The Phase Space of Gravity on Null Hypersurfaces

Abstract: We construct the bulk kinematical Poisson structure of general relativity on a caustic-free null segment, prior to imposing the Raychaudhuri and Damour constraints. We first show that shifts of the Ehresmann connection, also known as Carroll boosts, are pure gauge in gravity. This allows the Ehresmann connection to be fixed as a background structure and leads to the prime phase space of null gravitational data. The resulting phase space comprises of a spin-0 pair that includes the area and its conjugate, a spin-1 pair including the null direction and its conjugate, and the spin-2 sector encoded in the unimodular transverse metric. We then construct the brackets through a three-stage Dirac reduction implementing a normalization, horizontality, and shear constraints. We find that the spin-2 bracket is non-local and involves an antisymmetric Green kernel for a first-order transport operator along the null generators. The self-brackets of the spin-0 and spin-1 momenta involve, respectively, the vertical and horizontal derivatives of the transverse metric contracted with the spin-2 propagator. We finally provide an independent derivation of the brackets from the explicit realization of the Hamiltonian vector fields of the symplectic form. These results provide the classical canonical arena for studying the null constraint algebra, observables, and quantization.

• The paper constructs the complete kinematical Poisson structure before imposing the Raychaudhuri and Damour constraints, organizing gravitational data into spin-0, spin-1, and spin-2 canonical sectors.
• The analysis shows that all nonlocality is governed by an antisymmetric bilocal Green kernel for transport along null generators, with results independently confirmed through Dirac reduction and Hamiltonian vector fields.
• The paper establishes that shifts of the Ehresmann connection are pure gauge, leaving a nondegenerate phase space of physical gravitational data while identifying constraint algebras, boundary effects, caustics, and quantum anomalies as open problems.

The paper constructs the complete kinematical Poisson structure of general relativity on a caustic-free null hypersurface, prior to imposing the Raychaudhuri and Damour constraints. The analysis is carried out intrinsically, in the language of ruled Carrollian geometry, and yields a phase space organized into spin-0 (area/surface tension), spin-1 (null generator/momentum aspect), and spin-2 (unimodular metric/shear) canonical pairs. The central technical result is a set of reduced Dirac brackets in which all non-locality is encoded in an antisymmetric bilocal Green kernel for a first-order transport operator along the null generators. The brackets are derived twice — by a three-stage Dirac reduction and independently from the Hamiltonian vector fields of the symplectic form — providing a non-trivial consistency check.

Geometric setting

A null hypersurface $\mathcal{N}$ is described intrinsically by a Carrollian structure $(q_{ab},\ell^a)$ with $q_{ab}$ degenerate of rank 2 and $\ell^a q_{ab}=0$, decomposed as $q_{ab}=\Omega\,\bar q_{ab}$ into an area density $\Omega$ and a unimodular transverse metric. A torsionless, minimally non-metrical Carrollian connection $D_a$ exists only because the expansion tensor $\theta_{ab}=\tfrac12\mathcal{L}_\ell q_{ab}$ is nonzero; the key identity relating expansion to non-metricity and torsion shows that one cannot have both vanishing. To split tangent directions into vertical and horizontal parts, an Ehresmann connection $k_a$ with $\iota_\ell k=1$ is introduced, completing the ruled Carrollian structure $(q_{ab},\ell^a)$0. The connection coefficients are packaged in $(q_{ab},\ell^a)$1 (inaffinity plus Hájíček connection) and a horizontal symmetric tensor $(q_{ab},\ell^a)$2.

The intrinsic null Brown–York stress tensor $(q_{ab},\ell^a)$3 decomposes as $(q_{ab},\ell^a)$4 with momenta $(q_{ab},\ell^a)$5 and $(q_{ab},\ell^a)$6, where $(q_{ab},\ell^a)$7 is the surface tension canonically conjugate to $(q_{ab},\ell^a)$8. Its covariant divergence reproduces the Raychaudhuri and Damour equations as vertical and horizontal projections; the vorticity $(q_{ab},\ell^a)$9 drops out of both, while the acceleration $q_{ab}$0 enters only the Damour equation.

