Partially massless gravity

Partially massless gravity is a spin-2 field on de Sitter space, the spacetime of a universe expanding at a constant rate under a positive cosmological constant, whose mass takes the one value at which an extra gauge symmetry appears and removes the field’s helicity-0 polarization [T]. That value is the edge of the Higuchi bound, below which a massive spin-2 field on de Sitter space carries a ghost. Bimetric gravity has a parameter point, studied as partially massless bimetric gravity, at which the theory has de Sitter solutions in which the two metrics are proportional, with a ratio that the equations leave undetermined, and its potential is symmetric under exchange of the two metrics. The point matters to the program because at α = 1, which in the literature’s usual notation means equal Planck masses for the two metrics, it lies inside the palindromic branch that Born reciprocity selects. That raises the open question whether reciprocity is the origin of the partially massless symmetry, and it means the program’s vacuum-energy identity must be compared with this literature before it can be called new.

Status: Established — [T] — the linear theory is standard, and the bimetric parameter point is cited in the checkpoint as [24S], retrieved by live search. Its relation to the program is open: whether reciprocity is the origin of the partially massless symmetry is [O], part of open problem 2 of #11 §14, the partially-massless comparison (OP2).

A massive spin-2 field on de Sitter space#

De Sitter space#

De Sitter space is the spacetime of a universe expanding at a constant Hubble rate H under a positive cosmological constant Λ = 3H² [T], the relation the round-3 audit uses to convert Higuchi’s bound below (review/round3/referee/audit-notes.md, Step 1). On this page H is the Hubble rate, not the program’s phase-space metric H.

Five polarizations#

A massive spin-2 field in four dimensions has five polarizations — calc note #6 counts “its five polarizations” for the momentum graviton X (#6 §3) — with helicities ±2, ±1 and 0 [T]. A massless graviton keeps only the helicity ±2 pair; its gauge symmetry, linearized diffeomorphisms, removes the others [T] (general relativity and the graviton).

The Higuchi bound#

In flat space any positive mass is allowed. On de Sitter space a massive spin-2 field must satisfy the Higuchi bound [T]:

m² ≥ 2H²

The withdrawn manuscript #14 cites it as reference [6], A. Higuchi, Nucl. Phys. B 282, 397 (1987). The round-3 audit read the bound from Higuchi’s abstract, “M^2 cannot be between 0 and 2/3 x (cosmological constant)”, converted it with “Lambda = 3H^2” to “m^2 >= 2H^2”, and found that the reference supports the statement (audit notes, Step 1).

For masses between zero and the bound, the helicity-0 polarization has negative norm: it is a ghost, and the theory is not unitary [T]. A field much heavier than H satisfies the bound trivially, which is how #14 treated the momentum graviton; see the Higuchi bound under bimetric cosmology.

The partially massless value#

At the edge of the bound, m² = 2H², the field’s linear equations acquire a scalar gauge symmetry [T]:

δh_μν = (∇_μ∇_ν + H² g_μν) ξ          (ξ any function; mostly-plus sign convention)

Here h_μν is the spin-2 field, g_μν the de Sitter metric and ∇ its covariant derivative. The symmetry removes the helicity-0 polarization, so the field propagates helicities ±2 and ±1 only. It has a mass and more polarizations than a massless graviton, yet it carries a gauge symmetry as a massless field does: hence “partially massless” [T]. The point needs a cosmological constant. With H = 0 the edge of the bound is m = 0, the ordinary massless graviton.

The partially massless point of bimetric gravity#

Proportional de Sitter backgrounds#

Hassan–Rosen bimetric gravity has two dynamical metrics, g and f, each with its own Einstein–Hilbert term and Planck mass, M_g and M_f, and a potential whose five coefficients β₀ to β₄ multiply the elementary symmetric polynomials of √(g⁻¹f) [T]. Among its solutions are proportional backgrounds, f = c²g with c a constant, in which each metric solves Einstein’s equations with an effective cosmological constant [T]; the case that concerns this page is the one in which both are de Sitter. Around such a background the fluctuations separate into a massless spin-2 mode and a massive one [T].

The parameter point#

The checkpoint states the point from the partially massless bimetric literature [24S]:

β₀ = 3α⁻²β₂,      β₁ = β₃ = 0,      β₄ = 3α²β₂

In its words, “the PM point β₀ = 3α⁻²β₂, β₁ = β₃ = 0, β₄ = 3α²β₂ makes the potential g↔f symmetric, with proportional de Sitter solutions of undetermined ratio [24S]” (#11 §7.4). PM stands for partially massless. The undetermined ratio is the constant c: at this point the field equations do not fix it.

The checkpoint does not define α or c. In the bimetric literature’s usual notation α is the ratio of the two Planck masses, M_f/M_g, and c is the constant in f = c²g [T]; the wiki reads #11 with that notation. The round-3 audit’s quotation from a bimetric paper uses α in a way consistent with this reading: “the alpha->0 limit brings bimetric theory arbitrarily close to GR [general relativity]”, and heavy spin-2 fields are “not excluded as long as … M_f is sufficiently small” (audit notes, Step 1; see bimetric cosmology). The program’s documents do not discuss how far the extra gauge symmetry extends beyond the linear theory, a question studied in the partially massless literature [T].

The reference, as the checkpoint lists it:

  • [24][S] Partially massless bimetric gravity and its exchange-symmetric parameter point, arXiv:1609.09515.

