QiqtGrPPWave · section of the QIQT-H book

QIQTH.QiqtGrPPWave

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QiqtGrPPWave · entries 557–557 of 1000

Lemma 557 (qiqt_gr_ppwave). source ↗

QIQT→GR for the explicit pp-wave spacetime. The Einstein equations a·kgStress = G + Λg with g = ppMetric H (a curved pp-wave), every geometric premise discharged concretely (metric/inverse symmetry + g·gi=I, smoothness, the explicit tetrad’s congruence and invertibility). Carries — honestly, per plan §0 — the matter field + EOM hKG (the curved KG field is the documented Stage-4 frontier), the FQ/realization inputs (hbound/hcap/hK/hS/hA), and a covariantly-constant congruence W. Axiom-free.

({H})\in C^{\infty} \to \forall (\varphi : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}} \to \mathbb{R}) (m \eta \hbar a : \mathbb{R}), \hbar \ne 0 \to 0 < \hbar \to \eta \ne 0 \to a = 2 \cdot \pi / (\hbar \cdot \eta) \to ({\varphi})\in C^{\infty} \to (\forall (x : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}), \href{/browser/qiqth-kgstressconservation#d-qiqth-curvature-boxfield}{(\Box {\varphi})({x})} = {m}^{2} \cdot \varphi\,x) \to \forall (A : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}} \to (\mathrm{Fin}\,4 \to \mathbb{R}) \to \mathbb{R} \to \mathbb{R}) (\mathrm{sd} : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}} \to (\mathrm{Fin}\,4 \to \mathbb{R}) \to \mathbb{R}) (p : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}} \to (\mathrm{Fin}\,4 \to \mathbb{R}) \to \mathbb{R} \to \iota \to \mathbb{R}), (\forall (x : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (v : \mathrm{Fin}\,4 \to \mathbb{R}) (t : \mathbb{R}) (r : \iota), 0 \le p\,x\,v\,t\,r) \to (\forall (x : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (v : \mathrm{Fin}\,4 \to \mathbb{R}) (t : \mathbb{R}), \sum_{r} p\,x\,v\,t\,r = 1) \to (\forall (x : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (v : \mathrm{Fin}\,4 \to \mathbb{R}), p\,x\,v\,0 = \lambda x \mapsto {((\#\,\iota))}^{-1}) \to (\forall (x : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (v : \mathrm{Fin}\,4 \to \mathbb{R}), \eta \cdot A({x},{v},{0}) = \log\,(\#\,\iota)) \to \forall (W : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}} \to (\mathrm{Fin}\,4 \to \mathbb{R}) \to \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}} \to \mathrm{Fin}\,4 \to \mathbb{R}), (\forall (x : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (v : \mathrm{Fin}\,4 \to \mathbb{R}), ({\href{/browser/qiqth-ppwavemetric#d-qiqth-curvature-ppmetric}{g^{\mathrm{pp}}}\,H\,x})({v},{v}) = 0 \to W\,x\,v\,x = v) \to (\forall (x : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (v : \mathrm{Fin}\,4 \to \mathbb{R}) (\mu : \mathrm{Fin}\,4), ({\lambda y \mapsto W\,x\,v\,y\,\mu})\in C^{\infty}) \to (\forall (x : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (v : \mathrm{Fin}\,4 \to \mathbb{R}) (a b : \mathrm{Fin}\,4) (y : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}), \href{/browser/qiqth-curvature#d-qiqth-curvature-covderivvec}{(\nabla {W\,x\,v})_{{a}}{}^{{b}}({y})} = 0) \to (\forall (x : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (v : \mathrm{Fin}\,4 \to \mathbb{R}), ({\href{/browser/qiqth-ppwavemetric#d-qiqth-curvature-ppmetric}{g^{\mathrm{pp}}}\,H\,x})({v},{v}) = 0 \to ({\lambda t \mapsto \href{/browser/qiqth-branchledger#d-qiqth-branchledger-shannon}{S({p\,x\,v\,t})}})'({0})={\dot{S}({x},{v})}) \to (\forall (x : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (v : \mathrm{Fin}\,4 \to \mathbb{R}), ({\href{/browser/qiqth-ppwavemetric#d-qiqth-curvature-ppmetric}{g^{\mathrm{pp}}}\,H\,x})({v},{v}) = 0 \to ({\lambda t \mapsto \href{/browser/qiqth-branchledger#d-qiqth-branchledger-shannon}{S({p\,x\,v\,t})} + \href{/browser/qiqth-relentpositivity#d-qiqth-relentpositivity-kl}{D_{\mathrm{KL}}({p\,x\,v\,t}\,\|\,{p\,x\,v\,0})}})'({0})={2 \cdot \pi / \hbar \cdot ({\href{/browser/qiqth-kgstressconservation#d-qiqth-curvature-kgstress}{T({x})}})({v},{v})}) \to (\forall (x : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (v : \mathrm{Fin}\,4 \to \mathbb{R}), ({\href{/browser/qiqth-ppwavemetric#d-qiqth-curvature-ppmetric}{g^{\mathrm{pp}}}\,H\,x})({v},{v}) = 0 \to ({A\,x\,v})'({0})={-\sum_{\nu} W\,x\,v\,x\,\nu \cdot \partial_{{\nu}}({\lambda y \mapsto \href{/browser/qiqth-raychaudhuri#d-qiqth-curvature-expansion}{\theta({y})}})({x})}) \to (\forall (x : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (v : \mathrm{Fin}\,4 \to \mathbb{R}), ({\href{/browser/qiqth-ppwavemetric#d-qiqth-curvature-ppmetric}{g^{\mathrm{pp}}}\,H\,x})({v},{v}) = 0 \to \text{for }t\text{ near }0,\; \href{/browser/qiqth-branchledger#d-qiqth-branchledger-shannon}{S({p\,x\,v\,t})} \le \eta \cdot A({x},{v},{t})) \to \forall (\mathrm{mw} : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}} \to (\mathrm{Fin}\,4 \to \mathbb{R}) \to \mathbb{R}), (\forall (x : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (v : \mathrm{Fin}\,4 \to \mathbb{R}), 0 < \mathrm{mw}\,x\,v) \to \exists \Lambda, \forall (x : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (\mu \nu : \mathrm{Fin}\,4), a \cdot \href{/browser/qiqth-kgstressconservation#d-qiqth-curvature-kgstress}{T({x})\,\mu\,\nu} = \href{/browser/qiqth-curvature#d-qiqth-curvature-einsteintensor}{G_{{\mu}{\nu}}({x})} + \Lambda \cdot \href{/browser/qiqth-ppwavemetric#d-qiqth-curvature-ppmetric}{g^{\mathrm{pp}}}\,H\,x\,\mu\,\nu

Proof. By ppMetric_symm, ppMetric_inv, ppMetricInv_symm, ppMetric_contDiff, ppMetricInv_contDiff, ppFrame, ppFrameInv, ppFrame_cong, ppFrame_pp, ppFrame_pp', qiqt_gr_freefield_complete_covCong. $\square$

Used by qiqt_gr_ppwave_showcase.

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