A local, generally covariant field theory of modified inertia

Abstract

Preprint. This deposit exists to timestamp a construction and to carry its reproducibility scripts; it is not a journal publication. It completes the worldline action of DOI 10.5281/zenodo.21845411 into a LOCAL, GENERALLY COVARIANT field theory with three healthy propagating modes. It does NOT derive the MOND acceleration scale, and it makes NO claim about particle physics, the Standard Model or unification. The theory. S = SEH[g] + Skh[g,T] + Sχ[g,u,χ] + Sm[g,T,χ,x], with the khronon sector N√h[KijKij − λK2 + ξR(3) + η aiai], ai = ∂i ln N, and nμ = −∂μT/√(−(∂T)2). 1. THE MEMORY KERNEL BECOMES A LOCAL FIELD. The general-orbit form Θ = ∫ds K(s) s |a(τ−s/2)|/c carries both a factor s and a lag s/2; the substitution s = 2u collapses both into one kernel G(u) = 4u K(2u), and for the minimal causal K = (N/λ)e−s/λ this is G = g u e−mu with m = 2/λ, g = 4N/λ. That IS the retarded Green's function of (d/dτ + m)2, so χ'' + 2mχ' + m2χ = g|a|/c: a damped oscillator whose FRICTION term is the memory. Verified in the transform domain, by the jump conditions, and closed NUMERICALLY to 1e-23 against the nonlocal convolution. The damping ratio is EXACTLY 1 (critical; zero discriminant, double root), and the order is FORCED: K ~ ske−s/λ gives order k+2, so the exponential kernel is the MINIMAL localisation. The extra factor of u traces to the rapidity gap being LINEAR in s -- the same fact that, in the companion no-go (DOI 10.5281/zenodo.21854464), puts the required kernel moment exactly on the pole of ζ. 2. AND a0 BECOMES A COUPLING RATIO: a0 = (2/3) c m2/g -- the ratio of the auxiliary field's MASS SQUARED to its COUPLING, rebuilt to 1e-25 on BOTH a0 footings, with the implied kernel weight N = 2.11e6 matching the independently derived bound of the earlier paper. Combined with the companion no-go (M1 is a logarithmically divergent moment), g/m2 is a RENORMALISED COUPLING -- which is what every coupling in every local field theory is. 3. THE GHOST QUESTION BECOMES A COMPUTATION, AND RESOLVES FAVOURABLY. The multiplier obeys the adjoint (d/dτ − m)2 -- anti-damped, the Bateman mirror -- and the (Θ,π) kinetic form is off-diagonal with signature (+,−): a ghost by the naive criterion. It is NOT a new degree of freedom: π is a COSTATE carrying a FINAL-value condition and no Cauchy data, it decays backward, and the count closes exactly at 2+2 = 4 = the order of the fourth-order equation of motion already derived. The indefinite metric is the Keldysh (−) branch of the in-in formalism already in use. This REPLACES the earlier 'Ostrogradsky is silent because it needs a local Lagrangian' hedge with a positive statement. 4. THE PREFERRED FRAME BECOMES DYNAMICAL, and the vorticity vanishes IDENTICALLY -- a theorem. For any gradient-built n, ∂[μnν] = A[μnν] with A = −∂ ln N, verified termwise, and the spatial projector annihilates n; the Christoffels drop out, so ω = 0 in EVERY metric. Consequence: this is the HYPERSURFACE-ORTHOGONAL (Horava) sub-case of Einstein-aether and the spin-1 sector is removed entirely; the four aether couplings collapse to THREE, (c1+c3)K.K + (c4−c1)a.a + c2(tr K)2, Jacobian rank 3. Nothing is lost: in the gauge |∂T| = 1 with T = t the coupling √((u.n)2) equals γ EXACTLY, so the earlier action is the UNITARY GAUGE of this one. Still CPT-even. 5. AND THE KHRONON'S OWN ACCELERATION IS THE NEWTONIAN FIELD: aμ[n] = ∂μΦ at linear order in a static weak field, computed from the Christoffels and with three prespecified decoys rejected. Before covariantisation the only acceleration in the theory was the particle's own |a|; MOND relates gobs to gbar, and gbar now exists as a GEOMETRIC OBJECT. 6. THE SPIN-0 MODE IS HEALTHY, in an explicit nonempty window. The quadratic scalar action is DERIVED from the ADM pieces. The GR check passes first: at η = 0 the α constraint degenerates to ζ = 0, so general relativity has no propagating scalar -- reproduced, not assumed. Then cs2 = ξ(2ξ−η)(1−λ)/[η(1−3λ)] = (2−η)(λ−1)/[η(3λ−1)] at ξ = 1, with the tensor sector giving graviton speed2 = ξ and a POSITIVE kinetic term -- so GW170817 makes ξ = 1 a MEASUREMENT to ~1e-15, not a choice. NO GHOST ⇔ λ > 1 or λ < 1/3; NO GRADIENT INSTABILITY ⇔ 0 < η < 2 -- and the two conditions COINCIDE, both excluding exactly the band 1/3 < λ < 1. In the PPN-safe corner cs2 → (λ−1)/η EXACTLY, so only the RATIO survives and the vacuum Cherenkov bound reduces to the single inequality λ−1 ≥ η > 0; superluminal is SAFE because there is a preferred frame. PPN and Cherenkov make INDEPENDENT demands: no conflict. 7. AND MOND FALLS OUT OF THE JOINT FIELD EQUATIONS, non-circularly. Modified inertia PRESUPPOSES an unmodified gravitational field, and the same PPN bound that made the khronon healthy is what guarantees it: khronon corrections to Φ are O(η, λ−1) ≤ 1e-7 while MOND needs Φbar only to ~1% -- FIVE ORDERS of margin. With the Newtonian metric established independently, the worldline equation gives μ(gobs/a0) gobs = gbar, hence gobs = √(a0gbar) and v4 = G M a0 exactly -- flat rotation curves and the baryonic Tully-Fisher relation -- with NO new parameter. Propagating content 2 (graviton) + 1 (khronon) + 0 (χ) = 3. 