Asymptotic Observer Research Articles

Revisiting the fastest way to circle a black hole

The shortest time for a null particle traveling between two arbitrary points outside a static spherically symmetric black hole is revisited. We introduce a functional for the time taken by a null particle in traveling on the path between the two points. By variating the time functional, we analyze the possible path with the shortest travel time for the null particle. It is analytically proven that the Euler–Lagrange equation corresponding to the time-functional for finding the path with the shortest traveling time is equivalent to the geodesics equation. This is in agreement with Hod’s conjecture on the fastest way to circle a black hole. We apply the formalism to the dirty black hole in Einstein-square-root nonlinear electrodynamics-dilaton theory. We calculate explicitly the time measured by an asymptotic observer which is needed for a null particle to circle the dirty black hole. Accordingly, a null particle circling the dirty black hole on an almost circular path of radius infinity achieves the shortest time.

Pinning Asymptotic Observability of Distributed Boolean Networks.

Asymptotic observability of distributed Boolean networks (DBNs) is studied in this article. Via a parallel extension method, asymptotic observability of the original system is converted to reachability at a fixed point of the extended system. Based on the structure matrix of the extended system, a necessary and sufficient condition is presented for asymptotic observability. Further, for unobservable systems, mode-dependent pinning control is first introduced and applied to achieve asymptotic observability, including the selections of pinning nodes, the design of output feedback controls, and the adding approaches. Then, a set of matrices is defined for the construction of the desired structure matrix. Based on it, a necessary condition is given to guarantee the solvability of the corresponding output feedback controls and the adding approaches. Finally, a numerical example is presented to show the effectiveness of the obtained results.

Out-of-time-order asymptotic observables are quasi-isomorphic to time-ordered amplitudes

Asymptotic observables in quantum field theory beyond the familiar S-matrix have recently attracted much interest, for instance in the context of gravity waveforms. Such observables can be understood in terms of Schwinger-Keldysh-type ‘amplitudes’ computed by a set of modified Feynman rules involving cut internal legs and external legs labelled by time-folds.In parallel, a homotopy-algebraic understanding of perturbative quantum field theory has emerged in recent years. In particular, passing through homotopy transfer, the S-matrix of a perturbative quantum field theory can be understood as the minimal model of an associated (quantum) L∞-algebra.Here we bring these two developments together. In particular, we show that Schwinger-Keldysh amplitudes are naturally encoded in an L∞-algebra, similar to ordinary scattering amplitudes. As before, they are computed via homotopy transfer, but using deformation-retract data that are not canonical (in contrast to the conventional S-matrix). We further show that the L∞-algebras encoding Schwinger-Keldysh amplitudes and ordinary amplitudes are quasi-isomorphic (meaning, in a suitable sense, equivalent). This entails a set of recursion relations that enable one to compute Schwinger-Keldysh amplitudes in terms of ordinary amplitudes or vice versa.

An automata theoretic approach to observer design for switched linear systems

We present an approach for designing asymptotic observers for discrete-time switched linear systems. We first give an automata theoretic characterization of switching signals containing an infinite number of reconstructible sequences, i.e. sequences allowing to estimate the state of the system. We show that such switching signals can be generated by a deterministic Büchi automaton whose construction is given in the paper. Then, we present a methodology to design switched observers. These observers have an internal discrete state variable whose dynamics is given by the transition map of the Büchi automaton. We then present two approaches to design observer gains such that the observer is convergent for all switching signals whose occurrence rate of reconstructible sequences is higher than a tunable threshold. The first approach gives an explicit construction of the observer gains while the second one is based on linear matrix inequalities. For switched systems with invertible state matrices, we show that the proposed observer structure is universal in the sense that it is always possible to design an observer of the proposed form. We use a simple example to illustrate our methodology and then consider a case study in which we design an observer for a multicellular converter.

