The Noether Symmetry Approach: Foundation and Applications: The Case of Scalar-Tensor Gauss–Bonnet Gravity
We sketch the main features of the Noether Symmetry Approach, a method to reduce and solve dynamics of physical systems by selecting Noether symmetries, which correspond to conserved quantities. Specifically, we take into account the vanishing Lie derivative condition for general canonical Lagrangia...
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MDPI AG
2023-08-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/15/9/1625 |
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| author | Francesco Bajardi Salvatore Capozziello Tiziana Di Salvo Francesca Spinnato |
| author_facet | Francesco Bajardi Salvatore Capozziello Tiziana Di Salvo Francesca Spinnato |
| author_sort | Francesco Bajardi |
| collection | DOAJ |
| description | We sketch the main features of the Noether Symmetry Approach, a method to reduce and solve dynamics of physical systems by selecting Noether symmetries, which correspond to conserved quantities. Specifically, we take into account the vanishing Lie derivative condition for general canonical Lagrangians to select symmetries. Furthermore, we extend the prescription to the first prolongation of the Noether vector. It is possible to show that the latter application provides a general constraint on the infinitesimal generator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ξ</mi></semantics></math></inline-formula>, related to the spacetime translations. This approach can be used for several applications. In the second part of the work, we consider a gravity theory, including the coupling between a scalar field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϕ</mi></semantics></math></inline-formula> and the Gauss–Bonnet topological term <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">G</mi></semantics></math></inline-formula>. In particular, we study a gravitational action containing the function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>(</mo><mi mathvariant="script">G</mi><mo>,</mo><mi>ϕ</mi><mo>)</mo></mrow></semantics></math></inline-formula> and select viable models by the existence of symmetries. Finally, we evaluate the selected models in a spatially flat cosmological background and use symmetries to find exact solutions. |
| first_indexed | 2024-03-10T21:54:54Z |
| format | Article |
| id | doaj.art-ce3f394a4dde4f1a93594e4f7a48312c |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T21:54:54Z |
| publishDate | 2023-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-ce3f394a4dde4f1a93594e4f7a48312c2023-11-19T13:10:23ZengMDPI AGSymmetry2073-89942023-08-01159162510.3390/sym15091625The Noether Symmetry Approach: Foundation and Applications: The Case of Scalar-Tensor Gauss–Bonnet GravityFrancesco Bajardi0Salvatore Capozziello1Tiziana Di Salvo2Francesca Spinnato3Scuola Superiore Meridionale, Largo San Marcellino 10, 80138 Napoli, ItalyScuola Superiore Meridionale, Largo San Marcellino 10, 80138 Napoli, ItalyDipartimento di Fisica e Chimica “Emilio Segré”, Università degli Studi di Palermo, 90123 Palermo, ItalyDipartimento di Fisica e Chimica “Emilio Segré”, Università degli Studi di Palermo, 90123 Palermo, ItalyWe sketch the main features of the Noether Symmetry Approach, a method to reduce and solve dynamics of physical systems by selecting Noether symmetries, which correspond to conserved quantities. Specifically, we take into account the vanishing Lie derivative condition for general canonical Lagrangians to select symmetries. Furthermore, we extend the prescription to the first prolongation of the Noether vector. It is possible to show that the latter application provides a general constraint on the infinitesimal generator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ξ</mi></semantics></math></inline-formula>, related to the spacetime translations. This approach can be used for several applications. In the second part of the work, we consider a gravity theory, including the coupling between a scalar field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϕ</mi></semantics></math></inline-formula> and the Gauss–Bonnet topological term <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">G</mi></semantics></math></inline-formula>. In particular, we study a gravitational action containing the function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>(</mo><mi mathvariant="script">G</mi><mo>,</mo><mi>ϕ</mi><mo>)</mo></mrow></semantics></math></inline-formula> and select viable models by the existence of symmetries. Finally, we evaluate the selected models in a spatially flat cosmological background and use symmetries to find exact solutions.https://www.mdpi.com/2073-8994/15/9/1625Noether symmetriesmodified theories of gravityexact solutions |
| spellingShingle | Francesco Bajardi Salvatore Capozziello Tiziana Di Salvo Francesca Spinnato The Noether Symmetry Approach: Foundation and Applications: The Case of Scalar-Tensor Gauss–Bonnet Gravity Symmetry Noether symmetries modified theories of gravity exact solutions |
| title | The Noether Symmetry Approach: Foundation and Applications: The Case of Scalar-Tensor Gauss–Bonnet Gravity |
| title_full | The Noether Symmetry Approach: Foundation and Applications: The Case of Scalar-Tensor Gauss–Bonnet Gravity |
| title_fullStr | The Noether Symmetry Approach: Foundation and Applications: The Case of Scalar-Tensor Gauss–Bonnet Gravity |
| title_full_unstemmed | The Noether Symmetry Approach: Foundation and Applications: The Case of Scalar-Tensor Gauss–Bonnet Gravity |
| title_short | The Noether Symmetry Approach: Foundation and Applications: The Case of Scalar-Tensor Gauss–Bonnet Gravity |
| title_sort | noether symmetry approach foundation and applications the case of scalar tensor gauss bonnet gravity |
| topic | Noether symmetries modified theories of gravity exact solutions |
| url | https://www.mdpi.com/2073-8994/15/9/1625 |
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