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BMS Supertranslations and Not So Soft Gravitons

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Eduardo Conde, Pujian Mao

High Energy Physics - Theory · Submitted 25 Dec 2016 · 51 references

Pith's verdict: A clean, honest extension of the authors' 1/r trick to gravity: supertranslations alone reproduce all three tree-level soft graviton orders, with one acknowledged unfixed constant at third order.

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arXiv ID1612.08294 (version 2)
Pith IDQSZYLWOM, Pith's permanent record of this paper and its review
CitationEduardo Conde, Pujian Mao. "BMS Supertranslations and Not So Soft Gravitons." arXiv:1612.08294 (2016). https://pith.science/paper/QSZYLWOM
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The paper

The paper claims

All three tree-level soft graviton theorems follow from one symmetry, BMS supertranslations, the paper argues, with no need for superrotations.

The reading

When a graviton is emitted with vanishingly small energy, the scattering amplitude factorizes into a universal soft factor times the amplitude without the graviton. This paper argues that all three orders of that factorization at tree level — the leading $1/E_q$ term, the constant term, and the term linear in $E_q$ — are Ward identities of one and the same symmetry: BMS supertranslations, the angle-dependent time shifts that survive at null infinity. The mechanism is a dictionary between two expansions: the energy expansion of the soft theorem is matched to a $1/r$ expansion of the supertranslation charge, so that each additional order in the soft energy corresponds to one more power of the inverse radius in the asymptotic charge. If the argument holds, the sub-leading and sub-sub-leading soft graviton theorems need neither superrotations nor the extended, asymptotic-flatness-violating transformations that earlier work invoked for them. The demonstration is carried out in linearized gravity coupled to a massless complex scalar, with the graviton three-point vertex left out.

What carries the argument

The carrier of the argument is the asymptotic charge $Q = Q^{(0)} + Q^{(1)}/r + Q^{(2)}/r^2 + O(r^{-3})$, which is the leading supertranslation charge generalized to higher orders in the inverse radius. Its non-linear part creates a zero-frequency graviton on the vacuum, while its linear part acts on hard matter through the stress-energy tensor; the Ward identity $Q = Q^{(0)} + Q^{(1)}/r + Q^{(2)}/r^2 + O(r^{-3})$ then turns into the soft theorem. The supporting device is the Newman-Penrose tower of conserved quantities, which the authors show can be repackaged as the single integral $Q = Q^{(0)} + Q^{(1)}/r + Q^{(2)}/r^2 + O(r^{-3})$ over a cross section of null infinity, with $f(z, -z)$ a spin-weighted spherical harmonic or, for the soft theorem, $Q = Q^{(0)} + Q^{(1)}/r + Q^{(2)}/r^2 + O(r^{-3})$.

Extended reading notes

Core claim

The paper's claim is that the BMS supertranslation charge alone suffices to produce the leading, sub-leading and sub-sub-leading tree-level soft graviton theorems. Working in linearized Einstein-Hilbert gravity on Minkowski space in Newman-Unti gauge, the authors solve the Pauli-Fierz equations in a $1/r$ expansion around future null infinity and write the asymptotic charge as $Q = -Q^{(1)}/r + Q^{(2)}/r^2 + O(r^{-3})$, with $Q^{(0)}$ the familiar supertranslation charge that yields Weinberg's leading soft theorem. They then decompose each order into a linear piece, built from the stress-energy tensor, and a non-linear piece, built from the radiative news, and evaluate both on asymptotic states using Fourier relations such as $Q = Q^{(0)} + Q^{(1)}/r + Q^{(2)}/r^2 + O(r^{-3})$ and canonical boundary commutators. For the supertranslation $f(z, -z) = 1/(w-z)$ the resulting Ward identities reproduce, order by order, the standard soft factors of equations (57), (75) and (85), including the helicity flip between the leading and the two sub-leading orders. The authors note that the $1/r$ expansion of the charge continues further, but argue that fall-off conditions at null infinity prevent the integrations by parts needed to reorganize those higher orders into soft factors, so the symmetry fixes exactly three orders and no more.

