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Off-Shell Noether Currents and Potentials for First-Order General Relativity

1
Departamento de Física, Cinvestav, Avenida Instituto Politécnico Nacional 2508, San Pedro Zacatenco, 07360 Gustavo A. Madero, Mexico
2
Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México, UNAM-Campus Morelia, Apartado Postal 61-3, Morelia, 58090 Michoacán, Mexico
*
Author to whom correspondence should be addressed.
Academic Editor: Ignatios Antoniadis
Symmetry 2021, 13(2), 348; https://doi.org/10.3390/sym13020348
Received: 6 February 2021 / Revised: 16 February 2021 / Accepted: 17 February 2021 / Published: 20 February 2021
(This article belongs to the Section Physics and Symmetry/Asymmetry)
We report off-shell Noether currents obtained from off-shell Noether potentials for first-order general relativity described by n-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 SO(n) or SO(n1,1) 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 not 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 SO(3,1) or SO(4) 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. View Full-Text
Keywords: Noether potentials; Noether currents; Noether charges; Palatini Lagrangian; Holst Lagrangian Noether potentials; Noether currents; Noether charges; Palatini Lagrangian; Holst Lagrangian
MDPI and ACS Style

Montesinos, M.; Gonzalez, D.; Romero, R.; Celada, M. Off-Shell Noether Currents and Potentials for First-Order General Relativity. Symmetry 2021, 13, 348. https://doi.org/10.3390/sym13020348

AMA Style

Montesinos M, Gonzalez D, Romero R, Celada M. Off-Shell Noether Currents and Potentials for First-Order General Relativity. Symmetry. 2021; 13(2):348. https://doi.org/10.3390/sym13020348

Chicago/Turabian Style

Montesinos, Merced, Diego Gonzalez, Rodrigo Romero, and Mariano Celada. 2021. "Off-Shell Noether Currents and Potentials for First-Order General Relativity" Symmetry 13, no. 2: 348. https://doi.org/10.3390/sym13020348

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