Stationary covariant Noether boundary data

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.