ИСТИНА

Conserved currents and related superpotentials for perturbations on arbitrary backgrounds in the Lovelock theory are constructed. We use the Lagrangian based field-theoretical methods where perturbations are considered as dynamical fields propagating on a given background. Such a formulation is exact (not approximate) and equivalent to the theory in the original metric form, for a more detail see [1]. From the very start, using Noether theorem, we derive the Noether-Klein identities and adopt them for the purposes of the current work. Applying these identities in the framework of Lovelock theory, we construct conserved currents, energy-momentum tensors out of them, and related superpotentials with arbitrary displacement vectors, not restricting to Killing vectors. A comparison with the well known Abbott-Deser-Tekin approach is given. The developed general formalism is applied to give conserved quantities for perturbations on anti-de Sitter (AdS) backgrounds. As a test we calculate mass of the Schwarzschild-AdS black hole in the Lovelock theory in arbitrary D dimensions. This part of the results can be found in [2]. Formalism [2] adopted to the case of a pure Lovelock gravity (with only one polynomial in Riemannian tensor in the Lagrangian) is used for constructing conserved quantities for static [3] and dynamic of the Vaidya type [4] black holes with AdS, dS and flat asymptotics. Global energy as well as quasi-local energy and fluxes of these quantities are calculated on AdS, dS and flat backgrounds in correspondence with asymptotics of black holes. In the case of the dynamic black holes, geometries of related static black holes are considered as backgrounds as well. For the dynamic black holes energy densities and energy flux densities are constructed in the frame of a freely falling observer on backgrounds of the related static black holes. The results are new and clarify new properties of the solutions under consideration, e.g., they allow us to correct parameters of the null liquid for the Vaidya type solutions. This part of the results can be found in [5]. [1] Petrov A N, Kopeikin S M, Lompay R R and Tekin B 2017 Metric Theories of Gravity: Perturbations and Conservation Laws (de Gruyter: Germany) [2] Petrov A N 2019 Class. Quantum Grav. 36, 235021 [3] Cai R-G and Ohta N 2006 Phys. Rev. D 74, 064001 [4] Cai R-G, Cao L-M, Hu Y-P and Kim S P 2008 Phys. Rev. D 78, 124012 [5] Petrov A N Conserved quantities for black hole solutions in pure Lovelock gravity; arXiv:2010.07196