(x) }{\partial\phi^{\alpha} (x)} - d_{\mu} \left(\frac{\partial{\cal L}(x) }{\partial\partial_{\mu}\phi^{\alpha} (x)} \right)+\ldots, $$ which obscures/betrays its variational origin, but is often used for notational convenience. See e.g. this, this, & this Phys.SE posts and links therein.

• Eq. (3) is the same as eq. (1), except the Lagrangian density ${\cal L}=\sqrt{|g|}L$ is written$^1$ as a product of a density $\sqrt{|g|}$ and a scalar function $L$.

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$^1$ As always, be aware that that different authors use different notation.