On symmetries of gravitational on-shell boundary action at null infinity

This paper resolves corner ambiguities in the gravitational boundary action at null infinity by constraining it to reproduce tree-level 5-point eikonal amplitudes, thereby naturally deriving the subleading soft graviton theorem and proposing an infinite tower of Goldstone modes via a generalized Geroch tensor to account for higher-order soft insertions and tree-level symmetries.

The Gist

Imagine the universe as a giant, invisible ocean. When massive objects like black holes or stars crash into each other, they create ripples in this ocean called gravitational waves. These waves travel outward forever, eventually reaching the very edge of the universe, a place physicists call "Null Infinity."

For a long time, scientists thought they understood the rules of these ripples. But recently, a new paper by Shivam Upadhyay suggests that the rules are much more complex and beautiful than we thought. Here is a simple breakdown of what this paper is about, using everyday analogies.

1. The "Receipt" Problem (The Boundary Action)

Imagine you are at a restaurant. You eat a meal (the gravitational event), and you get a bill (the Action). In physics, to calculate what happened during a collision, you need to look at this "bill."

Usually, physicists calculate the bill by looking at the food eaten in the middle of the restaurant (the bulk of space). But in gravity, there's a catch: the bill is actually written entirely on the receipt at the edge of the table (the boundary).

Upadhyay's paper focuses on this receipt. He says, "Wait a minute, the receipt has some smudges and ambiguities." Specifically, there are "corner terms"—like extra charges for the napkins or the tip—that weren't clearly defined. If you get these corners wrong, your total bill is wrong, and your prediction of the future is wrong.

2. Fixing the Smudges (The 5-Point Amplitude)

How do you fix a smudged receipt? You compare it to a known truth.

Upadhyay uses a specific test: The 5-Point Amplitude.
Think of this as a "standard meal" that we know exactly how much it costs. It involves two particles crashing and one extra, very tiny, almost invisible particle (a "soft graviton") popping out.

Upadhyay says: "We will adjust the smudges on our receipt until the total cost matches the known price of this 5-particle meal."

By doing this, he fixes the "corner ambiguities." He proves that if you fix the receipt correctly, the math naturally explains why that tiny extra particle appears. This tiny particle is the universe's way of remembering that a collision happened, even after the waves have passed. This is called Gravitational Memory.

3. The "Infinite Library" of Ripples (The Infinite Tower)

Here is the most exciting part.

For a long time, we thought there were only two main types of ripples:

1. The Big Wave: The main crash (Leading order).
2. The Small Ripple: The memory of the crash (Subleading order).

But Upadhyay proposes something wild. He suggests there is an infinite library of ripples.

Imagine the gravitational wave isn't just a single sound, but a symphony.

• The first note is the crash.
• The second note is the memory.
• But Upadhyay says there are notes 3, 4, 5, and so on, forever.

He calls these "Goldstone Modes." Think of them like the "ghosts" of the symmetry. When the universe breaks a rule (like when two black holes merge), it leaves behind a "ghost" of that rule. Upadhyay found a way to generalize a mathematical tool (the Geroch tensor) to catch not just the first ghost, but an infinite tower of ghosts.

4. The "Universal Translator" (Symmetries)

Why does this matter? Because these ripples are actually symmetries.

In physics, a "symmetry" is a rule that says, "If I change the view, the laws of physics stay the same."

• Supertranslations: Imagine shifting the time on a clock. The universe looks the same.
• Superrotations: Imagine spinning the sky like a globe. The universe looks the same.

Upadhyay shows that by fixing the "receipt" (the boundary action) correctly, we can see that the universe has an infinite number of these rules. Every time you look at a deeper level of the "soft" ripples (the faintest whispers of gravity), you find a new symmetry rule.

The Big Picture Analogy

Imagine the universe is a giant drum.

• When you hit it, it makes a loud boom (the collision).
• Then it hums a low note (the memory).
• Upadhyay's paper is like discovering that the drum doesn't just make a boom and a hum. It actually plays a complex, infinite jazz solo.

Every note in that solo corresponds to a fundamental law of the universe. By listening to the faintest, quietest notes (the "soft theorems"), we can decode the entire song of the universe.

Why is this a big deal?

1. It connects the dots: It links the math of how we calculate particle collisions (Quantum Mechanics) with the math of how space bends (General Relativity).
2. It finds hidden rules: It suggests the universe has an infinite number of hidden conservation laws (symmetries) that we haven't fully understood yet.
3. It fixes the math: It solves a long-standing problem about how to correctly calculate the "cost" of gravity at the edge of the universe.

In short, Upadhyay cleaned up the math at the edge of the universe, fixed the "receipt," and in doing so, discovered that the universe is playing a much more complex and beautiful song than we ever imagined.

Technical Summary

1. Problem Statement

The paper addresses the formulation of the gravitational on-shell boundary action at null infinity ($\mathcal{I}^\pm$) for asymptotically flat spacetimes. While the connection between asymptotic symmetries (BMS group), soft theorems, and infrared physics is well-established, there are specific challenges in deriving these results directly from the path integral formulation of the S-matrix (the Arefeva-Faddeev-Slavnov or AFS approach):

• Corner Ambiguities: The boundary action on null hypersurfaces contains "corner terms" arising from the junction of null boundaries with spacelike or timelike segments. These terms are ambiguous due to the freedom in parametrizing null generators and choosing the conformal factor.
• Memory Effects: Previous derivations of the tree-level S-matrix from boundary actions (e.g., by Fabbrichesi et al.) often assumed a trivial vacuum (vanishing memory). The paper seeks to construct the action in a generic supertranslated vacuum where non-trivial gravitational memory exists.
• Subleading Symmetries: While the leading soft graviton theorem (associated with supertranslations) is well-understood, the derivation of subleading soft theorems (associated with superrotations) and the potential existence of an infinite tower of higher-order soft theorems from the boundary action remains an open problem.

