Constrained bimetric gravity

Constrained bimetric gravity is the picture of the program’s gravity that the tuning test leaves. Hassan–Rosen bimetric gravity is a ghost-free theory of two interacting metrics, each with its own dynamics. In the program’s version the second frame is not a free field: it is the polarization frame, built from the polarization J, and J ranges only over the compact space Λ(d) of splittings of phase space into position and momentum, which is ten-dimensional at d = 4. Calc note #7 calls the result a predictive truncation of bimetric theory, with fewer free functions. In it the massless combination of the two spin-2 fields is the ordinary graviton, the massive one is the momentum graviton X, and the light direction of the compact space is the dark-energy field τ. Reciprocity then restricts the potential to its palindromic branch, and with the condition for a flat vacuum two free parameters remain. The picture is a structural derivation; the standard bimetric facts it uses come from the literature, and nothing in it is machine-checked.

Status: Alive — [D — structural] in calc note #7 §4, with the standard bimetric facts [T] — not machine-checked; carried into the technical abstract (§1) and §7 of checkpoint #11 with d = 4, as Λ(4).

The claim#

Calc note #7 (the Debt-1 calc notes) heads its §4 “The picture that emerges: constrained bimetric gravity [D — structural]”:

J carries only the 10 compact moduli of U(d)/O(d) — not a full second metric. So the “second vielbein” E₀[J] is confined to a compact 10-dimensional moduli space. BBT’s geometric sector is therefore

Hassan–Rosen bimetric gravity with the second frame constrained to a compact moduli space Λ(d) — a predictive truncation of bimetric theory: fewer free functions, with the light direction of the compact space being τ (the dark-energy field) and the massive spin-2 being X.

BBT is the Broken-Born program. Checkpoint #11 states the current form in its technical abstract: a chain of structural results “forces X to be a massive spin-2 field in the interchange-symmetric branch of Hassan–Rosen bimetric gravity [10†, 11†] with the second frame confined to Λ(4)” (#11 §1). The interchange-symmetric branch is the restriction that reciprocity adds afterwards (the palindrome βₙ = β₄₋ₙ).

Bimetric gravity [T]#

Hassan–Rosen bimetric gravity has two dynamical metrics, g and f, each with its own Einstein–Hilbert kinetic term. They interact through a potential that is a linear combination of the elementary symmetric polynomials eₙ of the matrix √(g⁻¹f), with constant coefficients βₙ, the form that avoids the Boulware–Deser ghost (Hassan–Rosen bimetric gravity; the Fierz–Pauli mass term and the Boulware–Deser ghost). #11 cites the theory from model memory as [10†, 11†]. In the general theory the second metric is a full dynamical field in its own right, with its own value at each point, and the potential has five constant coefficients, β₀ to β₄.

#7 §4 takes over the theory’s spectrum: “Standard bimetric facts then slot in [T]: one massless combination (our graviton, coupling to SM stress) and one massive (X)”. SM is the Standard Model. The fluctuation of each metric is a mixture of the two mass eigenstates, one massless spin-2 field and one massive.

Why the second frame is confined [D — structural]#

The argument has three steps, each in the notes.

  1. The second frame is a polarization frame. In the frame tier the passing potential is a wedge polynomial of the symplectic vielbein E and the polarization frame E₀[J] (#7 §3; the frame tier). In calc note #8’s notation the two frames of the bimetric potential are “(e, f = E₀[J])” (#8 §1). The second frame is a function of J alone.
  2. J lives on the Lagrangian Grassmannian. Calc note #5: “The space of H-compatible real structures on (ℝ^{2d}, I) is {J} ≅ U(d)/O(d) = Λ(d), the Lagrangian Grassmannian” (#5 §B). The lab notebook gives its size: “compact, of dimension d(d+1)/2 = 10 for d=4” (#1 §2). A compact space is closed and bounded, like a circle or a sphere [T].
  3. So the second sector is truncated. A free second metric is replaced by a frame that moves only over this compact space: “not a full second metric” (#7 §4). #7 calls this “a predictive truncation of bimetric theory: fewer free functions”.

The notes write Λ(d). #11 writes Λ(4) where it confines the second frame (§1, §7) and Λ(d) for the Grassmannian in general (§1, §6). They agree at d = 4, the physical case (the Lagrangian Grassmannian Λ(d)).

What fills the picture#

  • The massless combination: the graviton. It couples to the stress of Standard Model matter (#7 §4). Reciprocity later identifies it as the reciprocity-even combination, which is exactly massless (#8 §3; Row 6: the exactly massless graviton; on the parity under #11 §6’s Z → −Z⁻¹, see flag 39 in review/wiki-build-flags.md).
  • The massive combination: X. The momentum graviton, the real part of the modular field Z, acquires its mass when the vacuum’s choice of polarization breaks its gauge symmetry, with “M_X = the polarization-breaking scale” (#11 §7.1; the gravitational Higgs mechanism).
  • The light direction: τ. τ is the Maslov phase, the angle along the one circle direction of Λ(d). It is not among the modes that X absorbs in acquiring its mass: “τ is a J-modulus on the compact Λ(d), a different sector with different origin; it survives as the light field” (#8 §3, [D — leading order]; τ is not eaten). Calc note #6 had flagged this as a check still to be made (#6 §3).
  • The other nine directions. That τ is the lightest modulus, “with the other nine directions heavy, is assumed [P]” (#1 §2). Calc note #5 expected constraints of the inverse-Higgs type to “likely eliminate most” of the nine [O] (#5 §D); no later document works this out (Open: field content of the broken vacuum).

