Paper 8: Interior Matching for the Linearised Rotating Exterior Perturbation of the Four-Dimensional Scale Space Framework: Mode Selection, Amplitude Normalisation, and the Scale Separation of Binary Pulsars

Paper 6 of this series derived the linearised rotating exterior perturbation of the four-dimensional scale space framework — the scale-Kerr solution to first order in angular momentum — but left three items unresolved: the separation constant k (which determines the spatial scale of the off-diagonal field), the amplitude normalisation (stated heuristically as−2GJ/(ρc^2)), and the correct value of the scale separation ∆s for real binary pulsars. We resolve all three.

First, we derive the interior scale-velocity distribution of a rigidly rotating neutron star: ˙s(r) = 2GM(r)/(rc) = ˙ssurf(r/R)^2 for a uniform sphere, with surface value ˙ssurf = 2GM/(Rc) and volume average (3/5)˙ssurf. This distribution is the source for the interior off-diagonal field h^int ϕs.

Second, we perform the interior–exterior matching at the stellar surface ρ= R. The matching condition selects the DC mode (k→0), corresponding to the slowly rotating limit in which the off-diagonal field has no spatial oscillation and decays as 1/ρ for ρ>R. This is the exact scale-space analogue of the Hartle (1967) slowly rotating star result in GR. 

Third, the DC-mode matching fixes the amplitude: the factor −2GJ/(ρc^2) in the near-zone metric of Paper 6 is the correct Hartle–Thorne normalisation, not a heuristic estimate. The scale-dragging field extends to all spatial distances as −2GJ/(ρc^2) e^(λexact(s−sM)).

Fourth, we resolve the scale separation of binary pulsars. The scale coordinate s is the logarithm of a physical length, not an accumulated displacement; for two neutron stars with the same radius, ∆s = ln(MA/MB), confirming the Paper 2 estimate from a first-principles derivation that avoids the dimensional confusion between ˙s (metres per second) and ds/dt (nats per second). For PSR J0737−3039, ∆s= ln(1.338/1.249) = 0.0688 nats, and all Paper 7 numerical results are confirmed.