Nonlocal Time Gradient Gravity A Global Smoothing Interpretation of Spacetime

Geometric Restorativity of Spacetime Jianheng Huang Linyi University November 7, 2025 Abstract We propose a Geometric Restorativity (GRS) framework in the weak, static limit, (1 − ℓ2∇2 ) ∇2 Φ = 4 πGρ , with an environment-dependent recovery length ℓ . Without invoking dark matter, GRS unifies disk rotation curves and merging-cluster lensing: in the inner disk, nearly constant ℓ yields a central analytic kernel and a solid-body rise v∝R aligned with the exponential-disk peak; in the outer disk, setting ℓ = λR produces a radial invariant Q and v2 ( R ) = Q + C Rn with n = 2 −k+ ( λ ) < 0, explaining flat/mildly declining/rising types and enabling a direct extraction of Q and λ from derivatives of v ( R ). When ℓ→ℓsat , the curve approaches a Yukawa/Kepler tail. Projecting to 2D gives (1 −ℓ2 θ∇2 θ ) κ = Σ / Σ crit [ 1 ]: differential smoothing (small ℓ for galaxies, large ℓ for gas) generically yields mass–gas peak offsets and equipotential/shear bridges. The framework reduces to Newton/GR for ℓ/r ≪ 1and provides falsifiable slope/scale relations tied to radius, diffuseness, and merger phase. Keywords: geometric restorativity; nonlocal weak gravity; unified rotation curves; outer-disk plateau and slope; Yukawa kernel; radial invariant Q ; ℓ = λR ; Bullet Cluster; peak offset; equipotential/shear bridge Introduction In the weak, static limit we propose a geometric restorativity framework governed by an environmentdependent recovery length ℓ, with field equation (1 −ℓ2∇2)∇2Φ=4πGρ. (1) Equation (1) is equivalent to a Newtonian kernel dressed by an adaptive low-pass (Yukawa) operator[ 2 , 3 ]: ℓ is small in high-curvature/dense regions and large in low-curvature/diffuse regions. A unified rotation-curve narrative then follows naturally: in the inner disk with ℓ≈const the convolved kernel is analytic at the center, yielding a solid-body rise v∝R and the standard exponential-disk peak[4, 5]; in the outer disk taking ℓ∝Rrecasts the equation into a conservativeflux form so that v2 approaches a constant plateau (allowing small power-law deviations); farther out, if ℓ saturates to a constant the solution transitions to a finite-scale Yukawa tail that smoothly tends to a Kepler fall-off[6]. Projecting (1) along the line of sight gives the modified lensing relation (1 −ℓ2 θ∇2 θ)κ= Σ/Σcrit,(2) 1 so the same “kernel as low-pass” mechanism preserves dense galaxy peaks (small ℓθ ) while suppressing diffuse, ram-stripped gas (large ℓθ ), thereby predicting both a mass–gas peak offset and a shear/equipotential bridge along the merger axis[ 7 – 9 ]. The framework automatically reduces to Newton/GR in the ℓ/r≪ 1regime and unifies disk rotation curves with merging-cluster lensing into a single, observation-quantifiable picture. Contents 1 Two Postulates of Geometric Restorativity 3 1.1 P1 — Geometric Restorativity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 P2 — Curvature–Recovery Closure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Immediate corollaries (not additional postulates). . . . . . . . . . . . . . . . . . . . . 3 2 Unified Rotation Curves from Inner to Outer Galactic Disks: The Mainline of Geometric Restorativity 4 2.1 Inner disk: central analyticity of the convolved kernel and determination of the peak 4 2.2 Solar-system limit: why strictly Newtonian . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Transition zone: slowly varying ℓ(R)and continuity conditions . . . . . . . . . . . . 5 2.4 Outer plateau (ρb≃0,ℓ=λR): conserved quantity and small deviations . . . . . . . 6 2.5 Farther vacuum: saturation of ℓand the Yukawa/Kepler tail . . . . . . . . . . . . . 6 2.6 Unified picture and testable predictions . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 Merging-Cluster Lensing: Bullet-Cluster Peak Offset and Shear Bridge 8 3.1 Setup&projection...................................... 