The Hilbert stress-energy tensor of a metric-dependent action

Let $(M,g,N)$ be an oriented time-oriented smooth Lorentzian four-manifold (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, Oriented smooth manifolds and oriented charts), let $c>0$ be a fixed real constant, let $\mathcal{X}$ be a set, and let

$S:\mathcal{D}⟶\mathbb{R}$

be a real-valued functional defined on a subset $\mathcal{D}$ of the set of pairs $(\tilde{g},\phi )$ with $\tilde{g}$ a smooth Lorentzian metric on $M$ of signature $(-,+,+,+)$ and $\phi \in \mathcal{X}$. Fix $(\tilde{g},\phi )\in \mathcal{D}$; below we always take $\tilde{g}=g$. The elements $\phi$ are the model's matter configurations; nothing about their nature is assumed here. Assume that $\mathcal{D}$ admits every compactly supported metric direction at $(g,\phi )$: for each smooth symmetric covariant $2$-tensor field $h$ with compact support, there is ${\epsilon }_h>0$ such that $(g+sh,\phi )\in \mathcal{D}$ for every $\mathrm{\mid }s\mathrm{\mid }<{\epsilon }_h$. Since $\mathcal{D}$ contains only pairs whose metric is Lorentzian, these admitted linear families are Lorentzian throughout that interval. This condition makes every compactly supported inverse-metric variation available as well, since $h\mapsto -g^{ac}{g}^{bd}{h}_{cd}$ is a bijection on compactly supported smooth symmetric tensor fields.

(a) Metric families. A smooth metric family at $g$ with compactly supported variation is a map $s\mapsto g_s$, $\mathrm{\mid }s\mathrm{\mid }<\epsilon$ for some $\epsilon >0$, such that $(g_s,\phi )\in \mathcal{D}$ for all $s$, such that $g_s$ is a smooth Lorentzian metric of signature $(-,+,+,+)$ for every $s$, such that in every chart $(U,x)$ the component functions $(s,x(p))\mapsto (g_s{)}_{ab}(p)$ are jointly $C^{\mathrm{\infty }}$ on $(-\epsilon ,\epsilon )\times x(U)$, such that $g_0=g$, and such that there is a compact subset $K\subseteq M$ with $g_s=g$ outside $K$ for every $\mathrm{\mid }s\mathrm{\mid }<\epsilon$. For such a family define

$q^{ab}:=\frac{d}{ds}{g}_s^{ab}{\mid }_{s=0},$

the derivative at $s=0$ of the components of the inverse metric of $g_s$ in any smooth chart; the family of functions $q^{ab}$ is a smooth symmetric contravariant $2$-tensor field of compact support, well defined because the components $g_s^{ab}$ depend smoothly on $s$ (The inverse metric, musical isomorphisms and index raising, Tensor fields of type (r,s) on a Lorentzian four-manifold and their components).

(b) Hilbert tensor. A smooth symmetric covariant $2$-tensor field $T$ on $M$ is a Hilbert stress-energy tensor of $S$ at $(g,\phi )$ with respect to $c$ when, for every smooth metric family $s\mapsto g_s$ at $g$ with compactly supported variation,

$\frac{d}{ds}S(g_s,\phi ){\mid }_{s=0}=-\frac{1}{2c}{\int }_M{T}_{ab}{q}^{ab}{\mathrm{v}\mathrm{o}\mathrm{l}}_g,$

where ${\mathrm{v}\mathrm{o}\mathrm{l}}_g$ is the metric volume form of The metric volume form, current flux through 3-faces and finite-chain charges and ${\int }_M\omega$ is the integral of the compactly supported smooth top form $\omega :=T_{ab}{q}^{ab}{\mathrm{v}\mathrm{o}\mathrm{l}}_g$ over the oriented manifold $M$ (Finite chart localization gives choice-free integration and compact Stokes). The right-hand side is defined because $q^{ab}$ has compact support, so $\omega$ is compactly supported. If such a $T$ exists it is unique, by Uniqueness of the Hilbert stress-energy tensor ↗; one then calls it the Hilbert stress-energy tensor of $S$ at $(g,\phi )$.

(c) Conventions and scope. The sign and the factor $1\mathrm{/}(2c)$ are fixed conventions of this library: they are the ones that turn the framework's adopted Einstein-Hilbert normalization into the field equation displayed in the framework's physical postulate. The condition is a property of the functional $S$ at the point $(g,\phi )$; it is not automatic, and this definition does not assert that any functional has a Hilbert tensor. The definition uses the orientation of $M$; a general functional $S$ has no orientation-reversal law, so no transformation of $S$ or $T$ is asserted. When $S$ is given by integrating $L(g,\phi ){\mathrm{v}\mathrm{o}\mathrm{l}}_g$ with an orientation-independent scalar coefficient $L$ and the integral exists (or compact action differences are used), define the reversed-orientation action by the same construction. Reversing orientation replaces ${\mathrm{v}\mathrm{o}\mathrm{l}}_g$ by $-{\mathrm{v}\mathrm{o}\mathrm{l}}_g$ and reverses the sign of integration of a fixed top form. These two signs cancel, so both the action variation and the integral in the defining identity are unchanged, as is its tensor $T$.

• Relation to the invariant-coordinate form. In a chart, writing $\delta$ generically for the derivative at $s=0$, the defining identity reads $\delta S=-\frac{1}{2c}\int T_{ab}\delta g^{ab}w{d}^{4}x$ with $w$ the metric density of The metric volume form, current flux through 3-faces and finite-chain charges. The normalization must be compared at the same functional $S$. When its smooth symmetric coordinate variational derivative $D_{ab}:=\delta S\mathrm{/}\delta g^{ab}$ exists, meaning $\delta S=\int D_{ab}\delta g^{ab}{d}^{4}x$ for compactly supported variations, our convention gives $T_{ab}=-2c{w}^{-1}{D}_{ab}$. Carroll's (4.70) instead gives $T_{ab}^{\mathrm{C}}=-w^{-1}{D}_{ab}$, so $T_{ab}=2c{T}_{ab}^{\mathrm{C}}$ for the same $S$. Tong's (4.46) gives $T_{ab}^{\mathrm{T}}=-2w^{-1}{D}_{ab}$; Reall's (9.28)-(9.29), after $\delta g_{ab}=-g_{ac}{g}_{bd}\delta g^{cd}$, gives that same coefficient. Thus our tensor is $c$ times their tensors for the same $S$. Rescaling $S$ also rescales its tensor; matching physical normalizations therefore requires specifying the action normalization as well as the factor $c$.

• The matter configuration is held fixed. The definition concerns the partial derivative of $S$ in the metric direction at fixed $\phi$; the matter equations of motion are a separate condition on the same functional and are not part of this definition.