On time-dependent backgrounds in 1 + 1 dimensional string theory

In perturbative string theory, one is generally interested in asymptotic observables, such as the S-matrix in flat spacetime, and boundary correlation functions in anti-de Sitter spacetime. However, there are backgrounds in which such observables do not exist. We study examples of such backgrounds in 1 + 1 dimensional string theory. In these examples, the Liouville wall accelerates and can become spacelike in the past and/or future. When that happens, the corresponding null infinity, at which the standard scattering states are defined, is shielded by the Liouville wall. We compute scattering and particle production amplitudes in these backgrounds in the region in parameter space where the wall remains timelike, and discuss the continuation of this picture to the spacelike regime. We also discuss the physics from the point of view of the dynamics of free fermions in backgrounds with a time-dependent Fermi surface.

Robust asymptotic super twisting sliding mode observer for non-linear uncertain biochemical systems

The problem of state estimation (reconstruction of the state vector) for a given class of biochemical systems under uncertain system dynamics has been addressed in this paper. In detail, the bioreactor at a water resource recovery facility represents the considered biochemical systems. The biochemical processes taking place in the bioreactor have been modelled using an activated sludge model. Based on this model, an appropriate utility model has been derived for estimation purposes. The internal dynamics of the model have been burdened with unstructured and parametric uncertainty due to the unknown reaction kinetics functions. Taking this uncertainty into account, an analysis of the observability and detectability of the utility model has been carried out. The utility model and the available set of inputs and measured outputs have been used to design a new robust non-linear observer that allows the estimation of state variables in the presence of uncertainty. In the synthesis of the observer, the asymptotic observer methodology has been combined with a second-order sliding mode observer, a so-called super twisting algorithm. The developed observer generates not only robust and stable estimates of the state variables but also enables the reconstruction of unknown kinetic functions. The stability of the designed observer has been proven using the Lyapunov stability theory. The observer has been implemented in the Matlab/Simulink environment. The comprehensive simulation studies carried out show the high efficiency of the estimation process using the developed state observer.

A relativistic appearance of hotspots on an ultra-compact dark star

We depict the deflection of light near ultra-compact dark energy stars with a photon sphere, drawing inspiration from modern confirmations of general relativity. Utilizing the description of dark energy through a phantom scalar field, we model the spacetime outside the star using an exact dark energy solution metric. We provide a summary of the properties of photon orbits. Additionally, we illustrate the relativistic appearance of a hot spot on the surface of a dark energy star and investigate both the optical appearance of the surface and the star’s visual size on the screen of an asymptotic observer.

What can be measured asymptotically?

We consider asymptotic observables in quantum field theories in which the S-matrix makes sense. We argue that in addition to scattering amplitudes, a whole compendium of inclusive observables exists where the time-ordering is relaxed. These include expectation values of electromagnetic or gravitational radiation fields as well as out-of-time-order amplitudes. We explain how to calculate them in two ways: by relating them to amplitudes and products of amplitudes, and by using a generalization of the LSZ reduction formula. As an application, we discuss one-loop master integrals contributing to gravitational radiation in the post-Minkowski expansion, emphasizing the role of classical cut contributions and highlighting the different infrared physics of in-in observables.

Fractional adaptive observer for variable structure high cell density fed-batch cultures

This paper presents the design and application of a fractional order asymptotic adaptive observer coupled to an adaptive controller for the robust operation of high-cell density cultures in fed-batch mode. The control goal is to maximize biomass productivity by controlling the culture’s estimated specific growth rate. Since the specific growth rate cannot be measured, a fractional order asymptotic adaptive observer is proposed, based on the equivalent integer order asymptotic observer proposed before. Simulations are performed to validate the observer and controller, under the assumption that the system is in the oxidative regime under aerobic conditions. Obtained results show that, in close loop operation, the fractional adaptive observer behaves better than the integer order observer in the presence of measurement noise. For fractional orders of the observer in the range α G [0.6,0.8], it was observed a 51.71% increase in biomass concentration, compared to the biomass obtained with the classic integer-order observer. Furthermore, the controlled system reaches very low ethanol concentrations (< 1 grams per liter), which is desirable in this process.

