Atlas VI Goal: Einstein–Maxwell certificate of the equal-spin ansatz ⇔ seven reduced radial equations

Atlas VI Goal: Einstein–Maxwell certificate of the equal-spin ansatz ⇔ seven reduced radial equations

For radial data B, D, A, H, W, Φ, Ψ : ℝ → ℝ and R : Set ℝ with ESData R B D A H W Φ Ψ (R open; the seven functions C^∞ on R; B, D, A, H ≠ 0 on R), if R is nonempty and 0 < A r, 0 < H r, 0 < B r · D r for every r ∈ R, then EMCertificateD (esMetric B D A H W) (esInverse B D A H W) (esFaraday Φ Ψ) (esDomain R) holds if and only if ReducedSystem B D A H W Φ Ψ R, i.e. the five Einstein equations redE_tt, redE_tσ, redE_rr, redE_uu, redE_σσ and the two Maxwell equations redJ_t, redJ_σ vanish at every r ∈ R. Here esMetric = −B dt² + D dr² + A[du²/(4u(1−u)) + u(1−u)(dφ−dψ)²] + H(σ − W dt)², σ = u dφ + (1−u) dψ, esInverse is its explicit frame-form inverse, esFaraday Φ Ψ = d(Φ dt + Ψ σ), and esDomain R = {x | x 1 ∈ R ∧ 0 < x 2 < 1}. Both directions are proved.

Scope. All radial data (B, D, A, H, W, Φ, Ψ : ℝ → ℝ) and every R : Set ℝ with ESData R B D A H W Φ Ψ (R open, the seven functions C^∞ on R, B, D, A, H ≠ 0 on R); domain esDomain R = {x | x 1 ∈ R ∧ 0 < x 2 ∧ x 2 < 1}. The Goal adds exactly R.Nonempty and ∀ r ∈ R, 0 < A r ∧ 0 < H r ∧ 0 < B r * D r.

中文摘要 等自旋五维拟设：在 ESData（R 开、七个径向函数在 R 上 C^∞、B,D,A,H 非零）且 R 非空、A>0、H>0、BD>0 的前提下，esDomain R 上的 Einstein–Maxwell 证书 EMCertificateD 成立当且仅当 R 上五个 Einstein 与两个 Maxwell 约化径向方程全部为零。不假设 B>0、D>0、连通性或仅外域。