How far the truncation goes#

Calc note #8 quantifies the truncation once reciprocity has acted (#8 §3, tagged [D + T]):

Parameter count: five βₙ → palindrome (−2) → T = 0 (−1) → two: the Fierz–Pauli mass combination m_FP² ∝ β₁ + 2β₂ + β₃ = 2(β₁+β₂), i.e. M_X, and one orthogonal shape parameter controlling cubic interactions. The “predictive truncation” of Chunk A-iii is now quantified: HR bimetric’s five-function freedom collapses to two numbers.

HR is Hassan–Rosen, and Chunk A-iii is calc note #7; on “five-function freedom”, see flag 8 in review/wiki-build-flags.md. The palindrome βₙ = β₄₋ₙ removes two coefficients, and the condition T = 0 from a flat aligned vacuum removes one more (the palindrome; the vacuum-energy identity). #11 §7.5 carries the count, “Five βₙ → two parameters”, tagged [D] (spectrum, parameters and reciprocity parity).

Superseded: safety through a dominant massless combination#

#7 §4 went on to say how ordinary matter is protected from X: “safety requires the massless combination to dominate SM couplings — a parametric check to log”, together with the bounds on a graviton mass from gravitational-wave observations, “m_g ≲ 10⁻²³ eV territory”, read as a bound on the mixing angle.

Calc note #8 replaced this reading. “With equal effective Planck masses (forced by the same symmetry), SM matter — coupled to g — sees both eigenstates with comparable strength; safety is by range, not decoupling: X-mediated forces on the SM are Yukawa-cut at 1/M_X, and gravitational-strength short-range forces are experimentally unconstrained down to far below any plausible M_X” (#8 §3; Yukawa forces). The graviton-mass bounds from gravitational waves apply to the reciprocity-even eigenstate, which is exactly massless, so they hold identically rather than constraining a mixing angle (#8 §3; #11 §7.5 and §13 row 6). The current argument is on safety by range.

Before the test: the conditional form#

Calc note #6 had stated the picture with a condition attached (#6 §3):

Restated as a selling line: BBT’s geometric sector is bimetric gravity with the second metric’s kinetic structure and couplings dictated by modular (Siegel-domain) symmetry rather than chosen by hand — if the dRGT tuning emerges from the symmetry.

dRGT is de Rham–Gabadadze–Tolley (dRGT massive gravity). The note adds that “bimetric gravity was the right neighborhood from the start; the modular structure explains which bimetric.” The tuning test met the condition in frame variables, which is the sense of its conditional pass (the verdict: a conditional pass). #7 §5 then records “the resulting physical picture (constrained HR bimetric with compact second sector)” as “sharper and more predictive than generic bimetric gravity”.

Scope and what is open#

History#

  • Calc note #6 and the v0.4 addendum: the conditional form, “bimetric gravity with structure dictated by modular symmetry” (notebook ‘Addendum (v0.4)’).
  • Calc note #7 and the v0.5 addendum: “BBT’s geometric sector is Hassan–Rosen bimetric gravity with the second frame confined to the compact moduli space Λ(d) — a predictive truncation of bimetric theory. One massless graviton (ours), one massive spin-2 (X), and the compact sector’s light direction is τ” (the lab notebook).
  • Calc note #8 and the v0.6 addendum: the truncation quantified, and the picture folded into the program’s answer to Born: X’s “self-consistent dynamics is Hassan–Rosen bimetric gravity restricted to the reciprocity-symmetric branch with the second frame confined to the compact moduli space Λ(d)” (#8 §4; Born’s answer).
  • The v1 program paper and checkpoint #11 write Λ(4): “the reciprocity-symmetric branch of Hassan–Rosen bimetric with the second frame confined to the compact Λ(4)” (#9 §5) and “the reciprocity-symmetric branch of bimetric gravity with the second frame confined to compact Λ(4)” (#11 §7).
  • The handoff: “Geometric sector = interchange-symmetric (palindromic βₙ = β₄₋ₙ) Hassan–Rosen bimetric, second frame confined to compact Λ(4)” (HANDOFF.md §2; the handoff).

Sources#

  • notes/debt1a-iii-tuning-test.md §4 — the picture [D — structural], the bimetric facts [T] and the superseded safety reading; §3 — the polarization frame in the frame tier; §5 — “sharper and more predictive”.
  • papers/broken-born-checkpoint-v2.md §1, §7 — the current statement with Λ(4); §7.1 — M_X; §7.5 and §13 row 6 — the two parameters and the massless even eigenstate; §14 — open problem 6.
  • notes/debt1a-ii-X-classification.md §3 — the conditional form before the test; τ un-eaten flagged.
  • notes/debt1a-iv-beta-coefficients.md §1 — the two frames (e, f = E₀[J]); §3 — the parameter count, equal Planck masses, safety by range and τ not eaten; §4 — Born’s answer with Λ(d).
  • notes/debt1a-geometric-core.md §B — the space of real structures is Λ(d); §D — the ten σ-modes.
  • notes/broken-born-theory.md §2 — Λ(d) compact, of dimension 10 for d = 4, and the nine heavy directions assumed; ‘Addendum (v0.4)’, ‘Addendum (v0.5)’, ‘Addendum (v0.6)’ — the notebook’s statements.
  • notes/broken-born-program-v1.md §5 — the v1 statement with Λ(4).
  • notes/knot-dark-matter-standalone.md §1 — stability at equal Planck masses as an open condition.
  • HANDOFF.md §2 — the handoff’s one-screen statement.
  • review/wiki-build-flags.md item 8 — “five-function” for five constant coefficients; awaiting Paul’s ruling.