Higher spin dynamics in Weyl reference frame

Abstract: In this manuscript we derive the $w_{1+\infty}$ algebra on an arbitrary null hypersurface located at finite distance. A straightforward integration of the evolution Bianchi identities is performed in a suitable dynamical reference frame, dubbed the Weyl reference frame. In this reference frame, the extraction of the (linear) higher spin charge bracket is similar to the asymptotic null case, except that the corner metric evolves non trivially in time. In other words, boundary degrees of freedom that are frozen at null infinity due to the boundary conditions now become part of the dynamics. This feature is reflected in the appearance of non-local terms in the brackets. In order to avoid the presence of these non-local terms in the charge bracket, we absorb this non-locality - which encodes nothing other than the history of the corner metric - into the definition of the higher spin charges. Moreover, a new perspective on the memory effect at finite distance is suggested. In conclusion, we believe that this treatment can also be applied to asymptotically (A)dS$_{4}$ spacetimes, where the presence of a non-vanishing cosmological constant leads to the introduction of a cosmological reference frame.

• The paper derives the wedge algebra of w_{1+∞} on arbitrary finite-distance null hypersurfaces by using a Weyl reference frame that restores asymptotic-like Bianchi identities and symplectic dynamics.
• Non-local memory terms arise because the inverse dressing-time derivative does not commute with the evolving corner metric’s Weyl-covariant angular derivative, requiring redefined charges and additional fall-off conditions.
• Renormalized charges reproduce finite-distance soft-graviton actions and memory effects, while the discarded corner symplectic term, strong fall-offs, and unresolved physical meaning of charge redefinitions remain important limitations.

The paper derives the $w_{1+\infty}$ higher spin charge algebra on an arbitrary null hypersurface at finite distance, rather than at future null infinity $\mathcal{I}^+$. The central obstacle is that, unlike the asymptotic case where boundary conditions freeze the corner metric, a generic null surface carries genuine dynamical boundary degrees of freedom that complicate both the evolution Bianchi identities and the symplectic structure. The author's strategy is to trade these unfrozen degrees of freedom for a field-dependent coordinate system — the Weyl reference frame (WRF) — in which the asymptotic computational machinery can be reproduced almost verbatim, at the cost of introducing non-local corrections to the charges that are interpreted as memory terms.

Geometry of null hypersurfaces and weighted operators

The setting is a null hypersurface $\mathscr{H} \simeq \mathscr{S}\times\mathbb{R}$ embedded in spacetime via $\Pi$, with Carrollian data $(q_{ab}, \ell)$ and null normals $(\ell, n)$ defined up to boost rescaling. Working in the Newman–Penrose/GHP formalism near a horizon ($n = -\partial_\rho$), the geometry is encoded in spin coefficients with definite $\{p,q,\omega\}$ weights under Lorentz class III transformations and Weyl rescalings. The leading orders of the Weyl scalars define spin-weighted charge aspects $Q_s$ for $s = -2,\dots,2$ (with $\mathcal{I}^+$0, $\mathcal{I}^+$1, $\mathcal{I}^+$2), obeying the recursive Bianchi identities

$\mathcal{I}^+$3

where $\mathcal{I}^+$4 and $\mathcal{I}^+$5 are Weyl-covariantized GHP operators. These identities transform covariantly under the near-horizon symmetry group $\mathcal{I}^+$6. The canonical pre-symplectic potential is the null Brown–York potential; its spin-2 sector involves the longitudinal shear $\mathcal{I}^+$7 and the expansion $\mathcal{I}^+$8.

A key structural point established here: because the corner metric evolves non-trivially in time at finite distance, the operators $\mathcal{I}^+$9 and $\mathscr{H} \simeq \mathscr{S}\times\mathbb{R}$0 do not commute. This is the source of all qualitative differences from the asymptotic treatment.

The Weyl reference frame

Two existing dressing-time proposals are reviewed for comparison: the RZ dressing time defined by $\mathscr{H} \simeq \mathscr{S}\times\mathbb{R}$1, which absorbs $\mathscr{H} \simeq \mathscr{S}\times\mathbb{R}$2 and $\mathscr{H} \simeq \mathscr{S}\times\mathbb{R}$3 into a dynamical coordinate but whose status as a dynamical reference frame in the sense of Goeller–Höhn–Kirklin is left open; and the CFL dressing time, defined by vanishing of the boost connection, which does follow the relational construction.

The paper introduces a new frame combining a class III transformation with a Weyl rescaling (with $\mathscr{H} \simeq \mathscr{S}\times\mathbb{R}$4, $\mathscr{H} \simeq \mathscr{S}\times\mathbb{R}$5) chosen so that

$\mathscr{H} \simeq \mathscr{S}\times\mathbb{R}$6

with dressing time satisfying $\mathscr{H} \simeq \mathscr{S}\times\mathbb{R}$7, i.e. $\mathscr{H} \simeq \mathscr{S}\times\mathbb{R}$8. In this frame $\mathscr{H} \simeq \mathscr{S}\times\mathbb{R}$9, so the Bianchi identities take exactly the asymptotic form. Two consequences deserve emphasis. First, since the null Raychaudhuri equation is not Weyl-covariant, setting $\Pi$0 does not force $\Pi$1 to vanish — the longitudinal shear survives as the radiative mode. Second, the construction is formulated through embedding fields $\Pi$2, so the extended phase space acquires a corner symplectic term encoding edge modes; the paper explicitly retains only the bulk term and defers the edge-mode analysis.

