$Γ$-convergence of a diffeomorphism-natural MDL functional to Einstein-Hilbert with Gibbons-Hawking-York boundary term
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909901078396928 |
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| author | Lela, Marko |
| author_facet | Lela, Marko |
| contents | We prove a \(Γ\)-convergence result for a diffeomorphism-natural discrete MDL-type functional to the Einstein-Hilbert action with the Gibbons-Hawking-York boundary term. On boundary-fitted, shape-regular meshes we establish interior and boundary blow-ups, identify the Carathéodory densities \(f_{\mathrm{in}}=α_0+α_1 R\) and \(f_{\mathrm{bdry}}=β_1 K\), and obtain the \(\liminf/\limsup\) bounds via a recovery sequence based on reflected Fermi smoothing. A boundary first-layer asymptotics shows that boundary cells contribute at order \(h^{d-1}\), yielding a global \(O(h)\) boundary remainder, while the interior remainder is \(O(h^2)\). The paper is foundational; Appendix~E specifies a reproducible protocol for rate checks and calibration of \(α_0,α_1,β_1\). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_10867 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $Γ$-convergence of a diffeomorphism-natural MDL functional to Einstein-Hilbert with Gibbons-Hawking-York boundary term Lela, Marko Mathematical Physics General Relativity and Quantum Cosmology Analysis of PDEs Differential Geometry Primary 49J45, Secondary 53C80, 53C21, 83C99 We prove a \(Γ\)-convergence result for a diffeomorphism-natural discrete MDL-type functional to the Einstein-Hilbert action with the Gibbons-Hawking-York boundary term. On boundary-fitted, shape-regular meshes we establish interior and boundary blow-ups, identify the Carathéodory densities \(f_{\mathrm{in}}=α_0+α_1 R\) and \(f_{\mathrm{bdry}}=β_1 K\), and obtain the \(\liminf/\limsup\) bounds via a recovery sequence based on reflected Fermi smoothing. A boundary first-layer asymptotics shows that boundary cells contribute at order \(h^{d-1}\), yielding a global \(O(h)\) boundary remainder, while the interior remainder is \(O(h^2)\). The paper is foundational; Appendix~E specifies a reproducible protocol for rate checks and calibration of \(α_0,α_1,β_1\). |
| title | $Γ$-convergence of a diffeomorphism-natural MDL functional to Einstein-Hilbert with Gibbons-Hawking-York boundary term |
| topic | Mathematical Physics General Relativity and Quantum Cosmology Analysis of PDEs Differential Geometry Primary 49J45, Secondary 53C80, 53C21, 83C99 |
| url | https://arxiv.org/abs/2511.10867 |