Gravitational Gauss Laws and Boundary Anchoring

A gravitational excitation with nonzero energy cannot be represented by an exactly compactly supported gauge-invariant operator. The Hamiltonian and other asymptotic charges are surface terms on the constraint surface, so an operator that changes them must also change boundary-visible gravitational data. Dressing a matter field to an asymptotic anchor supplies that data. Different dressings can have identical local matter content but distinct radiative or Coulombic tails; what is unavoidable is the charge flux, not a unique spatial profile. We work perturbatively about Lorentzian asymptotically global AdS with $g_{\mu\nu}=\bar g_{\mu\nu}+\kappa h_{\mu\nu}$, $\kappa^2=32\pi G_N$, and standard reflecting boundary conditions.

Required background. Interacting reconstruction and dressing supplies the first-order invariant operator, while gauge constraints and edge data supplies the algebraic role of boundary flux.

Helpful background. Dirac reduction explains observables on a constrained phase space, and the gravitational constraints and Bianchi identity relate constraint propagation to stress-tensor conservation.

The gravitational constraint reaches the boundary

Let $\Sigma$ be a bulk Cauchy slice with induced metric $q_{ij}$, momentum $\pi^{ij}$, unit normal $n^\mu$, and asymptotic boundary $\partial\Sigma$. The ADM generator with lapse and shift $\xi^\mu$ has the form

$$H[\xi]=\int_\Sigma d^dx\, \big(\xi^\perp\mathcal H+\xi^i\mathcal H_i\big) +Q_{\partial\Sigma}[\xi].$$

Physical configurations satisfy $\mathcal H=\mathcal H_i=0$. Consequently,

$$H[\xi]\big|_{\rm phys}=Q_{\partial\Sigma}[\xi].$$

For the asymptotic time translation $\xi=\partial_t$, this surface term is the total energy. At linear order the same fact follows by integrating the $0\mu$ Einstein constraint,

$$\delta G_{0\mu}[h]+\Lambda h_{0\mu}=8\pi G_N T_{0\mu},$$

over $\Sigma$ and applying the divergence theorem. The bulk stress-energy is tied to a boundary integral of derivatives of $h_{\mu\nu}$. This is the gravitational analog of Gauss’s law, with energy-momentum playing the role of charge.

If a gauge-invariant operator $A$ created positive energy but commuted with every asymptotic metric observable, then

$$[H,A]=[Q_{\partial\Sigma},A]=0.$$

It could not change energy. Thus a nontrivial energy-creating observable must be visible in the asymptotic gravitational field. The conclusion uses the constraint and boundary conditions; it is not an assertion that every component of a chosen coordinate metric is observable.

Two dressings of one matter insertion

For a scalar insertion at $x$, write the first-order invariant operator as

$$\Phi_V(x)=\phi(x)+V^\mu[h](x)\partial_\mu\phi(x)+O(\kappa^2), \qquad \delta_\xi V^\mu=-\kappa\xi^\mu.$$

A line dressing fixes $x$ by following a specified geodesic from a boundary point and concentrates the initial gravitational flux along that direction. A Coulomb dressing averages the line dressing over directions and yields a rotationally symmetric constraint field. The matter insertion is the same; the gravitational coherent state attached to it is not.

The distinction can be computed without choosing all metric components. In a local asymptotically flat region of a $d$-dimensional spatial slice, let $\mathcal F_r$ denote the radial constraint flux normalized so that

$$\int_{S_r^{d-1}}r^{d-1}d\Omega\,\mathcal F_r=E.$$

For an excitation of energy $E$, Coulomb and idealized line data are

$$\mathcal F_r^{\rm C}(r,\Omega) =\frac{E}{\Omega_{d-1}r^{d-1}}, \qquad \mathcal F_r^{\rm L}(r,\Omega) =\frac{E}{r^{d-1}}\delta^{(d-1)}(\Omega,\Omega_0).$$

Both give the same integrated charge:

$$\int r^{d-1}d\Omega\,\mathcal F_r^{\rm C} =\int r^{d-1}d\Omega\,\mathcal F_r^{\rm L}=E,$$

but their angular multipoles differ. Linearized Einstein evolution maps these initial constraint data to a smooth Coulomb tail in the first case and a narrow, radiating stringlike field that relaxes toward a Coulomb component in the second. In global AdS, the reflecting boundary and discrete normal modes modify the propagation, while the equality of the boundary charge remains. Explicit AdS line and Coulomb dressings exhibit precisely this shared charge and different field profile (Giddings and Kinsella 2018, §§2–4).

The difference $V_{\rm L}-V_{\rm C}$ is itself gauge invariant at this order. It is a homogeneous gravitational excitation, not a gauge transformation. Therefore spacelike commutators of $\Phi_{V_{\rm L}}$ and $\Phi_{V_{\rm C}}$ may differ even though correlators of the undressed scalar core agree.

Compact-support adversarial test

Assume, for contradiction, that a gauge-invariant operator $A_K$ is supported in a compact bulk set $K$, creates a state with energy shift $\Delta E\neq0$, and has no action on boundary data. Equal-time locality would give

$$[Q_{\partial\Sigma}[\partial_t],A_K]=0,$$

because the charge density is entirely at the boundary. But the Heisenberg relation for an energy-changing insertion requires

$$\langle\psi|[H,A_K]|0\rangle =(E_\psi-E_0)\langle\psi|A_K|0\rangle,$$

which is nonzero for a component with $E_\psi\neq E_0$. Since $H=Q_{\partial\Sigma}$ on physical states, the two statements conflict. One must abandon exact compact support, nonzero charge, gauge invariance, or the stated boundary conditions.

The same argument does not prohibit neutral relational composites whose leading asymptotic multipoles cancel, nor does it say that all such composites are exactly local. Gravitational binding energy and higher multipoles still obey the constraints. It also does not establish that boundary measurements can efficiently decode the bulk state: nonzero asymptotic imprint and practical reconstruction are different claims. Donnelly and Giddings formulate the resulting obstruction to sharply localized subsystems in perturbative gravity (Donnelly and Giddings 2016, §§III–V).

What remains local

Perturbative gravity still admits useful approximate localization. Choose a dressing, a low-energy code sector, a spacetime region well inside the AdS curvature scale, and an error norm. Matter correlators can then reproduce local QFT to leading order, while dressing commutators are suppressed by $G_NE/r^{d-2}$ or the corresponding AdS Green function. The exact algebra, however, includes asymptotic gravitational data and does not factor into strictly independent compact bulk regions.

Relational bulk observables explain how an anchor specifies an event and where caustics spoil that specification. Reconstruction error norms quantify the residual nonlocality on a chosen state set. Entanglement-wedge recovery later concerns a code-subspace representation of an algebra; it does not remove the Gauss-law tail of the physical observable.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Donnelly, W., and Giddings, S. B. (2016). “Observables, gravitational dressing, and obstructions to locality and subsystems.” Physical Review D 93, 024030. DOI.
• Giddings, S. B., and Kinsella, A. (2018). “Gauge-invariant observables, gravitational dressings, and holography in AdS.” Journal of High Energy Physics 2018(11), 074. DOI.
• Regge, T., and Teitelboim, C. (1974). “Role of surface integrals in the Hamiltonian formulation of general relativity.” Annals of Physics 88, 286–318. DOI.