A local, generally covariant field theory of modified inertia

Abstract

VERSION 2: the strong-coupling scale -- v1's sharpest named risk -- is COMPUTED, and it does NOT threaten the phenomenology. Restoring the khronon fluctuation T = t + π gives ln N = −πdot + πdot2/2 + (∂π)2/2, so ai = −∂iπdot and Kij = −∂i∂jπ. At λ = ξ = 1, η = 0 the khronon action VANISHES IDENTICALLY (K.K − K2 = k4π2 − k4π2 = 0), so π is pure gauge in general relativity, as it must be -- and a prespecified λ = 3/2 decoy does NOT cancel. Everything surviving is therefore proportional to the small parameters, S2 ~ MPl2[−δ(∂2π)2 + η(∂iπdot)2]. That is the entire origin of the worry, now stated precisely. AN INDEPENDENT CROSS-CHECK FALLS OUT. That action's dispersion ηω2k2 = δk4 gives cs2 = (λ−1)/η -- EXACTLY the PPN-corner limit the spin-0 section obtained from the UNITARY-GAUGE ζ with the ADM constraints eliminated, here from the STUECKELBERG π in flat space. Two gauges, two variables, one answer; a prespecified decoy δ/(2η) is rejected. AND THE STATIC NONLINEARITY OF THE η SECTOR VANISHES. The leading self-interaction is −2πdot(∂iπdot)2 − 2∂iπdot ∂jπ ∂i∂jπ, and every term carries a time derivative, so the whole cubic part vanishes for static configurations -- no static Vainshtein-type screening radius from that sector (a decoy cubic (∂xπ)3 does NOT vanish, so this is a property of the actual terms). Derivative counting then gives Λsc ~ √η MPl/cs ≈ 7.7e14 GeV at η = 1e-7: the PPN-preferred corner does lower the cutoff, but only as η1/2, and from the Planck mass. AND THE CONCLUSION IS ROBUST TO THE POWER. Scanning Λsc = ηpMPl over p = 1/2, 1, 2, 3, 4 -- deliberately allowing powers far worse than the derived one -- Λsc exceeds EVERY scale at which the theory is APPLIED in every case: the Milky Way's orbital frequency (5.7e-31 eV), its inverse size (7.8e-28 eV), 1/AU (1.3e-18 eV) and 1/metre (2.0e-7 eV). At the pessimistic p = 4 there are still 26 orders of galactic margin and 6 against the laboratory; at the derived p = 1/2, fifty. So the strong-coupling scale bears on whether this is a UV-complete QUANTUM theory -- which it never claimed to be -- and NOT on the phenomenology. AGAINST INTEREST, AND CORRECTING A CLAIM OF THE AUTHOR'S OWN RATHER THAN SOFTENING IT. A first draft asserted the cutoff clears every scale INCLUDING the LHC (1.4e13 eV). That is FALSE: at p ≥ 3 the cutoff falls BELOW collider energies, so the khronon effective theory would not cover the LHC at those powers. Harmless only because the matter-khronon coupling there is |B| ~ a02/8g2 = 1.1e-23, and the DERIVED p = 1/2 clears the LHC by 10 orders. Two further caveats: only the SCALING is computed, not the coefficient (the comparison is built to be insensitive to it, but a factor of 100 in the prefactor is excluded by nothing); and the δ-SECTOR'S STATIC NONLINEARITY IS NOT ANALYSED -- the η sector's static cubic dies, but the (∂2π)2 sector's need not, so a Vainshtein radius from THAT sector is unknown. That is now the sharpest gap in the construction, and it replaces the strong-coupling item in the limitations list. Flat space throughout. A second self-correction: a first cubic-order counter returned ZERO terms, which made a check pass VACUOUSLY on an empty list; it was found and replaced with explicit order bookkeeping. A NOTE ON WHICH THEORY THIS IS. The notorious λ → 1 strong coupling belongs to PROJECTABLE Horava gravity. The non-projectable 'healthy extension' carrying the aiai term is the known repair -- and this construction landed on it BY THEOREM rather than by choice, since the vorticity of a gradient-built n vanishes identically. NOTHING ELSE CHANGES IN v2. Every equation, result and caveat of v1 stands, including that a0's VALUE is NOT derived, that κ = 1/2 remains FITTED, that the theory GAINED two free parameters rather than fewer, and that NO claim is made regarding particle physics, the Standard Model or unification. One further verification script is added (24/24). 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 i