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If the fields $\phi^j$ are all scalar and $L(\phi,g)$ does not depend on derivatives of $g$, then $T^{\mu\nu}$ coincides with the canonical stress-energy tensor. This is no longer the case for spinor fields, whose Lagrangian usually also depends on the first derivatives of the metric through the spin connection, for scalar fields with non-minimal curvature coupling, or for the electromagnetic field.

If the fields $\phi^j$ are all scalar and $L(\phi,g)$ is a Lagrangian of first order in $\phi$ with a Klein-Gordon-like kinetic part and not depending on derivatives of $g$, then $T^{\mu\nu}$ coincides with the canonical stress-energy tensor. This is no longer the case for spinor fields, whose Lagrangian usually also depends on the first derivatives of the metric through the spin connection, for scalar fields with non-minimal curvature coupling, or for the electromagnetic field.

Let $L=L(\phi,g)$ be a local field Lagrangian in the space-time $(M,g)$, and $$S_K[\phi,g]=\int_K L(\phi,g)\sqrt{|\det g|}\mathrm{d}x\ ,\quad K\subset M\text{ any bounded region}$$ the corresponding (family of) action functional(s indexed by $K$ as above). We allow $L$ to have finite but otherwise arbitrary order dependence on $\phi$ and $g$, and no explicit space-time dependence since we want it not to depend on any other fields. The infinitesimal variation of $S_K$ with respect to a vector field $X$ on $M$ (i.e. an infinitesimal diffeomorphism) is then given by $$\delta_X S_K[\phi,g]=\int_K\left(\frac{\delta L(\phi,g)}{\delta g_{\mu\nu}}\delta_X g_{\mu\nu}+\frac{\delta L(\phi,g)}{\delta \phi^j}\delta_X \phi^j+\nabla_\mu(T^{\mu\nu}X_\nu)\right)\sqrt{|\det g|}\mathrm{d}x\ ,\quad X_\rho=g_{\rho\sigma}X^\sigma\ ,$$ where $\frac{\delta L(\phi,g)}{\delta g_{\mu\nu}}$ and $\frac{\delta L(\phi,g)}{\delta \phi^j}$ are respectively the Euler-Lagrange (i.e. variational) derivatives of $L(\phi,g)$ with respect to $g$ and $\phi$, $\nabla$ is the Levi-Civita covariant derivative associated to $g$, $T^{\mu\nu}$ is the (canonical or improved) stress-energy tensor, $$\delta_X g_{\mu\nu}=\nabla_\mu X_\nu+\nabla_\nu X_\mu$$ is the Lie derivative of $g$ along $X$ and $\delta_X\phi^j$ depends on the particular way we lift $X$ to a projectable vector field on the total space of the fiber bundle over $M$ where the fields $\phi^j$ live (for instance, if they are all scalar fields, we simply have $\delta_X\phi^j=-X\phi^j=-X^\mu\nabla_\mu\phi^j$).

There is an implicit but crucial requirement on the admissible improvements for $T^{\mu\nu}$ - namely, the improved Noether current $j^\mu(L,X)=T^{\mu\nu}X_\nu$ associated with the would-be symmetry $X$ of the action functional should not only be linear in $X$ but depend only on the point values of $X$ (we call this property ultralocality) - therefore, we wrote it already as a tensor contraction. This requirement also affects to a certain extent the definition of the infinitesimal field variation $\delta_X\phi^j$, but the details of this are not important in what follows. Why do we insist on this requirement? As we shall see below, ultralocality singles out a unique improvement prescription for $T^{\mu\nu}$ which in addition satisfies all physical desiderata.

Let $L=L(\phi,g)$ be a local field Lagrangian in the space-time $(M,g)$, and $$S_K[\phi,g]=\int_K L(\phi,g)\sqrt{|\det g|}\mathrm{d}x\ ,\quad K\subset M\text{ any bounded region}$$ the corresponding (family of) action functional(s indexed by $K$ as above). We allow $L$ to have finite but otherwise arbitrary order dependence on $\phi$ and $g$, and no explicit space-time dependence since we want it not to depend on any other fields. The infinitesimal variation of $S_K$ with respect to a vector field $X$ on $M$ (i.e. an infinitesimal diffeomorphism) is then given by $$\delta_X S_K[\phi,g]=\int_K\left(\frac{\delta L(\phi,g)}{\delta g_{\mu\nu}}\delta_X g_{\mu\nu}+\frac{\delta L(\phi,g)}{\delta \phi^j}\delta_X \phi^j+\nabla_\mu(T^{\mu\nu}X_\nu)\right)\sqrt{|\det g|}\mathrm{d}x\ ,\quad X_\rho=g_{\rho\sigma}X^\sigma\ ,$$ where $\frac{\delta L(\phi,g)}{\delta g_{\mu\nu}}$ and $\frac{\delta L(\phi,g)}{\delta \phi^j}$ are respectively the Euler-Lagrange (i.e. variational) derivatives of $L(\phi,g)$ with respect to $g$ and $\phi$, $\nabla$ is the Levi-Civita covariant derivative associated to $g$, $T^{\mu\nu}$ is the (canonical or improved) stress-energy tensor, $$\delta_X g_{\mu\nu}=\nabla_\mu X_\nu+\nabla_\nu X_\mu$$ is the Lie derivative of $g$ along $X$ and the infinitesimal field variation $\delta_X\phi^j$ depends on the particular way we lift $X$ to a projectable vector field on the total space of the fiber bundle over $M$ where the fields $\phi^j$ live (for instance, if they are all scalar fields, we simply have $\delta_X\phi^j=-X\phi^j=-X^\mu\nabla_\mu\phi^j$).

There is an implicit but crucial requirement on the admissible improvements for $T^{\mu\nu}$ - namely, the improved Noether current $j^\mu(L,X)=T^{\mu\nu}X_\nu$ associated with the would-be symmetry $X$ of the action functional should not only be linear in $X$ but depend only on the point values of $X$ (we call this property ultralocality) - therefore, we wrote it already as a tensor contraction. This requirement also affects to a certain extent the definition of $\delta_X\phi^j$, but the details of this are not important in what follows. Why do we insist on this requirement? As we shall see below, ultralocality singles out a unique improvement prescription for $T^{\mu\nu}$ which in addition satisfies all physical desiderata. This idea applies more generally to any local symmetry - for instance, it may be used to improve the canonical Noether current associated with local gauge symmetries.

Actually, the metric variational definition for the stress-energy tensor is an universal improvement procedure for the canonical stress-energy tensor (and hence not always concides with the latter), in a sense which will be made precise below. Such a procedure is necessary because the canonical stress-energy tensor, although always conserved, often fails to satisfy other physical requirements like gauge invariance (since it is an observable quantity), symmetry (needed if we want it to be a source for the gravitational field) and tracelessness (for locally scale invariant theories). For example, all three requirements fail for pure electrodynamics in four space-time dimensions.

Actually, the metric variational definition for the stress-energy tensor (due to Hilbert, as remarked by Qmechanic) is an universal improvement procedure for the canonical stress-energy tensor (and hence not always concides with the latter), in a sense which will be made precise below. Such a procedure is necessary because the canonical stress-energy tensor, although always conserved, often fails to satisfy other physical requirements like gauge invariance (since it is an observable quantity), symmetry (needed if we want it to be a source for the gravitational field) and tracelessness (for locally scale invariant theories). For example, all three requirements fail for pure electrodynamics in four space-time dimensions.