Load-bearing premise

The derivation of the third, sub-sub-leading order requires setting two integration constants in the shifted stress-energy tensor to zero, a choice the framework does not justify.

Editorial extensions

If this is right

  • The sub-leading soft graviton theorem becomes a Ward identity of the same symmetry that controls the leading one, so the two need not be attributed to different asymptotic symmetry groups.
  • Superrotations, and the extended superrotations that do not preserve asymptotic flatness, become unnecessary for the tree-level soft graviton story at these three orders.
  • The correspondence between soft-energy orders and inverse-radius orders of an asymptotic charge is promoted from a photon-only observation to a statement that also holds in gravity.
  • Because the argument stops at $O(r^{-2})$ for reasons of fall-off and integration by parts, it predicts that supertranslations cannot yield universal soft graviton behavior beyond sub-sub-leading order at tree level.
  • The charge algebra remains Abelian in the linearized theory, so no central extension or Virasoro structure appears in this treatment.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the dictionary between energy order and inverse-radius order is general, the same bookkeeping should apply to Yang-Mills, where the leading soft gluon theorem already has an asymptotic-symmetry interpretation; the non-Abelian multiple-soft ordering is the natural place to test whether the correspondence survives.
  • The authors' remark that the unfixed integration constants in the sub-sub-leading charge might be tied to loop corrections suggests a check: at one loop the soft graviton theorem is expected to be modified first at sub-sub-leading order, exactly where the constants enter.
  • The paper's claim that only three orders are physically meaningful because only finitely many boundary fields live at infinity is a statement about fall-off conditions; weakening those conditions would be the natural way to ask whether a fourth-order soft theorem has any symmetry origin.
  • Treating the charge itself, rather than the soft theorem, as the fundamental object suggests looking for observables that depend on the higher $1/r$ orders even when they cannot be reorganized into soft factors.
Original abstract

In a previous article, we have argued that Low's sub-leading soft photon theorem can be recovered as a Ward identity associated to the same large gauge transformations that control the leading piece of the theorem. The key for that was to link the energy expansion displayed in the soft theorem to a $\frac{1}{r}$ expansion that we can perform in the associated asymptotic charge. We expect this idea to be valid in general, and here we provide compelling evidence for it by showing how the same method works in the case of Einstein-Hilbert gravity. More precisely, we are able to derive the three orders of the tree-level soft graviton theorem simply from the BMS supertranslation charge, known to give rise to the leading soft graviton theorem. In particular, we do not need to invoke superrotations (nor extended superrotations) at any point of the argument.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,352 input tokens.

What the abstract itself states, as probabilities: claims first 0.07 · states limitation 0.18 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.15 · compares with prior results 0.88 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

In a previous article, we have argued that Low's sub-leading soft photon theorem can be recovered as a Ward identity associated to the same large gauge transformations that control the leading piece of the theorem. The key for that was to link the energy expansion displayed in the soft theorem to a $\frac{1}{r}$ expansion that we can perform in the associated asymptotic charge. We expect this idea to be valid in general, and here we provide compelling evidence for it by showing how the same method works in the case of Einstein-Hilbert gravity. More precisely, we are able to derive the three orders of the tree-level soft graviton theorem simply from the BMS supertranslation charge, known to give rise to the leading soft graviton theorem. In particular, we do not need to invoke superrotations (nor extended superrotations) at any point of the argument.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 In a previous article, we have argued that Low's sub-leading soft photon theorem can be recovered as a Ward identity associated to the same large gauge transformations that control the leading piece of the theorem. background 0.98 0.59 0.08 asserted 1.00
s1 The key for that was to link the energy expansion displayed in the soft theorem to a $\frac{1}{r}$ expansion that we can perform in the associated asymptotic charge. method 0.86 0.80 0.06 asserted 1.00
s2 We expect this idea to be valid in general, and here we provide compelling evidence for it by showing how the same method works in the case of Einstein-Hilbert gravity. contribution 0.95 0.70 0.20 asserted 0.62
s3 More precisely, we are able to derive the three orders of the tree-level soft graviton theorem simply from the BMS supertranslation charge, known to give rise to the leading soft graviton theorem. result 0.89 0.96 0.78 asserted 1.00
s4 In particular, we do not need to invoke superrotations (nor extended superrotations) at any point of the argument. result 0.69 0.88 0.19 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

