Existence of conserved quantities and their algebra in curved spacetime

In general relativity, finding out the geodesics of a given spacetime manifold is an important task because it determines which classical processes are dynamically forbidden. Conserved quantities play an important role in solving the geodesic equations of a general spacetime manifold. Furthermore, knowing all possible conserved quantities of a system gives information about the hidden symmetries of that system since conserved quantities are deeply connected with the symmetries of the system. These are very important in their own right. Conserved quantities are also useful to capture certain features of spacetime manifold for an asymptotic observer. In this article, we show the existence of these conserved charges and their algebra in a generic curved spacetime for a class of dynamical systems with the Hamiltonians quadratic and linear in momentum and spin.

Classical and quantum equations of motion of a 4-dimensional Schwarzschild–AdS and Reissner–Nordström–AdS black hole

We investigate the gravitational collapse of both a massive (Schwarzschild–AdS) and a massive-charged (Reissner–Nordström–AdS) 4-dimensional domain wall in AdS space. Here, we consider both the classical and quantum collapse, in the absence of quasi-particle production and backreaction. For the massive case, we show that, as far as the asymptotic observer is concerned, the collapse takes an infinite amount of time to occur in both the classical and quantum cases. Hence, quantizing the domain wall does not lead to the formation of the black hole in a finite amount of time. For the infalling observer, we find that the domain wall collapses to both the event horizon and the classical singularity in a finite amount of proper time. In the region of the classical singularity, however, the wave function exhibits both nonlocal and nonsingular effects. For the massive-charged case, we show that, as far as the asymptotic observer is concerned, the details of the collapse depend on the amount of charge present; that is, the extremal, nonextremal and overcharged cases. In the overcharged case, the collapse never fully occurs since the solution is an oscillatory solution which prevents the formation of a naked singularity. For the extremal and nonextremal cases, it takes an infinite amount of time for the outer horizon to form. For the infalling observer in the nonextremal case, we find that the domain wall collapses to both the event horizon and the classical singularity in a finite amount of proper time. In the region of the classical singularity, the wave function also exhibits both nonlocal and nonsingular effects. Furthermore, in the large energy density limit, the wave function vanishes as the domain wall approaches classical singularity implying that the quantization does not rid the black hole of its singular nature.

Quantum collapse of a charged n-dimensional BTZ-like domain wall

We investigate both the classical and quantum gravitational collapse of a massive, charged, nonrotating [Formula: see text]-dimensional Bañados–Teitelboim–Zanelli (BTZ)-like domain wall in AdS space. In the classical picture, we show that, as far as the asymptotic observer is concerned, the details of the collapse depend on the amount of charge present in the domain wall; that is, if the domain wall is extremal, nonextremal or overcharged. In both the extremal and nonextremal cases, the collapse takes an infinite amount of observer time to complete. However, in the over-charged case, the collapse never actually occurs, instead one finds an oscillatory solution which prevents the formation of a naked singularity. As far as the infalling observer is concerned, in the nonextremal case, the collapse is completed within a finite amount of proper time. Thus, the gravitational collapse follows that of the typical formation of a black hole via gravitational collapse.Quantum mechanically, we take the absence of induced quasi-particle production and fluctuations of the metric geometry; that is, we ignore the effect of radiation and back-reaction. For the asymptotic observer, we find that, near the horizon, quantization of the domain wall does not allow the formation of the black hole in a finite amount of observer time. For the infalling observer, we are primarily interested in the quantum mechanical effect as the domain wall approaches the classical singularity. In this region, the main result is that the wave function exhibits nonlocal effects, demonstrated by the fact that the Hamiltonian depends on an infinite number of derivatives that cannot be truncated after a finite number of terms. Furthermore, in the large energy density limit, the wave function vanishes at the classical singularity implying that quantization does not rid the black hole of its singularity.

Application of super-twisting observers to the estimation of state and unknown inputs in an anaerobic digestion system

This paper presents the estimation of the unknown states and inputs of an anaerobic digestion system characterized by a two-step reaction model. The estimation is based on the measurement of the two substrate concentrations and of the outflow rate of biogas and relies on the use of an observer, consisting of three parts. The first is a generalized super-twisting observer, which estimates a linear combination of the two input concentrations. The second is an asymptotic observer, which provides one of the two biomass concentrations, whereas the third is a super-twisting observer for one of the input concentrations and the second biomass concentration.