8 3.2 Unified picture of the offset. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.3 Two-component, piecewise-constant model. . . . . . . . . . . . . . . . . . . . . . . . 8 3.4 Implementation & falsifiable trends. . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4 Conclusions and Outlook 9 4.1 Conclusions.......................................... 9 4.2 Outlook............................................ 9 5 Refsection 10 6 Appendix A (Detailed): Step-by-step derivation from the action to the modified Poisson equation 11 2 7 Appendix B — Weak-Field Static Limit and the Newton–Yukawa Green Function 14 8 Appendix C: Unified Derivation of Galactic Rotation Curves from Inner to Outer Disk in GRS 19 9 Appendix D:Bullet Cluster: Peak Offset and Equipotential Bridges (Derivation) 27 1 Two Postulates of Geometric Restorativity Scope and regime. Throughout this section we work in the weak-field, static limit and adopt the Newtonian gauge with negligible anisotropic stress so that Φ ≃ Ψ. Sources are non-relativistic (T00 ≈ρ), and all operators are Euclidean in space. 1.1 P1 — Geometric Restorativity. Spacetime restores local geometric balance by smoothing the Newtonian source u≡ ∇2 Φover an environment-dependent recovery length ℓ .Operationally, this “restorativity” is implemented by a single-scale, linear, translationand rotation-invariant smoothing of u that: (i) reduces to Newtonian gravity as ℓ→ 0; and (ii) remains bounded and causal at all scales. Under these minimal clauses, the response is uniquely ueff = (1 −ℓ2∇2)−1u, ⇐⇒ (1 −ℓ2∇2)∇2Φ=4πG ρ, equivalently a real-space Yukawa-type kernel Gℓ ( s ) ∝ (1 −e−s/ℓ ) /s or a Fourier multiplier µ ( k ) = 1/(1+ℓ2k2). 1.2 P2 — Curvature–Recovery Closure. The recovery length ℓ is set by the local geometric environment (“curvature/compactness”): it is small in high-curvature or compact regions and large in low-curvature or diffuse regions. The main text leaves the functional form ℓ = ℓ (x)unspecified; illustrative closures (e.g. ℓ∝K−1/4 or ℓ∝u−1/2 ), as well as piecewise-constant or ℓ∝R ansätze for applications, are examined in the Appendices. 1.3 Immediate corollaries (not additional postulates). • Newtonian limit / Solar-system safety. ℓ→ 0 ⇒ Poisson’s equation; ℓ/r ≪ 1recovers Keplerian dynamics. • Galaxy disks (used later). Inner disk ℓ≈const ⇒ peak +quasi-flat band; outer disk ℓ∝R⇒v2→Q; far outer region where ℓsaturates ⇒mild decline. • Merging-cluster lensing. Line-of-sight projection gives (1 −ℓ2 θ∇2 θ ) κ = Σ / Σ crit ; smaller ℓ (galaxies) is preserved while larger ℓ (diffuse gas) is suppressed ⇒ peak offset +shear bridge. 3 2 Unified Rotation Curves from Inner to Outer Galactic Disks: The Mainline of Geometric Restorativity Overview and claim. This chapter uses the recovery length ℓ as the organizing thread to present asingle mechanism for rotation curves from the inner disk to the outermost vacuum. The core conclusion is: in high-curvature regions (inner disk) ⇒ℓ is nearly constant, the convolved kernel is analytic at the center, producing a solid-body rise v∝R ; across the mid–outer transition ⇒ ℓ ( R )increases smoothly with radius and matches curvature self-consistently, so the curve changes continuously with no discontinuity; in the near-vacuum outer region ⇒ taking ℓ≃λR produces a conserved quantity Q and a small deviation v2 = Q + CRn with n < 0; and farther out, if ℓ saturates to a constant, the speed transitions to a Yukawa/Kepler tail ( v∝R−1/2 ). The entire “rise–plateau–mild tilt–Kepler tail” morphology requires no dark matter and follows solely from the curvature dependence of ℓtogether with the conservation structure. 