Einstein Equations from Information-Geometric Variational Principle: A Rigorous Derivation with Explicit Commutable Limit and Radon-Type Closure
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2025
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| _version_ | 1866901605343821824 |
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| author | Ma, Haobo Zhang, Wenlin |
| author_facet | Ma, Haobo Zhang, Wenlin |
| contents | <div> <div>\textbf{We derive Einstein's equations and the Hollands--Wald stability condition from a single information-geometric variational principle.}</div> <br> <div>Key features: Fixed-volume duality + explicit commutable limit (small diamond limit with directly invocable constant families); Radon-type closure: pushing family constraints down to pointwise equations (area identity $\Rightarrow$ pointwise); Second-order layer = Hollands--Wald canonical energy (relative entropy non-negativity and stability closed in a single variational chain).</div> <br> <div>On small-scale causal diamond $\mathcal{D}_\ell(p)$ at each manifold point, taking generalized entropy</div> <div>$$</div> <div>S_{\rm gen}= \frac{A(\text{waist surface})}{4G\hbar}+S_{\rm out}^{\rm ren}+S_{\rm ct}^{\rm UV}-\frac{\Lambda}{8\pi G}\frac{V(B_\ell)}{T}</div> <div>\qquad\big(T=\hbar|\kappa_\chi|/2\pi\big)</div> <div>$$</div> <div>as basic variational functional, we propose Information-Geometric Variational Principle (IGVP): first-order layer takes stationarity under fixed-volume constraint, second-order layer requires relative entropy non-negativity. This work provides four directly-invocable technical pillars: (i) Based on Raychaudhuri--Sachs--Gr\"onwall, \textbf{explicit commutable limit inequality} and \textbf{boundary layer estimate}, writing shear/twist control in geometric constants; (ii) Via \textbf{weighted ray transform} and \textbf{test function localization lemma}, realize ``family constraint $\Rightarrow$ pointwise'' \textbf{Radon-type closure}, then configure with ``null-cone characterization lemma'' and Bianchi identity to obtain tensorial closure; (iii) Under OS reflection positivity and KMS strip analyticity, establish sufficient condition and \textbf{lower bound} for Fisher--Rao metric via analytic continuation to be \textbf{real, non-degenerate, Lorentzian signature}, giving \textbf{operational criterion} for cross-component vanishing; (iv) In covariant phase space framework, provide \textbf{standard null boundary and corner term prescription}, proving symplectic flux no-outflow and Hamiltonian variation integrability, explicitly verified on Minkowski small diamond. From first-order layer obtain</div> <div>$$</div> <div>G_{ab}+\Lambda g_{ab}=8\pi G\,T_{ab},</div> <div>$$</div> <div>from second-order layer under JLMS and $\mathcal{F}_Q=\mathcal{E}_{\rm can}$ condition obtain non-negativity of Hollands--Wald canonical energy; in no-duality context use QNEC/ANEC as fallback. This work also elucidates rescaling and orientation-invariance of $\delta Q/T$, $\delta A/(4G\hbar)$, and shows that $V/T$ scales with rescaling; at first-order extremum layer adopting fixed temperature scale ($\delta T=0$) avoids its gauge dependence.</div> <div> </div> </div> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17706884 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
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| spellingShingle | Einstein Equations from Information-Geometric Variational Principle: A Rigorous Derivation with Explicit Commutable Limit and Radon-Type Closure Ma, Haobo Zhang, Wenlin <div> <div>\textbf{We derive Einstein's equations and the Hollands--Wald stability condition from a single information-geometric variational principle.}</div> <br> <div>Key features: Fixed-volume duality + explicit commutable limit (small diamond limit with directly invocable constant families); Radon-type closure: pushing family constraints down to pointwise equations (area identity $\Rightarrow$ pointwise); Second-order layer = Hollands--Wald canonical energy (relative entropy non-negativity and stability closed in a single variational chain).</div> <br> <div>On small-scale causal diamond $\mathcal{D}_\ell(p)$ at each manifold point, taking generalized entropy</div> <div>$$</div> <div>S_{\rm gen}= \frac{A(\text{waist surface})}{4G\hbar}+S_{\rm out}^{\rm ren}+S_{\rm ct}^{\rm UV}-\frac{\Lambda}{8\pi G}\frac{V(B_\ell)}{T}</div> <div>\qquad\big(T=\hbar|\kappa_\chi|/2\pi\big)</div> <div>$$</div> <div>as basic variational functional, we propose Information-Geometric Variational Principle (IGVP): first-order layer takes stationarity under fixed-volume constraint, second-order layer requires relative entropy non-negativity. This work provides four directly-invocable technical pillars: (i) Based on Raychaudhuri--Sachs--Gr\"onwall, \textbf{explicit commutable limit inequality} and \textbf{boundary layer estimate}, writing shear/twist control in geometric constants; (ii) Via \textbf{weighted ray transform} and \textbf{test function localization lemma}, realize ``family constraint $\Rightarrow$ pointwise'' \textbf{Radon-type closure}, then configure with ``null-cone characterization lemma'' and Bianchi identity to obtain tensorial closure; (iii) Under OS reflection positivity and KMS strip analyticity, establish sufficient condition and \textbf{lower bound} for Fisher--Rao metric via analytic continuation to be \textbf{real, non-degenerate, Lorentzian signature}, giving \textbf{operational criterion} for cross-component vanishing; (iv) In covariant phase space framework, provide \textbf{standard null boundary and corner term prescription}, proving symplectic flux no-outflow and Hamiltonian variation integrability, explicitly verified on Minkowski small diamond. From first-order layer obtain</div> <div>$$</div> <div>G_{ab}+\Lambda g_{ab}=8\pi G\,T_{ab},</div> <div>$$</div> <div>from second-order layer under JLMS and $\mathcal{F}_Q=\mathcal{E}_{\rm can}$ condition obtain non-negativity of Hollands--Wald canonical energy; in no-duality context use QNEC/ANEC as fallback. This work also elucidates rescaling and orientation-invariance of $\delta Q/T$, $\delta A/(4G\hbar)$, and shows that $V/T$ scales with rescaling; at first-order extremum layer adopting fixed temperature scale ($\delta T=0$) avoids its gauge dependence.</div> <div> </div> </div> |
| title | Einstein Equations from Information-Geometric Variational Principle: A Rigorous Derivation with Explicit Commutable Limit and Radon-Type Closure |
| url | https://doi.org/10.5281/zenodo.17706884 |