Maxwell kinematical algebras and 3D gravities

This paper presents a unified semigroup expansion framework to systematically derive non- and ultra-relativistic Maxwell kinematical algebras and their invariant tensors, enabling the construction of corresponding three-dimensional Chern-Simons gravity theories.

The Gist

Imagine the universe as a giant, complex dance floor. For a long time, physicists have been trying to write the "rules of the dance" that govern how different observers (dancers) move relative to each other.

In the past, we had a few basic rulebooks:

1. The Relativistic Rulebook (Einstein): The speed of light is the ultimate speed limit. Nothing goes faster.
2. The Non-Relativistic Rulebook (Newton): The speed of light is infinite. You can move as fast as you want.
3. The Ultra-Relativistic Rulebook (Carroll): The speed of light is zero. You can't move at all; time moves, but space is frozen.

In 1967, two physicists named Bacry and Lévy-Leblond drew a cube to show how all these different rulebooks are connected. They showed that you could turn one rulebook into another by "crushing" certain parts of the math (a process called contraction), kind of like squishing a balloon until it changes shape.

The Problem: The "Leaky" Rulebooks

The authors of this paper, Patrick Concha and his team, noticed a problem. When they tried to use these rulebooks to build a theory of gravity (specifically in 3D space, which is like a flat sheet), some of the rulebooks were "leaky."

In physics, to write a consistent theory of gravity, you need a special mathematical tool called an invariant bilinear form. Think of this as a perfectly sealed waterproof suit.

• If the suit is sealed (non-degenerate), the theory works, and you can predict how gravity behaves.
• If the suit has holes (degenerate), the theory falls apart, and the equations don't make sense.

Many of the "non-relativistic" and "ultra-relativistic" rulebooks had holes in their suits. They couldn't be used to build a stable theory of gravity without adding extra patches.

The Solution: The "Expansion" Machine

Instead of squishing the rulebooks (contraction), the authors decided to expand them. They used a mathematical tool called a Semigroup Expansion.

The Analogy: The LEGO Tower
Imagine the original rulebooks are small LEGO towers.

• Contraction is like taking the tower apart and throwing away some bricks to make it smaller.
• Expansion is like taking the tower and adding new, specialized bricks to make it taller and more complex.

The authors took the original "cube" of rulebooks and used this expansion machine to build a Maxwellian Cube.

• Maxwell here doesn't refer to electricity directly, but to a specific type of mathematical extension (named after James Clerk Maxwell) that adds new "generators" (new types of moves) to the dance.
• By adding these new moves, they were able to patch the holes in the waterproof suits. Suddenly, the non-relativistic and ultra-relativistic rulebooks became "leak-proof."

What Did They Build?

They created a new, unified framework that connects:

1. The Standard Rulebooks: The old Newton and Einstein rules.
2. The New Maxwell Rulebooks: These are the "patched" versions that work perfectly for gravity.
3. The Infinite Hierarchy: They didn't just stop at one level. They showed that you can keep expanding these rulebooks forever, creating an infinite family of new, complex rulebooks (called $B_k$ algebras).

Think of it like a family tree. The original cube is the great-grandparents. The Maxwell versions are the parents. But the authors showed there are great-great-grandchildren, great-great-great-grandchildren, and so on, all the way up to infinity. Each generation has more complex rules but still fits together perfectly.

Why Does This Matter?

1. Gravity in 3D: They showed how to write down the exact equations (Chern-Simons actions) for gravity using these new, patched rulebooks. This is like finally having the blueprints to build a stable house out of materials that were previously too wobbly.
2. New Physics: These new rulebooks might help us understand weird physical situations, like:
   • The Quantum Hall Effect: How electrons behave in thin layers.
   • Tensionless Strings: Theoretical strings that have no tension.
   • Black Holes: Understanding how they behave in extreme, non-standard universes.
3. The "Hidden" Fields: The expansion added new "fields" (like new instruments in an orchestra). We don't fully know what these instruments play yet, but they are necessary to keep the music (the universe) in tune. The authors suggest these might be related to "gravito-magnetic" effects or forces we haven't fully understood in non-relativistic settings.