Symmetries and the purity of the shift

Three symmetries act on the ruled Carrollian structure: diffeomorphisms of $q_{ab}$1, rescalings $q_{ab}$2, and shifts $q_{ab}$3 with $q_{ab}$4 horizontal. The paper fixes the transformation of the connection under each symmetry so that both the stress tensor and the equations of motion $q_{ab}$5 are covariant. Under rescaling, the Raychaudhuri constraint has weight 2 and the Damour constraint weight 1, suggesting that the latter will play the role of a CFT current upon quantization. Under shifts, the stress tensor is strictly invariant, and the combined action on the densitized constraints $q_{ab}$6 is lower-triangular, consistent with the semi-direct product structure $q_{ab}$7.

The pivotal result of the symplectic analysis is that the shift symmetry is pure gauge: its Noether charge vanishes off shell and it lies in the kernel of the presymplectic form, $q_{ab}$8. Consequently, the Ehresmann connection carries no independent bulk degree of freedom and can be promoted to background structure without freezing physical diffeomorphisms. This is achieved by compensating each diffeomorphism $q_{ab}$9 with field-dependent rescaling and shift parameters $\ell^a q_{ab}=0$0 and $\ell^a q_{ab}=0$1, which enforce $\ell^a q_{ab}=0$2 while keeping $\ell^a q_{ab}=0$3 horizontal. This defines the prime phase space, generalizing earlier work that had imposed the more restrictive condition $\ell^a q_{ab}=0$4 and thereby decoupled the spin-1 sector.

The resulting charges take the expected form: the smeared generators $\ell^a q_{ab}=0$5 and $\ell^a q_{ab}=0$6 vanish on shell of the Raychaudhuri and Damour constraints respectively, up to corner terms. When the total flux at adapted cuts vanishes (which requires vanishing vorticity and appropriate boundary conditions on $\ell^a q_{ab}=0$7), the charge algebra closes as

$\ell^a q_{ab}=0$8

with $\ell^a q_{ab}=0$9.

Three-stage Dirac reduction

Starting from the bare symplectic potential over unconstrained pairs $q_{ab}=\Omega\,\bar q_{ab}$0 and $q_{ab}=\Omega\,\bar q_{ab}$1, the reduction proceeds through three sets of second-class constraints:

1. Normalization: $q_{ab}=\Omega\,\bar q_{ab}$2 together with $q_{ab}=\Omega\,\bar q_{ab}$3, fixing the vertical component of the momentum.
2. Horizontality: $q_{ab}=\Omega\,\bar q_{ab}$4 and $q_{ab}=\Omega\,\bar q_{ab}$5, imposing degeneracy of the metric and orthogonality of the momentum.
3. Shear: $q_{ab}=\Omega\,\bar q_{ab}$6, identifying the traceless part of the momentum with the shear $q_{ab}=\Omega\,\bar q_{ab}$7.

Steps 1 and 2 have algebraically invertible constraint matrices. Step 3 requires inverting the kernel $q_{ab}=\Omega\,\bar q_{ab}$8, whose inverse is the antisymmetric bilocal tensor $q_{ab}=\Omega\,\bar q_{ab}$9 satisfying the transport equation

$\Omega$0

In adapted coordinates with vanishing shear and expansion, $\Omega$1 reduces to a Heaviside-function retarded kernel along each generator; in general it is dressed by the holonomy $\Omega$2 of the operator $\Omega$3. The existence and uniqueness of this inverse depend on boundary conditions for the first-order transport problem on the null segment between caustics; different admissible prescriptions may yield different Green kernels, and the skew-symmetry required of a Poisson bracket selects one compatible choice.

The final step-3 brackets among $\Omega$4 include the local canonical pairs $\Omega$5 and $\Omega$6, together with the non-local sector

$\Omega$7

plus brackets of $\Omega$8 and $\Omega$9 with $D_a$0 involving derivatives of $D_a$1 contracted with $D_a$2 and horizontal derivatives $D_a$3. Structurally, each insertion of $D_a$4 inserts a shear (vertical evolution of the metric), each insertion of $D_a$5 inserts a horizontal derivative of the metric, and all index contractions and bilocal terms are mediated by $D_a$6 — a diagrammatic organization the authors suggest could be developed further. All brackets involving $D_a$7 follow consistently from those involving $D_a$8 via $D_a$9, checked explicitly against the transport equation.