8. THE MASS QUESTION, CONFRONTED. Rest energy mc2 (μ-independent), kinetic mμv2/2, so mgrav = m and minert = mμ, AND THEY DIFFER -- not a defect to explain away, it IS the definitional content of modified inertia. The WEAK EQUIVALENCE PRINCIPLE SURVIVES: m cancels, giving μa = gbar with no reference to mass or composition, so universality of free fall is exact; what is violated is the EQUALITY mi = mg. And a galaxy's GRAVITATING mass is its BARYONIC mass. 9. NEW: A FORK THE COVARIANTISATION EXPOSES. Since aμ[n] = ∂μΦ, the theory possesses TWO acceleration scalars, and Θ may be sourced by either: Θ[|a|] is pure modified inertia, while Θ[|a[n]|] is a THIRD theory, external-field-driven, neither pure MI nor AQUAL, sharing the same interpolation function and the same a0. The worldline formulation could not even express this fork. They are distinguishable by a directional external-field test, where pure modified inertia predicts EXACTLY ZERO aligned rotation-curve asymmetry. Exposed here, NOT resolved here. AGAINST INTEREST, in the paper and in this record rather than buried. a0's VALUE is still not derived and κ = 1/2 remains FITTED. The companion no-go shows WHY a free-field correlator cannot supply it. The theory GAINED two free parameters (λ, η), not fewer. General covariance and ghost-freedom were bought with them, and nothing in modified inertia forces either -- so 'the scalar is healthy' means 'a viable region exists', much weaker than a prediction. The η → 0 corner that PPN prefers is the known STRONG-COUPLING corner, and that scale is NOT computed here. It is the sharpest remaining gap. Ostrogradsky is NOT evaded in the LOCALISED writing (the source is |a|, so the system is second-derivative in x); the exact action contains only u, which is why the two statements coexisted rather than contradicting. Exact localisation of the FULL bilocal cosh-1(−u.u'/c2) is NOT available by this route -- the auxiliary-propagator trick needs linearity in the delayed field. The localisation is exact for the MIDPOINT form only; the relative error against the exact bilocal is O((λΩ)2) = 7.7e-7 galactically. BTFR normalisation BOTH WAYS: Afw = 1/(Ga0) = 80.5 vs McGaugh's fitted 47 M⊙(km/s)-4, a 0.234 dex mass offset; the ALT footing fits BETTER (1.42 vs 1.71); but A = 47 was set at Υ3.6 = 0.5 so the framework needs 0.856, high yet inside the spread and near the 0.70 an independent RAR fit prefers, and the Υ-FREE gas-dominated a0-line box [0.84, 1.36]e-10 CONTAINS 9.3619e-11. A mass-to-light question, not an a0 question: neither a win nor a deficit. A first draft called the single-stream restriction the sharpest gap. That was wrong and is corrected in the paper rather than absorbed: Θ integrates over the particle's OWN past, so χ is a per-worldline INTERNAL variable and two stars crossing at a point simply carry different χ; and the metric never sees χ because mgrav = m is μ-independent, the first χ-dependent source being suppressed by (v/c)2 = 5.4e-7. The restriction binds only a continuum rewriting undertaken for its own sake. Strong fields, black-hole universal horizons and nonlinear stability: not addressed. The spin-0 analysis is flat-space, quadratic order, scalar sector. Energy-momentum conservation for a time-nonlocal modified-inertia theory is CITED (Milgrom 1994), not proved here. NO claim regarding particle physics, the Standard Model, or unification. The author publicly withdrew earlier statements of that kind and does not restate them. A complete field theory of modified inertia is NOT a theory of everything. ν = √(1+1/y) is IDENTICAL to Eq. (9) of Milgrom, Phys. Lett. A 253, 273 (1999). The khronon sector is standard technology; what is new here is its use to complete THIS construction, the acceleration theorem, and the fork. CREDIT. Khronon / hypersurface-orthogonal aether and the infrared limit of non-projectable Horava gravity: Jacobson 2010 PRD 81:101502; Blas, Pujolas & Sibiryakov 2010 PRL 104:181302 and 2011 JHEP 1104:018; Jacobson & Mattingly 2001 PRD 64:024028; Horava 2009 PRD 79:084008. Second-order relaxation: Israel & Stewart 1979 Ann.Phys. 118:341. Time-nonlocality of modified inertia and its conservation laws: Milgrom 1994 Ann.Phys. 229:384. Vacuum Cherenkov: Elliott, Moore & Stoica 2005 JHEP 0508:066. PPN: Will. Graviton speed: LIGO/Virgo GW170817. BTFR: McGaugh 2012 AJ 143:40. Auxiliary-field localisation of nonlocal actions, the Bateman dual and costate boundary conditions are classical. AI-assistance disclosure. Portions of the analysis, numerical verification and drafting were carried out with the assistance of a large language model (Anthropic Claude). The author directed the work, specified and reviewed every load-bearing calculation, and takes full responsibility including for any errors. No AI system satisfies the criteria for author