The Bottom Line

The authors took an old, famous diagram (the cube of kinematical algebras) and realized that instead of just squishing it to get new physics, we should stretch and expand it. By doing so, they fixed the broken parts of the theory, allowing us to build consistent models of gravity for a wide variety of universes, from the slow-moving (Newtonian) to the frozen (Carrollian), and even created a whole new infinite family of possibilities for future physicists to explore.

They turned a static cube into a dynamic, expanding universe of mathematical possibilities.

Technical Summary

1. Problem Statement

The paper addresses the challenge of constructing consistent, non-degenerate gravitational theories in non-Lorentzian regimes (non-relativistic/Galilean and ultra-relativistic/Carrollian) within the Chern-Simons (CS) framework.

• The Core Issue: In 3D gravity formulated as a CS gauge theory, the existence of a non-degenerate invariant bilinear form on the underlying symmetry algebra is a prerequisite for deriving well-defined field equations.
• The Limitation: Standard kinematical algebras (classified by Bacry and Lévy-Leblond) and their naive Maxwell extensions often suffer from degeneracies in their invariant tensors when applied to non-Lorentzian limits. This prevents the construction of consistent CS gravity actions.
• Previous Work: While specific non-degenerate algebras (like the Extended Bargmann and Extended Carroll algebras) have been constructed via specific contractions or ad-hoc extensions, a unified systematic framework to generate the entire "Maxwellian" hierarchy of these algebras was lacking.

2. Methodology

The authors employ the $S$-expansion method (specifically using semigroups) to generalize the construction of kinematical algebras. This approach replaces the traditional Inönü-Wigner contraction with an expansion procedure that increases the number of generators while preserving the algebraic structure.

• Semigroup Expansion ($S_E^{(N)}$): The authors utilize the abelian semigroup $S_E^{(N)} = \{\lambda_0, \lambda_1, \dots, \lambda_{N+1}\}$ with a specific multiplication law.
• Resonant Subalgebra Construction:
  1. They decompose a parent Lie algebra $\mathfrak{g}$ into subspaces $V_0 \oplus V_1$ satisfying a $\mathbb{Z}_2$-graded structure.
  2. They decompose the semigroup into subsets $S_0$ and $S_1$ that are "resonant" with the algebraic subspaces.
  3. They construct the expanded algebra $\mathfrak{G} = (S_0 \times V_0) \oplus (S_1 \times V_1)$.
  4. A $0_S$-reduction is applied (setting the zero element of the semigroup to zero) to obtain the final algebra.
• Application to Kinematical Cubes:
  • Step 1: Start with the relativistic AdS algebra ($\mathfrak{so}(2,2)$).
  • Step 2: Apply an $S_E^{(2)}$-expansion to generate the Maxwell algebra (relativistic).
  • Step 3: Apply the same expansion logic to the non-Lorentzian limits (Newton-Hooke, Galilei, AdS-Carroll, Carroll) to generate their Maxwellian extensions.
  • Step 4: Generalize the process using an arbitrary semigroup $S_E^{(N)}$ to create an infinite hierarchy of algebras known as $B_k$ algebras.

3. Key Contributions

A. The Maxwellian Kinematical Cube

The authors successfully construct a "Maxwellian generalization" of the Bacry and Lévy-Leblond cube. By replacing contraction arrows with expansion arrows, they demonstrate that:

• Non-Relativistic (NR) Maxwell Algebras: The Maxwellian Extended Bargmann (MEB) algebra is derived. It can be obtained either from the Extended Newton-Hooke algebra or directly from the relativistic Maxwell algebra.
• Ultra-Relativistic (UR) Maxwell Algebras: The Maxwellian Extended Carroll (MEC) algebra is derived. It arises from the Extended AdS-Carroll or the relativistic Maxwell algebra.
• Static Algebras: A Maxwellian Extended Static (MES) algebra is introduced, which requires specific central charges to ensure non-degeneracy.

