Covariant stationary Noether charge and Wald entropy

Let $n\ge 2$, let $M$ be an oriented, time-oriented smooth $n$-manifold, and let $g$ be a smooth Lorentzian metric of signature $(-,+,\dots ,+)$. Write ${\mathit{ϵ}}_g$ for its oriented metric volume form. Let $\mathrm{\nabla }$ be its Levi-Civita connection, the unique torsion-free, metric-compatible connection. Let $\varphi =(g,\psi )$ denote a smooth section of a specified field bundle $\pi :E\to M$, including the metric and any other dynamical fields $\psi$ that are smooth tensor fields with the pullback action of spacetime diffeomorphisms. An admitted field family is a smooth map $\mathrm{\Phi }:P\times M\to E$ with $\pi (\mathrm{\Phi }(p,x))=x$, where $P$ is a finite-dimensional smooth parameter manifold; for $v\in T_{p}P$, ${\delta }_v\varphi$ is the derivative of $\mathrm{\Phi }$ along any smooth parameter curve with initial velocity $v$. A finite-jet Lagrangian $n$-form is a smooth spacetime $n$-form $\mathcal{L}(\varphi )$ whose coordinate coefficients depend smoothly on the field components and finitely many spacetime derivatives. It is diffeomorphism-covariant when pullback by every admitted local diffeomorphism $\mathrm{\Psi }$ satisfies $\mathcal{L}({\mathrm{\Psi }}^{\ast }\varphi )={\mathrm{\Psi }}^{\ast }\mathcal{L}(\varphi )$.

A chosen covariant first-variation decomposition is data consisting of an Euler–Lagrange expression $E_{\varphi }[\eta ]$, the $n$-form linear in a field variation $\eta$ left after spacetime derivatives have been moved off $\eta$ by integration by parts, and a covariant $(n-1)$-form $\theta (\varphi ;\eta )$, also linear in $\eta$, such that $\delta \mathcal{L}=E_{\varphi }[\delta \varphi ]+d\theta (\varphi ;\delta \varphi )$ for every admitted smooth finite-dimensional family of fields. Here covariance means that for every admitted local diffeomorphism $\mathrm{\Psi }$ and field variation $\eta$, $E_{{\mathrm{\Psi }}^{\ast }\varphi }[{\mathrm{\Psi }}^{\ast }\eta ]={\mathrm{\Psi }}^{\ast }{E}_{\varphi }[\eta ],\theta ({\mathrm{\Psi }}^{\ast }\varphi ;{\mathrm{\Psi }}^{\ast }\eta )={\mathrm{\Psi }}^{\ast }\theta (\varphi ;\eta )\mathrm{.}$ The choice of $\theta$ is part of the data and need not be unique. A solution has $E_{\varphi }=0$; a linearized solution $\delta \varphi$ at $\varphi$ has $\delta E_{\varphi }[\delta \varphi ]=0$.

A smooth embedded submanifold is spacelike when its induced metric is positive definite. A smooth embedded codimension-one submanifold $N\subset M$ is a null hypersurface when the pullback metric $g{\mathrm{\mid }}_{TN}$ is degenerate with a one-dimensional kernel; that kernel is the null line distribution tangent and normal to $N$.

A stationary solution with generator $\xi$ is a solution $\varphi$ and a fixed smooth spacetime vector field $\xi$ such that ${\mathcal{L}}_{\xi }\varphi =0$. Thus ${\mathcal{L}}_{\xi }g=0$, so $\xi$ is a Killing field of $g$. A bifurcate Killing horizon is a pair of smooth embedded null hypersurfaces $N_{+}$ and $N_{-}$, each invariant under the local flow of $\xi$, meeting transversely in a compact codimension-two submanifold $B$, such that away from $B$ the nonzero field $\xi$ spans the null line distribution of each hypersurface and $\xi =0$ on $B$. Fix $B$ to be oriented, boundaryless, and spacelike. Choose a future-directed unit timelike normal $u$ and a chosen unit spacelike normal $n_B$ along $B$, orthogonal to each other, and define the binormal ${ϵ}_{ab}=2u_{[a}(n_B{)}_{b]}$, so ${ϵ}_{ab}{ϵ}^{ab}=-2$. Orient $B$ compatibly with the spacetime orientation and this normal frame so that, on $B$, $({\mathit{ϵ}}_g{)}_{ab{a}_3\dots a_n}={ϵ}_{ab}(d{A}_B{)}_{a_3\dots a_n}$; here $d{A}_B$ is the positive volume form of the induced metric with that orientation. The horizon is nondegenerate when the fixed normalization of $\xi$ gives a constant $\kappa >0$ on $B$ with ${\mathrm{\nabla }}_a{\xi }_b=\kappa {ϵ}_{ab}$ there; $\kappa$ is its surface gravity. Variations keep $\xi$ and the embedded surface $B$ fixed.