Independent verification from Hamiltonian generators

The second derivation recasts the symplectic two-form so as to isolate the three spin sectors, with the spin-2 variables shifting the spin-0 and spin-1 momenta by $\theta_{ab}=\tfrac12\mathcal{L}_\ell q_{ab}$0 and $\theta_{ab}=\tfrac12\mathcal{L}_\ell q_{ab}$1. Five families of canonical transformations are then constructed: the trivial spin-0 action generated by $\theta_{ab}=\tfrac12\mathcal{L}_\ell q_{ab}$2; the nontrivial spin-0 action generated by $\theta_{ab}=\tfrac12\mathcal{L}_\ell q_{ab}$3, which rescales the area while adding a nonlocal compensating traceless deformation $\theta_{ab}=\tfrac12\mathcal{L}_\ell q_{ab}$4 to the metric; the trivial spin-1 action generated by $\theta_{ab}=\tfrac12\mathcal{L}_\ell q_{ab}$5; the nontrivial spin-1 action generated by $\theta_{ab}=\tfrac12\mathcal{L}_\ell q_{ab}$6, which shifts $\theta_{ab}=\tfrac12\mathcal{L}_\ell q_{ab}$7 horizontally with a compensating $\theta_{ab}=\tfrac12\mathcal{L}_\ell q_{ab}$8 term; and the spin-2 action generated by $\theta_{ab}=\tfrac12\mathcal{L}_\ell q_{ab}$9, acting on the metric through the propagator map $k_a$0. In each case the spin-2 contribution cancels the non-exact part of the lower-spin contribution by virtue of the Green equation satisfied by $k_a$1, proving canonicity. Unsmearing the identities $k_a$2 reproduces every bracket of the Dirac derivation, including the self-bracket $k_a$3 obtained from $k_a$4.

Relation to prior formulations

The construction occupies a distinct position relative to two closely related programs. Reisenberger's double-null analysis solves the null Einstein equations first, obtaining unconstrained free data parametrizing solutions; there the bulk of each null sheet carries only spin-2 data, and his non-local conformal-metric self-bracket is the closest precursor of the covariant kernel $k_a$5 used here. The present work instead retains the Raychaudhuri and Damour equations as dynamical constraints acting on a kinematical phase space containing all three spin sectors simultaneously. Similarly, the solution-space phase space of Adami et al. corresponds to imposing the null constraints and gauge-fixing to the integrable Ehresmann gauge $k_a$6 with $k_a$7; the Gaussian-null line element matches their parametrization under direct identifications ($k_a$8, $k_a$9, $\iota_\ell k=1$0), so imposing the constraints and co-rotating gauge provides a direct route from the kinematical arena constructed here to that solution-space description.

Limitations and open questions

Several restrictions are stated explicitly. The analysis is confined to a single caustic-free null segment; behavior near or across caustics, and the treatment of endpoint cuts and their edge degrees of freedom (needed when fluxes do not vanish), are deferred. The invertibility of the third-step constraint matrix assumes a well-posed transport problem with chosen boundary conditions, and possible zero modes must be fixed or projected out; different admissible prescriptions yield different Green kernels. The derivation assumes three-dimensional $\iota_\ell k=1$1, though the authors note the extension to general $\iota_\ell k=1$2 via $\iota_\ell k=1$3 leaves the third-step brackets unchanged. Dynamically, the complete algebra of the Raychaudhuri and Damour constraints, their reductions, and the associated observables remain to be worked out. At the quantum level, the paper poses a specific open question: whether the anomaly found previously in the quantization of the Raychaudhuri constraint persists, is modified, or becomes part of a larger anomalous constraint algebra once the Damour constraint and transverse dynamics are included.

Conclusion

This work completes the local kinematical canonical analysis of gravity on a null hypersurface. By establishing that Ehresmann-connection shifts are pure gauge, the prime phase space acquires a non-degenerate symplectic structure on genuine gravitational data spanning the spin-0, spin-1, and spin-2 sectors. The full bracket structure, with its characteristic bilocal Green-kernel non-locality transported along the null generators, is established by two independent derivations that agree in detail. This supplies the classical arena on which the null constraint algebra, observables, dressed quantization, and the fate of the known Raychaudhuri anomaly in the coupled multi-generator system can now be systematically investigated.