2.1 Inner disk: central analyticity of the convolved kernel and determination of the peak 2.1.1 (i) The point-source near field is only a reference. For a point mass Mwith constant ℓ, the midplane response v2(r) = GM r1−(1 + x)e−x, x =r/ℓ (C-1-1) at x≪ 1gives a v∝r1/2 rise, via (C-1-2)–(C-1-4). This is merely a toy near-field for a point source and does not describe a real thin disk. 2.1.2 (ii) A real thin (exponential) disk necessarily yields v∝Rat small R. Convolving K ( s ) = (1 −e−s/ℓ0 ) /s (C-1-5) with Σ( R′ )(C-1-7), performing the vector Taylor expansion for small R, and angle-averaging the first order, the quadratic coefficient is 1 2κ, leading to g(R) = −Φ′(R) = κ R +O(R3), v(R)∝R. (C-1-11) Using the identity (C-1-12) and integration by parts gives κ as in (C-1-13). For an exponential disk Σ = Σ 0e−R/Rd we have d Σ /dR < 0and dK/dR < 0, hence κ > 0 ⇒g∝R, v ∝R (solid-body rise). 2.1.3 (iii) The peak follows from the stationarity of the Hankel form. With the k-space low-pass F(k) = 1/(1 + ℓ2 0k2)and v2(R) = 2πG RZ∞ 0 ˜ Σ(k)J1(kR)k dk 1+ℓ2 0k2[5],(C-1-16) together with ˜ Σ ( k )of the exponential disk (C-1-20), one obtains the full-radius expression (C-1-22). Differentiating with respect to R and imposing dv2/dR = 0 gives the peak condition (C-1-23). In the limit β=ℓ0/Rd→0we recover the Freeman peak Rpeak ≈2.2Rd. 4 2.2 Solar-system limit: why strictly Newtonian 2.2.1 (i) k-space view: modifications are suppressed at short scales. In the weak, static limit the scalar equation (C-2-1) shows that the model equals the Newtonian kernel multiplied by F ( k ) = 1 / (1 + ℓ2k2 ). When the problem’s characteristic scale L satisfies kℓ ≪ 1 (or ℓ≪L), the Newtonian limit (C-2-2) follows. 2.2.2 (ii) Real-space view: the correction factor is exponentially small. For a point mass, g(r) = GM r21−(1 + x)e−x=gN(r) [1 −δ(x)], δ(x) = (1 + x)e−x, x =r/ℓ, (C-2-3) and at x≫ 1we have g ( r ) ≃gN ( r )(C-2-4). If one requires |δ|≲ 10 −9∼−10 , it suffices to have x≳ 25, i.e. ℓ≲r/ 25. On AU scales the high curvature drives ℓ≪AU , making the correction unobservable. 2.2.3 (iii) Conclusion. The solar system adheres to Newton/GR because ℓ is far smaller than orbital scales in a highcurvature environment, whereas only in low-curvature outer disks do we see the plateau and small slopes (quantified by (C-2-5)). 2.3 Transition zone: slowly varying ℓ(R)and continuity conditions 2.3.1 (i) Slow variable and conservative form. Define ε(R)≡dln ℓ/d ln R(≪1,≲O(0.1)).(C-3-1) The governing equation can be recast on the midplane into the conservative form ∇·∇Φ−∇(ℓ2u)= 4πGρ, u ≡ ∇2Φ.(C-3-2) 2.3.2 (ii) Ring-integration continuity conditions. Integrating over a thin coaxial ring of radius Rtr gives [RΦR]+ −= 0,[R(ℓ2u)R]+ −= 0,(C-3-3) ensuring seamless matching to the outer-zone invariants. 5 2.4 Outer plateau (ρb≃0,ℓ=λR): conserved quantity and small deviations 2.4.1 (i) Source-free equation and flux conservation. In the outer disk the baryonic source is exhausted ( ρb≃ 0) and ℓ is set by the environment scale (take ℓ=λR). The equation becomes ∇·∇Φ−∇(ℓ2u)= 0,(C-4-1) so that the radial invariant Q≡RΦR−(ℓ2u)R=const, v2(R) = RΦR=Q+R(ℓ2u)R(C-4-3) holds. 2.4.2 (ii) Large-radius ansatz and admissible exponent. Assuming ℓ(r) = λr, u(r) = A r−k, λ > 0,(C-4-4) and substituting into the outer equation yields λ2(k−2)(k−3) = 1, k±(λ) = 5±p1+4/λ2 2, k+(λ)>3,(C-4-6, C-4-7) so that v2(R) = Q+C Rn, n = 2 −k+<0.(C-4-13) The sign (and amplitude) set by the matching constant A gives two morphologies: A < 0 ⇒ positive correction and gentle decline to Q ; A > 0 ⇒ negative correction and gentle rise to Q . These correspond to the observed “flat/mildly declining/mildly rising” types. 2.4.3 (iii) Inferring Qand λfrom observations. Let W(R)≡v2(R) = RΦR.(1) Then using (C-4-11) one obtains Q=W−2λ2R W′−λ2R2W′′ ≈const, vflat =pQ. (C-4-12) This provides a direct outer-disk recipe: from observed v ( R ), construct Q using smoothed W′ and W′′, and estimate λ(hence k+and n). 