A Covariant Formulation of Logarithmic Supertranslations at Spatial Infinity

This paper proposes a new covariant symplectic structure and boundary conditions at spatial infinity that extend the BMS algebra to include regular log-translations and log-supertranslations, yielding finite conserved charges with a central extension and revealing novel physical information for future observables at null and timelike infinity.

The Gist

Imagine the universe as a giant, infinite ocean. In physics, we often study what happens right at the "horizon" of this ocean—where the water meets the sky. In the context of gravity, these horizons are called infinity.

For decades, physicists have studied two main horizons:

1. Null Infinity: Where light rays (photons) go to die. This is where we see gravitational waves.
2. Spatial Infinity: The point where you would be if you traveled infinitely far away in a straight line, but stopped time.

This paper is about Spatial Infinity. It's a bit like the "backstage" of the universe. While the "front stage" (where light goes) is well-known, the "backstage" has been a bit of a mystery.

Here is the story of what the authors discovered, explained without the heavy math.

1. The Problem: The Universe is Messy

Imagine you are trying to describe the shape of a perfectly smooth table. But, if you look really closely, you realize the table isn't just smooth; it has tiny scratches, bumps, and even some weird, jagged edges that only show up if you zoom in enough.

In physics, when we try to describe the universe far away from stars and black holes, we usually assume it's perfectly smooth. But the authors realized: It's not. When you solve the equations of gravity, "logarithmic" terms (weird, slow-growing bumps) naturally appear. They are like the "static" or "noise" in a radio signal that you can't just ignore.

Previous theories tried to smooth these bumps out by forcing the universe to be "even" on one side and "odd" on the other (like a mirror image). The authors said, "No, let's keep the mess. Let's see what happens if we accept the universe is a bit jagged."

2. The Discovery: New "Moves" for the Universe

When you look at a smooth, perfect table, you can only push it in straight lines or spin it. These are the standard rules of physics (translations and rotations).

But when the authors looked at the "jagged" table (the universe with logarithmic terms), they found new ways to move it.

• Supertranslations: Imagine the table isn't rigid. You can push the left side up and the right side down, creating a wave. This is a "supertranslation."
• Log-Supertranslations: This is the new discovery. Imagine the table has a special property where, if you push it, it doesn't just move; it lingers. The movement grows slowly over time, like a whisper that gets louder and louder the longer you listen.

The authors found that the universe has a hidden symmetry that allows for these "lingering" movements. They call this the Log-BMS Algebra. It's like discovering that the universe has a secret dance move that no one knew existed before.

3. The "Charge" (The Scorecard)

In physics, every time you have a symmetry (a way to move the universe without breaking it), there is a "charge" associated with it. Think of a charge like a score on a scoreboard.

• If you rotate the universe, the score is Angular Momentum (how much it spins).
• If you move it, the score is Energy (how much mass it has).

The authors calculated the scores for these new "Log-Supertranslation" moves. They found that:

1. The scores are finite (they don't blow up to infinity).
2. They are conserved (they stay the same over time).
3. The Twist: There is a "central extension." This is a fancy way of saying that if you try to do a "Supertranslation" move and then a "Log-Supertranslation" move, the order matters. It's like putting on socks and shoes: socks then shoes is different from shoes then socks. This interaction creates a new, fundamental link between these two types of movements.

4. Why Does This Matter? (The "Goldstone" Analogy)

Why should we care about these weird, lingering moves?

Imagine a ball sitting at the bottom of a valley. It's in a stable position. But if the valley is actually a flat plain (which is what the universe looks like at infinity), the ball can roll anywhere without losing energy.

The authors suggest that the "jaggedness" of the universe (the logarithmic terms) creates a landscape with many different "valleys."

• The Log-Supertranslations are the keys that let the universe roll from one valley to another.
• The "charges" they calculated are the coordinates telling us which valley the universe is currently in.

This is huge because it means the universe has a memory. Even if nothing is happening right now, the universe "remembers" all the past moves it made. This memory is encoded in these new charges.

