Relational Bulk Observables and Dressing Choices

A bulk coordinate is not an observable in gravity. A usable insertion must specify an event through relations—for example, a fixed proper distance along a geodesic shot inward from a labeled boundary point—and must include the gravitational field that makes this specification invariant. Such observables are perturbatively well defined only where the reference construction is unique. Different dressings need not be gauge-equivalent: once their asymptotic or radiative fields differ, they are different physical operators. We use Lorentzian asymptotically AdS gravity, standard reflecting boundary conditions, and work to first order in $\kappa=\sqrt{32\pi G_N}$ in a low-energy code sector.

Required background. Gravitational Gauss laws establish the necessary boundary tail, and type-III local algebras prevent a naive identification of a continuum region with a finite-dimensional tensor factor.

Helpful background. Relational gravitational observables supplies the general construction; interacting dressing tracks perturbative commutators; gauge-invariant response kernels identify measurable perturbations; and localization cost explains why increasingly sharp reference systems are not free.

A boundary-anchored geodesic defines an event locally

Choose a boundary event $p=(t_0,\Omega_0)$ and an inward unit normal frame $e_a^\mu(p)$ fixed by the asymptotic AdS structure. For a specified initial direction $u^a$, let $\gamma_{p,u}(s;g)$ solve

$$\frac{D^2\gamma^\mu}{Ds^2}=0, \qquad \gamma(0)=p, \qquad \dot\gamma^\mu(0)=u^a e_a^\mu(p),$$

with a regulated boundary start and renormalized affine or proper parameter $s$. The relational scalar is

$$\Phi_{\rm geo}[p,u,s;g,\phi] =\phi\!\left(\gamma_{p,u}(s;g)\right).$$

Under a diffeomorphism that dies off at the boundary, both $g$ and the geodesic move, so their composition is invariant. A large diffeomorphism acting nontrivially on the boundary changes the anchor and is an asymptotic symmetry, not a redundancy to quotient away.

Perturb around $\bar g$. If $X_0^\mu=\gamma^\mu(s;\bar g)$ and $\delta X^\mu[h]$ is the geodesic displacement, then

$$\Phi_{\rm geo} =\phi(X_0)+\delta X^\mu[h]\,\partial_\mu\phi(X_0) +O(\kappa^2).$$

The displacement solves the forced Jacobi equation

$$\frac{D^2\delta X^\mu}{Ds^2} +\bar R^\mu{}_{\nu\alpha\beta} \dot X_0^\nu\delta X^\alpha\dot X_0^\beta =-\delta\Gamma^\mu_{\alpha\beta}[h] \dot X_0^\alpha\dot X_0^\beta,$$

with anchor and frame conditions at $p$. This equation makes the dressing concrete: $\delta X[h]$ is a line integral of the metric perturbation along the reference geodesic.

Geodesic and Coulomb dressings compared

The leading geodesic operator has the generic dressed form

$$\Phi_{\rm geo}(X_0)=\phi(X_0)+V_{\rm geo}^\mu[h](X_0)\partial_\mu\phi(X_0).$$

A Coulomb-dressed insertion at the same background point is

$$\Phi_{\rm C}(X_0)=\phi(X_0)+V_{\rm C}^\mu[h](X_0)\partial_\mu\phi(X_0).$$

Both satisfy $\delta_\xi V^\mu=-\kappa\xi^\mu$, so

$$\Delta V^\mu=V_{\rm geo}^\mu-V_{\rm C}^\mu, \qquad \delta_\xi\Delta V^\mu=0.$$

Their difference is therefore

$$\Phi_{\rm geo}-\Phi_{\rm C} =\Delta V^\mu\partial_\mu\phi+O(\kappa^2).$$

This is an auditable physical comparison, not merely two coordinate formulas. The geodesic dressing initially concentrates gravitational data along the anchor direction. The Coulomb dressing distributes the constraint field over angles. They have the same total energy charge and local scalar core, but $\Delta V$ creates a source-free gravitational excitation with different multipoles or radiation. Boundary stress-tensor one-point functions and suitably defined response observables can distinguish them. Explicit perturbative AdS constructions give both dressings and their boundary behavior (Giddings and Kinsella 2018, §§3–5).

One should consequently state a reconstructed operator as $\Phi_V$, not as an undressed $\phi(x)$. An equality between two boundary representations is meaningful only after matching the anchor, asymptotic charges, perturbative order, and state domain.

The caustic and reference-structure test

The map $(p,u,s)\mapsto\gamma_{p,u}(s)$ is locally invertible only while its Jacobi fields do not develop a zero mode. Let $J(s)$ be the transverse Jacobi matrix. A conjugate point satisfies

$$\det J(s_c)=0.$$

At or beyond $s_c$, more than one geodesic with the prescribed boundary data may reach the same neighborhood, or nearby anchor data may produce a large displacement. The label $(p,u,s)$ then fails to define a unique smooth event. The correct response is to restrict the normal neighborhood, choose an additional branch rule, or replace the reference construction—not to declare the coordinate label globally gauge invariant.

A second adversarial test perturbs the reference structure. Shift the boundary anchor by $\delta p$ while holding the coordinate point $X_0$ fixed. The relational insertion changes by

$$\delta_p\Phi_{\rm geo} =\delta_pX^\mu\partial_\mu\phi+\cdots.$$

That change is physical because a different anchor was selected. Conversely, a compactly supported diffeomorphism that moves both the fields and the geodesic gives zero through the retained order. This contrast is the practical gauge-invariance check.

Locality, commutators, and state domain

Two background points may be spacelike while their dressings overlap at the boundary. The commutator then has the structure

$$[\Phi_V(X),\Phi_W(Y)] =[\phi(X),\phi(Y)] +[V^\mu(X),W^\nu(Y)] \partial_\mu\phi(X)\partial_\nu\phi(Y)+\cdots.$$

The first term vanishes for a free scalar at spacelike separation; the dressing term need not. It is suppressed perturbatively but is required by the gravitational constraints. Perturbative analyses use this structure to show why exact local commuting subalgebras do not survive unchanged in gravity (Donnelly and Giddings 2016, §§IV–V; Marolf 2015, §§2–3).

The controlled claim is an approximate one on a specified low-energy state family, with fixed dressing and error norm. Large excitations can move the reference geodesic, create new caustics, or backreact enough that the background label ceases to track the intended event. A relational definition is not automatically state-independent or nonperturbative.

Controlled limits and handoff

The geodesic construction is reliable inside a convex normal neighborhood, before caustics, at the retained order in $\kappa$, and for boundary conditions that keep the anchor meaningful. Averaging line dressings can approach a Coulomb choice, but the averaging prescription changes the gravitational state. In the $G_N\to0$ limit with fixed low energy, dressing commutators vanish and local QFT is recovered; at finite $G_N$, they should be bounded rather than erased.

Causal wedges next identify what a boundary domain accesses by propagation. Finite-$N$ limits explain why perturbative relational observables do not automatically extend across all states or behind horizons.

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.
• Marolf, D. (2015). “Comments on microcausality, chaos, and gravitational observables.” Classical and Quantum Gravity 32, 245003. DOI.