B. Systematic Construction of Invariant Tensors

A major technical achievement is the automatic generation of non-degenerate invariant bilinear forms for all these new algebras. The $S$-expansion method naturally provides the components of the invariant tensor (e.g., $\langle J, S \rangle$, $\langle G_a, P_b \rangle$, etc.) in terms of the parameters of the parent algebra. This solves the degeneracy problem that plagued previous attempts to define Maxwellian non-Lorentzian gravity.

C. Generalized $B_k$ Hierarchy

The paper extends the $S_E^{(2)}$ expansion to an arbitrary $S_E^{(N)}$ expansion.

• This generates an infinite hierarchy of algebras denoted as $B_k$ (where $k = N+2$).
• $N=1$: Recovers the standard Bacry-Lévy-Leblond cube (Poincaré, Galilei, Carroll, etc.).
• $N=2$: Recovers the Maxwellian extensions discussed above.
• $N \geq 3$: Generates new families of algebras (e.g., $B_5$) that are neither post-Newtonian nor post-Carrollian in the traditional sense but represent higher-order generalizations.

D. Chern-Simons Gravity Actions

For each constructed algebra, the authors explicitly write down the corresponding 3D Chern-Simons gravity action.

• The actions are constructed using the gauge connection one-forms associated with the expanded generators (e.g., spin connections, vielbeins, and new Maxwell fields).
• The field equations are derived as the vanishing of the curvature two-forms.
• The paper highlights that the non-degeneracy of the invariant tensor ensures that the equations of motion are well-posed, unlike in degenerate cases where constraints might be lost.

4. Key Results

1. Unified Framework: The paper proves that the non-degenerate Maxwell algebras found in literature (MEB, MEC) are not isolated cases but are specific elements of a unified expansion scheme rooted in the relativistic Maxwell algebra.
2. Isomorphisms: The authors identify isomorphisms between seemingly different algebras. For instance, the Maxwell algebra is isomorphic to the Extended AdS-Carroll algebra, and the MEB algebra is isomorphic to the Extended AdS-Static algebra (renamed "Para-MEB").
3. New Generators and Fields: The expansion introduces new generators (e.g., $Z, Z_a, T, L, S, B, Y$) which correspond to new gauge fields in the gravity action. These fields are essential for maintaining non-degeneracy.
   • In the MEC case, the presence of a central charge $L$ is crucial for the non-degeneracy of the invariant tensor.
4. Recursive Structure: The CS action for the $B_k$ algebra is shown to be the CS action for $B_{k-1}$ plus an additional contribution proportional to a new coupling constant. This provides a recursive method for building higher-order gravity theories.
5. Explicit Commutation Relations: The paper provides complete tables of commutation relations for the extended kinematical algebras (Appendix A) and the $B_5$ generalizations (Appendix B).

5. Significance and Future Directions

• Theoretical Unification: This work unifies the study of relativistic, non-relativistic, and ultra-relativistic gravity under a single algebraic expansion framework. It clarifies the structural relationship between Maxwell algebras and kinematical limits.
• New Gravity Models: The construction of non-degenerate invariant tensors allows for the formulation of consistent 3D CS gravity theories in non-Lorentzian regimes with Maxwell fields. This opens the door to studying black hole solutions, thermodynamics, and asymptotic symmetries in these new theories.
• Physical Interpretation: The authors suggest that the new gauge fields introduced by the expansion could be interpreted as post-Newtonian or post-Carrollian corrections, or as sources for gravito-magnetic effects and torsion in non-relativistic geometries.
• Holography and Asymptotic Symmetries: The paper proposes that these algebras may correspond to asymptotic symmetries of gravitational models (generalizing the $BMS_3$ algebra) and could play a role in flat holography or AdS-Carroll holography.
• Extensions: The framework is readily extensible to supersymmetry and higher-spin theories, suggesting a path toward non-degenerate supergravity and higher-spin gravity in non-Lorentzian limits.

In summary, the paper provides a robust algebraic machinery ($S$-expansion) to systematically generate a "Maxwellian" version of the kinematical universe, ensuring mathematical consistency (non-degeneracy) and providing the necessary tools (CS actions) to explore the physics of these generalized gravitational theories.