2.5 Farther vacuum: saturation of ℓand the Yukawa/Kepler tail 2.5.1 (i) Saturation case and local slope. If at larger radii ℓ stops growing and saturates to ℓsat (beyond Rsat ), the exterior solution returns to the constant-ℓYukawa form. Define y≡R/ℓsat, B(y)≡1−(1+y)e−y,(2) 6 and by continuity at Rsat with v2(Rsat)=Qwe fix an effective enclosed mass Meff, giving v(R) = sGMeff RB(y).(C-5-2) The local logarithmic slope is sV(R) = 1 2 y2e−y B(y)−1!.(C-5-4) 2.5.2 (ii) Asymptotic Kepler limit and a 1% criterion. As y→ ∞, sV(R) = −1 2+1 2y2e−y+O(e−2y)→ −1 2.(C-5-7) Requiring |sV + 1 / 2 |< δ gives 1 2y2e−y< δ . For δ = 0 . 01 we obtain R≳ 8 . 1 ℓsat , where the fractional speed error is ∼1 2(1+y)e−y≲1.5×10−3—practically a Kepler regime. 2.6 Unified picture and testable predictions 2.6.1 (i) Unified picture. High curvature ⇒ short ℓ , central analyticity ⇒v∝R ;decreasing curvature ⇒ℓ∝R , giving v2 = Q + CRn with a power-law deviation that decays with radius; near vacuum ⇒ℓ saturates and the speed approaches a Yukawa/Kepler tail. The chain is closed by (C-1-1)–(C-5-7). 2.6.2 (ii) Testable predictions and data workflow. • Peak–scale alignment: In the β→ 0limit, Rpeak ≈ 2 . 2 Rd constrains the upper bound of ℓ0/Rd; • Plateau value Q :Construct Q from outer-disk v ( R )via smoothed derivatives using (C-4-12), and test its constancy with R; • Mild-tilt index n :Fit the outer-disk slope s ( R ) = dln v/d ln R≃ ( n/ 2) ε ( R )to infer λ and k+; • Far-region criterion: Use |sV + 1 / 2 | together with 1 2 (1 + y ) e−y from (C-5-4)–(C-5-7) to mark the onset of the Kepler zone. Summary. This chapter stitches together “central analyticity of the convolved kernel — continuity across the transition — outer invariant with a power-law deviation — saturated Yukawa/Kepler tail” into a single mechanism that needs no dark matter. The key quantities ( ℓ0, λ, Q, n )are directly extractable from observed curves and match one-to-one to the analytic structure (C-1-1)–(C-5-7). 7 3 Merging-Cluster Lensing: Bullet-Cluster Peak Offset and Shear Bridge 3.1 Setup & projection. Starting from the weak, static field equation of geometric restorativity, (1 −ℓ2∇2)∇2Φ(x) = 4πG ρ(x),(3) line-of-sight (LOS) projection gives the modified 2D lensing relation (1 −ℓ2 θ∇2 θ)κ(θ) = Σ(θ) Σcrit , ℓθ≡ℓ/DL,Σcrit ≡c2 4πG DS DLDLS .(4) Equivalently, lensing is a Yukawa-type smoothing of Σ/Σcrit: κ(θ) = Zd2θ′ 2π ℓ2 θ K0|θ−θ′| ℓθΣ(θ′) Σcrit ,(5) or, in Fourier space, a strict low-pass filter, ˜κ(k) = 1 1+ℓ2 θk2 ˜ Σ(k) Σcrit .(6) 3.2 Unified picture of the offset. During a merger, galaxy-dominated components are denser (higher curvature) and thus have smaller ℓ ; ram-pressure–stripped X-ray gas is diffuse and has larger ℓ . The low-pass operator therefore preserves sharp galaxy peaks (small ℓ ) and suppresses diffuse gas peaks (large ℓ ), naturally pulling the lensing peak toward the galaxies and away from the gas. 3.3 Two-component, piecewise-constant model. Split Σ=Σgal + Σgas and assume ℓgal ≪ℓgas in their dominant regions: κ(θ) = Σgal/Σcrit∗Hℓgal +Σgas/Σcrit∗Hℓgas , Hℓ(∆θ)∝K0(|∆θ|/ℓ).(7) Along the merger axis (1D cut), a sufficient condition for the lensing peak to shift toward the galaxies is Ag K1(a/ℓg) ℓ3 g > Ab K1(a/ℓb) ℓ3 b ,(ℓb≫ℓg),(8) where a is the angular separation of the centroids. The bridge amplitude at the midpoint scales as the sum of two Yukawa tails, κbridge ≈Ag 2πℓ2 g K0 a ℓg+Ab 2πℓ2 b K0 a ℓb,(9) decaying approximately as e−a/ℓ/√a. 8 3.4 Implementation & falsifiable trends. • Real-space: convolve Σ / Σ crit with Hℓθ ; for spatially varying ℓ ( θ ), iterate a closure (guess ℓ , compute κ, update ℓ[κ]). •Fourier-space: apply the multiplier (1 + ℓ2 θk2)−1patchwise, then inverse-transform. • Trends: larger gas diffuseness/stronger shocks ⇒ larger ℓgas , larger peak offset, and a broader/stronger bridge; late-time relaxation (smaller ℓ) diminishes both. 4 