In this paper, we review the issue of defining energy for test particles on a background stationary spacetime. We revisit different notions of energy as defined by different observers. As is well-known, the existence of a timelike isometry allows for the notion of total conserved energy to be well defined. We use this well-known quantity to show that a gravitational potential energy can be consistently defined. As examples, we study the case of the exterior regions of an asymptotically flat black hole and of the [Formula: see text] Schwarzschild–de Sitter (SdS) case, where an asymptotic region is not available. We then consider the situation in which the test particle is absorbed by the black hole and analyze the energetics in detail. In particular, we show that the notion of horizon energy as defined by the isolated horizons formalism provides a satisfactory notion of energy compatible with the particle’s total conserved energy. With these choices, there is a global conservation of energy. Finally, we comment on a recent proposal to define energy of the black hole as seen by a nearby observer at rest, for which this feature is lost.
Dynamics and fundamental frequencies of test particles orbiting Kerr–Newman–NUT–Kiselev black hole in Rastall gravity