For such data define the Noether current $(n-1)$-form $J_{\xi }:=\theta (\varphi ;{\mathcal{L}}_{\xi }\varphi )-{\iota }_{\xi }\mathcal{L}\mathrm{.}$ Choose a smooth covariant $(n-2)$-form $Q_{\xi }$ on a neighborhood of the admitted field data and define the residual constraint form $C_{\xi }:=J_{\xi }-d{Q}_{\xi }$. Require $C_{\xi }=0$ on the selected solution and, for each admitted linearized perturbation used below, require its fixed-$\xi$ variation $\delta C_{\xi }=0$. These are explicit on-shell and linearized-constraint conditions; no existence or global exactness of $Q_{\xi }$ follows merely from closure of a presymplectic current. The pullback of $Q_{\xi }$ is required to be globally compatible on the integration region.

Here a smooth solution space $\mathcal{S}$ is a specified smooth manifold whose points are the admitted smooth solution fields satisfying the chosen boundary falloff. A smooth curve in $\mathcal{S}$ is viewed as a family of sections of $E$; its tangent vector $v\in T_{\varphi }\mathcal{S}$ induces a field variation ${\delta }_v\varphi$. Every admitted perturbation used in the constraint-variation and boundary-charge statements below is such a tangent variation, and $d_{\mathcal{S}}{H}_{\xi }$ acts on these tangent vectors.

Boundary data also include a family of closed codimension-two surfaces $S_R$ approaching the asymptotic boundary, and finite oriented smooth singular hypersurface chains $C_R$ with $\mathrm{\partial }{C}_R=S_R-B$. With a fixed boundary-term convention, define the charge-variation one-form by ${\alpha }_{\xi }(\delta \varphi ):={\lim }_{R\to \mathrm{\infty }}{\int }_{S_R}(\delta Q_{\xi }-{\iota }_{\xi }\theta (\varphi ;\delta \varphi ))\mathrm{.}$ Require this limit to exist and be globally integrable: there is a smooth Hamiltonian charge $H_{\xi }$ on the specified solution space with $d_{\mathcal{S}}{H}_{\xi }={\alpha }_{\xi }$, where $d_{\mathcal{S}}$ is the exterior derivative on that solution space. This exactness, and the chosen boundary formula itself, are hypotheses. Other boundary actions or falloff conditions can add terms and require a separately stated charge formula.

For the algebraic-curvature subclass used in the entropy formulas, write $\mathcal{L}=\mathrm{\ell }{\mathit{ϵ}}_g$, with $\mathrm{\ell }=\mathrm{\ell }(g_{ab},R_{abcd},\psi ,\mathrm{\nabla }\psi ,\dots ,{\mathrm{\nabla }}_{(a_1}\cdots {\mathrm{\nabla }}_{a_m)}\psi )$, $m<\mathrm{\infty }$, and no covariant derivatives of curvature. Use $R(X,Y)Z={\mathrm{\nabla }}_X{\mathrm{\nabla }}_{Y}Z-{\mathrm{\nabla }}_Y{\mathrm{\nabla }}_{X}Z-{\mathrm{\nabla }}_{[X,Y]}Z$, $R_{abcd}=g(R(e_c,e_d)e_b,e_a)$ in a local frame, and $R=g^{ac}{g}^{bd}{R}_{abcd}$. At fixed metric and matter-jet arguments, define $P^{abcd}$ as the Riemann-symmetric contravariant tensor representing the differential in the curvature argument: $d_R\mathrm{\ell }[K]=P^{abcd}{K}_{abcd}$ for every algebraic curvature variation $K_{abcd}$, where $K$ and $P$ have antisymmetry in each index pair, symmetry under exchange of the pairs, and the first Bianchi identity; explicitly, $P^{abcd}=-P^{bacd}=-P^{abdc}=P^{cdab}$ and $P^{a[bcd]}=0$. The general finite-jet boundary data above also admit curvature-derivative theories; extending the entropy formula to them requires a separate treatment of the curvature-derivative terms.