Conclusions and Outlook 4.1 Conclusions. We systematize Geometric Restorativity (GRS) in the weak, static regime via (1 −ℓ2∇2)∇2Φ = 4πG ρ, (10) where the environment-dependent recovery length ℓ implements a controlled low-pass filtering of the Newtonian source u≡ ∇2 Φ. This single mechanism yields a unified account of two ostensibly separate phenomena: (i) Unified disk rotation curves. In inner disks where ℓ varies slowly, aligning the kernel’s intrinsic extremal scale with the exponential-disk peak gives an “upturn–peak–quasi-flat” pattern; in outer disks, ℓ∝R produces a conserved flux and drives v2 ( R ) →Q ; in far outskirts a saturated ℓ recovers a finite-length Yukawa tail, yielding a mild negative slope and asymptotic Keplerian falloff. (ii) Merging-cluster lensing (Bullet Cluster). LOS projection leads to the modified 2D lensing equation (1 −ℓ2 θ∇2 θ)κ(θ) = Σ(θ) Σcrit ,(11) equivalent to an ℓθ -scale Yukawa smoothing of Σ / Σ crit . Since galaxies are denser (smaller ℓ ) while ram-pressure–stripped gas is diffuse (larger ℓ ), the lensing map preserves galaxy peaks and suppresses gas peaks, naturally producing the observed mass–gas peak offset[ 7 , 8 ] and an equipotential/shear bridge[ 8 ] along the merger axis. Analytical offset criteria and bridge scalings follow from the same kernel. The framework recovers the Newtonian limit for ℓ/r ≪ 1and remains compatible with first-order cosmological constraints under a controlled k -dependence. Overall, a single environmental scale ℓ and one Yukawa-type kernel organize disk kinematics and merging-cluster lensing into a coherent picture[9]. 4.2 Outlook. We outline a data–theory program to render the above mechanism falsifiable: • Disk-level fits and diagnostics: Hierarchical fits of ℓ ( R )closures; direct construction of the outer-disk invariant Q from derivatives of V ( R )and population statistics of quasi-flat radii and outer slopes[10]. • Merging-cluster morphology: Regress the offset–diffuseness (or Mach-number) correlation and bridge width vs. ℓ across a sample; validate analytic scalings using joint X-ray/lensing reconstructions. 9 B.2 Verification: (1 −ℓ2∇2)∇2ˆ G(r)=−δ(3) Write ˆ G(r) = −1 4πr +e−βr 4πr , β =1 ℓ>0, r =|r|.(B-2-1) Use the distributional identities ∇21 4πr=−δ(3)(r),(B-2-2) ∇2 e−βr 4πr !=β2e−βr 4πr −δ(3)(r).(B-2-3) Thus ∇2ˆ G=−−δ(3)+β2e−βr 4πr −δ(3)=β2e−βr 4πr −2δ(3).(B-2-4) It is convenient to introduce Y ( r ) := e−βr 4πr . From the Fourier derivation we also have the compact identity ∇2ˆ G=−β2Y. (B-2-5) Applying the operator (1 −ℓ2∇2)and using ℓ2β2= 1: (1 −ℓ2∇2)∇2ˆ G=−β2Y+ℓ2β2∇2Y=−β2Y+∇2Y=−δ(3)(r),(B-2-6) since ∇2Y=β2Y−δ(3). This confirms Lˆ G=−δ(3). B.3 Angular averaging for spherical symmetry and exterior solution Starting from the convolution Φ(x) = −GZd3x′ρ(x′)1−e−|x−x′|/ℓ |x−x′|,(B-3-1) let r = | r | , r′ = | r ′| , and assume ρ = ρ ( r′ ). Decompose Φ = Φ N + Φ Y into Newton and Yukawa parts: ΦN(r) = −GZd3r′ρ(r′)1 |r−r′|,(B-3-2) ΦY(r) = + GZd3r′ρ(r′)e−|r−r′|/ℓ |r−r′|.(B-3-3) Choose the polar axis along r. With µ= cos θ′, |r−r′|=qr2+r′2−2rr′µ. (B-3-4) and the angular average of the Yukawa kernel is 1 4πZdΩ′e−|r−r′|/ℓ |r−r′|=1 2Z1 −1 dµ e−√r2+r′2−2rr′µ/ℓ pr2+r′2−2rr′µ =ℓ 2rr′e−|r−r′|/ℓ −e−(r+r′)/ℓ.(B-3-5) 16 Similarly, for the Newton kernel, 1 4πZdΩ′1 |r−r′|=1 max(r, r′).(B-3-6) Substituting and using d3r′= 4πr′2dr′, one obtains the unified 1D kernel form Φ(r) = −GZ∞ 0 dr′4πr′2ρ(r′)1 max(r, r′)−ℓ 2rr′e−|r−r′|/ℓ −e−(r+r′)/ℓ.(B-3-7) Define the angle-averaged Newton–Yukawa kernel K(r, r′;ℓ) := 1 max(r, r′)−ℓ 2rr′e−|r−r′|/ℓ −e−(r+r′)/ℓ.(B-3-8) The first term (Newton) satisfies the shell theorem; the second (Yukawa) does not. B.4 Piecewise expressions and exterior solution B.4.1 Newton part. Splitting r′at r, ΦN(r) = −G1 rZr 0 4πr′2ρ(r′)dr′+Z∞ r 4πr′ρ(r′)dr′.