5. The Big Picture: A New Definition of Spin

One of the most practical results of this paper is about Angular Momentum (spin).
Currently, if you try to measure how much a black hole is spinning, the answer can change depending on how you look at it (a problem called "frame dependence"). It's like trying to measure the speed of a car while you are also running alongside it; the number changes based on your speed.

The authors showed that by using these new "Log-Supertranslation" charges, we can redefine Angular Momentum in a way that is absolute. It doesn't matter how you look at it; the "center-of-mass spin" is now a fixed, unchangeable number. It's like finding a universal ruler that works everywhere, no matter how you tilt it.

Summary

• The Old View: The universe at infinity is smooth and simple.
• The New View: The universe at infinity is "polyhomogeneous" (it has logarithmic bumps and jagged edges).
• The Result: This jaggedness reveals a new set of symmetries called Log-Supertranslations.
• The Payoff: These symmetries give us a way to track the universe's "memory" and define a perfect, unchangeable measure of how much the universe is spinning.

In short, the authors opened the door to a new room in the house of the universe, found a new set of keys (symmetries), and realized that the locks (charges) on the doors hold the secrets to the universe's spin and history.

Technical Summary

Here is a detailed technical summary of the paper "A Covariant Formulation of Logarithmic Supertranslations at Spatial Infinity" by Girelli et al.

1. Problem Statement

The paper addresses the asymptotic symmetries of asymptotically flat spacetimes specifically at spatial infinity ($i^0$). While the Bondi-Metzner-Sachs (BMS) group is well-established at null infinity ($\mathscr{I}^\pm$), the structure of symmetries at spatial infinity has been a subject of debate, particularly regarding the inclusion of logarithmic terms in the metric expansion.

Key issues addressed include:

• Logarithmic Terms: Generic solutions to Einstein's equations often contain logarithmic terms in their radial expansions (polyhomogeneous expansions), yet many previous analyses imposed parity conditions to eliminate them or restrict the symmetry algebra.
• Log-Supertranslations: Recent work by Fuentealba, Henneaux, and Troessaert (FHT) identified "log-supertranslations" as asymptotic symmetries but relied on specific parity conditions to regularize the theory and define finite charges within a Hamiltonian framework.
• Covariance vs. Hamiltonian: There was a need to realize these extended symmetries (including log-supertranslations) within a covariant phase space formalism without relying on parity conditions, which are often viewed as artificial constraints to ensure smoothness at null infinity.
• Charge Finiteness and Conservation: Constructing finite, conserved charges at spatial infinity without imposing parity conditions has been a significant technical hurdle.

2. Methodology

The authors employ a covariant phase space formalism combined with a polyhomogeneous Beig–Schmidt expansion.

• Metric Expansion: They generalize the Beig–Schmidt metric form to include polyhomogeneous terms (powers of $\rho$ and $\log \rho$) up to sub-subleading orders:
  $$ds^2 = \sigma^2 d\rho^2 + 2\Sigma_a d\rho dx^a + \rho^2 h_{ab} dx^a dx^b$$
  where $\sigma, \Sigma_a, h_{ab}$ contain terms like $\log \rho$ and $\log^2 \rho$. This allows for the action of log-supertranslations.
• Symplectic Regularization: Instead of imposing parity conditions to cancel divergences, they utilize the inherent ambiguities of the covariant phase space formalism (specifically the freedom to add boundary terms to the symplectic potential). They adopt a regularization prescription (following [47, 48]) to render the symplectic form finite without restricting the field configurations' angular dependence.
• Boundary Conditions: They derive a new set of boundary conditions that ensure:
  1. The symplectic form is finite.
  2. The symplectic potential vanishes on-shell (ensuring charge conservation).
  3. The conditions do not restrict the angular dependence of supertranslations or log-supertranslations.
  4. The Weyl tensor admits a purely polynomial asymptotic expansion in $1/\rho$ (with logs appearing only at sub-subleading order).
• Charge Construction: Using the Iyer-Wald method, they compute the canonical charges associated with the residual diffeomorphisms preserving these fall-offs.

