Spin-2 fields

A spin-2 field is a symmetric tensor field, h_μν, whose quanta carry two units of angular momentum; the graviton of general relativity is the massless case. Such a field can be given a mass, or joined by a second one, only in narrow ways, because a generic mass term or coupling adds a ghost, a mode whose energy has the wrong sign. This section collects the established results that define those narrow ways, from the Fierz–Pauli mass term to the ghost-free de Rham–Gabadadze–Tolley (dRGT) and Hassan–Rosen theories, because the program’s momentum graviton X is held to them. Everything here is literature; the program’s own claims about X are in the neighbouring sections.

Literature only: all fifteen pages are established physics or literature, so the section ranks no claims of the program’s own; those are tracked in the registers of claims that stand, the graveyard, open problems and the queue, conjectures and superseded statements. Where a page says how the program uses a result, it gives the program’s tags and its machine checks (battery tests T4, T6 and T7). One superseded statement is folded into a page here: the vacuum-energy cancellation framed as a cosmological-constant result, on the cosmological-constant problem.

Map#

The section runs from the massless graviton, the strength and universality of its coupling and the range a mass would give it, through what makes a mode a ghost and the conditions under which a spin-2 field may carry a mass without one, to the ghost-free theories and their algebra, and ends with the literature that bears on the program’s later results.

General relativity and the graviton comes first: the metric, the Einstein–Hilbert action, minimal coupling and the equivalence principle, then the linearized theory, whose gauge symmetry δh_μν = ∂_μξ_ν + ∂_νξ_μ makes the graviton a massless spin-2 field, and the result that one consistently interacting massless graviton is general relativity. X is defined by analogy with this graviton: its local shifts are a second copy of the same linearized diffeomorphisms.

Planck units and the two Planck masses gives the units built from ℏ, c and G, and the Planck mass as the strength with which a spin-2 field couples to matter. It sets out the two conventions, M_Pl = G^{−1/2} and the reduced M̄_Pl = M_Pl/√(8π), with the rule that a formula keeps one of them throughout, the rule whose breach produced the retired soliton candidate’s reported spread between the conventions. It then lists where the program’s documents use each, from #13’s Born length at the Planck length to the equal Planck masses of the bimetric sector.

The equivalence principle: free fall and local inertial frames states that gravity acts on every body alike: the universality of free fall, Einstein’s form with its local inertial frames, free-falling and Rindler frames, the tidal effects that confine it to small regions, and its tests. The program postulates it for both kinds of matter through P4, so conjugate matter falls, orbits and bends light like any other dark matter, and #13 reads it as the relativity principle of the momentum polarization.

Yukawa forces and their range gives the force carried by a field whose quanta have a mass M: the potential e^(−Mr)/(4πr) with range ℏ/(Mc), the same law in momentum as 1/(q² + M²), the Yukawa term beside Newton’s potential that fifth-force searches test, and scattering through a Yukawa mediator in its contact and light-mediator limits. X’s safety by range and the standard expectations of the dark-matter scattering rows rest on it.

Ghosts and perturbative instabilities defines the failure the rest of the section guards against. A ghost is a mode whose kinetic term has the wrong sign, so that its excitations carry negative energy or negative norm; once it interacts with ordinary matter, as gravity ensures it does, the vacuum can decay. The page separates ghosts from tachyons, gradient instabilities and strong coupling, shows how ghost freedom is checked, from constraints and the kinetic matrix to the Hamiltonian constraint whose loss brings back the Boulware–Deser ghost, and covers the Higuchi ghost on de Sitter space and phantom fields. It then gathers the ghosts that bear on the program: the exclusion of an X without gauge symmetry, the tuning test, the kinetic term of model I’s real structure on its de Sitter target, and the lab notebook’s superseded minisuperspace argument.

The Fierz–Pauli mass term and the Boulware–Deser ghost gives the only healthy quadratic mass term, Tr(h²) − (Tr h)², and shows why its relative weight is special. It holds the Boulware–Deser ghost that generic nonlinear completions revive, and records how the program’s metric-built potential misses the Fierz–Pauli weight, the comparison checked by battery test T4. The signature obstruction explains why the theory keeps the gauge-invariant kinetic term of the massless graviton: without that gauge symmetry no two-derivative kinetic term for a symmetric tensor field is free of ghosts, because in the signature of spacetime the mixed time–space components enter with the wrong sign, while the gauge-invariant term keeps h₀₀ and h₀ᵢ from propagating. Calc note #6 tags the statement [T — standard] without citing a source, and the program uses it to exclude an X without gauge symmetry. The multi-graviton no-go theorem shows that several massless gravitons cannot interact with one another. Since X must couple to the metric and to conjugate matter, it cannot stay massless; with the signature obstruction, this forces a mass.

dRGT massive gravity is the ghost-free nonlinear completion of the Fierz–Pauli term with one dynamical metric, a potential linear in the elementary symmetric polynomials of √(g⁻¹f). The program took its class as the health criterion for X, and the page follows the criterion through the tuning test. Hassan–Rosen bimetric gravity makes the second metric dynamical. Its potential, V = 2m²M² Σ βₙ eₙ(√(g⁻¹f)), and the exchange identity that reads the list of coefficients backwards when the two metrics trade places, are what reciprocity later acts on; its spectrum is one massless and one massive spin-2 field. Elementary symmetric polynomials gives the algebra of that potential, with battery tests T6 and T7. The vielbein formulation and wedge potentials gives its frame form, ghost-free by construction, in which the program’s tuning test passed.

The last three pages are context for the program’s results. Partially massless gravity is a massive spin-2 field on de Sitter space at the mass where a gauge symmetry removes its helicity-0 mode, together with the bimetric parameter point studied as partially massless bimetric gravity, which lies inside the program’s palindromic branch at α = 1. Bimetric cosmology and its instabilities records the known instabilities of bimetric cosmologies, the heavy spin-2 field shown to be linearly stable at late times when the second metric’s Planck mass is much smaller than the first, and the Higuchi bound; they bear on X’s open stability at equal Planck masses. The cosmological-constant problem records the vacuum-energy problem that the program’s vacuum-energy identity does not by itself address.

Boundaries#