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ANISOTROPIC SPACE DYNAMICS · OPEN RESEARCH

Complete spatial structure of rotating compact objects

Determine the azimuthal response, detector coupling and strong-field solution from longitudinal–transverse coupling and the rotating source; then calculate directed propagation and critical capture. Distinguish source rotation, exterior field angular velocity and the boundary energy conjugate.

Based on the final manuscript · Chapter 2

TheoryTheory · OPENReviewMathematical review · OPEN REVIEWEvidenceEvidence · UNTESTEDCanonicalNoStatus, review and sources
OP-GRV-003 · P2

NEEDS THEORY · Needs theory

AstrophysicsPDE
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01Task

An open research task extracted from the final Chapter 2 manuscript, not a new canonical derivation.

02Current state

The stationary-current Green solution, far-field azimuthal dipole and first-order opposite-direction difference for a fixed coaxial loop have been obtained; ζ=4 is empirically calibrated. Remaining work is causal source dynamics, strong-field continuation, detector coupling and a complete apparatus observable.

ASD-WEB-GRV-008
dℓopt2=Φ−2dℓs2=X−2dr2+X−1r2dφ2,dt=dℓoptcd\ell_{\mathrm{opt}}^2=\Phi^{-2}d\ell_s^2=X^{-2}dr^2+X^{-1}r^2d\varphi^2,\quad dt=\frac{d\ell_{\mathrm{opt}}}{c}

Mathematical result
Conditional on the stated inputs and domain. Manuscript numbering is separate from the website ID.
Chapter reference: Final collected manuscript · Chapter 2 §IV.1 (2.26) · ASD-BOOK-ddbade997b45

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ASD-WEB-GRV-045
rph=3ℓg=3G0Mc2=32rhr_{\mathrm{ph}}=3\ell_g=\frac{3G_0M}{c^2}=\frac32r_h

Mathematical result
Conditional on the stated inputs and domain. Manuscript numbering is separate from the website ID.
Chapter reference: Final collected manuscript · Chapter 2 §V.3 (2.70) · ASD-BOOK-ddbade997b45

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ASD-WEB-GRV-046
bc=B(rph)=33G0Mc2=332rhb_c=\mathcal B(r_{\mathrm{ph}})=3\sqrt3\frac{G_0M}{c^2}=\frac{3\sqrt3}{2}r_h

Mathematical result
Capture requires an absorbing boundary; b_c/Dobs is not a complete observed emission-ring model.
Chapter reference: Final collected manuscript · Chapter 2 §V.3 (2.71) · ASD-BOOK-ddbade997b45

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ASD-WEB-GRV-048
∇⋅urot=0,∇2urot=−4πζG0c2jm,urot(x)=ζG0c2∫jm(x′)∣x−x′∣ d3x′,urotdip=ζG02c2J×rr3,ωrot=ζG0J2c2r3.\begin{aligned}\nabla\cdot\mathbf u_{\rm rot}&=0,\quad\nabla^2\mathbf u_{\rm rot}=-\frac{4\pi\zeta G_0}{c^2}\mathbf j_m,\\\mathbf u_{\rm rot}(\mathbf x)&=\frac{\zeta G_0}{c^2}\int\frac{\mathbf j_m(\mathbf x')}{|\mathbf x-\mathbf x'|}\,d^3x',\\\mathbf u_{\rm rot}^{\rm dip}&=\frac{\zeta G_0}{2c^2}\frac{\mathbf J\times\mathbf r}{r^3},\quad\omega_{\rm rot}=\frac{\zeta G_0J}{2c^2r^3}.\end{aligned}

Mathematical result
Stationary localized source, no bulk translation, decay at infinity and the far-field dipole approximation; ζ=4 is calibrated, not independently predicted microscopically.
Chapter reference: Final collected manuscript · Chapter 2 §V.4 (2.72)—(2.77) · ASD-BOOK-ddbade997b45

