Black-Hole Models in Loop Quantum Gravity

Dynamical black-hole scenarios have been developed in loop quantum gravity in various ways, combining results from mini and midisuperspace models. In the past, the underlying geometry of space-time has often been expressed in terms of line elements with metric components that differ from the classical solutions of general relativity, motivated by modified equations of motion and constraints. However, recent results have shown by explicit calculations that most of these constructions violate general covariance and slicing independence. The proposed line elements and black-hole models are therefore ruled out. The only known possibility to escape this sentence is to derive not only modified metric components but also a new space-time structure which is covariant in a generalized sense. Formally, such a derivation is made available by an analysis of the constraints of canonical gravity, which generate deformations of hypersurfaces in space-time, or generalized versions if the constraints are consistently modified. A generic consequence of consistent modifications in effective theories suggested by loop quantum gravity is signature change at high density. Signature change is an important ingredient in long-term models of black holes that aim to determine what might happen after a black hole has evaporated. Because this effect changes the causal structure of space-time, it has crucial implications for black-hole models that have been missed in several older constructions, for instance in models based on bouncing black-hole interiors. Such models are ruled out by signature change even if their underlying space-times are made consistent using generalized covariance. The causal nature of signature change brings in a new internal consistency condition, given by the requirement of deterministic behavior at low curvature. Even a causally disconnected interior transition, opening back up into the former exterior as some kind of astrophysical white hole, is then ruled out. New versions consistent with both generalized covariance and low-curvature determinism are introduced here, showing a remarkable similarity with models developed in other approaches, such as the final-state proposal or the no-transition principle obtained from the gauge-gravity correspondence.

On the Choice of Momentum Control Variables and Covariance Modeling for Mesoscale Data Assimilation

For mesoscale variational data assimilation with high-resolution observations, there has been an issue concerning the choice of momentum control variables and related covariance modeling. This paper addresses the theoretical aspect of this issue. First, relationships between background error covariance functions for differently chosen momentum control variables are derived, and different choices of momentum control variables are proven to be theoretically equivalent in the sense that they lead to the same optimally analyzed incremental wind field in the limit of infinitely high spatial resolution provided their error covariance functions satisfy the derived relationships. It is then shown that when the velocity potential χ and streamfunction ψ are used as momentum control variables with their background error autocovariance functions modeled by single-Gaussian functions, the derived velocity autocovariance functions contain significant negative sidelobes. These negative sidelobes can represent background wind error structures associated with baroclinic waves on the synoptic scale but become unrepresentative on the mesoscale. To reduce or remove these negative sidelobes for mesoscale variational data assimilation, Gaussian functions are used with two types of modifications to model the velocity covariance functions in consistency with the assumed homogeneity and isotropy in variational data assimilation. In this case, the random (χ, ψ) background error fields have no classically valid homogeneous and isotropic covariance functions, but generalized (χ, ψ) covariance functions can be derived from the modified velocity covariance functions for choosing (χ, ψ) as momentum control variables. Mathematical properties of generalized covariance functions are explored with physical interpretations. Their important implications are discussed for mesoscale data assimilation.

Constrained Multi-Sensor Control Using a Multi-Target MSE Bound and a δ-GLMB Filter

The existing multi-sensor control algorithms for multi-target tracking (MTT) within the random finite set (RFS) framework are all based on the distributed processing architecture, so the rule of generalized covariance intersection (GCI) has to be used to obtain the multi-sensor posterior density. However, there has still been no reliable basis for setting the normalized fusion weight of each sensor in GCI until now. Therefore, to avoid the GCI rule, the paper proposes a new constrained multi-sensor control algorithm based on the centralized processing architecture. A multi-target mean-square error (MSE) bound defined in our paper is served as cost function and the multi-sensor control commands are just the solutions that minimize the bound. In order to derive the bound by using the generalized information inequality to RFS observation, the error between state set and its estimation is measured by the second-order optimal sub-pattern assignment metric while the multi-target Bayes recursion is performed by using a δ-generalized labeled multi-Bernoulli filter. An additional benefit of our method is that the proposed bound can provide an online indication of the achievable limit for MTT precision after the sensor control. Two suboptimal algorithms, which are mixed penalty function (MPF) method and complex method, are used to reduce the computation cost of solving the constrained optimization problem. Simulation results show that for the constrained multi-sensor control system with different observation performance, our method significantly outperforms the GCI-based Cauchy-Schwarz divergence method in MTT precision. Besides, when the number of sensors is relatively large, the computation time of the MPF and complex methods is much shorter than that of the exhaustive search method at the expense of completely acceptable loss of tracking accuracy.