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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2402.17429 |
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| _version_ | 1866912125543251968 |
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| author | Peng, Jun-Jin Li, Hua |
| author_facet | Peng, Jun-Jin Li, Hua |
| contents | In this paper, we aim to perform a systematical investigation on the field equations and Noether potentials for the higher-order gravity theories endowed with Lagrangians depending on the metric and the Riemann curvature tensor, together with $i$th ($i=1,2,\cdot\cdot\cdot$) powers of the Beltrami-d'Alembertian operator $\Box$ acting on the latter. We start with a detailed derivation of the field equations and the Noether potential corresponding to the Lagrangian $\sqrt{-g}L_R(R,\Box R,\cdot\cdot\cdot,\Box^m R)$ through the direct variation of the Lagrangian and a method based upon the conserved current. Next the parallel analysis is extended to a more generic Lagrangian $\sqrt{-g}L_{\text{Ric}}(g^{μν}, R_{μν},\Box R_{μν}, \cdot\cdot\cdot,\Box^m R_{μν})$, as well as to the generalization of the Lagrangian $\sqrt{-g}L_{\text{Ric}}$, which depends on the metric $g^{μν}$, the Riemann tensor $R_{μνρσ}$ and $\Box^i R_{μνρσ}$s. Finally, all the results associated to the three types of Lagrangians are extended to the Lagrangian relying on an arbitrary tensor and the variables via $\Box^i$ acting on such a tensor. In particular, we take into consideration of equations of motion and Noether potentials for nonlocal gravity models. For Lagrangians involving the variables $\Box^i R$, $\Box^i R_{μν}$ and $\Box^i R_{μνρσ}$, our investigation provides their concrete Noether potentials and the field equations without the derivative of the Lagrangian density with respect to the metric. Besides, the Iyer-Wald potentials associated to these Lagrangians are also presented. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_17429 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Field equations and Noether potentials for higher-order theories of gravity with Lagrangians involving $\Box^i R$, $\Box^i R_{μν}$ and $\Box^i R_{μνρσ}$ Peng, Jun-Jin Li, Hua General Relativity and Quantum Cosmology High Energy Physics - Theory In this paper, we aim to perform a systematical investigation on the field equations and Noether potentials for the higher-order gravity theories endowed with Lagrangians depending on the metric and the Riemann curvature tensor, together with $i$th ($i=1,2,\cdot\cdot\cdot$) powers of the Beltrami-d'Alembertian operator $\Box$ acting on the latter. We start with a detailed derivation of the field equations and the Noether potential corresponding to the Lagrangian $\sqrt{-g}L_R(R,\Box R,\cdot\cdot\cdot,\Box^m R)$ through the direct variation of the Lagrangian and a method based upon the conserved current. Next the parallel analysis is extended to a more generic Lagrangian $\sqrt{-g}L_{\text{Ric}}(g^{μν}, R_{μν},\Box R_{μν}, \cdot\cdot\cdot,\Box^m R_{μν})$, as well as to the generalization of the Lagrangian $\sqrt{-g}L_{\text{Ric}}$, which depends on the metric $g^{μν}$, the Riemann tensor $R_{μνρσ}$ and $\Box^i R_{μνρσ}$s. Finally, all the results associated to the three types of Lagrangians are extended to the Lagrangian relying on an arbitrary tensor and the variables via $\Box^i$ acting on such a tensor. In particular, we take into consideration of equations of motion and Noether potentials for nonlocal gravity models. For Lagrangians involving the variables $\Box^i R$, $\Box^i R_{μν}$ and $\Box^i R_{μνρσ}$, our investigation provides their concrete Noether potentials and the field equations without the derivative of the Lagrangian density with respect to the metric. Besides, the Iyer-Wald potentials associated to these Lagrangians are also presented. |
| title | Field equations and Noether potentials for higher-order theories of gravity with Lagrangians involving $\Box^i R$, $\Box^i R_{μν}$ and $\Box^i R_{μνρσ}$ |
| topic | General Relativity and Quantum Cosmology High Energy Physics - Theory |
| url | https://arxiv.org/abs/2402.17429 |