Metric-affine cosmological models and the inverse problem of the calculus of variations. Part 1: variational bootstrapping -- the method
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866911916596658176 |
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| author | Ducobu, Ludovic Voicu, Nicoleta |
| author_facet | Ducobu, Ludovic Voicu, Nicoleta |
| contents | The method of variational completion allows one to transform an (in principle, arbitrary) system of partial differential equations -- based on an intuitive ``educated guess'' -- into the Euler-Lagrange one attached to a Lagrangian, by adding a canonical correction term. Here, we extend this technique to theories that involve at least two sets of dynamical variables : we show that an educated guess of the field equations with respect to one of these sets of variables only is sufficient to variationally complete these equations and recover a Lagrangian for the full theory, up to terms that do not involve the respective variables.
Applying this idea to natural metric-affine theories of gravity, we prove that, starting from an educated guess of the metric equations only, one can find the full metric equations, together with a generally covariant Lagrangian for the theory, up to metric-independent terms; the latter terms (which can only involve the distortion of the connection) are then completely classified over 4-dimensional spacetimes, by techniques pertaining to differential invariants. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_15564 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Metric-affine cosmological models and the inverse problem of the calculus of variations. Part 1: variational bootstrapping -- the method Ducobu, Ludovic Voicu, Nicoleta Mathematical Physics General Relativity and Quantum Cosmology Differential Geometry The method of variational completion allows one to transform an (in principle, arbitrary) system of partial differential equations -- based on an intuitive ``educated guess'' -- into the Euler-Lagrange one attached to a Lagrangian, by adding a canonical correction term. Here, we extend this technique to theories that involve at least two sets of dynamical variables : we show that an educated guess of the field equations with respect to one of these sets of variables only is sufficient to variationally complete these equations and recover a Lagrangian for the full theory, up to terms that do not involve the respective variables. Applying this idea to natural metric-affine theories of gravity, we prove that, starting from an educated guess of the metric equations only, one can find the full metric equations, together with a generally covariant Lagrangian for the theory, up to metric-independent terms; the latter terms (which can only involve the distortion of the connection) are then completely classified over 4-dimensional spacetimes, by techniques pertaining to differential invariants. |
| title | Metric-affine cosmological models and the inverse problem of the calculus of variations. Part 1: variational bootstrapping -- the method |
| topic | Mathematical Physics General Relativity and Quantum Cosmology Differential Geometry |
| url | https://arxiv.org/abs/2403.15564 |