2. Methodology

The author employs a combination of geometric analysis, conformal compactification, and the path integral formalism:

• LMPS Formalism: The paper utilizes the construction by Lehner, Myers, Poisson, and Sorkin (LMPS) to define the gravitational action with null boundaries, specifically addressing the corner terms and flux contributions.
• Conformal Compactification: The analysis is performed in the unphysical spacetime (Penrose diagram) where null infinity $\mathcal{I}$ is a codimension-1 null hypersurface. The physical metric $\hat{g}_{ab}$ is related to the unphysical metric $g_{ab}$ via a conformal factor $\Omega$ ($g_{ab} = \Omega^2 \hat{g}_{ab}$).
• Fixing Ambiguities via Physical Constraints: Instead of arbitrarily fixing the corner terms, the author imposes a physical constraint: the exponential of the on-shell action must reproduce the 5-point tree-level eikonal amplitude with an outgoing soft graviton in a generic supertranslated vacuum. This fixes the undetermined coefficient ($\zeta$) in the corner term.
• Extension to Superrotations: The framework is extended to include holomorphic superrotations. The author analyzes the flux term's behavior under conformal transformations and introduces a "generalized" Geroch tensor to account for subleading and sub-subleading memory modes.

3. Key Contributions and Results

A. Fixing Corner Ambiguities

The boundary action at null infinity contains an ambiguous corner term proportional to the integral of the news tensor $N_{AB}$ and the shear $C_{AB}$.

• The action takes the form:
  $$S_k \sim \int_{\mathcal{I}^\pm} m_B + \zeta \int_{S^2} N_{AB} \hat{C}^{AB}$$
• By demanding that the variation of the S-matrix with respect to the soft mode reproduces the Weinberg soft graviton theorem (specifically the leading soft factor), the author determines that the coefficient must be $\zeta = -2$.
• This fixes the corner term uniquely, ensuring the action is invariant under supertranslations and consistent with the conservation of supermomentum.

B. Subleading Soft Theorem and Superrotations

The paper investigates the action in an arbitrary conformal frame (not restricted to the Bondi frame).

• Flux Covariantization: The flux term in the action is not conformally invariant. By covariantizing the flux term, an extra contribution arises involving the Geroch tensor ($T_{AB}$), which is a symmetric, trace-free tensor related to the boundary metric.
• Subleading Insertion: The modified boundary action includes a term:
  $$S_{sub} \propto \int d^2z \, N_{AB}^{(1)} T^{AB}$$
  where $N_{AB}^{(1)}$ is the subleading soft news.
• Derivation: A functional derivative of the S-matrix with respect to the Geroch tensor (interpreted as a conjugate variable to the subleading memory) yields the subleading soft graviton theorem. This establishes a direct link between the AFS path integral approach and the Ward identities of the extended BMS (eBMS) group.

C. Infinite Tower of Soft Theorems

The most significant theoretical proposal is the generalization of the Geroch tensor to an infinite series:
$$T_{AB} \to T_{AB}^{(g)} = \sum_{n=0}^{\infty} u^n T_{AB}^{(n+1)}$$
where $T_{AB}^{(n)}$ are divergence-free symmetric traceless tensors on the sphere (allowing for singularities/punctures to accommodate meromorphic superrotations).

• Subn-leading Modes: This generalization introduces an infinite tower of terms in the on-shell action corresponding to subn-leading soft insertions ($O(\omega^{n-1})$).
• Goldstone Modes: The author proposes that these divergence-free tensors act as Goldstone modes for an infinite tower of tree-level symmetries.
• Connection to $w_{1+\infty}$: These modes are conjectured to be the classical realization of the infinite tower of soft graviton symmetries governed by the $w_{1+\infty}$ algebra, as proposed by Strominger and others.

4. Significance

• Unification of Formalisms: The paper successfully bridges the gap between the geometric path integral approach (AFS/LMPS) and the asymptotic symmetry approach (Strominger's infrared triangle). It demonstrates that the soft theorems are not just Ward identities but are encoded in the structure of the on-shell boundary action itself.
• Resolution of Ambiguities: It provides a rigorous physical criterion (reproduction of the 5-point amplitude) to resolve the long-standing ambiguity in corner terms for null boundaries.
• New Symmetry Structure: The proposal of an infinite tower of Goldstone modes associated with subn-leading soft theorems offers a concrete mechanism within the AFS framework to understand the recently discovered $w_{1+\infty}$ celestial symmetries.
• Future Directions: The work highlights the need to relax fall-off conditions on the news tensor to accommodate these infinite towers in classical scattering and suggests that loop corrections (logarithmic soft theorems) will require modifications to this tree-level framework.

Conclusion

Shivam Upadhyay's paper provides a robust derivation of the gravitational boundary action at null infinity, fixing its ambiguities through physical scattering constraints. It successfully recovers the leading and subleading soft theorems and proposes a mechanism for an infinite hierarchy of soft symmetries, thereby deepening the understanding of the infrared structure of quantum gravity and its connection to celestial holography.