Under the assumptions $\Pi$3 and $\Pi$4, the bulk symplectic form becomes

$\Pi$5

the finite-distance analogue of the Ashtekar–Streubel form, with fundamental bracket $\Pi$6.

Non-local charges and memory

Integration of the Bianchi identities requires fall-off conditions on $\Pi$7, $\Pi$8, $\Pi$9, $(q_{ab}, \ell)$0:

Quantity	Fall-off
$(q_{ab}, \ell)$1	$(q_{ab}, \ell)$2
$(q_{ab}, \ell)$3	$(q_{ab}, \ell)$4, $(q_{ab}, \ell)$5
$(q_{ab}, \ell)$6	$(q_{ab}, \ell)$7
$(q_{ab}, \ell)$8	$(q_{ab}, \ell)$9

These additional conditions on $(\ell, n)$0, $(\ell, n)$1, $(\ell, n)$2 are necessary — not merely convenient — for recovering the algebra, and constitute an assumption beyond what is imposed asymptotically. Because $(\ell, n)$3, direct integration produces non-local terms of the form $(\ell, n)$4, where $(\ell, n)$5 encodes the history of the corner metric. The author absorbs these into redefined ("dressed") charges $(\ell, n)$6, with the memories $(\ell, n)$7 given by a closed recursion involving commutators $(\ell, n)$8. With this redefinition the integrated charges recover precisely the asymptotic-flat structure, with linear part $(\ell, n)$9 and quadratic part involving powers of $n = -\partial_\rho$0. The paper concedes that absorbing the non-locality into the charges was motivated by mathematical convenience, and states that the physical implications of this procedure remain to be understood.

In the WRF the non-radiative condition $n = -\partial_\rho$1 becomes simply $n = -\partial_\rho$2, and the linear $n = -\partial_\rho$3 action reproduces a finite-distance version of the near-horizon memory effect, with the asymptotic transversal shear replaced by $n = -\partial_\rho$4. This offers a new perspective on memory at finite distance, distinct from the black hole memory tensor of Rahman–Wald, whose relation to near-horizon supertranslations remains unestablished.

Charge actions and renormalization

The actions of $n = -\partial_\rho$5, $n = -\partial_\rho$6, $n = -\partial_\rho$7 on $n = -\partial_\rho$8 are computed explicitly using pseudo-differential calculus (Leibniz rules for fractional derivatives, Vandermonde-type identities). A notable feature is that the unrenormalized brackets diverge as $n = -\partial_\rho$9 for $\{p,q,\omega\}$0, requiring counter-term subtractions:

$\{p,q,\omega\}$1

The general renormalized linear charge satisfies $\{p,q,\omega\}$2, where $\{p,q,\omega\}$3 is the finite-distance analogue of the (sub)$\{p,q,\omega\}$4-leading soft graviton operator. The quadratic action on the opposite-spin soft modes takes the compact form

$\{p,q,\omega\}$5

structurally identical to the result at null infinity after the substitutions $\{p,q,\omega\}$6, $\{p,q,\omega\}$7.

The $\{p,q,\omega\}$8 bracket

Since the quadratic action is independent of $\{p,q,\omega\}$9, the linear-level bracket computation reduces exactly to the asymptotic one, yielding

$Q_s$0

i.e., the wedge algebra of $Q_s$1 holds on any null hypersurface once expressed in the WRF. The author argues that this method should transfer to asymptotically (A)dS$Q_s$2: there, $Q_s$3 behaves as a weakly isolated horizon with time-dependent boundary metric, so the same mechanism for handling unfrozen degrees of freedom applies — though the definition of an appropriate news tensor in the presence of $Q_s$4 is acknowledged to be less straightforward and deserving of separate treatment.

Limitations and open questions

Several caveats qualify the results. The corner contribution to the extended symplectic form, which encodes the edge modes exchanged between physical and reference manifolds, is discarded throughout; a complete phase-space treatment requires it. The fall-off conditions on $Q_s$5, $Q_s$6, $Q_s$7 are assumptions strong enough to guarantee stabilization of the geometry to a spherical configuration at late dressing time, and their relaxation may alter the bracket. The physical meaning of the non-local charge redefinition is unresolved. Whether the RZ dressing time can be derived from field-dependent diffeomorphisms, and how it relates to the CFL and Weyl frames, is left open. Finally, the relation between the finite-distance memory terms identified here and the horizon memory tensor of Rahman–Wald, as well as extensions to Einstein–Maxwell theory and to the algebroid/twistor formulations of the celestial $Q_s$8 symmetry, remain future work.

Conclusion

This work establishes that the $Q_s$9 higher spin symmetry, previously understood at null infinity, persists on arbitrary null hypersurfaces provided one works in a suitably dressed dynamical frame. The price of retaining genuine finite-distance degrees of freedom is twofold: non-local (memory-like) corrections to the charges arising from the non-commutativity of the Weyl covariant derivative with time integration, and additional fall-off conditions on the transversal shear and expansion. The result sharpens the distinction between asymptotic and finite-distance gravitational phase spaces and provides a concrete framework — the Weyl reference frame — likely applicable to cosmological settings where the boundary metric is inherently dynamical.