(B-4-1) For r > R when the source is fully inside radius R,ΦN(r) = −GM/r. B.4.2 Yukawa part. Using the averaged Yukawa kernel, ΦY(r) = GZr 0 dr′4πr′2ρ(r′)ℓ 2rr′e−(r−r′)/ℓ −e−(r+r′)/ℓ +GZ∞ r dr′4πr′2ρ(r′)ℓ 2rr′e−(r′−r)/ℓ −e−(r+r′)/ℓ.(B-4-2) Factorizing yields ΦY(r) = G ℓ re−r/ℓ Zr 0 dr′4πr′ρ(r′) sinhr′ ℓ+ sinhr ℓZ∞ r dr′4πρ(r′)e−r′/ℓ.(B-4-3) For the exterior region r > R with ρ(r′)=0for r′> R, the second integral vanishes and ΦY(r) = G ℓ re−r/ℓ ZR 0 dr′4πr′ρ(r′) sinhr′ ℓ,(r > R).(B-4-4) The full exterior solution is therefore Φ(r) = −GM r+G ℓ re−r/ℓ ZR 0 dr′4πr′ρ(r′) sinhr′ ℓ,(r > R).(B-4-5) 17 Remark (not a point-source equivalence). Because the Yukawa kernel violates the shell theorem, the exterior potential depends in general on the interior mass distribution. Only in the compact-source limit R≪ℓ, where sinh(r′/ℓ)≃r′/ℓ, one finds ZR 0 4πr′ρ(r′) sinhr′ ℓdr′≃1 ℓZR 0 4πr′2ρ(r′)dr′=M ℓ,(B-4-6) hence ΦY(r)≃GM re−r/ℓ,(B-4-7) Φ(r)≃ −GM rh1−e−r/ℓi,(R≪ℓ, r > R).(B-4-8) B.5 Rotation curves B.5.1 Exterior region r > R (compact-source approximation) With R≪ℓ, the point-source approximation gives Φ(r)≃ −GM rh1−e−r/ℓi,(B-5-1) g(r) = −Φ′(r) = GM r21−1 + r ℓe−r/ℓ,(B-5-2) v2(r) = r g(r) = GM r1−1 + r ℓe−r/ℓ,(B-5-3) consistent with the main-text behaviour (Newtonian at large r , near-flat with a mild negative slope at intermediate radii). B.5.2 General spherical ρ(r)(no r > R assumption) From Φ(r) = −GZ∞ 0 dr′4πr′2ρ(r′)1 max(r, r′)−ℓ 2rr′e−|r−r′|/ℓ −e−(r+r′)/ℓ,(B-5-4) differentiate with respect to rand use the angle averages to obtain the 1D kernel form g(r) = −Φ′(r) = GM(< r) r2 | {z } Newton −G ℓ r2e−r/ℓZr 0 dr′4πr′ρ(r′)ℓcoshr′ ℓ−r′sinhr′ ℓ−coshr ℓZ∞ r dr′4π ρ(r′)e−r′/ℓ | {z } Yukawa , (B-5-5) where M ( < r ) = Rr 0 4 πr′2ρ ( r′ ) dr′ . Then v2 ( r ) = r g ( r )follows directly. Numerically, this reduces to a single 1D quadrature. 18 8 Appendix C: Unified Derivation of Galactic Rotation Curves from Inner to Outer Disk in GRS C.1 Inner Disk: Point-Source Limit and Constant ℓApproximation C.1.1 Near-field expansion for a point mass with constant ℓ In the high-curvature central region of a galactic disk, the geometric recovery length ℓ can be regarded as constant. For a point mass M, the geometric-restorative potential gives v2(r) = GM r1−(1 + x)e−x, x ≡r ℓ.(C-1-1) For x≪1, e−x= 1 −x+x2 2−x3 6+O(x4),(C-1-2) 1−(1 + x)e−x=x2 2−x3 3+O(x4).(C-1-3) Substituting back yields v2(r) = GM 2ℓ2r−GM 3ℓ3r2+···, v(r)≃sGM 2ℓ2r1/21−r 3ℓ+···.(C-1-4) This near-field result indeed produces a v∝r1/2 rise, but it corresponds only to the response of a point source. Real galaxies are distributed thin disks; to describe a real disk one must convolve the above kernel with the surface density Σ(R′)in an axisymmetric manner. C.1.2 Convolution for an exponential thin disk and central expansion Let the observation point be Rwith R = | R | , and a source point on the disk be R 1 with R1 = | R 1| . Write the point-source midplane kernel as the angle-averaged form K(s) = 1−e−s/ℓ0 s, s =|R−R1|,(ℓ≃ℓ0).(C-1-5) Its small-sexpansion is finite and analytic: K(s) = 1 ℓ0−s 2ℓ2 0 +s2 6ℓ3 0 +O(s3).(C-1-6) The potential is the convolution Φ(R) = −GZd2R1Σ(R1)K(|R−R1|).(C-1-7) For small R , the vector Taylor expansion (the first-order term angle-averages to zero by axisymmetry) gives to second order K(|R1−R|)=K(R1) + RaRb 2∂a∂bK(R1)+O(R4).(C-1-8) 19 For any radial function f(r), RaRb∂a∂bf(r)=R2 2f′′(r) + f′(r) rr=R1 .(C-1-9) Substituting back yields Φ(R) = Φ(0) −GR2 4Zd2R1Σ(R1) d2K dR2 1 +1 R1 dK dR1!+O(R4).(C-1-10) Denoting the quadratic coefficient by 1 2κ, we obtain g(R) = −Φ′(R) = κR +O(R3), v(R)∝R. (C-1-11) Using the identity d2K dR2 1 +1 R1 dK dR1 =1 R1 d dR1R1 dK dR1,(C-1-12) an integration by parts (vanishing boundary term) gives κ=πGZ∞ 0 dΣ dR1 R1 dK dR1 dR1.