Carrollian Electric Limit of Gravity

• Carrollian Electric Limit is the ultra-relativistic contraction of GR where only the electric tidal components remain, while magnetic modes vanish.
• The framework employs a 1+3 covariant decomposition with constraints that freeze temporal dynamics, resulting in an ultra-local static gravitational field.
• This limit is pivotal for analyzing black hole horizons and null infinity, offering insights into gravitational memory and holographic boundary data.

The Carrollian Electric Limit is the ultra-relativistic ($c \to 0$) contraction of General Relativity in which only the “electric” components of the gravitational field—those encoding static tidal forces—survive, while “magnetic” components responsible for radiative and frame-dragging effects are suppressed. This construction parallels the distinction between “electric” and “magnetic” Carrollian limits familiar from Carrollian electromagnetism. In the most precise covariant (1+3) language, the Carrollian Electric Limit defines a static, constraint-based gravitational theory whose natural setting is null hypersurfaces such as black hole event horizons and future (or past) null infinity (Patil et al., 9 Sep 2025).

1. Covariant Structure and Weyl Decomposition

In the covariant “1+3” formalism, the free gravitational field is captured by the decomposition of the Weyl tensor $C_{\mu\nu\lambda\rho}$ into its electric and magnetic parts with respect to a timelike 4-velocity $u^\mu$:

• Electric part: $E_{ab} = C_{acbd}u^c u^d$ (tidal tensor)
• Magnetic part: $$H_{ab} = \frac{1}{2}\epsilon_{acpq} C^{pq}{}_{bd} u^c u^d$$ (gravito-magnetic/radiative tensor)

The Carrollian Electric Limit arises by sending $c \to 0$ so that dynamical time derivatives are frozen out:

$\dot{E}^{ab} = 0$

$H^{ab} = 0$

$$(\mathrm{curl}\ E)^{ab} = \eta^{cd\langle a}\widetilde{\nabla}_c E^{b\rangle}{}_d = 0$$

where $\dot{E}^{ab}$ is the derivative along $u^\mu$, $\widetilde{\nabla}_a$ is the spatially projected derivative, and $\eta^{abc}$ is the volume element on the spatial hypersurface.

2. Constraint System and Elimination of Dynamics

The vanishing of $\dot{E}^{ab}$ means that the evolution of the tidal field is completely arrested; gravitational “memory” imprints or gravitational wave propagation are absent. Together with $H^{ab}=0$ and the constraint on the curl, the system is ultra-local: there are no propagating degrees of freedom. The remaining gravitational field is determined solely by static spatial constraints supplied by matter distribution.

The gravitational Gauss law for $E_{ab}$ becomes

$$\widetilde{\nabla}_b E^{ab} - \frac{1}{3}\widetilde{\nabla}^a\mu = 0$$

where $\mu$ is the local energy density of matter. Thus, $E_{ab}$ encodes all non-trivial gravitational tidal information, but responds instantaneously (in Carrollian “time”) to matter.

An additional constraint arises from requiring invariance under Carrollian boosts (wherein time and space play dual, degenerate roles):

$-\frac{1}{2} (\mu + p) \sigma_{ab} = 0$

If $(\mu + p) \ne 0$, the shear tensor $\sigma_{ab}$ must vanish: $\sigma_{ab} = 0$ Thus, fluid congruences must be shear-free in the electric limit. This further enforces the static character of the solution space.

3. Transformation Properties and Carrollian Symmetry

Under Carrollian boosts, the 1+3 decomposition ensures that the electric limit is preserved only if the special constraints above are imposed. The theory does not arise from simply contracting every component of linearized Einstein's equations, but rather through a careful extraction of a self-consistent subtheory—one in which transformations generate no magnetic (radiative or frame-dragging) modes, and in which remaining variables are invariant under the associated degenerate time-direction.

4. Physical Interpretation: Horizons, Memory, and Holography

The Carrollian Electric Limit provides a geometric and dynamical framework for the near-horizon structure of black holes and for null infinity in asymptotically flat space:

• Black Hole Horizons: The event horizon is a null surface, naturally associated with the degeneration of the metric characteristic of Carrollian structure. On the horizon, only static tidal components described by $E_{ab}$ survive, serving as boundary data (“soft hair”) that encode information about the gravitational field impinging on the horizon.
• Gravitational Memory: Because $\dot{E}^{ab}=0$, the only form of “memory” is in a permanent change to the static tidal field sourced by matter. A burst of radiation alters the boundary conditions for $E_{ab}$ permanently, but there is no propagating dynamical response.
• Flat Space Holography: The symmetry group at null infinity (the BMS group) is naturally realized in Carrollian geometry. The Electric Limit provides the static regime in which holographic data are encoded in boundary values of $E_{ab}$, where static “soft charge” configurations or “hair” are meaningful, but radiative data (associated to $H_{ab}$) vanish.

This setup thus explains why Carrollian geometry and, in particular, the Electric Limit, are relevant in encoding the “soft sector” of gravity at null infinity and black hole horizons.

5. Implications and Further Context

The systematic isolation of the Carrollian Electric Limit:

• Offers a rigorous covariant framework for studying static, ultra-local limits of General Relativity and underpins the geometric structure of null boundaries;
• Clarifies why naive Carrollian contractions of linearized Einstein equations fail to capture consistent gravitational dynamics unless the above constraints on $E_{ab}$ and $H_{ab}$ are imposed;
• Provides the foundation for further development of Carrollian gravity theories, including coupling to matter or extensions incorporating BMS symmetry or subleading corrections appropriate to near-horizon and holographic contexts.

In summary, the Carrollian Electric Limit is a rigid, static, constraint-based sector of General Relativity characterized by $c \to 0$, the vanishing of the Weyl magnetic part, frozen tidal evolution, and strict spatial constraints on $E_{ab}$, with direct relevance to gravitational horizons, memory effects, and holography in asymptotically flat spacetimes (Patil et al., 9 Sep 2025).