The checkpoint gives this descriptive title and no authors (see flag 9 in review/wiki-build-flags.md), and the handoff cites the arXiv number without a mark (HANDOFF §2, §8). The wiki does not supply the authors.

Inside the palindromic branch#

The branch#

The program’s frame-tier potential is a Hassan–Rosen potential whose coefficients reciprocity restricts to βₙ = β₄₋ₙ: β₀ = β₄ and β₁ = β₃, with β₂ free (#8 §1; the palindrome βₙ = β₄₋ₙ).

At α = 1#

Setting α = 1 in the three conditions gives β₀ = β₄ = 3β₂ and β₁ = β₃ = 0, which satisfies βₙ = β₄₋ₙ. The checkpoint: “At α = 1 the PM point lies inside our palindromic branch”, so partially massless gravity “is a special point of the reciprocity-symmetric family, raising the open question of whether reciprocity is the origin of the PM symmetry [O]” (#11 §7.4).

On the literature’s reading of α, α = 1 means equal Planck masses. Calc note #8 states that reciprocity forces “equal effective Planck masses” (#8 §3), so on this reading α = 1 is the case that applies to the program; #11 does not draw this link itself. The handoff states the finding without the condition: the partially massless bimetric point “is g↔f symmetric and sits inside the palindromic branch” (HANDOFF §2). The checkpoint, which ranks above the handoff, adds “At α = 1”.

A different point of the same branch#

The program’s vacuum-energy identity selects its own point of the branch. “Note the PM point sits at Λ ≠ 0 with c undetermined; our flat-aligned-vacuum identity selects a different point of the same branch (β₁ ≠ 0 generically, β₀ = −4β₁ − 3β₂)” (#11 §7.4). The identity concerns a flat vacuum at the aligned point, where the two frames coincide; the partially massless point concerns de Sitter backgrounds with c left free. See the vacuum-energy identity V(1) = 2T.

The comparison still to be made#

The literature check that found this point downgraded the identity’s novelty. The handoff’s one-screen state records that the partially massless bimetric point “(arXiv:1609.09515) is g↔f symmetric and sits inside the palindromic branch; Hossenfelder arXiv:0807.2838 has a different exchange-symmetric cancellation. Detailed comparison still owed.” (HANDOFF §2). The checkpoint downgraded the novelty claim “from ’likely standalone result’ to ‘requires detailed comparison with the PM and exchange-symmetric literature’” (#11 §7.4). Hossenfelder’s theory, reference [25S] of the checkpoint, cancels vacuum energy by a different mechanism, “where twin-sector source terms cancel identically”; the program’s identity “is sector-internal (no twin matter required) and, to our current knowledge, distinct” (#11 §7.4).

The comparison is open problem 2 of the checkpoint: “PM-adjacency: detailed comparison of the palindromic branch with partially-massless bimetric [24S]; is reciprocity the origin of the PM symmetry? Convention-careful check of the vacuum-energy identity against [24S, 25S] — the queued novelty audit [O]” (#11 §14). The handoff’s queue named three comparisons: “arXiv:1609.09515 / 0807.2838 / the Schmidt-May–von Strauss review’s no-protecting-symmetry statement” (HANDOFF §8 item 5); the third belongs to the cosmological-constant problem. Since the handoff the comparison is part of item 1 of the current queue (CLAUDE.md, “Since the handoff”). It has its own page, Open: the partially-massless comparison (OP2), and is listed with the program’s open problems.

Standard references#

  • A. Higuchi, Nucl. Phys. B 282, 397 (1987): the bound m² ≥ 2H². It is #14’s reference [6], unmarked there; the round-3 audit found that it supports the statement.
  • [24][S] arXiv:1609.09515: the partially massless bimetric parameter point, cited by #11 with a descriptive title.
  • [25][S] S. Hossenfelder, “A bi-metric theory with exchange symmetry,” Phys. Rev. D 78, 044015 (2008), arXiv:0807.2838: the exchange-symmetric cancellation by a different mechanism, cited by #11.
  • Added by the wiki from model memory (†): S. Deser and R. I. Nepomechie (1984)†, on gauge invariance and masslessness of fields on de Sitter space; S. Deser and A. Waldron (2001)†, on partial masslessness of higher-spin fields; S. F. Hassan, A. Schmidt-May and M. von Strauss (2012)†, on partially massless bimetric gravity.

Used by#

Sources#

  • papers/broken-born-checkpoint-v2.md §7.4 — the parameter point, its place in the palindromic branch at α = 1, the open question, the different point the identity selects, and the novelty downgrade.
  • papers/broken-born-checkpoint-v2.md §14 item 2, refs [24], [25] — the open comparison; the citations, marked [S].
  • HANDOFF.md §2, §8 — the literature check that downgraded the identity’s novelty; the three comparisons in queue item 5.
  • CLAUDE.md ‘Since the handoff’ — the comparison in item 1 of the current queue.
  • notes/knot-dark-matter-standalone.md §1, ref [6] — the Higuchi bound and its citation (manuscript withdrawn for an unrelated error: its Kibble estimate of the relic abundance had been normalized too low).
  • review/round3/referee/audit-notes.md Step 1 — Higuchi’s abstract and the conversion with Λ = 3H².
  • notes/debt1a-ii-X-classification.md §3 — the five polarizations of a massive spin-2 field.
  • notes/debt1a-iv-beta-coefficients.md §1, §3 — the palindromic constraint; equal effective Planck masses forced by reciprocity.
  • review/wiki-build-flags.md item 9 — reference [24][S], with no authors and a title that may be descriptive; awaiting Paul’s ruling.