(C-1-13) For an exponential disk Σ( R1 )=Σ 0e−R1/Rd we have d Σ /dR1< 0and dK/dR1< 0, hence κ > 0⇒g∝R, v ∝R(solid-body rise). C.1.3 Fourier (Hankel) form and the inner peak In the inner disk take ℓ≃ℓ0(constant). The point-kernel in k-space acts as a low-pass filter F(k) = 1 1+ℓ2 0k2. The zeroth-order Hankel transform is defined by ˜ Σ(k) = Z∞ 0 Σ(R1)J0(kR1) 2πR1dR1.(C-1-14) The midplane tangential speed over all radii is v2(R) = 2πGRZ∞ 0 ˜ Σ(k)J1(kR)k dk 1+ℓ2 0k2.[5](C-1-15) For small R, with J1(z)=z/2+O(z3), one gets v2(R) = πGR2Z∞ 0 ˜ Σ(k)k2dk 1+ℓ2 0k2+O(R4), κ =πGZ∞ 0 ˜ Σ(k)k2dk 1+ℓ2 0k2.(C-1-16) Key integral (stated and proved): Z∞ 0 e−arJ0(br)r dr =a (a2+b2)3/2, a > 0.(C-1-17) 20 Setting a= 1/Rd, b =kgives Z∞ 0 e−R1/RdJ0(kR1)R1dR1=R2 d (1 + (kRd)2)3/2.(C-1-18) Therefore ˜ Σ(k) = 2πΣ0R2 d (1 + (kRd)2)3/2.[11](C-1-19) Substituting into κ, κ=2π2GΣ0 RdZ∞ 0 q2dq (1+q2)3/2(1+β2q2), β =ℓ0 Rd .(C-1-20) The full rotation curve is v2(R) 2πGΣ0Rd = 2yZ∞ 0 J1(2yq) (1+q2)3/2(1+β2q2)q dq, y =R 2Rd .(C-1-21) Differentiating with respect to R and setting dv2/dR = 0 (using J′ 1 = 1 2 ( J0−J2 )) gives the peak condition: Z∞ 0 J1(2yq) (1+q2)3/2(1+β2q2)q dq +yZ∞ 0 J0(2yq)−J2(2yq) (1+q2)3/2(1+β2q2)q2dq = 0.(C-1-22) When β=ℓ0/Rd→0(i.e., ℓ0≪Rd), the numerical solution yields Rpeak ≈2.2Rd[4]. C.2 Solar-system limit: verification of Newtonian recovery C.2.1 Weak-field equation and k-space filtering In the static weak-field limit, the scalar potential obeys (1 −ℓ2∇2)∇2Φ=4πGρ, ˜ Φ(k) = −4πG k2(1+ℓ2k2)˜ρ(k).(C-2-1) When the problem’s characteristic scale L≫ℓ(equivalently kℓ ≪1), 1 1+ℓ2k2≃1−ℓ2k2+···, so the model immediately reduces to the Newtonian kernel ∝1/k2. C.2.2 Point-mass response and the suppression factor For a point mass M, g(r) = GM r21−(1 + x)e−x, x =r ℓ=gN(r) [1 −δ(r/ℓ)] , δ = (1 + x)e−x.(C-2-2) When x≫1(i.e. ℓ≪r), the correction factor δis exponentially suppressed, hence g(r)≃gN(r) [1 −O(e−r/ℓ)] ≃gN(r).(C-2-3) To require |δ|< 10 −9∼−10 , one needs (1 + x ) e−x≪ 10 −9∼−10 , implying x≳ 25, i.e. ℓ≲r/ 25. At r∼1AU, an ℓof order 106km or smaller renders the deviation practically unobservable. 21 C.2.3 Curvature dependence and Newtonian recovery By the second postulate, the recovery length ℓ decreases in stronger curvature and increases in weaker curvature. In the corona and planetary-orbit scales of the solar system, ℓ is driven far below AU, yielding the Newtonian/GR weak-field limit; in the low-curvature outer disks of galaxies, ℓ increases with radius (approximately ℓ∝R ), producing the observed flat/mildly varying rotation curves. In k -space, the model multiplies the Newtonian kernel by F ( k ) = 1 / (1 + ℓ2k2 ). For solar-system wavenumbers k∼ 1 /r , if ℓ≪r then kℓ ≪ 1 ⇒F ( k ) ≃ 1, so orbital, geodetic and radar-time observables agree with Newtonian/GR predictions to very high precision. C.2.4 Remark on the near-field toy limit The x≪ 1expansion leading to g≃GM/ (2 ℓ2 )and v∝r1/2 is merely a point-source, constantℓ ultra-near-field toy limit. In the solar system of interest we have r≳AU and actually r/ℓ ≫ 1, entering the Newtonian limit. For a distributed thin disk, the convolved central potential is quadratic, giving g∝rand v∝r, independent of any constant-gassumption. C.2.5 Summary The solar system strictly follows Newtonian gravity because in a high-curvature environment ℓ is far below orbital scales, with the fractional correction (1 + r/ℓ ) e−r/ℓ ≪ 1. In k -space this is kℓ ≪ 1 ⇒F ( k ) ≃ 1. Thus the model naturally recovers the Newtonian/GR weak-field limit in the solar system, while exhibiting flat/weakly varying behavior only in low-curvature galactic outer disks. C.3 From Inner Disk to Transition Zone: Slowly Varying ℓ and Geometric Restorativity C.3.1 Slowly varying ℓand conservative form As curvature weakens outward, ℓgrows slowly with radius. Define the relative variation rate ε(R)≡dln ℓ dln R, ε ≪1 (typically O(0.1)).(C-3-1) The governing equation (1 −ℓ2∇2)∇2Φ=4πGρ can be recast into a flux-conservative divergence form: ∇·∇Φ−∇(ℓ2u)= 4πGρ, u ≡ ∇2Φ.