Compact stars: To cross or go around? That is the question

The travel times of light signals between two antipodal points on the surface of a compact star are calculated for two different trajectories: a straight line that passes through the center of the star and a semi-circular trajectory that connects the antipodal points along the surface of the star. Interestingly, it is explicitly proved that, for highly dense stars, the longer trajectory (the one that goes along the surface of the star) may be characterized by the shorter travel time as measured by asymptotic observers. In particular, for constant density stars we determine analytically the critical value of the dimensionless density-area parameter Λ≡4πR2ρ that marks the boundary between situations in which a direct crossing of the star through its center has the shorter travel time and situations in which the semi-circular trajectory along the surface of the star is characterized by the shorter travel time as measured by asymptotic observers [here {R,ρ} are respectively the radius of the star and its density].

Universal lower bound on orbital periods

It is proved, using the curved line element of a spherically symmetric charged object in general relativity and the Schwinger discharge mechanism of quantum field theory, that the orbital periods T_{infty } of test particles around central compact objects as measured by flat-space asymptotic observers are fundamentally bounded from below. The lower bound on orbital periods becomes universal (independent of the mass M of the central compact object) in the dimensionless ME_{text {c}}gg 1 regime, in which case it can be expressed in terms of the electric charge e and the proper mass m_{e} of the lightest charged particle in nature: T_{infty }>{{2pi ehbar }over {sqrt{G}c^2 m^2_{e}}} (here E_{text {c}}=m^2_{e}/ehbar is the critical electric field for pair production). The explicit dependence of the bound on the fundamental constants of nature {G,c,hbar } suggests that it may reflect a fundamental physical property of the elusive quantum theory of gravity.

On the visibility of singularities in general relativity and modified gravity theories

We investigate the global causal structure of the end state of a spherically symmetric marginally bound Lemaitre–Tolman–Bondi (LTB) (Lemaitre 1933 Ann. Soc. Sci. Brux. A 53 51, Tolman 1934 Proc. Natl Acad. Sci. USA 20 169–76, Bondi 1947 Mon. Not. Astron. Soc. 107 410) collapsing cloud (which is well studied in general relativity) in the framework of modified gravity having the generalized Lagrangian in the action. Here R is the Ricci scalar, and is a constant. By fixing the functional form of the metric components of the LTB spacetime, using up the available degree of freedom, we realize that the matching surface of the interior and the exterior metric are different for different values of α. This change in the matching surface can alter the causal property of the first central singularity. We depict this by showing a numerical example. Additionally, for a globally naked singularity to have physical relevance, a congruence of null geodesics should escape from such singularity to be visible to an asymptotic observer for infinite time. For this to happen, the first central singularity should be a nodal point. We here give a heuristic method to show that this singularity is a nodal point by considering the above class of theory of gravity, of which general relativity is a particular case.

Holography of information in massive gravity using Dirac brackets

The principle of holography of information states that in massless gravity, it is possible to extract bulk information using asymptotic boundary operators. In our work, we study this principle in a linearized setting about empty flat space and formulate it using Dirac brackets between boundary Hamiltonian and bulk operators. We then address whether the storage of bulk information in flat space linearized massive gravity resembles that of massless gravity. For linearized massless gravity, using Dirac brackets, we recover the necessary criteria for the holography of information. In contrast, we show that the Dirac bracket of the relevant boundary observable with bulk operators vanishes for massive gravity. We use this important distinction to outline the canonical Hilbert space. This leads to split states, and consequently, one cannot use asymptotic boundary observables to extract bulk information in massive gravity. We also argue the split property directly without an explicit reference to the Hilbert space. The result reflects that we can construct local bulk operators in massive gravity about the vacuum, which are obscured from boundary observables due to the lack of diffeomorphism invariance. Our analysis sheds some light on evaporating black holes in the context of the islands proposal.

