Off-Shell Noether Currents and Potentials for First-Order General Relativity
We report off-shell Noether currents obtained from off-shell Noether potentials for first-order general relativity described by <i>n</i>-dimensional Palatini and Holst Lagrangians including the cosmological constant. These off-shell currents and potentials are achieved by using the corre...
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MDPI AG
2021-02-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/13/2/348 |
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| author | Merced Montesinos Diego Gonzalez Rodrigo Romero Mariano Celada |
| author_facet | Merced Montesinos Diego Gonzalez Rodrigo Romero Mariano Celada |
| author_sort | Merced Montesinos |
| collection | DOAJ |
| description | We report off-shell Noether currents obtained from off-shell Noether potentials for first-order general relativity described by <i>n</i>-dimensional Palatini and Holst Lagrangians including the cosmological constant. These off-shell currents and potentials are achieved by using the corresponding Lagrangian and the off-shell Noether identities satisfied by diffeomorphisms generated by arbitrary vector fields, local <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>O</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></semantics></math></inline-formula> or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>O</mi><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula> transformations, ‘improved diffeomorphisms’, and the ‘generalization of local translations’ of the orthonormal frame and the connection. A remarkable aspect of our approach is that we do <i>not</i> use Noether’s theorem in its direct form. By construction, the currents are off-shell conserved and lead naturally to the definition of off-shell Noether charges. We also study what we call the ‘half off-shell’ case for both Palatini and Holst Lagrangians. In particular, we find that the resulting diffeomorphism and local <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>O</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula> or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>O</mi><mo>(</mo><mn>4</mn><mo>)</mo></mrow></semantics></math></inline-formula> off-shell Noether currents and potentials for the Holst Lagrangian generically depend on the Immirzi parameter, which holds even in the ‘half off-shell’ and on-shell cases. We also study Killing vector fields in the ‘half off-shell’ and on-shell cases. The current theoretical framework is illustrated for the ‘half off-shell’ case in static spherically symmetric and Friedmann–Lemaitre–Robertson–Walker spacetimes in four dimensions. |
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| spelling | doaj.art-d380d2f6eb284875ad4188fddb628ce82023-12-11T17:46:47ZengMDPI AGSymmetry2073-89942021-02-0113234810.3390/sym13020348Off-Shell Noether Currents and Potentials for First-Order General RelativityMerced Montesinos0Diego Gonzalez1Rodrigo Romero2Mariano Celada3Departamento de Física, Cinvestav, Avenida Instituto Politécnico Nacional 2508, San Pedro Zacatenco, 07360 Gustavo A. Madero, MexicoDepartamento de Física, Cinvestav, Avenida Instituto Politécnico Nacional 2508, San Pedro Zacatenco, 07360 Gustavo A. Madero, MexicoDepartamento de Física, Cinvestav, Avenida Instituto Politécnico Nacional 2508, San Pedro Zacatenco, 07360 Gustavo A. Madero, MexicoCentro de Ciencias Matemáticas, Universidad Nacional Autónoma de México, UNAM-Campus Morelia, Apartado Postal 61-3, Morelia, 58090 Michoacán, MexicoWe report off-shell Noether currents obtained from off-shell Noether potentials for first-order general relativity described by <i>n</i>-dimensional Palatini and Holst Lagrangians including the cosmological constant. These off-shell currents and potentials are achieved by using the corresponding Lagrangian and the off-shell Noether identities satisfied by diffeomorphisms generated by arbitrary vector fields, local <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>O</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></semantics></math></inline-formula> or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>O</mi><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula> transformations, ‘improved diffeomorphisms’, and the ‘generalization of local translations’ of the orthonormal frame and the connection. A remarkable aspect of our approach is that we do <i>not</i> use Noether’s theorem in its direct form. By construction, the currents are off-shell conserved and lead naturally to the definition of off-shell Noether charges. We also study what we call the ‘half off-shell’ case for both Palatini and Holst Lagrangians. In particular, we find that the resulting diffeomorphism and local <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>O</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula> or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>O</mi><mo>(</mo><mn>4</mn><mo>)</mo></mrow></semantics></math></inline-formula> off-shell Noether currents and potentials for the Holst Lagrangian generically depend on the Immirzi parameter, which holds even in the ‘half off-shell’ and on-shell cases. We also study Killing vector fields in the ‘half off-shell’ and on-shell cases. The current theoretical framework is illustrated for the ‘half off-shell’ case in static spherically symmetric and Friedmann–Lemaitre–Robertson–Walker spacetimes in four dimensions.https://www.mdpi.com/2073-8994/13/2/348Noether potentialsNoether currentsNoether chargesPalatini LagrangianHolst Lagrangian |
| spellingShingle | Merced Montesinos Diego Gonzalez Rodrigo Romero Mariano Celada Off-Shell Noether Currents and Potentials for First-Order General Relativity Symmetry Noether potentials Noether currents Noether charges Palatini Lagrangian Holst Lagrangian |
| title | Off-Shell Noether Currents and Potentials for First-Order General Relativity |
| title_full | Off-Shell Noether Currents and Potentials for First-Order General Relativity |
| title_fullStr | Off-Shell Noether Currents and Potentials for First-Order General Relativity |
| title_full_unstemmed | Off-Shell Noether Currents and Potentials for First-Order General Relativity |
| title_short | Off-Shell Noether Currents and Potentials for First-Order General Relativity |
| title_sort | off shell noether currents and potentials for first order general relativity |
| topic | Noether potentials Noether currents Noether charges Palatini Lagrangian Holst Lagrangian |
| url | https://www.mdpi.com/2073-8994/13/2/348 |
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