Adopt the $c=1$ action-unit convention: time is measured as length, $[\mathrm{\hslash }]=\text{action}$, and $[k_B]=\text{energy}\mathrm{/}\text{temperature}$; neither $\mathrm{\hslash }$ nor $k_B$ is set to $1$. Normalize the flow parameter of the fixed generator $\xi$ in length units. Then its Lie action and $\mathrm{\nabla }\xi$ have units $L^{-1}$, and the surface gravity $\kappa$ in ${\mathrm{\nabla }}_a{\xi }_b=\kappa {ϵ}_{ab}$ has units $L^{-1}$. The spacetime integral of $\mathcal{L}$ has units of action, so if $\mathcal{L}=\mathrm{\ell }{\mathit{ϵ}}_g$, then $[\mathrm{\ell }]=\text{action}{L}^{-n}$. Since curvature has units $L^{-2}$, the Einstein--Hilbert normalization ${\mathrm{\ell }}_{\mathrm{E}\mathrm{H}}=R\mathrm{/}(16\pi G)$ in $n$ spacetime dimensions has $[G]=L^{n-2}{\text{action}}^{-1}$. In the algebraic-curvature subclass, $d_R\mathrm{\ell }[K]=P^{abcd}{K}_{abcd}$ gives $[P]=\text{action}{L}^{2-n}$, and the downstream functional $-(2\pi k_B\mathrm{/}\mathrm{\hslash }){\int }_B{P}^{abcd}{ϵ}_{ab}{ϵ}_{cd}d{A}_B$ has entropy units $k_B$. The integrated Noether charge and $H_{\xi }$ have energy units, consistent with $\mathrm{\hslash }\kappa \delta S_W\mathrm{/}{k}_B$.

When SI units are restored for the $n$-dimensional Einstein--Hilbert normalization $c^{3}R\mathrm{/}(16\pi G)$, $[G]={\mathrm{m}}^{n-1}{\mathrm{k}\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$ and $[A_B]={\mathrm{m}}^{n-2}$, so the area expression is $S_W=k_B{c}^3{A}_B\mathrm{/}(4G\mathrm{\hslash })$. In four spacetime dimensions this is the usual SI Newton constant in ${\mathrm{m}}^3{\mathrm{k}\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$ and horizon area in ${\mathrm{m}}^2$; the SI acceleration surface gravity is ${\kappa }_{\mathrm{a}\mathrm{c}\mathrm{c}}=c^2{\kappa }_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}$.

Some Properties of Noether Charge, §§2–4

Advanced Lectures on General Relativity, §§1.3.1–1.3.3 and §1.4.2

Covariant phase space with boundaries, §§2.2–2.4

The Lie derivative of a tensor field

The Lie derivative of a differential form

The exact boundary charge and globally compatible $Q_{\xi }$ are inputs. The physical interpretation of a resulting Noether-charge entropy as a thermodynamic state function is a separate assumption; this definition does not derive that identification.

