QiqtGrThermo · section of the QIQT-H book

QIQTH.QiqtGrThermo

← all sections · ← QiqtGrShowcase · QiqtToGR →

QiqtGrThermo · entries 559–559 of 1000

Theorem 559 (qiqt_gr_freefield_thermo).  source ↗

The THERMODYNAMIC free-field QIQT→GR capstone. Einstein’s equations for the explicit free Klein–Gordon field, with the entropy/heat functionals CONSTRUCTED from a per-generator finite record law pp (a probability distribution for each deformation t, uniform at the equilibrium reference), and the saturation + relative-entropy premises (hsat/hDnn/hD0) DISCHARGED internally via clausius_package_from_finite_model (the axiom-free finite core: Gibbs/Jensen, uniform saturation, classical Klein). …

(∀(y:M4)(ab:Fin 4),gab(y)=gba(y))→(∀(y:M4)(ab:Fin 4),gab(y)=gba(y))→(∀(y:M4)(ab:Fin 4),∑σgaσ(y)⋅gσb(y)=δab)→(∀(ab:Fin 4),(λy↦gab(y))∈C∞)→(∀(ab:Fin 4),(λy↦gab(y))∈C∞)→∀(φ:M4→R)(mηℏa:R),ℏ≠0→η≠0→a=2⋅π/(ℏ⋅η)→(φ)∈C∞→(∀(x:M4),(□φ)(x)=m2⋅φ x)→∀(PPinv:M4→Fin 4→Fin 4→R),(∀(x:M4)(ij:Fin 4),∑kPik(x)⋅(P−1)kj(x)=δij)→(∀(x:M4)(ij:Fin 4),∑k(P−1)ik(x)⋅Pkj(x)=δij)→(∀(x:M4)(ij:Fin 4),gij(x)=∑k∑lPki(x)⋅ηkl⋅Plj(x))→∀(A:M4→(Fin 4→R)→R→R)(sd:M4→(Fin 4→R)→R)(p:M4→(Fin 4→R)→R→ι→R),(∀(x:M4)(v:Fin 4→R)(t:R)(r:ι),0≤p x v t r)→(∀(x:M4)(v:Fin 4→R)(t:R),∑rp x v t r=1)→(∀(x:M4)(v:Fin 4→R),p x v 0=λx↦((# ι))−1)→(∀(x:M4)(v:Fin 4→R),η⋅A(x,v,0)=log⁡ (# ι))→∀(W:M4→(Fin 4→R)→M4→Fin 4→R),(∀(x:M4)(v:Fin 4→R),(g x)(v,v)=0→W x v x=v)→(∀(x:M4)(v:Fin 4→R)(μ:Fin 4),(λy↦W x v y μ)∈C∞)→(∀(x:M4)(v:Fin 4→R)(y:M4)(μ:Fin 4),∑νW x v y ν⋅(∇W x v)νμ(y)=0)→(∀(x:M4)(v:Fin 4→R),(g x)(v,v)=0→∑μ∑ν(∇W x v)μν(x)⋅(∇W x v)νμ(x)=0)→(∀(x:M4)(v:Fin 4→R),(g x)(v,v)=0→(λt↦S(p x v t))′(0)=S˙(x,v))→(∀(x:M4)(v:Fin 4→R),(g x)(v,v)=0→(λt↦S(p x v t)+DKL(p x v t ∥ p x v 0))′(0)=2⋅π/ℏ⋅(T(x))(v,v))→(∀(x:M4)(v:Fin 4→R),(g x)(v,v)=0→(A x v)′(0)=−∑νW x v x ν⋅∂ν(λy↦θ(y))(x))→(∀(x:M4)(v:Fin 4→R),(g x)(v,v)=0→for t near 0,  S(p x v t)≤η⋅A(x,v,t))→∀(mw:M4→(Fin 4→R)→R),(∀(x:M4)(v:Fin 4→R),0<mw x v)→∀(ffff′:M4→(Fin 4→R)→R→C),(∀(x:M4)(v:Fin 4→R),MemLp (ff x v) 2 vol)→(∀(x:M4)(v:Fin 4→R),Integrable (ff x v) vol)→(∀(x:M4)(v:Fin 4→R)(θ:R),(ff x v)′(θ)=ff′ x v θ)→(∀(x:M4)(v:Fin 4→R),AEStronglyMeasurable (ff′ x v) vol)→∀(Bd:M4→(Fin 4→R)→R),(∀(x:M4)(v:Fin 4→R)(θ:R),∥ff′ x v θ∥≤Bd x v)→(∀(x:M4)(v:Fin 4→R),(g x)(v,v)=0→2⋅π/ℏ⋅⋯=⋯ .im)→∃Λ,∀(x:M4)(μν:Fin 4),a⋅T(x) μ ν=Gμν(x)+Λ⋅gμν(x)(\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 (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-einsteinequationofstate#d-qiqth-einsteineos-bl}{({g\,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}) (y : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (\mu : \mathrm{Fin}\,4), \sum_{\nu} W\,x\,v\,y\,\nu \cdot \href{/browser/qiqth-curvature#d-qiqth-curvature-covderivvec}{(\nabla {W\,x\,v})_{{\nu}}{}^{{\mu}}({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-einsteinequationofstate#d-qiqth-einsteineos-bl}{({g\,x})({v},{v})} = 0 \to \sum_{\mu} \sum_{\nu} \href{/browser/qiqth-curvature#d-qiqth-curvature-covderivvec}{(\nabla {W\,x\,v})_{{\mu}}{}^{{\nu}}({x})} \cdot \href{/browser/qiqth-curvature#d-qiqth-curvature-covderivvec}{(\nabla {W\,x\,v})_{{\nu}}{}^{{\mu}}({x})} = 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 ({\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-einsteinequationofstate#d-qiqth-einsteineos-bl}{({g\,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-einsteinequationofstate#d-qiqth-einsteineos-bl}{({g\,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-einsteinequationofstate#d-qiqth-einsteineos-bl}{({g\,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 \forall (\mathrm{ff} \mathrm{ff}^{\prime} : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}} \to (\mathrm{Fin}\,4 \to \mathbb{R}) \to \mathbb{R} \to \mathbb{C}), (\forall (x : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (v : \mathrm{Fin}\,4 \to \mathbb{R}), \mathrm{MemLp}\,(\mathrm{ff}\,x\,v)\,2\,\mathrm{vol}) \to (\forall (x : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (v : \mathrm{Fin}\,4 \to \mathbb{R}), \mathrm{Integrable}\,(\mathrm{ff}\,x\,v)\,\mathrm{vol}) \to (\forall (x : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (v : \mathrm{Fin}\,4 \to \mathbb{R}) (\theta : \mathbb{R}), ({\mathrm{ff}\,x\,v})'({\theta})={\mathrm{ff}^{\prime}\,x\,v\,\theta}) \to (\forall (x : \href{/browser/qiqth-curvature#d-qiqth-curvature-point}{M^{{4}}}) (v : \mathrm{Fin}\,4 \to \mathbb{R}), \mathrm{AEStronglyMeasurable}\,(\mathrm{ff}^{\prime}\,x\,v)\,\mathrm{vol}) \to \forall (\mathrm{Bd} : \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}) (\theta : \mathbb{R}), \|\mathrm{ff}^{\prime}\,x\,v\,\theta\| \le \mathrm{Bd}\,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 2 \cdot \pi / \hbar \cdot \cdots = \cdots .\mathrm{im}) \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 clausius_package_from_finite_model, qiqt_gr_freefield_geom. □\square

Used by qiqt_gr_freefield_complete.

← all sections · ← QiqtGrShowcase · QiqtToGR →