(C-3-2) When ρ is already small while ε remains mild, the effect of ℓ′ is an O ( ε )correction; the rotation curve varies smoothly across the transition and connects to the outer zone without any discontinuity. 22 C.3.2 Continuity conditions Integrating over a thin coaxial ring at radius Rtr and using the divergence theorem yields the continuity conditions [RΦR]+ −= 0,[R(ℓ2u)R]+ −= 0,(C-3-3) ensuring smooth matching to the invariants of the outer-zone equation. C.4 Outer Plateau (with ρb≃ 0, ℓ = λR ): Conserved Quantity and Small Deviations C.4.1 Equation and flux conservation Once in the outer disk, the baryonic source is essentially exhausted ( ρb≃ 0), and ℓ is set by the environment scale (take ℓ=λR). The equation becomes (1 −ℓ2∇2)∇2Φ=0 ⇔ ∇·(∇Φ−∇(ℓ2u)) = 0.(C-4-1) Defining the composite flux F≡ ∇Φ−∇(ℓ2u), its divergence vanishes. For a coaxial cylinder of radius R and height H , stationarity and symmetry leave only the lateral flux, hence Q≡RΦR−(ℓ2u)R=const, v2(R) = RΦR=Q+R(ℓ2u)R.(C-4-2) C.4.2 Large-radius ansatz and constraint At large radii, adopt ℓ(R) = λR, u(R)=A R−k, λ > 0.(C-4-3) Under spherical symmetry for any radial function f(r), ∇f=f′(r) ˆr, ∇2f=1 r2 d drr2f′(r). Thus u′(r) = −kA r−k−1,∇2u=k(k−1)A r−k−2, ℓ2(r) = λ2r2,∇2ℓ2= 6λ2.(C-4-4) Substituting into −u+ℓ2∇2u+ 2∇ℓ2·∇u+u∇2ℓ2= 0 and factoring out Ar−kgives λ2(k−2)(k−3) = 1,(C-4-5) with solutions k±(λ) = 5±p1+4/λ2 2, k+>3.(C-4-6) 23 C.4.3 Outer speed and power-law correction Since ℓ2u=λ2A r2−k, (ℓ2u)r=λ2A(2 −k)r1−k, R(ℓ2u)R=λ2A(2 −k)r2−k.(C-4-7) When k=k+>3, the exponent 2−k+<0, so for any sign of A, r2−k+−−−→ r→∞ 0, implying v2(R)−−−→ r→∞ Q. (C-4-8) The sign of A determines the morphology: A < 0gives a positive correction and a gentle decline toward Q (“slightly declining”); A > 0gives a negative correction and a gentle rise toward Q (“slightly rising”). These correspond to the three observed outer morphologies: flat, mildly declining, and mildly rising. C.4.4 Determining Qand λfrom observations Define v2(R) = RΦR≡W(R), Q =W−Rd dR(ℓ2u)=W−R2ℓℓ′u+ℓ2uR,(C-4-9) using d(ℓ2) dR = 2ℓℓ′. In the near-vacuum outer zone with nearly spherical equipotentials, u=∇2Φ = 1 R2 d dR R2ΦR,ΦR=W/R ⇒u=W+RW′ R2, uR=2W′+RW′′ R2−2(W+RW′) R3. (C-4-10) If ℓ=λR, then ℓ′=λ,2ℓℓ′/R = 2λ2, and ℓ2=λ2R2. Substituting into (C-4-9), Q=W−2λ2RW′−λ2R2W′′ ≈const, vflat =pQ. (C-4-11) Hence the outer speed can be written as v2(R) = Q+C Rn, n = 2 −k+<0,(C-4-12) and with ε(R)≡C QRn(|ε|≪1), v(R) = pQ√1+ε≃pQ1 + ε 2,(C-4-13) the speed deviation is ∆v(R) = v−pQ≃√Q 2ε(R) = C 2√QRn.(C-4-14) The dimensionless local slope s(R)≡dln v dln R=1 2 n ε 1+ε≃n 2ε(R).(C-4-15) Examples: λ = 1 ⇒k+≈ 3 . 618 , n ≈ − 1 . 618 (decay ∼R−1.62 ); λ = 0 . 5 ⇒k+≈ 4 . 56 , n ≈ − 2 . 56 (faster decay ∼R−2.56). 24 C.5 Far Vacuum: Saturation of ℓ and a Mild Downturn (Yukawa/Kepler Approach) C.5.1 Velocity form upon saturation When the radius grows further into the far outskirts, the recovery length ℓ ceases to grow and saturates to a constant ℓsat . The exterior solution then returns to the constantℓ Yukawa form. Define y≡R ℓsat , B(y)≡1−(1+y)e−y.(C-5-1) By continuity at Rsat with v2(Rsat)=Q, an effective enclosed mass Meff is fixed, giving v(R) = sGMeff RB(y).(C-5-2) C.5.2 Local logarithmic slope: closed form Define the local logarithmic slope: sV(R)≡dln v dln R.(C-5-3) From ln v=1 2(ln GMeff −ln R+ ln B(y)), we get sV(R) = 1 2−1 + dln B(y) dln R. By the chain rule (with ℓsat constant), dln B(y) dln R=B′(y) B(y) dy dln R=B′(y) B(y)y, and since B′(y) = d dy[1 −(1 + y)e−y]=y e−y, we obtain dln B(y) dln R=y2e−y B(y).(C-5-3) Therefore sV(R) = 1 2 y2e−y B(y)−1!.(C-5-4) Example: at y = 2, e−2≈ 0 . 135335, B (2) = 1 − 3 e−2≈ 0 . 593994, so y2e−y B(y)≈ 0 . 9114 and sV≈ − 0 . 044 (a mild negative slope just after saturation). C.5.3 Far-vacuum asymptotics As y→ ∞, B(y)=1−(1+y)e−y= 1 −ε, ε := (1 + y)e−y→0, 25