On the horizon area of effective loop quantum black holes

Effective models of quantum black holes inspired by loop quantum gravity (LQG) have had success in resolving the classical singularity with polymerisation procedures and by imposing the LQG area gap as a minimum area. The singularity is replaced by a hypersurface of transition from black to white holes, and a recent example is the Ashtekar, Olmedo and Singh (AOS) model for a Schwarzschild black hole. More recently, a one-parameter model, with equal masses for the black and white solutions, was suggested by Alonso-Bardaji, Brizuela and Vera (ABBV). An interesting feature of their quantisation is that the angular part of the metric retains its classical form and the horizon area is therefore the same as in the classical theory. In the present contribution we solve the dynamical equations derived from the ABBV effective Hamiltonian and, by applying the AOS minimal area condition, we obtain the scaling of the polymerisation parameter with the black hole mass. We then show that this effective model can also describe Planck scale black holes, and that the curvature and quantum corrections at the horizon are small even at this scale. By generating the exterior metric through a phase rotation in the dynamical variables, we also show that, for an asymptotic observer, the Kretschmann scalar is the same as in the classical Schwarzschild solution, but with a central mass screened by the quantum fluctuations.

Estimation of the solution of asymptotically observable linear completely regular differential-algebraic systems with delay

In the article, a problem of solution estimation for linear autonomous completely regular differential-algebraic systems with many commensurate delays is investigated. The class of completely regular differential-algebraic systems with delay under study includes the classes of linear systems of delayed and neutral types; in addition, the analysis of continuous-discrete systems is reduced to completely regular systems. For linear autonomous completely regular differential-algebraic systems with many commensurate delays, the property of asymptotic observability is determined, which are characterized by the fact that all solutions generating the same output signal are indistinguishable in the future. Conditions for asymptotic observability expressed in terms of the parameters of the original system are formulated and proved. For asymptotically observable systems, a solution estimation procedure is proposed, the implementation of which consists of the following steps. First, using the observed output, a linear autonomous non-homogeneous asymptotically observable retarded type system with a non-homogeneous part depending on the output is put in correspondence with the original system. The solution of the new system uniquely determines the solution of the original system. Then a transformation is constructed that reduces the matrices of the retarded type system to a certain form. After that, with the help of a finite chain of observers, the solution is evaluated. The results of the presented study are applicable to systems that do not have the property of final observability, which makes it possible to significantly reduce the requirements for observing organs when modeling the corresponding objects of the real world.

Designing asymptotic observers for linear completely regular differential algebraic systems with delay

We study the problem of generating a solution estimate based on the observed output signal data for linear autonomous completely regular differential algebraic systems with commensurate delays. To obtain an estimate for the solution, two types of observers are proposed: an asymptotic observer and an asymptotic observer with bounded error. An asymptotic observer is characterized by the fact that its error asymptotically approaches zero. In this case, if the original system has the property of final observability, then the rate of convergence of the estimation error to zero can be set in advance by choosing the characteristic quasi-polynomial of the homogeneous system that describes the behavior of the error. Otherwise, the estimation error is described by an inhomogeneous system, and the rate of its convergence to zero depends not only on the choice of the characteristic quasi-polynomial of the homogeneous system, but also on the behavior of the inhomogeneous part, the dynamics of which depends on the matrices that determine the structure of the output signal. A distinctive feature of an asymptotic observer with bounded error is that its estimation error remains bounded by some constant depending on the observer's initial condition. In this case, the conditions for the existence of such an observer impose weaker requirements on the parameters of the original system in comparison with the conditions for the existence of an asymptotic observer.

A hybrid observer for localization from noisy inertial data and sporadic position measurements

We propose an asymptotic position and speed observer for inertial navigation in the case where the position measurements are sporadic and affected by noise. We cast the problem in a hybrid dynamics framework where the continuous motion is affected by unknown continuous-time disturbances and the sporadic position measurements are affected by discrete-time noise. We show that the peculiar hybrid cascaded structure describing the estimation error dynamics is globally finite-gain exponentially ISS with gains depending intuitively on our tuning parameters. Experimental results, as well as the comparison with an Extended Kalman Filter (EKF), confirm the effectiveness of the proposed solution with an execution time two orders of magnitude faster and with a simplified observer tuning because our bounds are an explicit function of the observer tuning knobs.

