A dark-mass ladder in disk galaxies

The central relation of the directional-product model, N = S2−d2, describes the relativisticmomentum of a free particle as the product of two shifted factors, (pc)2 = K(K + 2mc2). Weapply the same structure to accelerations and propose the law g2 = gN (gN + a0), where gN isthe Newtonian acceleration of the baryons and a0 a characteristic acceleration; g is the squareroot of the determinant of a 2 × 2 matrix with eigenvalues gN and gN + a0. Fitted to theSPARC radial acceleration relation the law gives a0 = 1.50×10−10 m s−2(rms 0.027 dex), andthe baryonic TullyFisher relation of 118 galaxies gives 1.57 × 10−10 independently. The lawgives the dark mass in closed form, Md = Mb(√1 + x2 − 1) with x = r/ra, ra =pGMb/a0,and a ladder in steps of ra with the exact self-energy −Eq = −Mbv2fper step. In this revisedversion we add ve results. (i) We write the law as a eld equation, ∇ · g = ∇ · [νKgN ], whichreproduces it exactly, denes the dark mass as the phantom density of the eld, and gives theexternal-eld eect: with the Newtonian eld of the Galaxy at the Sun, 0.825 a0, wide binariesshould show a gravity raised by 7 to 49%, and the Solar System keeps the curvature of theSun to better than 10−11. (ii) The eld equation follows from an action whose Lagrangian isbuilt from the variables of the product, Q = S√N − d2cosh−1(S/d), twice the area under thehyperbola S2 − N = d2; Noether's theorem gives the conservation laws, the BohrSommerfeldcondition on the action gives the minimum quantum, and the epicycles of the action reproducethe Oort constants of Gaia (A = 14.5, B = −12.4, Ω0 = 27.0 km s−1 kpc−1 against 15.3, −11.9,27.1), where Newton with the same baryons is excluded at 3σ. (iii) The dark mass curvesspace-time: the ratio of its Ricci curvature to the Weyl curvature of the baryons is x2/√1 + x2,equal to one at x =√ϕ = 1.27 (ϕ the golden ratio) and to r/ra far out; the deection oflight tends to the constant 2πv2f/c2, 0.50′′ for the Milky Way, and weak lensing gives thedark mass 88 to 98% of the curvature at 115 kpc. (iv) The ladder has a minimum quantum,M∗ = (ℏ4a0/G3)1/7 = 3.49×10−17 kg, at which one quantum of angular momentum, of actionand of wavelength coincide; a system of mass M carries N = (M/M∗)7/4 quanta, and each stepholds N dark quanta md = ℏ/vf ra and N eld quanta ℏωa, so that dark matter and gravityare quantised by counting; md changes from system to system and lies below the Lyman-αbound for all SPARC galaxies, so it is a quantum of the eld, not a particle. (v) a0 from thebinned acceleration relation equals c2√Λ/2π = 1.493 × 10−10 m s−2(Planck) within 0.3%,which makes M∗ the weighted geometric mean of the Planck mass and the quantum of darkenergy. We test the results on independent observations: other samples, four cosmologies,weak lensing, the Oort constants and the Solar System agree; the free-electron template has amomentum form, which gives our law, and an energy form, KE = (pc)2/(1 + ε), which giveswith no new constant the simple function of the literature; read as an energy equation, withthe binding energy B = mv2/(1 + ε) of our atomic research as the half-separation, the centralrelation gives the energy form with λ = 1/(1+ε) = 12and nothing chosen, and the full SPARCrotation curves measure λ = 0.490.50; the function so derived is the one chosen empiricallysince 2005, an independent conrmation of the directional product (Table 18 compares ourroute with the others); we propagate the uncertainties of masses, distances, mass-to-lightratios and external elds by Monte Carlo, and nd three tensions: the redshift evolution ofa0, the decline of the outer Milky Way between 19.5 and 26.5 kpc (3.6σ with the eld equationsolved for the disk; the levels agree) and the trend of a0 with gas fraction (2.3σ in ts of all175 SPARC rotation curves, 3.1σ from the at velocities; not produced by the geometry of thedisks, present for every interpolating function but strongest for ours); the ts of the rotationcurves are as good as with the established functions, but give a0 = 1.17 × 10−10, so that theequality with c2√Λ/2π depends on the method; NGC 1052-DF2 is in tension at 20 Mpc (3.3σ)and consistent at 13 Mpc (1.4σ).