Gravity and the momentum graviton

Gravity in the Broken-Born program involves two spin-2 fields, symmetric tensor fields of the kind that carry gravity: the spacetime metric of general relativity, and a second tensor X, the momentum graviton, which carries the geometry of momentum space. A symmetry of the program’s phase-space geometry gives X a gauge symmetry, a redundancy in the description of the kind that keeps the ordinary graviton massless, so X behaves as a second graviton. Established results exclude a second massless graviton that interacts with the first. They also exclude a field of this kind whose kinetic term, the part of the theory that governs how it propagates, lacks the gauge symmetry, because such a field carries a ghost, a mode with negative energy that makes a theory unstable. That leaves X one option, a mass. Whether the potential that gives X its mass is free of ghosts was the program’s tuning test: potentials written in terms of metrics fail it, and potentials written in terms of frames, the sets of reference axes from which a metric is built, pass it conditionally. Reciprocity then restricts that potential to a palindromic branch, whose coefficients read the same forwards and backwards. On that branch, requiring a flat vacuum in which the two metrics coincide forces the vacuum energy of the spin-2 interaction potential to zero. This section sets out the established theory of massless and massive spin-2 fields first, then the program’s results on X, worked out as Debt 1, the program’s name for the missing field equation of the phase-space metric. Those results are structural derivations, partly machine-checked, and the checkpoint states that the word “theory” still waits on its first three open problems.

Stands: the established physics of spin-2 fields; the program’s chain on X — shift symmetry, dual diffeomorphisms, the symplectic partner of the graviton, why X must be massive, the gravitational Higgs mechanism, τ is not eaten, spectrum and reciprocity parity, safety by range and Born’s answer; the tuning test’s misalignment potential, the metric tier’s two failures, the frame tier, its conditional pass and the constrained bimetric gravity it yields; the palindrome βₙ = β₄₋ₙ and the vacuum-energy identity, whose novelty is under audit · Died: nothing · Open: deriving the wedge sector from S_H (OP1), the partially massless comparison (OP2), τ–Stückelberg mixing (OP6) and stability at equal Planck masses, which is not among #11 §14’s items; two more items of the Debt-1 residue live in other sections, Lorentzian dressing (OP3) and the dual-stress enhancement (OP5) (all in open problems) · Superseded: the geometric action S_H and the v0.1 action; earlier forms of several results are folded into the pages that replaced them, across the momentum graviton X, the tuning test, the palindromic branch and the cosmological-constant problem (all in superseded statements) · Conjectures: two gravitational charges and the conjugate field equations (chat-only), and one popsci-only claim, the “forty-year asymmetry”, folded into the symplectic partner of the graviton (both in conjectures)

Map#

Spin-2 fields comes first because it is the established literature that X is held to. It runs from general relativity’s graviton and its gauge symmetry, the Planck units that set its coupling, the equivalence principle and the Yukawa range of a massive mediator, through ghosts and perturbative instabilities, the Fierz–Pauli mass term, whose special weight keeps a massive spin-2 field free of ghosts, the signature obstruction, which requires the gauge-invariant kinetic term, and the multi-graviton no-go theorem, to the ghost-free nonlinear theories, dRGT massive gravity (de Rham–Gabadadze–Tolley) and Hassan–Rosen bimetric gravity, with their algebra and frame form. It ends with the literature behind the program’s later results: the partially massless point, bimetric cosmology and the cosmological-constant problem. Its pages are literature; where a page states how the program uses a result, it gives the program’s tags.

The momentum graviton X holds the program’s chain of results on X = Re Z: its shift symmetry and dual diffeomorphisms, why it must be massive, how the choice of polarization gives it a mass, why τ is not eaten, its spectrum and its safety for ordinary matter, the open questions, the chat-only results, and the program’s answer to Born’s question. These are program claims; the checkpoint tags the chain [D + T] (§7.1) and the spectrum [D] (§7.5).

The tuning test asks whether the potential that gives X its mass avoids the Boulware–Deser ghost. Potentials built from metrics fail twice, and each failure has a numerical check in the verification battery: their mass term lacks the Fierz–Pauli trace term (test T4), and their nonlinear structure lies outside the Hassan–Rosen family (test T7, which checks the power sum p₄). Potentials built from frames pass, which calc note #7 calls a conditional pass; the checkpoint tags the test [D][V] (§7.2). The section also keeps the superseded v0.1 action and the open derivation of the passing potential from the geometric action.

The palindromic branch comes last because it acts on the potential that passed: reciprocity forces βₙ = β₄₋ₙ, a flat aligned vacuum then forces the wedge sector’s vacuum energy to zero, and a comparison with the partially massless literature is still needed before that identity can be called new.

Boundaries#

Backward Open: the real model's one-loop preference (a lead) Spin-2 fields Forward