Carroll versus Newton and Galilei: two dual non-Einsteinian concepts of time

Abstract: The Carroll group was originally introduced by Levy-Leblond [1] by considering the limit of the Poincar\'e group as $c\to0$. In this paper an alternative definition, based on the geometric properties of a non-Minkowskian, non-Galilean but nevertheless boost-invariant, space-time structure is proposed. A "duality" with the Galilean limit $c\to\infty$ is established. Our theory is illustrated by Carrollian electromagnetism.

• The paper introduces dual non-Einsteinian time frameworks by deriving Carrollian and Galilean symmetries from distinct non-relativistic limits of the Poincaré group.
• It develops geometric structures through Newton-Cartan and Carroll manifolds, unified via Bargmann space, to rigorously establish the time duality.
• The analysis reveals practical insights including Carrollian electrodynamics and the immobility of free Carrollian particles, with potential applications in holography and metamaterials.

This paper explores the Carroll group and Carroll spacetime structures, presenting them as a dual counterpart to the well-known Galilei group and Newton-Cartan spacetime. The duality arises from considering different non-relativistic limits ($c \to \infty$ and $C \to \infty$) of the Poincaré group and the relativistic Maxwell equations, leading to two distinct notions of "time."

The Carroll Group and its Duality with the Galilei Group

The Carroll group, Carr(d+1), is introduced as a contraction of the Poincaré group E(d,1) in a limit where a new velocity-like constant $C$ goes to infinity. This contrasts with the standard Galilean limit where the speed of light $c$ goes to infinity. The paper highlights a fundamental duality between the absolute Newtonian time $t$ and a Carrollian "time" $s$. While $t$ has units of time, $s$ is shown to have units of action per mass ($L^2 T^{-1}$).

The defining transformations for flat (d+1)-dimensional spacetime are:

• Galilei Boosts ($c \to \infty$): $C \to \infty$0, $C \to \infty$1 (with $C \to \infty$2)
• Carroll Boosts ($C \to \infty$3): $C \to \infty$4, $C \to \infty$5 (with $C \to \infty$6, and $C \to \infty$7 is the Carrollian time coordinate)

The duality exchanges the roles of spatial and time coordinates in these boosts.

Geometric Structures

The paper defines geometric structures corresponding to these groups:

• Newton-Cartan (NC) manifolds: (N, $C \to \infty$8, $C \to \infty$9, $C$0), where N is a (d+1)-manifold, $C$1 is a degenerate contravariant metric (rank d), $C$2 is a nowhere-vanishing 1-form whose kernel generates the degeneracy of $C$3 ($C$4), and $C$5 is a compatible connection. $C$6 defines spatial slices. $C$7 is closed ($C$8), implying the existence of a global time function $C$9. The flat NC structure is $c$0, $c$1, $c$2.
• Carroll (C) manifolds: (C, g, $c$3, $c$4), where C is a (d+1)-manifold, g is a degenerate covariant metric (rank d), $c$5 is a nowhere-vanishing vector field whose kernel generates the degeneracy of g ($c$6), and $c$7 is a compatible connection. The flat Carroll structure is $c$8, $c$9, $t$0 (where $t$1 is the Carrollian time coordinate).

The geometric duality is evident: NC uses a degenerate contravariant metric and a special 1-form ($t$2, the clock), while Carroll uses a degenerate covariant metric and a special vector field ($t$3).

Unification via Bargmann Space

A key theoretical insight is that both structures can be unified and derived from a (d+2)-dimensional Minkowski space, referred to as Bargmann space (B, G, $t$4). The flat Bargmann metric is $t$5, with a null vector field $t$6.

• Newton-Cartan space is obtained by taking the quotient of Bargmann space by the translations generated by $t$7 ($t$8).
• Carroll space is obtained as a null hypersurface embedded in Bargmann space (e.g., defined by $t$9).

The duality $s$0 in Bargmann space corresponds to the duality between NC and Carroll structures. The Bargmann group (extended Galilei group) is a central extension of the Galilei group, and the Carroll group is the derived group of the Bargmann group.