ПРИЛИВНЫЕ СИЛЫ ВБЛИЗИ ЧЕРНЫХ ДЫР В ГРАВИТАЦИОННОЙ ТЕОРИИ С НАРУШЕНИЕМ СИММЕТРИИ ЛОРЕНЦА

It is generally accepted that the curvature of space-time is small near the event horizon of a large static black hole, and test particles can fall into it without being destroyed. However, in most cases, at a small distance from the event horizon, the curvature of the black hole can be so huge that the test particle will stretch and even collapse under the influence of tidal forces. Thus, any object that falls into black holes experiences huge tidal forces beyond the horizon. This is a result of the fact that the curvature is very large near black holes and almost zero near the horizon. When measurements are made in a static frame (which also becomes zero), the curvature components remain small. This means that all curvature invariants are small, and any perturbing corrections, such as charge, angular momentum, Ricci scalar, etc., can be neglected. However, in a freely falling frame (the reference frame of a distant asymptotic observer), the curvature components are very large. In our paper, we consider the influence of the perturbing parameter ℓ on the curvature invariants of black holes in the gravitational theory with Lorentz symmetry breaking. The results are compared with the Schwarzschild black hole solution, which can be obtained by substituting ℓ =0.

What can a detected photon with a given gravitational redshift tell us about the maximum density of a compact star?

Far away observers can in principle bound from below the dimensionless maximum-density parameter $\Lambda\equiv4\pi R^2\rho_{\text{max}}$ of a compact star by measuring the gravitational redshift factor $z\equiv\nu_{\text{e}}/\nu_{\infty}-1$ of photons that were emitted from the {\it surface} of the star: $\Lambda\geq{3\over2}[1-(1+z)^{-2}]$ [here $R$ is the radius of the star and $\{\nu_{\text{e}},\nu_{\infty}\}$ are respectively the frequency of the emitted light as measured at the location of the emission and by asymptotic observers]. However, if photons that were created somewhere {\it inside} the star can make their way out and reach the asymptotic observers, then the measured redshift parameter $z$ may not determine uniquely the surface properties of the star, thus making the above bound unreliable. In the present compact paper we prove that in these cases, in which the creation depth of a detected photon is not known to the far away observers, the empirically measured redshift parameter can still be used to set a (weaker) lower bound on the dimensionless density parameter of the observed star: $\Lambda\geq{3\over2}[1-(1+z)^{-2/3}]$.

Lower bound on the time-to-mass ratio of closed circular motions around spinning and charged naked singularities

It has recently been conjectured that circular trajectories (geodesic as well as non-geodesic) around central compact objects are characterized by the time-to-mass universal lower bound T_{infty }/M>{{mathcal {C}}}, where {T_{infty },M} are respectively the orbital period as measured by flat-space asymptotic observers and the mass of the central compact object, and mathcal{C}=O(1) is a dimensionless constant of order unity. In the present paper we prove that this dimensionless bound is respected by circular orbits around central naked singularities. Intriguingly, it is explicitly proved that the orbital-time-to-mass ratio T_{infty }/M around a central super-spinning singularity remains finite even in the rrightarrow 0 limit of circular orbits with infinitesimally small radial coordinates (which are characterized by a diverging time-to-radius ratio, T_{infty }/rrightarrow infty ). In particular, we reveal the fact that the shortest orbital period around a super-spinning naked singularity is given by the dimensionless relation T_{infty }/M=2pi . In addition, we explore the functional behavior of the time-to-mass ratio of circular trajectories around super-charged (non-vacuum) naked singularities and prove that the dimensionless ratio T_{infty }/M(r) is bounded from below, where M(r) is the gravitational mass contained within the orbital radius of the test particle.