QiqtGrExplicitKG · section of the QIQT-H book

QIQTH.QiqtGrExplicitKG

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QiqtGrExplicitKG · entries 548–548 of 1000

Theorem 548 (qiqt_gr_explicit_kg). source ↗

THE QIQT→GR EINSTEIN EQUATIONS FOR THE EXPLICIT FREE KLEIN–GORDON FIELD, axiom-free. Specialising the abstract qiqt_gr_from_wedge_kms_complete to T = kgStress (the concrete KG stress tensor): the matter-conservation input conserv is discharged INTERNALLY (kg_conserv_of_contDiff, from ContDiff smoothness of φ, g, gi + the equation of motion □φ = m²φ), and the stress-tensor symmetry is proved from metric symmetry. …

(\forall (y : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (a b : \mathrm{Fin}\,4), g_{{a}{b}}({y}) = g_{{b}{a}}({y})) \to (\forall (y : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (a b : \mathrm{Fin}\,4), g^{{a}{b}}({y}) = g^{{b}{a}}({y})) \to (\forall (y : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (a b : \mathrm{Fin}\,4), \sum_{\sigma} g_{{a}{\sigma}}({y}) \cdot g^{{\sigma}{b}}({y}) = \delta_{ab}) \to (\forall (a b : \mathrm{Fin}\,4), ({\lambda y \mapsto g_{{a}{b}}({y})})\in C^{\infty}) \to (\forall (a b : \mathrm{Fin}\,4), ({\lambda y \mapsto g^{{a}{b}}({y})})\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 \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 (P \mathrm{Pinv} : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}} \to \mathrm{Fin}\,4 \to \mathrm{Fin}\,4 \to \mathbb{R}), (\forall (x : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (i j : \mathrm{Fin}\,4), \sum_{k} P_{{i}{k}}({x}) \cdot (P^{-1})_{{k}{j}}({x}) = \delta_{ij}) \to (\forall (x : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (i j : \mathrm{Fin}\,4), \sum_{k} (P^{-1})_{{i}{k}}({x}) \cdot P_{{k}{j}}({x}) = \delta_{ij}) \to (\forall (x : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (i j : \mathrm{Fin}\,4), g_{{i}{j}}({x}) = \sum_{k} \sum_{l} P_{{k}{i}}({x}) \cdot \href{/browser/qiqth-einsteinequationofstate#d-qiqth-einsteineos-gm}{\eta_{{k}{l}}} \cdot P_{{l}{j}}({x})) \to \forall (\mathrm{Sf} \mathrm{KE} 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} \mathrm{kd} \mathrm{ad} : \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}), \href{/browser/qiqth-einsteinequationofstate#d-qiqth-einsteineos-bl}{({g\,x})({v},{v})} = 0 \to ({\mathrm{Sf}\,x\,v})'({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-einsteinequationofstate#d-qiqth-einsteineos-bl}{({g\,x})({v},{v})} = 0 \to ({\mathrm{KE}\,x\,v})'({0})={\dot{K}({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-einsteinequationofstate#d-qiqth-einsteineos-bl}{({g\,x})({v},{v})} = 0 \to ({A\,x\,v})'({0})={\dot{A}({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-einsteinequationofstate#d-qiqth-einsteineos-bl}{({g\,x})({v},{v})} = 0 \to \text{for }t\text{ near }0,\; \mathrm{Sf}\,x\,v\,t \le \eta \cdot A({x},{v},{t})) \to (\forall (x : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (v : \mathrm{Fin}\,4 \to \mathbb{R}), \href{/browser/qiqth-einsteinequationofstate#d-qiqth-einsteineos-bl}{({g\,x})({v},{v})} = 0 \to \mathrm{Sf}\,x\,v\,0 = \eta \cdot A({x},{v},{0})) \to (\forall (x : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (v : \mathrm{Fin}\,4 \to \mathbb{R}), \href{/browser/qiqth-einsteinequationofstate#d-qiqth-einsteineos-bl}{({g\,x})({v},{v})} = 0 \to \forall (t : \mathbb{R}), 0 \le \mathrm{KE}({x},{v},{t}) - \mathrm{Sf}\,x\,v\,t) \to (\forall (x : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (v : \mathrm{Fin}\,4 \to \mathbb{R}), \href{/browser/qiqth-einsteinequationofstate#d-qiqth-einsteineos-bl}{({g\,x})({v},{v})} = 0 \to \mathrm{KE}({x},{v},{0}) - \mathrm{Sf}\,x\,v\,0 = 0) \to \href{/browser/qiqth-wedgekmstogr#d-qiqth-wedgekmstogr-wedgekmsflux-complete}{\mathrm{WedgeKMSFlux\_complete}}\,g\,(\href{/browser/qiqth-kgstressconservation#d-qiqth-curvature-kgstress}{\mathrm{kgStress}}\,m\,\varphi\,g\,\mathrm{gi})\,\mathrm{kd}\,\hbar \to (\forall (x : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (v : \mathrm{Fin}\,4 \to \mathbb{R}), \href{/browser/qiqth-einsteinequationofstate#d-qiqth-einsteineos-bl}{({g\,x})({v},{v})} = 0 \to \dot{A}({x},{v}) = ({\lambda i j \mapsto \href{/browser/qiqth-curvature#d-qiqth-curvature-ricci}{R_{{i}{j}}({x})}})({v},{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 g_{{\mu}{\nu}}({x})

Proof. By pd, PdiffAt, scalarCurv, hreg_kg, kgLagr, kg_conserv_of_contDiff, qiqt_gr_from_wedge_kms_complete. $\square$

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