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ASD-WEB-GRV-049
Krot=1−urot⋅k^/c1−urot2/c2,T[Γ]=1c∫Γdℓopt−1c2∫Γurot⋅dx,T−−T+=2c2∮Curot⋅dx=4ωrotACc2=2ζG0JACc4r3.\begin{aligned}K_{\rm rot}&=\frac{1-\mathbf u_{\rm rot}\cdot\hat{\mathbf k}/c}{\sqrt{1-u_{\rm rot}^2/c^2}},\\T[\Gamma]&=\frac1c\int_\Gamma d\ell_{\rm opt}-\frac1{c^2}\int_\Gamma\mathbf u_{\rm rot}\cdot d\mathbf x,\\T_- -T_+&=\frac2{c^2}\oint_C\mathbf u_{\rm rot}\cdot d\mathbf x=\frac{4\omega_{\rm rot}A_C}{c^2}=\frac{2\zeta G_0JA_C}{c^4r^3}.\end{aligned}

Mathematical result
The optical path retains first order in weak-field rotation; the last two equalities require a coaxial circular loop at fixed radius and polar angle.
Chapter reference: Final collected manuscript · Chapter 2 §V.5 (2.78)—(2.82) · ASD-BOOK-ddbade997b45

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03Required work

Determine the azimuthal response, detector coupling and strong-field solution from longitudinal–transverse coupling and the rotating source; then calculate directed propagation and critical capture. Distinguish source rotation, exterior field angular velocity and the boundary energy conjugate.

04Deliverable and boundary

Causal field equations, domain, weak-field matching, boundary conditions and ray calculations including source and detector models.

A first-order far-field expression is not a complete rotating strong field or Kerr counterpart. No nonspherical-shadow amplitude is presumed.

05Sources and collaboration

Final collected manuscript · Chapter 2 §五.4—5 (2.72)—(2.82)；小结第四项

Gravity · Tests

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STATUS & REVIEW
Theory StatusTheory · OPEN
Mathematical ReviewMathematical review · OPEN REVIEW
Overall Evidence StatusEvidence · UNTESTEDOpen research task; no new confirmed observational result.
Canonical ASDNo · Website guideResearch task from Chapter 2
Mathematical Review Note

Results are conditional on the stated scale, path and coupling inputs; mathematical reproduction and microscopic closure are distinct.

Read the formal review items
Module version 1.0.0Reviewed 2026-09-14ASD-BOOK-ddbade997b45
Adopted source, conditions and previous versions

§V.4—5 (2.72)—(2.82); summary item 4

Update open-problem progress with established results; retain problem IDs and unresolved tasks.

Conditions retained in this adoption

  • Fixed G₀; isolated, static, spherically symmetric, uncharged exterior.
  • Areal radius, actual spatial length and reference optical path are defined separately.
  • Weak-field perturbation, slow-rotation far field and boundary statistics each have separate conditions.

Specific unresolved source issues

  • Strong-field, microscopic and full apparatus closure remain task-specific research.

Source locators: p0831 · p0838 · p0840 · p0849 · p0856 · p0937

Manuscript SHA-256: ddbade997b45913248e588953b1d27ccb9c7c206c3c8d992d3aef2364a64836e

Download source identity manifest

The manifest identifies the source; it is not the complete manuscript. Adopting this source does not upgrade evidence or grant a new licence.

Previous source, retained as history

FND-3dde8d8e081d · Chapter 2 / 第二章 · §V.4–5, (2.72)–(2.82), pp. 20–22; limitations p. 25

PREF-125df9543371 · 序章.pdf
125df9543371f89656a2c29d552e1166e5edcc9ff5f6ec1ef1e438c7688d6ec5

CH1-46d43c8c39bf · 第一章(2).pdf
46d43c8c39bfd47f06ad699a6780e1142adc14bdd3520e2d5603d37a696cd9e9

CH2-f9598318d123 · 第二章-ASD理论-修订书稿 (2).pdf
f9598318d123c1f85a7d720af9cecbed8392e0b399910e3f9bfb675d49494709

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Revision history

CONTENT REVISION

Gravity v1.0 · Canonical content release

Editorial record
SOURCE & SCOPE

Chapter 2 · §V.4—5 (2.72)—(2.82); summary item 4

This research task is based on the cited final Chinese manuscript. It is not a new canonical result. Both languages use the same source snapshot; original equation numbers and website IDs remain distinct.

Knowledge scope
CITATION & SOURCE FILE

ASD: Complete spatial structure of rotating compact objects. OP-GRV-003-EN. Version 1.0.0; 2026-10-24. https://asd-physics.org/en/open-problems/gravity/rotating-background. §V.4—5 (2.72)—(2.82); summary item 4. Source ASD-BOOK-ddbade997b45.

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