Use the metric, tensor fields, covariant first-variation decomposition, and Noether current of the preceding Definition. Suppose the Lagrangian is $\mathcal{L}=\mathrm{\ell }(g_{ab},R_{abcd},\psi ,\mathrm{\nabla }\psi ,\dots ,{\mathrm{\nabla }}_{(a_1}\cdots {\mathrm{\nabla }}_{a_m)}\psi ){\mathit{ϵ}}_g,$ where $m<\mathrm{\infty }$, all fields and coefficients are smooth, and the curvature argument is algebraic (there are no covariant derivatives of $R_{abcd}$). Use the curvature convention $R(X,Y)Z={\mathrm{\nabla }}_X{\mathrm{\nabla }}_{Y}Z-{\mathrm{\nabla }}_Y{\mathrm{\nabla }}_{X}Z-{\mathrm{\nabla }}_{[X,Y]}Z,$ and write $R_{abcd}:=g(R(e_c,e_d)e_b,e_a)$ in any local frame. Square brackets denote weight-one antisymmetrization. Define the Riemann-symmetric derivative $P^{abcd}$ by $\delta \mathrm{\ell }{\mid }_{g,\psi ,\mathrm{\nabla }\psi ,\dots }=P^{abcd}\delta R_{abcd},$ with $P^{abcd}=-P^{bacd}=-P^{abdc}=P^{cdab}$ and $P^{a[bcd]}=0$.

Use the covariant first-variation potential produced by the finite covariant integration-by-parts construction on the field jets, up to the stated potential ambiguities. For each smooth vector field $\zeta$, use the local Noether potential obtained by covariantly integrating the derivatives of $\zeta$ in $J_{\zeta }=\theta (\varphi ;{\mathcal{L}}_{\zeta }\varphi )-{\iota }_{\zeta }\mathcal{L}$ by parts. Call this the canonical local representative: it is defined on a neighborhood of the fields and generators, is a local covariant primitive of $J_{\zeta }$ on shell, and has no independent closed-form addition. The permitted representative changes are exactly those induced by $\mathcal{L}\mapsto \mathcal{L}+d\mu ,\theta \mapsto \theta +\delta \mu +d\mathbf{Y},$ and by an exact charge change; here $\mu$ is a covariant $(n-1)$-form and $\mathbf{Y}(\varphi ;\eta )$ is a covariant $(n-2)$-form linear in $\eta$. Assume the chosen representative is globally compatible on the integration region. Then it has the form $Q_{\zeta }=W^c{\zeta }_c+X^{cd}{\mathrm{\nabla }}_{[c}{\zeta }_{d]}+\mathbf{Y}(\varphi ;{\mathcal{L}}_{\zeta }\varphi )+dZ,$ where $W^c$ and $X^{cd}$ are local covariant form-valued coefficients, $Z$ is a local covariant $(n-3)$-form (the exact term is absent when $n=2$), and $(X^{cd}{)}_{a_3\dots a_n}=-P^{abcd}({\mathit{ϵ}}_g{)}_{ab{a}_3\dots a_n}\mathrm{.}$ For the canonical representative before the permitted changes one may take $\mathbf{Y}=0$ and $Z=0$.

Stationary covariant Noether boundary data

The global-compatibility condition is separate from the local integration by parts: the result does not assert that an arbitrary closed global charge shift is exact. The entropy-variation theorem uses only this representative class. Higher-derivative curvature terms require replacing $P$ by the full curvature Euler derivative and a separate statement.

