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
$${T}_{munu}=-frac{2}{sqrt{-g}} frac{delta{mathcal{L}}_M}{delta{g}^{munu}}, tag{1}$$
$${T}_{munu}=-frac{2}{sqrt{-g}} frac{delta{mathcal{S}}_M}{delta{g}^{munu}}, tag{2}$$ or even
$${T}_{munu}=-frac{2}{sqrt{-g}} frac{delta(mathcal{L}_Msqrt{-g})}{delta{g}^{munu}}. tag{3}$$
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 $(-,+,+,+)$.]

Qmechanic · Accepted Answer

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
$$ frac{delta {cal L}(x)}{deltaphi^{alpha} (x)}~:=~ 
frac{partial{cal L}(x) }{partialphi^{alpha} (x)} - d_{mu} left(frac{partial{cal L}(x) }{partialpartial_{mu}phi^{alpha} (x)} right)+ldots, $$
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 $L$.

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

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

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