Abstract
We report off-shell Noether currents obtained from off-shell Noether potentials for firstorder 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(n − 1, 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.
Original language | English |
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Article number | 348 |
Pages (from-to) | 1-38 |
Number of pages | 38 |
Journal | Symmetry |
Volume | 13 |
Issue number | 2 |
DOIs | |
State | Published - Feb 2021 |
Externally published | Yes |
Keywords
- Holst Lagrangian
- Noether charges
- Noether currents
- Noether potentials
- Palatini Lagrangian