Use the stationary Noether data of the preceding Definition, with fixed $\xi$, compact boundaryless bifurcation surface $B$, and surface gravity $\kappa >0$. Choose the covariant first-variation potential and $Q_{\xi }$ in the canonical local representative class of the preceding Noether-charge decomposition theorem. Let $\mathrm{\nabla }$ be the Levi-Civita connection of $g$, use the curvature convention $R(X,Y)Z={\mathrm{\nabla }}_X{\mathrm{\nabla }}_{Y}Z-{\mathrm{\nabla }}_Y{\mathrm{\nabla }}_{X}Z-{\mathrm{\nabla }}_{[X,Y]}Z,$ and write $R_{abcd}:=g(R(e_c,e_d)e_b,e_a)$ in any local frame and $R:=g^{ac}{g}^{bd}{R}_{abcd}$. Write $\mathcal{L}=\mathrm{\ell }(g,\psi ,\mathrm{\nabla }\psi ,\dots ,R){\mathit{ϵ}}_g$, where ${\mathit{ϵ}}_g$ is the oriented metric volume form and $\mathrm{\ell }$ depends smoothly on the Riemann tensor algebraically, with no covariant derivatives of $R$. Define $P^{abcd}$ to be the Riemann-symmetric metric representative of the differential of $\mathrm{\ell }$ in its curvature argument: $\delta \mathrm{\ell }{\mid }_{g,\psi ,\mathrm{\nabla }\psi ,\dots }=P^{abcd}\delta R_{abcd}\mathrm{.}$ Here $P$ satisfies $P^{abcd}=-P^{bacd}=-P^{abdc}=P^{cdab}$ and $P^{a[bcd]}=0$. Let $d{A}_B$ be the positive volume form of the induced Riemannian metric on $B$ (in local coordinates with induced metric $h$, $d{A}_B=\sqrt{\det h}d{y}^1\cdots d{y}^{n-2}$), and define the stationary Wald Noether-charge functional $S_W:=-\frac{2\pi k_B}{\mathrm{\hslash }}{\int }_B{P}^{abcd}{ϵ}_{ab}{ϵ}_{cd}d{A}_B\mathrm{.}$ For every admitted smooth linearized perturbation at the stationary solution, with $\xi$ and $B$ fixed and the residual constraint variation $\delta C_{\xi }=0$, the chosen Noether charge satisfies $\delta {\int }_B{Q}_{\xi }=\frac{\mathrm{\hslash }\kappa }{2\pi k_B}\delta S_W\mathrm{.}$ Here $\mathrm{\hslash }>0$ is Planck's constant, $k_B>0$ is Boltzmann's constant, and the action normalization is $c=1$. The identity is evaluated at the unperturbed value of $\kappa$; it does not include a $\delta \kappa$ term.

Stationary covariant Noether boundary data

Noether charge decomposition for algebraic-curvature Lagrangians

The sign is fixed by the displayed curvature, binormal, and Noether-current conventions. Higher-derivative curvature terms require replacing $P$ by the full curvature Euler derivative. This stationary identity provides no dynamical monotonicity theorem and no thermodynamic interpretation without a separate physical assumption.

For the stationary Noether data of the preceding Definition, choose $Q_{\xi }$ in the canonical local representative class of the preceding Noether-charge decomposition theorem. Let $\delta \varphi$ be an admitted linearized solution with fixed $\xi$ and $\delta C_{\xi }=0$. Let $C_R$ be finite oriented smooth singular hypersurface chains with $\mathrm{\partial }{C}_R=S_R-B$, where the boundary orientations are outward at $S_R$ and inward at $B$. Then $d(\delta Q_{\xi }-{\iota }_{\xi }\theta )=0$ on each $C_R$, and $\delta H_{\xi }={\lim }_{R\to \mathrm{\infty }}{\int }_{S_R}(\delta Q_{\xi }-{\iota }_{\xi }\theta )={\int }_B\delta Q_{\xi }=\frac{\mathrm{\hslash }\kappa }{2\pi k_B}\delta S_W\mathrm{.}$ The last equality uses the derivative-free curvature hypothesis and the Noether entropy functional of the preceding theorem. The identity is conditional on the exact boundary-charge and vanishing-constraint assumptions; $\kappa$ is the unperturbed surface gravity. For the Einstein-Hilbert Lagrangian ${\mathcal{L}}_{\mathrm{E}\mathrm{H}}=R{\mathit{ϵ}}_g\mathrm{/}(16\pi G)$ with Newton constant $G>0$ in the same $c=1$ action normalization, it gives $S_W=k_B{A}_B\mathrm{/}(4G\mathrm{\hslash })$, where $A_B:={\int }_{B}d{A}_B$.

Stationary covariant Noether boundary data

Noether charge decomposition for algebraic-curvature Lagrangians

The stationary relation is a model deduction. Calling $S_W$ the physical thermodynamic state entropy requires an additional physical identification; the restricted Kerr–Newman assumption on the earlier thermodynamic page is not generalized here.