Carrollian Electromagnetism

The paper analyzes the Carrollian limit of Maxwell's equations, analogous to the two Galilean limits (electric and magnetic types) found by Le Bellac and Lévy-Leblond. Starting from relativistic Maxwell equations and redefining fields, taking the $s$1 limit (where $s$2 is the time coordinate) yields two types of Carrollian electromagnetism:

1. Electric-like: Equations resemble $s$3, $s$4, $s$5, $s$6. Carroll boosts act via $s$7, $s$8.
2. Magnetic-like: Equations resemble $s$9, $t$0, $t$1, $t$2. Carroll boosts act via $t$3, $t$4.

These two types are related by an electric-magnetic duality transformation in the Carrollian setting, distinct from the relativistic duality. The geometric formulation distinguishes between theories based on a covariant 2-form $t$5 ("electric") and a contravariant bi-vector $t$6 ("magnetic"), reflecting the degenerate nature of the Carroll metric where "musical isomorphisms" (index raising/lowering) are not straightforward. Both Carroll electromagnetisms can be derived from Maxwell theory in the higher-dimensional Bargmann space.

For electromagnetism in a medium, the paper shows that the standard Maxwell equations can be made Carroll-invariant by choosing unconventional constitutive relations between $t$7 and $t$8. For the electric-type implementation, this leads to relations like $t$9 and $s$0, where $s$1 are functions of Carrollian field invariants.

Chaplygin Gas Example

The paper discusses the Chaplygin gas (a fluid model) as a physical system exhibiting Carroll symmetry, among others. The non-conventional, field-dependent nature of the symmetries for this system in the non-relativistic limit can be understood naturally by lifting the system to the higher-dimensional Bargmann space. The field $s$2 in the fluid dynamics can be related to the Bargmann 's' coordinate, and the strange field-dependent boosts in the original system lift to simple Carroll boosts in Bargmann space. The $s$3 duality in Bargmann space also relates the Galilean and Carroll boosts acting on this lifted system.

Carrollian Particles

Using the coadjoint orbit method (Souriau's method), the paper constructs elementary particles associated with the Carroll group. A key finding is that free massive Carrollian particles "cannot move." Their position and velocity remain constant with respect to the Carrollian time $s$4. This aligns with the Red Queen's description of needing "all the running you can do, to keep in the same place" in Through the Looking-Glass.

Practical Implications and Further Work

While highly theoretical, the paper suggests potential applications for the Carroll framework:

• Holography: The Carroll group is identified as the conformal extension of the Bondi-Metzner-Sachs (BMS) group, which appears in the study of asymptotically flat spacetimes and is relevant to holographic principles in quantum gravity.
• Brane Dynamics: Carroll groups emerge naturally in the limit where brane world volumes become lightlike.
• Spacetime Singularities and Strong Gravity: Carrollian spacetimes might be relevant near spacetime singularities or in strong coupling limits of General Relativity.
• Quantum Field Theory on Null Hypersurfaces: The structure of Carrollian space-times, particularly their relation to null hypersurfaces in Bargmann space, could be relevant for quantizing fields on causal horizons.
• Metamaterials: The discussion on non-Einsteinian electrodynamics in media raises the question of whether metamaterials could be engineered to exhibit the unconventional constitutive relations consistent with Galilean or Carrollian invariance, potentially leading to exotic light propagation phenomena.

In summary, the paper systematically develops the concept of Carroll geometry and its associated symmetry group as a mathematical dual to Newton-Cartan/Galilean geometry, primarily using group-theoretic contractions and higher-dimensional embedding via Bargmann space. It illustrates these concepts with theoretical examples in electromagnetism and fluid dynamics, revealing that different non-relativistic limits lead to distinct physical theories with unique symmetries and properties, such as the immobility of free Carrollian particles. The "practical" aspect lies in providing the theoretical framework for describing physical systems in these specific extreme non-relativistic regimes.