Carroll Bulk Geometry Framework

• Carroll bulk geometry is a degenerate spacetime framework characterized by a decoupling of time and space via an Ehresmann connection.
• It provides a precise mathematical basis for analyzing null hypersurfaces, capturing key features such as Carroll torsion and conformal symmetries.
• Its applications span flat space holography, black hole horizon analysis, and ultra-relativistic limits, linking gravitational asymptotics to Carrollian dynamics.

Carroll bulk geometry is a geometric framework that describes spacetime structures arising in the ultra-relativistic limit where the speed of light $c \to 0$. This leads to a decoupling of spatial and temporal directions, resulting in a degenerate metric whose kernel typically corresponds to a preferred "time" direction. Carrollian structures have become a central tool for the analysis of null hypersurfaces in Lorentzian manifolds, the paper of asymptotically flat limits (notably through their deep connections with BMS symmetry algebras), and the formulation of field and brane dynamics in non-Lorentzian regimes.

1. Fiber Bundle Structure and the Ehresmann Connection

Carrollian spacetimes are naturally described as $(d+1)$-dimensional fiber bundles, with a $d$-dimensional spatial base $S$ and a one-dimensional temporal fiber parametrized by the coordinate $t$:

• The canonical projection is $T: (t, x^i) \mapsto x^i$, so the vertical subbundle $V$ at each point is $\ker(dT) = \operatorname{span}\{\partial_t\}$.
• The tangent bundle decomposes as $T_{(t,x)}C = V \oplus H$, where $H$ is a horizontal subbundle defined by a chosen Ehresmann connection.

Locally, the Ehresmann connection is given by a one-form $b = b_i(t, x) dx^i$. The adapted horizontal basis vectors are

$E_i = \partial_i + b_i(t, x) \partial_t,$

so that an arbitrary spatial vector $W^i$ is lifted via $W^i E_i$. The Ehresmann connection governs the nontrivial separation between space and time, and defines a projection operator

$p = dt \otimes (\partial_t - b_i dx^i),$

implementing the splitting of vertical and horizontal directions.

Crucially, the Carrollian metric is degenerate, with form

$g = g_{ij}(t, x) dx^i \otimes dx^j,$

such that $\operatorname{ker}(g) = V$. For any vertical vector $V \in V$, $g(V, \cdot) = 0$, reinforcing that the metric measures "spatial" distances only.

This geometric structure is not only an intrinsic feature but is also the canonical induced geometry on null hypersurfaces in Lorentzian manifolds, where the degenerate direction is spanned by the null generator.

2. Carroll Geometry on Null Hypersurfaces

When a Lorentzian spacetime is foliated by null hypersurfaces (e.g., $u = \text{const}$), the pullback of the metric degenerates along the null generator $\ell$ which is both tangent and normal to the hypersurface:

• The induced metric is of Carrollian form.
• There exists a canonical one-form normal to the hypersurface, denoted $n = \Omega(dt - b_i dx^i)$, with $b_i$ matching the Ehresmann connection.
• The spatial tangent space is defined by $n(X) = 0$.

Intrinsic geometric quantities include:

• Carrollian Torsion (or Vorticity):

$W_{ij} = - Q(E_i (b_j) - E_j (b_i)),$

with $Q$ the appropriate projection, capturing the horizontal part of the curvature of the Ehresmann connection.

• Acceleration, Expansion, and Shear: Derived from time derivatives of the spatial metric $g_{ij}$ or from the curvature of the Levi–Civita–Carroll connection constructed from $g_{ij}$ and $b_i$.

Null hypersurfaces in Lorentzian spacetimes are thus natural settings for Carroll bulk geometry, with the degeneracy direction being the generator of the null surface.

3. Conformal Isometries and BMS Algebra

Because of the degeneracy, standard Killing vector definitions break down. The appropriate conformal isometry condition is

$\mathcal{L}_\xi g = \omega g,$

with $\omega$ an arbitrary function. A general Carrollian vector field is

$\xi = f(t, x) E + Y^i(x) E_i,$

where $E = \partial_t$ and $E_i$ are the horizontal lifts. The conformal Killing equations, in cases without shear ($S_{ij} = 0$), reduce to

$D_i Y_j + D_j Y_i = \frac{2}{d} (D_k Y^k) g_{ij},$

with $D_i$ the Levi–Civita–Carroll (spatial) covariant derivative.

Significantly:

• The solution space for $\xi$ forms a semi-direct sum algebra: the finite-dimensional conformal algebra of $g_{ij}$ and an infinite-dimensional "supertranslation" algebra generated by arbitrary functions shifting $t$.
• In $d = 1,2$, this algebra is isomorphic to the BMS algebra, $\mathfrak{bms}_{d+2}$, the asymptotic symmetry group of flat spacetime at null infinity.
• For $b_i$ pure gauge, or removed by a suitable Carrollian diffeomorphism, conformal Carrollian Killing vectors simplify:

$$\xi_{T,Y} = (T(x) + \tfrac{1}{2}\partial_t C(t,x) Y(x)) E + Y^i(x) E_i,$$

aligning the Carroll symmetry algebra with the structure of the BMS algebra, including supertranslations and (spatial) conformal isometries.

This deep connection underlies the "Carroll–null–BMS triangle," making Carroll geometry a foundational tool in flat space holography and in characterizing asymptotic symmetries of gravity.

4. Carroll Tensors, Connections, and Dynamics

Carrollian geometry supports a robust calculus:

• Tensors: All standard tensorial constructions are performed on the horizontal subbundle, with vertical vectors in the kernel of $g$ annihilated.
• Levi–Civita–Carroll Connection: An affine connection compatible with $g$ and $b_i$, allowing definitions of curvature and parallel transport on the spatial base.
• Projectors: The splitting of $T(C)$ into $V \oplus H$ is made explicit by $p$ and by the basis $\{E, E_i\}$.

Dynamic fields or geometries (such as those induced on evolving null hypersurfaces) are encoded via:

• Time derivatives along $E$,
• Horizontal derivatives via $E_i$,
• Structure equations involving torsion or other curvature measures built from $b_i$ and $g_{ij}$.

5. Applications and Significance

Carroll bulk geometry underpins several major developments:

• Null Surface/Boundary Holography: It gives the precise local description of the geometry at null infinity, providing the background for BMS-invariant field theories and their conformal Carrollian analogues in flat space holography.
• Membrane Paradigm and Black Hole Horizons: As null hypersurface geometry is Carrollian, analyses of isolated/dynamical horizons, their charges, and fluid limits naturally utilize Carrollian variables.
• Ultra-Relativistic Limits and Duality: Carrollian structures appear as ultra-relativistic limits of relativistic theories, dual to nonrelativistic (Galilei) geometries, enabling duality relations and symmetry exchanges (cf. Galilei/Carroll duality in brane theories (Bergshoeff et al., 2020)).
• Flat Space Holography and BMS Symmetry: The emergence of infinite-dimensional symmetry algebras for asymptotically flat gravity is best understood in the Carrollian framework, which organizes the supertranslations and the conformal isometries on the spatial sphere.

Carroll bulk geometry thus provides a flexible, mathematically rigorous scaffold for understanding both classical and quantum aspects of gravity in non-Lorentzian and boundary-dominated regimes. Its formalism is indispensable for any program aimed at a first-principles description of null hypersurface physics, asymptotic symmetries, and flat holography.