gr-qc / 12-LEVEL MATRIX FORMALISM

Semiclassical gravity as a transition system

CONDITIONAL CLOSURE · NOT AN EXPERIMENTAL CLAIMRe⁡λmax→0+⟺∫0t∗∥ω∥∞dt=∞⟺⟨0∣T^00∣0⟩ren<0\operatorname{Re}\lambda_{max}\to0^+\Longleftrightarrow\int_0^{t_*}\|\omega\|_\infty dt=\infty\Longleftrightarrow\langle0|\hat T_{00}|0\rangle_{ren}<0
FIELD / 01λ.41

General relativity

Metrics, causality, Riemann and Einstein tensors.

Rμν−½Rgμν
FIELD / 02λ.178

Semiclassical gravity

Backreaction of the renormalized quantum source.

Gμν=8πG⟨T̂μν⟩
FIELD / 03λ.315

QFT in curved spacetime

Vacua, effective action, heat kernels and trace anomaly.

Γeff[φ,g]
FIELD / 04λ.452

QEI

Negative-energy bounds and the χQEI ≤ 1 filter.

χQEI≤1
FIELD / 05λ.589

Wormholes

Throats, flare-out, horizons and quasi-black-hole regimes.

b(r₀)=r₀
FIELD / 06λ.726

Numerical relativity

Tensor pipeline, residuals and MATLAB convergence.

‖Eh‖→0
FIELD / 07λ.863

Spacetime topology

Blow-up, critical surfaces and invariants.

BlZ(M)
FIELD / 08λ.1000

Einstein–Maxwell

Gravitational-electromagnetic sources and fields.

Gμν=8πTμνEM

MATRIX / U1—U12

Formalism

U01Λtop=lim⁡x→0JTFJ\Lambda_{top}=\lim_{x\to0}J^TFJ
U02X=RFT/2hνNaRT[I+Sln⁡(T/T0)]X=R F^{T/2}\frac{h\nu N_a}{RT}[I+S\ln(T/T_0)]
U03Y˙=Iϵe2St−γY2\dot Y=I\epsilon e^{2St}-\gamma Y^2
U04ρexo=−∥Λtop∥F28πc4\rho_{exo}=-\frac{\|\Lambda_{top}\|_F^2}{8\pi c^4}
U05Gμν+Λgμν=8πGTμνG_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi GT_{\mu\nu}
U06∫0t∗∥ω∥∞dt=∞\int_0^{t_*}\|\omega\|_\infty dt=\infty
U07M=ΛtopXYTM=\Lambda_{top}XY^T
U08det⁡(M−λI)=0\det(M-\lambda I)=0
U09Lϕ=12(∂ϕ)2−V(ϕ)\mathcal L_\phi=\frac12(\partial\phi)^2-V(\phi)
U10Veff=V+ℏ64π2STr⁡[M4ln⁡(M2/μ2)]V_{eff}=V+\frac{\hbar}{64\pi^2}\operatorname{STr}[M^4\ln(M^2/\mu^2)]
U11H2=8πG3ρ−kc2a2H^2=\frac{8\pi G}{3}\rho-\frac{kc^2}{a^2}
U12Gμν=8πG⟨T^μν⟩renG_{\mu\nu}=8\pi G\langle\hat T_{\mu\nu}\rangle_{ren}