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not examined
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  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
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  • S sections
  • T body sentences
  • C citations
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References
How later papers used it

70 later papers cite it

Evidence: some. 54 of the citing papers share no author with it, and 4 of those carry a confident label from Pith's citation classifier.

Pith read 15 of the 93 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Uses its method or tool 1 of 4 papers with this label · 1 by the same authors

By the same authors

How the literature uses it

70 arXiv papers cite this in 93 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 55 sentences without a confident label

Contests it 1 paper
  • The supertranslation charge at each order proposed in [ref] is given by [A factor $1/2$ is missing on the righthand side of both eq.(2.26) and eq.(2.27) in [ref] .

    Remarks on infinite towers of gravitational memories · Infinite towers of conservation laws · contest 0.91

Builds on it 1 paper
Uses its method or tool 4 papers
Compares with it 2 papers
Cites it as background 28 papers
  • In four dimensions the leading soft theorem is equivalent to the supertranslation Ward identity [ref] , while the subleading soft theorem is equivalent to the Diff$(S^2)$ Ward identity [ref] .[An alternative proposal was given in [ref] where both the leading and subleading soft theorems were linked to supertranslations by expanding the associated charge in powers of $1/r$.] Together, supertranslations and Diff$(S^2)$ make up the asymptotic symmetry group.

    Asymptotic symmetries and subleading soft graviton theorem in higher dimensions · Introduction · 0.84

  • In recent years, soft graviton theorem has been studied from various perspectives – perturbative quantum field theory [ref] , perturbative string theory [ref] and BMS symmetry [ref] .

    Subleading Soft Graviton Theorem for Loop Amplitudes · Introduction · 1.00

  • It has recently become possible to make some of these statements more precise in four dimensions, due in large part to Strominger's infrared triangle that relates soft theorems, asymptotic symmetry groups and memory effects [ref] .

    A $d$-Dimensional Stress Tensor for Mink$_{d+2}$ Gravity · Introduction · 0.96

What it cites · 66 sentences without a confident label

Contests 9 papers
  • But actually, the asymptotic symmetries which are used in [ref] (and later in [ref] ) are not superrotations,[ Earlier in [ref] the sub-leading soft graviton theorem was recovered from twistor-string-like constructions, but the connection to the superrotations of [ref] is not completely clear to us. ] but a sort of generalization thereof, which actually do not preserve the asymptotic flatness condition.

    Symmetries of asymptotically flat 4 dimensional spacetimes at null infinity revisited · Introduction · contest 0.72

  • But actually, the asymptotic symmetries which are used in [ref] (and later in [ref] ) are not superrotations,[ Earlier in [ref] the sub-leading soft graviton theorem was recovered from twistor-string-like constructions, but the connection to the superrotations of [ref] is not completely clear to us. ] but a sort of generalization thereof, which actually do not preserve the asymptotic flatness condition.

    Aspects of the BMS/CFT correspondence · Introduction · contest 0.72

  • But actually, the asymptotic symmetries which are used in [ref] (and later in [ref] ) are not superrotations,[ Earlier in [ref] the sub-leading soft graviton theorem was recovered from twistor-string-like constructions, but the connection to the superrotations of [ref] is not completely clear to us. ] but a sort of generalization thereof, which actually do not preserve the asymptotic flatness condition.

    Supertranslations call for superrotations · Introduction · contest 0.72

Uses the method or tool of 1 paper
Compares with 2 papers
Cites as background 10 papers

Discussion

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