3. Key Contributions and Results

A. Extended Asymptotic Symmetry Algebra

The authors identify a new asymptotic symmetry algebra at spatial infinity that extends the BMS algebra. The algebra includes:

• Lorentz Transformations: $SO(1,3)$.
• Translations: Regular translations ($\omega_{reg}$).
• Supertranslations: Both even and odd parity modes ($\omega_{super}$).
• Log-Translations: Regular log-translations ($H_{reg}$).
• Log-Supertranslations: Both even and odd parity modes ($H_{super}$).

Crucially, this algebra is realized without imposing parity conditions on the fields. The authors show that the algebra is a semi-direct sum of the Lorentz algebra acting on these abelian sectors.

B. Central Extensions and Heisenberg Algebras

A major result is the discovery of non-trivial central extensions between specific sectors of the algebra:

• Between Supertranslations ($\omega$) and Log-Supertranslations ($H$).
• Between Singular Translations and Regular Log-Translations.

The algebraic structure is described as a direct sum of the Poincaré algebra and three Heisenberg algebras ($h_3$):
$$\text{Log-BMS} \cong \text{iso}(1,3) \oplus h_3(\omega_{sing}, H_{reg}) \oplus h_3(\omega_{super}^{even}, H_{super}^{odd}) \oplus h_3(\omega_{super}^{odd}, H_{super}^{even})$$
This contrasts with previous works where such central extensions were either absent or required specific parity restrictions to appear.

C. Finite and Conserved Charges

The authors construct explicit, finite, and conserved charges for all these symmetries:

• Translations: The charge associated with regular time-translation recovers the ADM mass.
• Log-Supertranslations: They identify a non-vanishing charge for regular log-translations, which were previously often treated as pure gauge.
• Goldstone Modes: The fields $\bar{\sigma}$ and $\bar{\beta}$ are identified as Goldstone modes associated with these symmetries, parametrizing the degenerate vacuum of the theory.

D. Redefinition of Angular Momentum

Due to the central extension between supertranslations and log-supertranslations, the standard Lorentz charges (angular momentum) depend on the "superframe" (the choice of supertranslation). The authors perform a redefinition of the Lorentz generators ($\tilde{Q}_Y$) using the central charge to make the Poincaré subalgebra an ideal.

• Result: The redefined angular momentum is BMS-frame invariant. It represents a "center-of-mass" angular momentum that remains unchanged under (log-)supertranslations, providing a preferred definition of angular momentum in asymptotically flat spacetimes.

E. Compatibility with Previous Work

• Compère-Dehouck (CD): The work extends CD's analysis by incorporating log-supertranslations.
• FHT: By imposing specific parity conditions on their general solution space, the authors recover the FHT log-BMS algebra and charge structure, demonstrating consistency. However, their general framework is more inclusive, allowing for non-smooth solutions (e.g., Kerr-Taub-NUT spacetimes) that do not satisfy standard peeling properties.

4. Significance and Outlook

• Generalization of Spatial Infinity: The paper demonstrates that the "standard" reliance on parity conditions to define spatial infinity is not forced by consistency in the covariant phase space but is a choice motivated by the desire for a smooth null infinity. This opens the door to studying "non-smooth" sectors of asymptotically flat spacetimes.
• New Observables: The identification of log-supertranslations and their associated charges suggests new physical observables at spatial infinity that encode information not visible at null infinity.
• Holography and Soft Theorems: The work suggests potential connections to holographic dualities (celestial or Carrollian holography) and implies the existence of new "logarithmic soft theorems" or Ward identities associated with these symmetries.
• Angular Momentum: The redefinition of angular momentum to be frame-independent resolves long-standing ambiguities in defining angular momentum in the presence of supertranslations.

In summary, this paper provides a robust, covariant framework for understanding the rich symmetry structure of spatial infinity, revealing that the inclusion of logarithmic terms leads to a larger, centrally extended symmetry algebra (Log-BMS) that naturally accommodates log-supertranslations and provides a frame-independent definition of angular momentum.